![Symbols
A0 Open loop system matrix
Am Closed loop system matrix
b Input matrix
c Output matrix
x(t) = [x1 x2 . . . xn]T
∈ n
System state
u1(t) L1 adaptive control signal
u2(t) State feedback control signal
r Input to the system
y Output of the system
e = r − y Tracking error
ω Unknown constant
θ Uncertainties
σ Disturbances
k Pre-filter gain
kg Feed-forward gain
K State feedback gain
C Adaptation gain
x̂ Predictor state
ŷ Predictor output
C(s) Low-pass filter
D(s) Strictly proper stable transfer function
Z Candidate solution vector
ξ Fuzzy basis function
t Time
h Time step
V Lyapunov function
Barbalat’s lemma operator
xxiii](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-25-320.jpg)
![Chapter 1
Introduction
Abstract This chapter introduces the book with brief description of the journey
towards modern control theories. Motivations of the book are portrayed owing to the
drawbacks of present literatures. The designed methodologies are described briefly
with an emphasis on contributions of the book. Then, the structure of the book is
provided.
1.1 Journey Towards Modern Control Theories
Classical controllers, viz., proportional integral derivative (PID) controller [1–3],
state feedback controller [4], observer based state feedback controller [5], output
feedbackcontroller[6,7],linearquadraticregulator(LQR)[8],linearquadraticGaus-
sian (LQG) [8, 9], etc., are suitable for linear time invariant (LTI) systems. Though,
they yield unsatisfactory control performances for systems with time varying quanti-
ties [10]. In various literatures, different modifications were furnished over classical
controllers to overcome their drawbacks. LQG controller was augmented with vibra-
tion compensator to suppress the vibration of a piezoelectric tube actuator [11]. LQG
was also combined with proportional integral (PI) controller to nullify steady state
error [12]. Grid voltage disturbance was tackled with the help of frequency-adaptive
multi-resonator augmented LQG current controller [13]. Two separately designed
PID controllers were brought together with resonant controller to suppress the current
harmonic components [14]. Data driven controller was augmented with d-step ahead
prediction optimal controller and PID controller to control a pulp neutralization
process by compensating unmodelled dynamics [15]. To tune the gains of the PID
controller, different improved strategies were adopted rather than the Ziegler-Nichols
method [10] in different literatures. Variable gains of PID controller were designed
for a linear model of unmanned aerial vehicle (UAV) [16]. Relay feedback tuning
method was also utilized to design the gains of the PID controller [17]. However, the
performances of the classical controllers were unsatisfactory to tackle systems with
uncertainties, disturbances, time varying parameters, nonlinearities and delays [18].
To overcome such drawbacks, modern control theories draw attention.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
R. Maiti et al., Hybrid L1 Adaptive Control, Studies in Systems, Decision
and Control 422, https://doi.org/10.1007/978-3-030-97102-1_1
3](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-44-320.jpg)
![4 1 Introduction
1.2 Overview of Modern Control Methodologies
Proper estimations of time varying uncertainties and disturbances are required to
eliminate these quantities, which can be obtained through adaptive controllers [19,
20]. To eliminate the drawbacks of conventional adaptive controller, a newly invented
adaptive controller, viz., L1 adaptive controller comes into place and increases its
applications in different fields. The parameters of different controllers are tuned
employing stochastic optimization and metaheuristics techniques to obtain optimized
performance of the system under control. In different literatures, fuzzy logic systems
are utilized to approximate nonlinearities [21], to produce controller by approxi-
mating uncertainties [22], disturbances [23], unmodelled dynamics [24], etc. Liter-
ature surveys of these modern control methodologies are provided in the following
sections.
1.2.1 Stochastic Optimization and Metaheuristics Techniques
Arbitrarily chosen parameter values of controllers, within the ranges obtained from
the stability condition, do not guarantee optimal performance. Optimization tech-
niques acquire great attention to choose the optimal parameter values of controller
during last two decades. Different optimization techniques, viz., genetic algorithm
(GA) [25, 26], particle swarm optimization (PSO) [27–29], ant colony optimization
(ACO) [30, 31], firefly (FF) algorithm [32–34], cuckoo search (CS) [35, 36], artificial
bee colony (ABC) [37, 38], harmony search algorithm (HS) [39, 40] were used in
different literatures to obtain optimized performance of the controllers. It is observed
from the literatures that, the computational cost reduces considerably in case of
harmony search algorithm compared to other optimization techniques [39]. Involve-
ment of lesser number of internal parameters and their easy selections increase the
use of HS algorithm in different optimization problems, viz., pipe network designing
[41], vehicle routing [42], structural designing [43], etc. Different parameter modi-
fications of the HS algorithm provides the easy enhancement of exploration and
exploitation capability of it. The advantages of HS algorithm over other optimization
process increase its utilization gradually as elaborated follows.
1.2.1.1 Harmony Search Algorithm
HS resembles with the group of musicians playing pitches with the aim to develop
an aesthetic harmony all together [40]. Being beneficial than other optimization
techniques, the utilization of HS algorithm increases gradually. The HS algorithm
was successfully utilized in low impact development and management of urban storm
water systems to avoid flooding risk and assure environmental interventions [44]. The
HS algorithm performed better than random sample consensus algorithm to detect](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-45-320.jpg)
![1.2 Overview of Modern Control Methodologies 5
vanishingpointpresentattheextendedlinesofroadlanesforself-drivingautonomous
vehicle [45]. In power system engineering, HS was employed in different literatures
[46–50]. A distributed-generation (DG) system was controlled through proportional
integral (PI) controller whose gains were tuned by means of HS algorithm [51].
The parameters of fuzzy controller were tuned by utilizing HS for air heater system
with transport-delay [52]. HS was also applied in medical fields to RNA secondary
structure prediction [53, 54], medical physics [55], hearing AIDS [56] etc.
It is of great importance to balance between exploration and exploitation capa-
bility of HS algorithm [57]. Good exploration phenomenon indicates finding more
search spaces which avoids local minima trapping problem. Good exploitation signi-
fies proper convergence of the optimization process [58]. Different modifications of
basic HS algorithm were manifested by designing its parameter values in different
ways to obtain optimal balance between diversification and intensification of the
optimization technique [59]. Dynamic pitch adjustment rate (PAR) [60], dynamic
bandwidth (bw) selection formulas [60, 61] were developed in different literatures.
Gaussian distribution was utilized to formulate the PAR with the aim to improvise
exploration at first and then exploitation at the end of the harmony improvisation
process of HS [62]. The HS algorithm was hybridized with hill climbing and global
best particle swarm optimization to improve local exploitation and global conver-
gence respectively which was employed to design university course timetable [63].
Basic HS algorithm was also combined with wavelet mutation and new harmony
was produced based on roulette wheel mechanism to improve convergence speed,
accuracy and robustness for economic load dispatch (ELD) problem [64]. Differen-
tial mutation in pitch adjustment rate was incorporated to solve job shop scheduling
problem [65]. Position update formula of PSO and mutation phenomenon of GA
replaced the harmony memory considering rate (HMCR) and pitch adjustment rate
(PAR) parameters of HS respectively in [66]. A detail review of harmony search
algorithm, its developments and applications can be found in [67] and the references
therein. To improvise the exploitation capability of basic HS algorithm, local search
based phenomenon was incorporated in it [68–70].
Literatures show that, HS algorithm and its different modifications can be used
efficiently to solve many problems. It can be used to tune the parameter values of the
controller to provide optimize performance of the system. However, these modified
HS may not find the proper system matrix values to model unmodelled dynamics,
nonlinearities, delays, etc. This phenomenon can be addressed properly employing
fuzzy logic. Literature reviews of fuzzy logic systems to model nonlinear systems,
unmodelled dynamics, delays, etc., are elaborated in the next section.
1.2.2 Fuzzy Logic Systems
To tackle nonlinearities, delays, unmodelled dynamics with the uncertainties and
disturbances, fuzzy logic based approaches gain much attention [71]. Uncertainties
and disturbances can be nullified through adaptive controllers but they fail to provide](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-46-320.jpg)
![6 1 Introduction
satisfactoryperformanceincaseofnonlinearities,delays,unmodelleddynamics,etc.,
present in the system simultaneously. Universal approximator fuzzy logic system
[72] was used extensively to tackle nonlinearities [73–77] during last few decades.
Fuzzy logic system can be utilized to approximate system models with nonlinearities,
unmodelled dynamics, uncertainties [78, 79] as well as to design different types of
controllers [80–84]. Approximation of nonlinear system and developing controller
for that system should be synchronized. With this aim, fuzzy parallel distributed
compensation (PDC) type controller was introduced [85, 86] and utilized in many
applications [87–90]. The nonlinear system can be approximated by fuzzy parallel
distributed compensation logic [91–94]. The nonlinear system is considered as the
fuzzy blending of finite number of linear systems. Then, for each linearized zone,
linear state feedback controller is designed and fuzzy augmented to produce overall
control action [95–97]. This is known as PDC control law which is nonlinear in nature
and can handle nonlinear systems efficiently [98, 99]. At first, fuzzy PDC logic is
employed to design the system and then same premise part is utilized to generate
PDC control law. Fuzzy PDC control law was applied in different cases, viz., atti-
tude control of spacecraft [99], speed control of surface-mounted permanent magnet
synchronous motor [100], wind energy system control [101], maximum power point
tracking control of solar photovoltaic electricity generation system [102], nonlinear
vehicle dynamics control [103], etc. Yet, the performance of PDC controller degrades
when uncertainties and disturbances appear in the system [101]. Fuzzy controller
can be efficiently hybridized with other controllers to tackle different quantities
consequently.
To tackle uncertainties and disturbances, a special type of adaptive controller, viz.,
L1 adaptive controller gains much attention during few years. The literature reviews
of the L1 adaptive controller are provided in the following section.
1.2.3 L1 Adaptive Control Methodologies
Systems with time varying uncertainties, disturbances can be controlled through
adaptive controllers [104–106]. Conventional adaptive controller utilizes low adap-
tation gain to retain the robustness of the system [107]. However, utilization of low
adaptation gain possesses sluggish transient performance [108]. To overcome this
drawback in 2006, L1 adaptive controller was proposed with high adaptation gain
to incur quick transient performance and a low-pass filter is attached in the control
channel to retain robust performance [109–111]. L1 adaptive controller comprises of
predictor, adaptation laws, state feedback control action, L1 control law and low-pass
filter. Predictor predicts the unknown constant, time varying uncertainties and time
varying disturbances present in the system and their values are adapted following the
adaptation laws. From these adaptive estimates, L1 adaptive control law is formulated
which contains high frequency component. These high frequencies, may yield insta-
bility of the system, are nullified by a low-pass filter attached after the control block.
The filtered output is augmented with state feedback control signal and provided to](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-47-320.jpg)
![1.2 Overview of Modern Control Methodologies 7
the system under control. Designing of these components and selection of param-
eters from L1 norm condition were elaborated in the seminal papers of L1 adap-
tive controller, published in 2006 by Chengyu Cao and Naira Hovakimyan [107,
108]. Despite of utilizing high adaptation gain, the guaranteed transient performance
was analyzed in [109–111]. The exponential decay of non-zero trajectory initial-
ization error, transient and steady state system performances with respect to the
reference system were also analyzed by Cao and Hovakimyan [112]. Stability anal-
ysis of L1 adaptive controller was performed in digital mode [113], linear matrix
inequality based approach [114, 115], etc. L1 adaptive controller can efficiently
handles systems with unknown time varying parameters [112], unmatched uncer-
tainties [116, 117], unmatched disturbances [118], unmodelled dynamics [119], etc.
Nonlinear systems with uncertainties and disturbances were also controlled through
L1 adaptive controller in different literatures [120–123]. Though, consideration of
uncertainties, disturbances, nonlinearities, unmodelled dynamics, delays, etc., simul-
taneously is quite difficult and missing in the literatures, as far as author’s belief and
knowledge goes.
Advantages of L1 adaptive controller made it so prevalent that, its applications
grow very fast and spread to different fields. L1 adaptive controller was successfully
employed to control different types of flight systems, viz., aircraft [124, 125], wing-
rock [126], X-wing tail-sitter micro aerial vehicle (MAV) [127], unmanned aerial
vehicle (UAV) [128, 129], fighter aircraft [130], autopilot [131, 132]. Pitch break
uncertainty as well as actuator failure of an unmanned military tailless unstable
aircraft were controlled through L1 adaptive controller and compared with model
reference adaptive controller (MRAC) to show the effectiveness of L1 adaptive
controller [133]. L1 adaptive controller was used to control magnetic torque coil in
attitude control system of a Pico-scale satellite test bed [134]. Depth and pitch angle
of a multi-input multi-output (MIMO) submarine system was controlled through this
controller [135]. Wind turbine was controlled by regulating the speed of a generator
for maximum power point tracking through the L1 adaptive controller [136]. Precise
controlling phenomenon of L1 adaptive controller extended its application in medical
field for delivering anaesthesia to the patient during surgery [137].
To enhance the performance of L1 adaptive controller, it is augmented with other
strategies which are elaborated in the subsequent section.
1.3 State of the Art of Hybrid L1 Adaptive Control
Methodologies
Practical systems contain nonlinearities, time varying structured and unstructured
uncertainties, disturbances, unknown constants, cross-couplings, delays, etc. To
tackle all of these quantities simultaneously, different controllers are hybridized
together. L1 adaptive controller provides fast transient performance as well as retains](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-48-320.jpg)
![8 1 Introduction
high robustness to tackle uncertainties and disturbances. Arbitrary parameter selec-
tion, within the ranges of L1 norm condition, does not guarantee optimal perfor-
mance of it, although ensures the stability of the closed loop system. Hence, proper
balance between quick transient performance and high robustness should be main-
tained by designing the components and selecting the parameter values of L1 adap-
tive controller appropriately. Different strategies were applied to design its different
components. A greedy randomized optimization with multi-criteria was utilized to
design the filter of L1 adaptive controller [138]. Fuzzy logic was employed to design
the pre-filter gain of L1 adaptive controller where the filter structure was kept fixed
[139]. Adaptive estimates of uncertainties, in designing L1 adaptive controller, were
also approximated utilizing fuzzy logic system for electropneumatic actuator [140]
and wind energy conversion systems [141]. Being beneficial in controlling systems
withtimevaryinguncertaintiesanddisturbanceswhichpresentinpracticalsystem,L1
adaptive controller was combined with other control strategies in many cases. Base-
line dynamic inversion controller was augmented with L1 adaptive controller to elim-
inate time-varying uncertainty of a combination of Novlit-3 micro aerial vehicle and
a novel non-orthogonal X shaped wing layout with a single propeller system [142].
Dynamic inversion based L1 adaptive controller was implemented for missile [143],
generic transport model [144], etc. L1 adaptive controller was augmented with differ-
ential proportional-integral (PI) baseline controller to tackle longitudinal dynamics of
a F16 aircraft [145]. L1 adaptive back-stepping controller was designed for unmanned
aerial vehicles (UAVs) with position kinematics and dynamics present in strict feed-
back form [146]. PID controller was appended with L1 adaptive controller to obtain
proper tracking by eliminating the time lag for depth control of an under-actuated
underwater vehicle [147].
Though in different literatures, different components of the L1 adaptive controller
were designed through optimization technique or utilizing fuzzy logic but the entire
controller structure design was absent there, as far as author’s knowledge goes. In
case of designing L1 adaptive controller for systems consisting of uncertainties,
disturbances and nonlinearities, separate consideration of nonlinearities were also
absent in the literatures.
After modelling the system and developing controller for that system, it is impor-
tant to analyze the stability conditions of it. There are different ways to investigate
the stability conditions of stochastic optimization and metaheuristics techniques,
fuzzy logic controllers, etc., whose literature surveys are provided in the subsequent
section.
1.4 Overview of Stability Conditions
It is essential to investigate the exploration (/intensification) and exploitation (/diver-
sification) capabilities in case of any optimization process. Good balance between
exploration and exploitation shows that, the optimization process searches huge
spaces with the aim to converge at the end of the process without diverging from the](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-49-320.jpg)
![1.4 Overview of Stability Conditions 9
stability region. Owing to the easy implementation and lower computational time of
HS, in this book, the literatures of stability analysis of HS algorithm are provided. It
is important to analyze mathematically the exploration and exploitation capabilities
of the HS algorithm which was performed in [148]. The authors of [148] showed
that, the population variance of the HS algorithm increased gradually with the itera-
tion which exhibited its better exploration capability. They also proved the spectral
radius convergence of the iterative matrix of the HS algorithm. This demonstrated
the better exploitation capability of the HS algorithm [148]. Stability analysis of
different modified HS algorithm was also provided in different literatures [149].
Mathematical analysis of the exploration and exploitation capabilities of the local
search based HS (lbest HS) algorithm was absent in the literatures to the best of
author’s understanding. The stochastic optimization and metaheuristics techniques
were utilized to provide the optimal parameter setting of the controllers. Therefore,
it is important to analyze the stability condition of the optimization techniques along
with the controller which was also not present in the literatures as per as author’s
knowledge goes.
In another case, for nonlinear system, a set of stability conditions have to be
derived which are difficult from a single Lyapunov function. In case of fuzzy PDC
logic, multiple number of fuzzy rules are formulated and combined to model the
system and controller [88, 150]. In case of such large number of fuzzy rules, use
of quadratic Lyapunov function [151, 152] introduces conservative conditions [153,
154]. To reduce conservatism, different approaches were made to formulate the
Lyapunov functions in different literatures, viz., piecewise Lyapunov function [153,
155–158], polynomial Lyapunov function [159, 160], multiple Lyapunov function
[161–165]. Based on this multiple Lyapunov function approach, fuzzy Lyapunov
function was introduced [166]. Fuzzy Lyapunov function is smooth in nature which
is advantageous over piecewise Lyapunov function. This fuzzy Lyapunov function
is developed by augmenting finite number of zone-wise Lyapunov functions where
in each zone, positive definite matrix is present [166, 167]. The same premise part
utilized to design the nonlinear system and controller is employed to formulate the
fuzzy Lyapunov function [168–171].
From these literature surveys of the existing controllers, optimization techniques,
fuzzy logic systems, etc., and their stability analysis, the motivations of designing
the control methodologies presented in this book arise which are as follows.
1.5 Aims and Scopes of the Book
1.5.1 Aims
System with uncertainties and disturbances can be tackled through adaptive
controllers. Though, conventional adaptive controllers provide sluggish transient
performance to retain robustness. To overcome such problem, in 2006, L1 adaptive](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-50-320.jpg)
![10 1 Introduction
controller originates with high adaptation gain to achieve quick transient perfor-
mance and a low-pass filter to retain the robust performance [109]. Literature shows
that, the components of L1 adaptive controller should be chosen sensibly within
the L1 norm bound to obtain fast transient performance with high robustness [111].
Different components of L1 adaptive controller were designed utilizing different
optimization techniques in different literatures [138, 139]. Nevertheless, designing
the overall controller optimally is of great importance to achieve the accurate param-
eter setting and optimal control performance of L1 adaptive controller. Again the
literatures show that, HS and its different modifications can be efficiently utilized to
choose the parameters of any controller. This motivates to design local best HS (lbest
HS) based L1 adaptive controller and to analyze overall stability condition [172].
On the other hand, though nonlinear systems with uncertainties and disturbances
are controlled through L1 adaptive controller but tackling nonlinearities separately
was absent in the literatures [120–123]. Decentralized L1 adaptive controller was
designed for large scale nonlinear system with unknown interconnection and unmod-
elled dynamics with the help of decentralized passive identifiers [120]. Non-affine
nonlinear systems, consist of unmeasured states, were controlled by implementing
piece-wise continuous adaptive laws of L1 adaptive controller [121]. Nonlinear time
varying reference system was also considered to control nonlinear system where an
upper bound had to be considered additionally with complicated calculation which
increased the conservatism [173, 174]. To deal with nonlinearities, in particular,
fuzzy parallel distributed compensation (PDC) logic gained much attention in last
two decades [175]. Motivated with these ideas, the L1 adaptive controller can be
hybridized with fuzzy PDC controller to handle system with nonlinearities, unmod-
elled dynamics, delays, uncertainties and disturbances all in conjunction. Stability
of the overall system can be analyzed utilizing fuzzy Lyapunov function due to its
less conservatism.
1.5.2 Scopes
Motivatedfromtheaforementionedconcerns,inthisbook,theadvantagesofdifferent
methodologies are hybridized to tackle different quantities simultaneously with the
aim to produce superior control performances for the practical systems. Owing to
the fast transient performance as well as high robustness, in this book, a special type
of adaptive controller, viz., L1 adaptive controller is utilized. All the components
of L1 adaptive controller are designed by tuning the parameters utilizing stochastic
optimization and metaheuristics techniques to obtain overall optimal performance.
Considering the advantages of HS algorithm and to enhance the exploration and
exploitation capabilities, a variant of HS algorithm, viz., metaheuristic lbest HS is
utilized in this book. The overall stability analysis of the concurrent lbest HS based
L1 (lbest HS-L1) adaptive controller is also performed. On the other hand, in many
literaturesL1 adaptivecontrollerwasemployedtotacklenonlinearsystemwithuncer-
tainties and disturbances where nonlinearity was not considered in a separate way. To](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-51-320.jpg)
![1.5 Aims and Scopes of the Book 11
tackle uncertainties, disturbances as well as nonlinearities, delays, etc., in this book,
fuzzy logic is utilized to design PDC controller augmented L1 adaptive controller.
This method is termed as fuzzy parallel distributed compensation type L1 (fuzzy
PDC-L1) adaptive controller. The stability of the system with controller is investi-
gated by means of fuzzy Lyapunov function. The proposals and the contributions of
this book are elaborated as follows.
The layout of the book proposals are depicted in Fig. 1.1. In this book, to obtain fast
transient performance by eliminating uncertainties and disturbances present in the
system, L1 adaptive controller with fast adaptation and high robustness is employed.
To obtain optimal performance from L1 adaptive controller, its parameters are tuned
utilizing optimization techniques. Being aware of easy implementation and good
balance between exploration and exploitation capability, a novel variant of HS algo-
rithm, viz., local best harmony search (lbest HS) algorithm is utilized. The values of
unknown constant, time varying uncertainties and time varying disturbances present
in the system are at first obtained utilizing lbest HS algorithm. Then, their values are
adapted concurrently following L1 adaptation laws to fine tune those values. In local
search based HS, the total harmony memory is divided into some groups and from
eachgrouplocalbestsolutionisfoundout.Fromthoselocalbestsolutions,globalbest
solution is obtained. In this local best search phenomenon, the exploitation capability
of the HS algorithm increases enormously. The exploration and exploitation capa-
bilities of the whole optimization technique augmented controller are investigated.
It is proved analytically that, the population variance of the lbest HS-L1 adaptive
controller increases gradually with iteration which shows its better exploration capa-
bility [172]. With good exploration capability, high exploitation capability of the
optimization process is also required. The exploitation capability of the designed
method is assured by means of spectral radius convergence phenomenon [172].
Again to handle nonlinearities present in the system along with uncertainties
and disturbances, in this book, fuzzy parallel distributed compensation (PDC) logic
Fig. 1.1 Layout of the book proposal](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-52-320.jpg)
![12 1 Introduction
is hybridized with L1 adaptive controller [175]. In this fuzzy PDC-L1 adaptive
controller, the nonlinear system is considered as the combination of finite number
of linear fuzzy systems. Then, the same fuzzy premise part, utilized to design the
system, is employed to design different components of L1 adaptive controller, viz.,
predictor, L1 adaptation law, L1 control law, PDC control law and low-pass filter. For
each linearized zone, fuzzy predictor is developed and combined to produce overall
predictor model. Uncertainties, disturbances present in these linearized zones are
adapted following zone-wise L1 adaptation laws and are accumulated to develop
whole L1 adaptation law. From zone-wise adaptation laws, L1 control signals are
developed keeping the zonal nature and then fuzzy blended to produce whole L1
control law. Due to the use of high adaptation gain, the control law of each zone
contains high frequency components. To nullify that high frequency from each of
the linearized zone, the produced control signals are filtered through low-pass filter
dedicated for each of those zones. The state feedback controller of basic L1 adaptive
controller is designed utilizing fuzzy PDC logic. The zone-wise developed fuzzy
PDC controllers are accumulated to produce overall fuzzy PDC control law. Filtered
L1 adaptive control law and fuzzy PDC control signal are augmented and provided
to the system under control. All of these components of the L1 adaptive controller
are designed separately from the fuzzy adaptation laws, developed from the stability
criteria of fuzzy Lyapunov functions. The overall stability of the closed loop system
is also investigated utilizing fuzzy Lyapunov function. The transient and steady state
stable performances of the system are guaranteed analytically.
The designed controllers are employed on four simulation case studies and three
experimental case studies to show their effectiveness. In simulation case studies,
two single-input single-output (SISO) system, viz., Duffing’s oscillatory system
with disturbance and nonlinear spring mass damper system are considered to show
the controller performance when disturbances are present in the system. Then, one
single-input multi-output (SIMO) system, viz., unstable non-minimum phase 4th
order inverted pendulum with cart is controlled and stabilized. A nonlinear, cross-
coupled twin rotor multi-input multi-output (MIMO) system (TRMS) is investigated
to show the controller performance in case of nonlinearities and cross-couplings.
Furthermore, in experimental case studies, speed control of a SISO system, viz., an
electrical actuator is performed owing to its large number of applications in practical
systems. Angular position control of a MIMO, nonlinear two link robot manipulator
(TLRM) with unknown dynamics is evaluated to examine the effectiveness of the
designed controllers in tackling system with nonlinearities, unmodelled dynamics,
etc. Finally, temperature control of a delay dependent air heater system with distur-
bance is performed to show the efficiency of the designed controllers to tackle large
delay. All of these systems are controlled through different types of controllers and
their performances are compared. At first, two classical controllers, viz., proportional
integral derivative (PID) controller, linear quadratic Gaussian (LQG) controller are
tested. Then, model reference adaptive controller (MRAC) and basic L1 adaptive
controller are imposed on them. After that, a stochastic optimization, viz., PSO
based L1 (PSO-L1) adaptive controller; metaheuristic, viz., basic HS based L1 (HS-
L1) adaptive controller and metaheuristic lbest HS based L1 (lbest HS-L1) adaptive](https://image.slidesharecdn.com/23376551-250522150804-2c28f326/85/Hybrid-L1-Adaptive-Control-Applications-Of-Fuzzy-Modeling-Stochastic-Optimization-And-Metaheuristics-Roshni-Maiti-53-320.jpg)
Hybrid L1 Adaptive Control Applications Of Fuzzy Modeling Stochastic Optimization And Metaheuristics Roshni Maiti
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- 5. Studies in Systems, Decision and Control 422 Roshni Maiti Kaushik Das Sharma Gautam Sarkar Hybrid L1 Adaptive Control Applications of Fuzzy Modeling, Stochastic Optimization and Metaheuristics
- 6. Studies in Systems, Decision and Control Volume 422 Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
- 7. The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science. More information about this series at https://link.springer.com/bookseries/13304
- 8. Roshni Maiti · Kaushik Das Sharma · Gautam Sarkar Hybrid L1 Adaptive Control Applications of Fuzzy Modeling, Stochastic Optimization and Metaheuristics
- 9. Roshni Maiti Department of Applied Physics University of Calcutta Kolkata, West Bengal, India Gautam Sarkar Department of Applied Physics University of Calcutta Kolkata, West Bengal, India Kaushik Das Sharma Department of Applied Physics University of Calcutta Kolkata, West Bengal, India ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-97101-4 ISBN 978-3-030-97102-1 (eBook) https://doi.org/10.1007/978-3-030-97102-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 10. Dedicated to my Parents Smt. Kajal Maiti and Mr. Asit Kumar Maiti. Also dedicated to the beloved students. —Roshni Maiti
- 11. Preface Conventional control methodologies, viz., Proportional Integral Derivative (PID) controller, Linear Quadratic Regulator (LQR), etc., can control Linear Time Invariant (LTI)systems.Though,theirperformancesdegradewhennonlinearities,timevarying uncertainties, time varying disturbances, delays, etc., present into the system. Adap- tive controllers arise to tackle time varying uncertainties and disturbances. The most extensively used adaptive control scheme, viz., Model Reference Adaptive Controller (MRAC) utilizes low adaptation gain to maintain robustness. Hence, MRAC possess sluggish transient performance. With the aim to overcome such problem, L1 adaptive controller was introduced to provide fast transient performance with high robustness. However, the performance of basic L1 adaptive controller degrades when nonlinear- ities, delays, etc., are present in the system. On the other hand, nonlinearities can be properly modelled through universal approximator fuzzy logic. Moreover, the parameters of the controllers can be chosen by employing different stochastic opti- mization and metaheuristics techniques to assure optimal performance. A number of controller designing schemes can be hybridized to control practical systems consist of nonlinearities, time varying uncertainties, disturbances, cross-couplings, unmodelled dynamics, delays, etc., simultaneously. This book deals with the designing of hybrid control strategies to control practical systems containing time varying uncertainties, disturbances, nonlinearities, unknown parameters, unmodelled dynamics, delays, etc., concurrently. In this book, the advan- tages of different controllers are brought together to produce superior control perfor- mance for the practical systems. Being aware of the advantages of adaptive controller totackleunknownconstant,timevaryinguncertaintiesandtimevaryingdisturbances, a newly invented adaptive controller, namely L1 adaptive controller is hybridized with other strategies. The parameters of L1 adaptive controller should be chosen sensibly to maintain proper balance between good transient performance and high robustness. In this book, to facilitate optimal parameter setting of the basic L1 adaptive controller, stochastic optimization and metaheuristics techniques are hybridized with it. A variant of Harmony Search (HS) algorithm, viz., Local Best Harmony Search (lbestHS) algorithm, which is a metaheuristic technique, is employed to tune the parameter values of L1 adaptive controller. This method is termed as lbest HS based vii
- 12. viii Preface L1 (lbest HS-L1) adaptive controller. At first, the parameter values of L1 adap- tive controller are tuned utilizing lbest HS algorithm within the ranges obtained from mathematical calculation of L1 norm condition. Then, the unknown constant, time varying uncertainties and time varying disturbances are adapted concurrently following adaptation laws to obtain fine-tuned values. The stability of the meta- heuristic technique along with the controller is guaranteed analytically with the help of spectral radius convergence. This method exhibits satisfactory exploration and exploitation capabilities. Its results are compared with stochastic optimization, viz., PSO-based L1 adaptive controller. Again, this book throws light on tackling nonlinearities along with uncertainties and disturbances by hybridized fuzzy logic with L1 adaptive controller. In this case, the nonlinear system is designed by combining finite number of linear fuzzy systems. Fuzzy logic-based L1 adaptive controller is implemented for each linearized zone with the same premise part utilized to design the system. The conventional state feed- back controller of the basic L1 adaptive controller is substituted with fuzzy Parallel Distributed Compensation (PDC) controller which is a nonlinear state feedback controller. The adaptive fuzzy logic systems are designed from the fuzzy Lyapunov function which is the combination of zone-wise Lyapunov functions. The overall stability of the nonlinear system with this controller is guaranteed with the help of fuzzy Lyapunov function to retain the zonal behaviour of the system. The fuzzy PDC-L1 adaptive controller is efficient to tackle nonlinearities and at the same time cancels unknown constant, time varying uncertainties and disturbances adequately. The performances of these two controllers are compared with different control methodologies to validate their effectiveness. At first, two classical controllers, viz., Proportional Integral Derivative (PID) controller, Linear Quadratic Gaussian (LQG) are examined. Then, conventional adaptive controller, viz., Model Reference Adap- tive Controller (MRAC) and basic L1 adaptive controller are tested. After those, a stochastic optimization-based L1 adaptive controller, viz., Particle Swarm Opti- mization (PSO) based L1 (PSO-L1) adaptive, metaheuristic HS-based L1 (HS-L1) adaptive and metaheuristiclbest HS-based L1 (lbest HS-L1) adaptive controller are employed. Finally, fuzzy PDC-L1 adaptive controller is evaluated. In simulation environment, two Single-Input Single-Output (SISO) systems, viz., Duffing’s oscil- latory system and nonlinear spring mass damper system are examined. Then, one Single-Input Multi-Output (SIMO) system, viz., 4th order inverted pendulum with cart system and one Multi-Input Multi-Output (MIMO) system, viz., Twin Rotor MIMO System (TRMS) are investigated. In experimental case studies, speed control of an electrical actuator, angular position control of a Two Link Robot Manipu- lator (TLRM) and temperature control of a delay dependent air heater system are performed. The results show that, the disturbance rejection phenomenon of basic L1 adaptive controller is better than the classical controllers and Model Reference Adaptive Controller (MRAC). Though, the tracking performance of basic L1 adaptive controller is not satisfactory. In case of the lbest HS-L1 adaptive controller, transient as well as tracking performances improve due to the optimal parameter setting of L1 adaptive controller. The fuzzy PDC-L1 adaptive controller provides better steady state tracking performance by tackling nonlinearities through fuzzy logic-based PDC
- 13. Preface ix controller as well as fast transient performance by properly eradicating uncertainties and disturbances by means of fuzzy logic-based L1 adaptive controller. Therefore, the salient features of the methods presented in this book can be summarized as follows. I Designing of local best harmony search-based L1 (lbest HS-L1) adaptive controller. (a) A newly developed adaptive controller, viz., L1 adaptive controller is hybridized with a metaheuristic local neighbourhood variant of HS algo- rithm, viz., local best harmony search (lbest HS) algorithm to obtain optimal balance between fast transient performance and high robustness by eliminating uncertainties and disturbances. (b) SatisfactoryexplorationphenomenonofthelbestHS-L1 adaptivecontroller is proved analytically by means of increasing population variance with iterations. (c) Stability of the lbest HS-L1 adaptive controller is guaranteed through math- ematical analysis of exploitation phenomenon in terms of spectral radius convergence of iterative matrix. (d) Satisfactory transient and steady state performance of this method are guaranteed. II Designing of fuzzy parallel distributed compensation type L1 (fuzzy PDC-L1) adaptive controller. (a) Fuzzy PDC strategy is augmented with L1 adaptive controller to tackle nonlinearities, unmodelled dynamics, delays, as well as uncertainties and disturbances, present in the system. (b) Fuzzy adaptive rules are formulated to design different components of fuzzy PDC-L1 adaptive controller, viz., predictor, L1 adaptation laws, L1 control law, PDC control law and low-pass filter. (c) The overall stability of the fuzzy PDC-L1 adaptive controller is assured with the help of fuzzy Lyapunov function. (d) The transient state and steady state stable performance of the fuzzy PDC- L1 adaptive controller are guaranteed analytically. III Evaluation of the control strategies on four simulation case studies and three experimental case studies. (a) Different types of controllers, viz., PID, LQG, MRAC, basic L1 adaptive, PSO-L1 adaptive, HS-L1 adaptive are compared with lbest HS-L1 adap- tive and fuzzy PDC-L1 adaptive controller in simulation case studies. In simulation, at first, two SISO systems, viz., chaotic Duffing’s oscillatory system and nonlinear spring mass damper system are examined. Then, a SIMO, non-minimum phase, unstable 4th order inverted pendulum with cart system is investigated. After that, a nonlinear, cross-coupled twin rotor MIMO system is considered.
- 14. x Preface (b) The control strategies are employed efficiently on three experimental case studies, viz., speed control of electrical actuator, angular position control of two link robot manipulator and temperature control of delay dependent air heater system. The performances of the lbest HS-L1 adaptive and fuzzy PDC-L1 adaptive control methodologies are compared with PID, LQG, MRAC, basic L1 adaptive, PSO-L1 adaptive and HS-L1 adaptive controller. (c) The controllers are at first tuned for systems with no disturbance or small disturbance and then subjected to the systems with large disturbance without further tuning to examine the robustness of the controllers. The results demonstrate that, the lbest HS-L1 adaptive controller provides better tracking and disturbance rejection phenomenon than the clas- sical controllers, MRAC, basic L1 adaptive controller as well as other optimization- based L1 adaptive controllers. The fuzzy PDC-L1 adaptive controller provides fast transient performance, better steady state tracking performance and high robustness than all other control strategies tested. This book is composed of total eight number of chapters. This book is organized as follows. Part-I: Prologue Prologue contains introduction of the book in Chap. 1. In this chapter, the journey towards modern control theory is portrayed. The state of the art of the modern control theories are elaborated next with three sub-sections describing stochastic optimiza- tion and metaheuristics techniques; fuzzy logic systems; and L1 adaptive controller. Then, the literatures of hybrid L1 adaptive controller are articulated. The literatures of stability analysis of the controllers, optimization techniques are provided in the next section. The motivations of this book from the drawbacks of existing litera- tures are discussed next. Then, the proposals of the book and main contributions are discussed. At last, the structure of the book is provided followed by the summary of the chapter. Part-II: Preliminaries In preliminary part, Chap. 2 explains the motivations of designing basic L1 adap- tive controller. The architecture and formulation of basic L1 adaptive controller are elaborated with its stability analysis. The controller is implemented and then
- 15. Preface xi the designed controller is subjected to unknown disturbances. Four simulation case studiesareperformedemployingbasicL1 adaptivecontrollertoshowitseffectiveness in elimination of unknown time varying uncertainties and time varying disturbances, compared to the classical controllers and MRAC. Part-III: Design of Hybrid L1 Adaptive Controller This part comprises of two chapters containing the theoretical formulations of the designed methodologies, viz., stochastic optimization and metaheuristics-based L1 adaptive controller and fuzzy PDC-L1 adaptive controller. The effectiveness of these methodologies are demonstrated through simulation case studies in these chapters. Stochastic optimization and metaheuristics-based design of L1 adaptive controller is provided in Chap. 3. In this chapter, the lbest HS algorithm is utilized to tune the parameters of L1 adaptive controller. The values of unknown constant, time varying uncertainties and time varying disturbances are first obtained from the lbest HS opti- mization technique. Then, their values are concurrently adapted following L1 adapta- tion laws. The superior exploration capability of the lbest HS-L1 adaptive controller is demonstrated analytically. The stability of the lbest HS-L1 adaptive controller is also guaranteed in terms of exploitation through spectral radius convergence. This method is employed on four simulation case studies and compared with basic L1 adaptive controller to show its fruitfulness. Chapter 4 describes the designing of fuzzy PDC-L1 adaptive controller. L1 adap- tive controller can handle system with uncertainties and disturbances very well. On the other hand, fuzzy PDC logic can handle nonlinear system well enough. Augmenting the advantages of L1 adaptive controller to handle uncertainties, distur- bances and fuzzy PDC logic to tackle nonlinearities present in the system, fuzzy PDC-L1 adaptive controller is designed. The nonlinear system is considered as the fuzzy blending of finite number of linear fuzzy systems. For each linearized zone, L1 adaptive controller and fuzzy PDC controller are developed with the same premise part, used to design the system. Utilization of same premise part keeps the zonal behaviour of the system in controller design which provides better controlling for nonlinear system. The stability of this method is guaranteed by means of fuzzy Lyapunov function. This method is employed on four simulation case studies and compared with basic L1 adaptive controller and lbest HS-L1 adaptive controller to demonstrate its superiority. Part-IV: Applications This part contains the validation of the lbest HS-L1 adaptive and fuzzy PDC-L1 adaptive control methods on three experimental case studies, viz., speed control
- 16. xii Preface of electrical actuator, angular position control of two link robot manipulator and temperature control of delay dependent air heater system. In Chap. 5, the lbest HS-L1 adaptive and fuzzy PDC-L1 adaptive control methods are employed to control the speed of an electrical actuator, viz., DC motor experi- mental setup. The controllers are designed as a way that the DC motor can track a variable step trajectory and then the tuned controllers are subjected to the 100% load disturbance without further tuning. The experimental results show that, the lbest HS- L1 adaptive controller performs better than the basic L1 adaptive controller. The fuzzy PDC-L1 adaptive controller performs better than both basic L1 adaptive controller and lbest HS-L1 adaptive controller. Angular position control of a Two Link Robot Manipulator (TLRM) experi- mental setup is examined in Chap. 6. This laboratory setup of TLRM is the emulated version of large scale industrial TLRM system. The system modelling and controller designing for this laboratory scale setup is performed as a way that it can be repli- cated in case of large scale TLRM system. The parameter values of industrial large scale TLRM may be unknown or unmeasurable which leads to unknown dynamics of it. Therefore, the nonlinear TLRM system is modelled utilizing fuzzy PDC logic. Then, different controllers are developed for this TLRM system. Experimental results show that, the lbest HS-L1 adaptive controller performs better than the basic L1 adap- tive controller and the fuzzy PDC-L1 adaptive controller outperforms the basic L1 adaptive controller as well as lbest HS-L1 adaptive controller. Chapter 7 portrays the control of an air heater system which contains time varying uncertainties, time varying disturbances, nonlinearities and large delay. lbest HS- L1 adaptive and fuzzy PDC-L1 adaptive control methodologies show better perfor- mances than other control methodologies tested in this book. In this chapter, the obtained results show that, these two designed controllers are very much efficient in tackling system with nonlinearities, delays, time varying uncertainties, time varying disturbances simultaneously. Part-IV: Epilogue In this part, concluding remarks, significant features of the book are summarized with the future research directions. Chapter 8 draws the conclusion with brief description of obtained results. The effectiveness of the lbest HS-L1 adaptive and fuzzy PDC-L1 adaptive control methods and key features of the book are summarized. The future prospective of the current works are also discussed in this chapter. Appendices and index terms are provided thereafter.
- 17. Preface xiii Kolkata, India Roshni Maiti Kaushik Das Sharma Gautam Sarkar
- 18. Acknowledgements Heartfelt gratitude of the authors go towards Prof. Anjan Rakshit, Retired Professor, Department of Electrical Engineering, Jadavpur University for fabricating air heater experimental setup which is utilized in this work. Author would like to express her gratefulness to Prof. Jitendra Nath Bera, Electrical Engineering Section, Depart- ment of Applied Physics, University of Calcutta, for building up the driver circuit of the DC motor experimental setup which is used in this work. The authors would like to extend their special thanks to all the Professors and research scholars of the Department of Applied Physics for their moral support. Last but not the least, authors would like to express heart-felt gratitude towards their family members for their selfless help, unflinching support and incessant encouragement during this work. Kolkata, India September 2021 Roshni Maiti Kaushik Das Sharma Gautam Sarkar xv
- 19. Contents Part I Prologue 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Journey Towards Modern Control Theories . . . . . . . . . . . . . . . . . . . . . 3 1.2 Overview of Modern Control Methodologies . . . . . . . . . . . . . . . . . . . 4 1.2.1 Stochastic Optimization and Metaheuristics Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Fuzzy Logic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 L1 Adaptive Control Methodologies . . . . . . . . . . . . . . . . . . . . 6 1.3 State of the Art of Hybrid L1 Adaptive Control Methodologies . . . . 7 1.4 Overview of Stability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Aims and Scopes of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 Scopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Part II Preliminaries 2 Basic L1 Adaptive Controller: A State of the Art Study . . . . . . . . . . . . 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Motivation of Designing L1 Adaptive Controller . . . . . . . . . . . . . . . . 26 2.2.1 Proportional Integral Derivative Controller . . . . . . . . . . . . . . . 26 2.2.2 Linear Quadratic Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Model Reference Adaptive Controller . . . . . . . . . . . . . . . . . . . 28 2.3 Architecture of Basic L1 Adaptive Controller . . . . . . . . . . . . . . . . . . . 31 2.4 Stability Analysis of Basic L1 Adaptive Controller . . . . . . . . . . . . . . 35 2.5 Transient and Steady State Performance Analysis of Basic L1 Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 Simulation Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 xvii
- 20. xviii Contents 2.6.1 Case Study-I: Duffing’s Oscillatory System . . . . . . . . . . . . . . 40 2.6.2 Case Study-II: Nonlinear Spring Mass Damper System . . . . 45 2.6.3 Case Study III: Inverted Pendulum with Cart . . . . . . . . . . . . . 47 2.6.4 Case Study-IV: Twin Rotor MIMO System . . . . . . . . . . . . . . 53 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Part III Design of Hybrid L1 Adaptive Controller 3 Hybrid L1 Adaptive Controller-I: Stochastic Optimization and Metaheuristics Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Description of Optimization Techniques Used in This Book . . . . . . 70 3.2.1 Particle Swarm Optimization: A Stochastic Optimization Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.2 Harmony Search Algorithm: A Metaheuristics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.3 Local Best Harmony Search Algorithm: An Advanced Metaheuristics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3 Designing of Lbest HS-L1 Adaptive Controller . . . . . . . . . . . . . . . . . 77 3.4 Stability Analysis of Lbest HS-L1 Adaptive Controller . . . . . . . . . . . 79 3.5 Transient and Steady State Performance Analysis of Lbest HS-L1 Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.6 Simulation Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.6.1 Case Study-I: Duffing’s Oscillatory System . . . . . . . . . . . . . . 91 3.6.2 Case Study-II: Nonlinear Spring Mass Damper System . . . . 95 3.6.3 Case Study-III: Inverted Pendulum with Cart . . . . . . . . . . . . . 97 3.6.4 Case Study-IV: Twin Rotor MIMO System . . . . . . . . . . . . . . 102 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4 Hybrid L1 Adaptive Controller-II: Fuzzy Parallel Distributed Compensation Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 Linear Consequence Rule Based T-S Fuzzy System Design . . . . . . . 114 4.2.1 Nonlinear System Approximation Utilizing T-S Fuzzy System with Linear Consequence . . . . . . . . . . . . . . . . . . . . . . 114 4.2.2 Fuzzy Parallel Distributed Compensation (PDC) Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3 Linear Consequence Rule Based Fuzzy PDC-L1 Adaptive Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3.1 Fuzzy Logic Based Predictor Design . . . . . . . . . . . . . . . . . . . . 117 4.3.2 Fuzzy Logic Based Adaptive Laws Formulation for Unknown Constant, Time Varying Uncertainties and Time Varying Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3.3 Fuzzy Logic Based Control Law Design . . . . . . . . . . . . . . . . . 123
- 21. Contents xix 4.3.4 Fuzzy Logic Based Filter Design . . . . . . . . . . . . . . . . . . . . . . . 128 4.4 Stability Analysis of Fuzzy PDC-L1 Adaptive Controller . . . . . . . . . 131 4.5 Transient and Steady State Performance Analysis of Fuzzy PDC-L1 Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.6 Simulation Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.6.1 Case Study-I: Duffing’s Oscillatory System . . . . . . . . . . . . . . 144 4.6.2 Case Study-II: Nonlinear Spring Mass Damper System . . . . 147 4.6.3 Case Study-III: Inverted Pendulum with Cart . . . . . . . . . . . . . 150 4.6.4 Case Study-IV: Twin Rotor MIMO System . . . . . . . . . . . . . . 153 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Part IV Applications 5 Speed Control of Electrical Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2 Dynamical Model of Electrical Actuator . . . . . . . . . . . . . . . . . . . . . . . 164 5.3 Description of Experimental Setup of Electrical Actuator . . . . . . . . . 168 5.4 System Identification of Electrical Actuator . . . . . . . . . . . . . . . . . . . . 168 5.5 Experimental Case Study of Electrical Actuator . . . . . . . . . . . . . . . . . 171 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6 Angular Position Control of Two Link Robot Manipulator . . . . . . . . . 181 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.2 Dynamical Model of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.3 Description of Experimental Setup of TLRM . . . . . . . . . . . . . . . . . . . 186 6.4 System Identification of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.5 Experimental Case Study of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7 Temperature Control of Air Heater System . . . . . . . . . . . . . . . . . . . . . . . 199 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.2 Dynamical Model of Air Heater System . . . . . . . . . . . . . . . . . . . . . . . 200 7.2.1 Padé Approximation Based Delay Modelling . . . . . . . . . . . . 203 7.2.2 Fuzzy Linear Consequence Rule Based Delay Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.3 Description of Experimental Setup of Air Heater System . . . . . . . . . 206 7.4 System Identification of Air Heater System . . . . . . . . . . . . . . . . . . . . . 207 7.5 Experimental Case Study of Air Heater System . . . . . . . . . . . . . . . . . 209 7.5.1 Experiment-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.5.2 Experiment-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
- 22. xx Contents Part V Epilogue 8 Future Research Directions of Hybrid Controller . . . . . . . . . . . . . . . . . . 223 8.1 Significance of the Methodologies Presented in This Book . . . . . . . 223 8.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Appendix A: Norms of Vector and Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Appendix B: Spectral Radius Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
- 23. About the Authors Roshni Maiti has completed her Ph.D. (Tech.) in Elec- trical Engineering from Department of Applied Physics, University of Calcutta, India in 2021 with University Research Fellowship. She received her B.Tech. and M.Tech. degrees in Electrical Engineering from the West Bengal University of Technology and University of Calcutta, India in 2012 and 2014 respectively. Her research interests include adaptive controller design, controller design for non-linear system, fuzzy controller design, stochastic optimization techniques (GA, PSO), metaheuristics techniques (HS), Control applications in robotics, etc. She has authored 5 SCI journals including 2 IEEE Transactions journal and many conferences. She actively serves as reviewer of different journals like IEEE Transactions on Circuits and Systems I: Regular Papers, IEEE Transactions on Neural Networks and Learning Systems, ISA Transac- tions, Measurement and Control (SAGE journals) etc. She is a member of IEEE (USA). xxi
- 24. xxii About the Authors Kaushik Das Sharma received his B.Tech and M.Tech degrees in Electrical Engineering from the University of Calcutta, India, in 2001 and 2004 respectively and Ph.D. (Engineering) degree from the Jadavpur Univer- sity, India in 2012. Presently he is Professor in Electrical Engineering Section, Department of Applied Physics, University of Calcutta, India. His research interests include fuzzy control system design, stochastic optimization applications, robotics etc. He has published more than 60 research arti- cles in international and national journals or confer- ences. He has authored a book titled ‘Intelligent Control: A Stochastic Optimization Based Adaptive FuzzyApproach’,SpringerNature,Singaporepublisher, 2018. He is a senior member of IEEE (USA), and life member of The Indian Science Congress Association (Engineering Section). Gautam Sarkar received his B.Tech, M.Tech and Ph.D. degrees from the University of Calcutta, India, in 1975, 1977 and 1991, respectively. He has retired as LD Chair Professor in Electrical Engineering Section, Department of Applied Physics, University of Calcutta, India. His research interests include control system design, smart grid technologies etc. He has published more than 80 research articles in international and national journals or conferences.
- 25. Symbols A0 Open loop system matrix Am Closed loop system matrix b Input matrix c Output matrix x(t) = [x1 x2 . . . xn]T ∈ n System state u1(t) L1 adaptive control signal u2(t) State feedback control signal r Input to the system y Output of the system e = r − y Tracking error ω Unknown constant θ Uncertainties σ Disturbances k Pre-filter gain kg Feed-forward gain K State feedback gain C Adaptation gain x̂ Predictor state ŷ Predictor output C(s) Low-pass filter D(s) Strictly proper stable transfer function Z Candidate solution vector ξ Fuzzy basis function t Time h Time step V Lyapunov function Barbalat’s lemma operator xxiii
- 26. Acronyms bw Bandwidth GMCR Group memory considering rate HM Harmony memory HMCR Harmony memory considering rate HS Harmony search IAE Integral absolute error lbest HS Local best harmony search MIMO Multi-input multi-output PAR Pitch adjustment rate PDC Parallel distributed compensation PI Performance index PSO Particle swarm optimization SIMO Single-input multi-output SISO Single-input single-output TLRM Two link robot manipulator TRMS Twin rotor MIMO system T-S Takagi-Sugeno xxv
- 27. List of Figures Fig. 1.1 Layout of the book proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Fig. 2.1 Architecture of basic L1 adaptive controller . . . . . . . . . . . . . . . . . 31 Fig. 2.2 Nature of the disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Fig. 2.3 Nature of 10 dB white Gaussian noise . . . . . . . . . . . . . . . . . . . . . . 41 Fig. 2.4 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Fig. 2.5 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for LQG . . . . . . 43 Fig. 2.6 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for MRAC . . . . . 43 Fig. 2.7 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Fig. 2.8 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Fig. 2.9 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Fig. 2.10 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 xxvii
- 28. xxviii List of Figures Fig. 2.11 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Fig. 2.12 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Fig. 2.13 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for LQG . . . . . 46 Fig. 2.14 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for MRAC . . . 46 Fig. 2.15 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Fig. 2.16 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Fig. 2.17 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Fig. 2.18 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Fig. 2.19 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Fig. 2.20 Schematic diagram of inverted pendulum with cart . . . . . . . . . . . 49 Fig. 2.21 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Fig. 2.22 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for LQG . . . . 50
- 29. List of Figures xxix Fig. 2.23 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Fig. 2.24 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Fig. 2.25 Evaluation period a system response and corresponding b control signal with tracking error of III (inverted pendulum with cart) with disturbance for PID controller . . . . . . . 51 Fig. 2.26 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for LQG . . . . . . 52 Fig. 2.27 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for MRAC . . . . . 52 Fig. 2.28 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Fig. 2.29 Schematic diagram of twin rotor MIMO system . . . . . . . . . . . . . 53 Fig. 2.30 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for PID controller . . . . . . . . . . . . . . . . . . . . . 56 Fig. 2.31 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Fig. 2.32 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Fig. 2.33 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for basic L1 adaptive controller . . . . . . . . . . 59
- 30. xxx List of Figures Fig. 2.34 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for PID controller . . . . . . . . . . . . . . . . . . . . . . 60 Fig. 2.35 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Fig. 2.36 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Fig. 2.37 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for basic L1 adaptive controller . . . . . . . . . . . 63 Fig. 3.1 Flowchart representation of PSO algorithm . . . . . . . . . . . . . . . . . 72 Fig. 3.2 Flowchart representation of HS algorithm . . . . . . . . . . . . . . . . . . 74 Fig. 3.3 Flowchart representation of lbest HS algorithm . . . . . . . . . . . . . . 75 Fig. 3.4 Flowchart representation of a number of group selection, b harmony selection into groups, c update of harmony . . . . . . . . 76 Fig. 3.5 Architecture of lbest HS-L1 adaptive controller . . . . . . . . . . . . . . 78 Fig. 3.6 Flowchart representation of lbest HS-L1 adaptive controller . . . . 79 Fig. 3.7 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Fig. 3.8 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Fig. 3.9 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Fig. 3.10 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
- 31. List of Figures xxxi Fig. 3.11 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Fig. 3.12 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for PSOL1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Fig. 3.13 Evaluation period a system response and corresponding b control signal with tracking error plot of case study-I (Duffing’s oscillatory system) with white Gaussian noise for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Fig. 3.14 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . 94 Fig. 3.15 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Fig. 3.16 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Fig. 3.17 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Fig. 3.18 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Fig. 3.19 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Fig. 3.20 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Fig. 3.21 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
- 32. xxxii List of Figures Fig. 3.22 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . 98 Fig. 3.23 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Fig. 3.24 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Fig. 3.25 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Fig. 3.26 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Fig. 3.27 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Fig. 3.28 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Fig. 3.29 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Fig. 3.30 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Fig. 3.31 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for basic L1 adaptive controller . . . . . . . . . . 104
- 33. List of Figures xxxiii Fig. 3.32 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for PSO-L1 adaptive controller . . . . . . . . . . . 105 Fig. 3.33 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for HS-L1 adaptive controller . . . . . . . . . . . . 106 Fig. 3.34 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for lbest HS-L1 adaptive controller . . . . . . . 107 Fig. 3.35 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance- I for basic L1 adaptive controller . . . . . . . . . . . 108 Fig. 3.36 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance- I for PSO-L1 adaptive controller . . . . . . . . . . . 109 Fig. 3.37 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance- I for HS-L1 adaptive controller . . . . . . . . . . . . 110 Fig. 3.38 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance- I for lbest HS-L1 adaptive controller . . . . . . . . 111 Fig. 4.1 Architecture of fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . 116 Fig. 4.2 Linear consequence rule based zone-wise fuzzy PDC-L1 adaptive control law design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Fig. 4.3 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
- 34. xxxiv List of Figures Fig. 4.4 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Fig. 4.5 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with disturbance for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Fig. 4.6 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Fig. 4.7 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . 146 Fig. 4.8 Evaluation period a system response and corresponding b control signal with tracking error of case study-I (Duffing’s oscillatory system) with white Gaussian noise for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . 147 Fig. 4.9 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Fig. 4.10 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Fig. 4.11 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with disturbance for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Fig. 4.12 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Fig. 4.13 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . 149 Fig. 4.14 Evaluation period a system response and corresponding b control signal with tracking error of case study-II (nonlinear spring mass damper) with white Gaussian noise for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . 150
- 35. List of Figures xxxv Fig. 4.15 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Fig. 4.16 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Fig. 4.17 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) without disturbance for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . 152 Fig. 4.18 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Fig. 4.19 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Fig. 4.20 Evaluation period a system response and corresponding b control signal with tracking error of case study-III (inverted pendulum with cart) with disturbance for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Fig. 4.21 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for basic L1 adaptive controller . . . . . . . . . . 155 Fig. 4.22 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for lbest HS-L1 adaptive controller . . . . . . . 155 Fig. 4.23 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) without disturbance for fuzzy PDC-L1 adaptive controller . . . . . 156 Fig. 4.24 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for basic L1 adaptive controller . . . . . . . . . . . 157
- 36. xxxvi List of Figures Fig. 4.25 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for lbest HS-L1 adaptive controller . . . . . . . . 158 Fig. 4.26 Evaluation period a pitch angle and corresponding b control signal with tracking error, c yaw angle and corresponding d control signal with tracking error of case study-IV (twin rotor MIMO system) with Disturbance-I for fuzzy PDC-L1 adaptive controller . . . . . . 158 Fig. 5.1 Experimental model of electrical actuator . . . . . . . . . . . . . . . . . . . 164 Fig. 5.2 Schematic diagram of electrical actuator . . . . . . . . . . . . . . . . . . . . 165 Fig. 5.3 Experimental setup of electrical actuator for speed control . . . . . 169 Fig. 5.4 Circuit layout of electrical actuator experimental setup . . . . . . . . 170 Fig. 5.5 a Input signal and corresponding b system output plot in open loop configuration for parameter estimation of electrical actuator experimental setup . . . . . . . . . . . . . . . . . . . . 170 Fig. 5.6 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for PID controller . . . . . . . . 172 Fig. 5.7 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for LQG . . . . . . . . . . . . . . . . 172 Fig. 5.8 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for MRAC . . . . . . . . . . . . . . 173 Fig. 5.9 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Fig. 5.10 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Fig. 5.11 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Fig. 5.12 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
- 37. List of Figures xxxvii Fig. 5.13 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-I for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Fig. 5.14 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for PID controller . . . . . . . . 175 Fig. 5.15 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for LQG . . . . . . . . . . . . . . . 175 Fig. 5.16 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for MRAC . . . . . . . . . . . . . 175 Fig. 5.17 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Fig. 5.18 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Fig. 5.19 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Fig. 5.20 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Fig. 5.21 Evaluation period a system response and corresponding b control signal with tracking error of electrical actuator experimental setup with reference-II for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Fig. 6.1 Experimental model of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Fig. 6.2 Schematic diagram of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Fig. 6.3 Experimental setup of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Fig. 6.4 Circuit layout of TLRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Fig. 6.5 a Input signal and corresponding b system output plot of first joint angle, c input signal and corresponding d system output plot of second joint angle in open loop configuration for parameter estimation of TLRM experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
- 38. xxxviii List of Figures Fig. 6.6 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Fig. 6.7 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Fig. 6.8 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Fig. 6.9 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for basic L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Fig. 6.10 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for PSO-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Fig. 6.11 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Fig. 6.12 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . 196 Fig. 6.13 Evaluation period a first joint angle response and corresponding b control signal with tracking error, c second joint angle response and corresponding d control signal with tracking error of TLRM experimental setup for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . 197 Fig. 7.1 Experimental model of air heater system . . . . . . . . . . . . . . . . . . . 201 Fig. 7.2 Front side view of a inlet, and b outlet of the air flow duct . . . . . 201 Fig. 7.3 Schematic diagram of air heater system . . . . . . . . . . . . . . . . . . . . 202 Fig. 7.4 Experimental setup of air heater system . . . . . . . . . . . . . . . . . . . . 206 Fig. 7.5 Layout of the driver circuit of air heater system . . . . . . . . . . . . . . 207
- 39. List of Figures xxxix Fig. 7.6 a Input signal and corresponding open loop system output plot of b Padé and c fuzzy PDC logic based delay modelling for parameter estimation of air heater experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Fig. 7.7 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for PID controller . . . . . . . . . . . . . . . . . . . . . . 210 Fig. 7.8 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Fig. 7.9 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Fig. 7.10 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for basic L1 adaptive controller . . . . . . . . . . . 211 Fig. 7.11 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for PSO-L1 adaptive controller . . . . . . . . . . . 211 Fig. 7.12 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for HS-L1 adaptive controller . . . . . . . . . . . . . 212 Fig. 7.13 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for lbest HS-L1 adaptive controller . . . . . . . . 212 Fig. 7.14 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 1 of air heater experimental setup for fuzzy PDC-L1 adaptive controller . . . . . . 213 Fig. 7.15 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for PID controller . . . . . . . . . . . . . . . . . . . . . . 213 Fig. 7.16 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for LQG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Fig. 7.17 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for MRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Fig. 7.18 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for basic L1 adaptive controller . . . . . . . . . . . 214 Fig. 7.19 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for PSO-L1 adaptive controller . . . . . . . . . . . 214
- 40. xl List of Figures Fig. 7.20 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for HS-L1 adaptive controller . . . . . . . . . . . . . 215 Fig. 7.21 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for lbest HS-L1 adaptive controller . . . . . . . . 215 Fig. 7.22 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup for fuzzy PDC-L1 adaptive controller . . . . . . 215 Fig. 7.23 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup with fixed disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Fig. 7.24 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup with fixed disturbance for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Fig. 7.25 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup with variable disturbance for lbest HS-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Fig. 7.26 Evaluation period a system response and corresponding b control signal with tracking error at Sensor 4 of air heater experimental setup with variable disturbance for fuzzy PDC-L1 adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
- 41. List of Tables Table 2.1 Comparative study of different control methodologies for case study-I (Duffing’s oscillatory system) . . . . . . . . . . . . . . . 42 Table 2.2 Comparative study of different control methodologies for case study-II (nonlinear spring mass damper system) . . . . . . 45 Table 2.3 Parameter values of inverted pendulum with cart . . . . . . . . . . . . . 49 Table 2.4 Comparative study of different control methodologies for case study-III (inverted pendulum with cart) . . . . . . . . . . . . . 53 Table 2.5 Parameter values of twin rotor MIMO system . . . . . . . . . . . . . . . 55 Table 2.6 Comparative study of different control methodologies for case study-IV (twin rotor MIMO system) without disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Table 2.7 Comparative study of different control methodologies for case study-IV (twin rotor MIMO system) with Disturbance-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Table 3.1 Comparative study of different control methodologies for case study-I (Duffing’s oscillatory system) . . . . . . . . . . . . . . . 91 Table 3.2 Comparative study of different control methodologies for case study-II (nonlinear spring mass damper) . . . . . . . . . . . . 95 Table 3.3 Comparative study of different control methodologies for case study-III (inverted pendulum with cart) . . . . . . . . . . . . . 99 Table 3.4 Comparative study of different control methodologies for case study-IV (twin rotor MIMO system) without disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Table 3.5 Comparative study of different control methodologies for case study-IV (twin rotor MIMO system) with Disturbance-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Table 4.1 Comparative study of different control methodologies for case study-I (Duffing’s oscillatory system) . . . . . . . . . . . . . . . 145 Table 4.2 Comparative study of different control methodologies for case study-II (nonlinear spring mass damper) . . . . . . . . . . . . 147 xli
- 42. xlii List of Tables Table 4.3 Comparative study of different control methodologies for case study-III (inverted pendulum with cart) . . . . . . . . . . . . . 151 Table 4.4 Comparative study of different control methodologies for case study-IV (twin rotor MIMO system) without disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Table 4.5 Comparative study of different control methodologies for case study-IV (twin rotor MIMO system) with Disturbance-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Table 5.1 Estimated nominal parameter values of DC motor . . . . . . . . . . . . 170 Table 5.2 Comparative study of different control methodologies for electrical actuator experimental setup . . . . . . . . . . . . . . . . . . . 172 Table 6.1 Comparative study of different control methodologies for TLRM experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Table 7.1 Estimated nominal parameter values of air heater system with Padé approximation of delay . . . . . . . . . . . . . . . . . . . . . . . . . 209 Table 7.2 Comparative study of different control methodologies in experiment-I for delay dependent air heater experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Table 7.3 Comparative study of different control methodologies in experiment-II for delay dependent air heater experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
- 43. Part I Prologue
- 44. Chapter 1 Introduction Abstract This chapter introduces the book with brief description of the journey towards modern control theories. Motivations of the book are portrayed owing to the drawbacks of present literatures. The designed methodologies are described briefly with an emphasis on contributions of the book. Then, the structure of the book is provided. 1.1 Journey Towards Modern Control Theories Classical controllers, viz., proportional integral derivative (PID) controller [1–3], state feedback controller [4], observer based state feedback controller [5], output feedbackcontroller[6,7],linearquadraticregulator(LQR)[8],linearquadraticGaus- sian (LQG) [8, 9], etc., are suitable for linear time invariant (LTI) systems. Though, they yield unsatisfactory control performances for systems with time varying quanti- ties [10]. In various literatures, different modifications were furnished over classical controllers to overcome their drawbacks. LQG controller was augmented with vibra- tion compensator to suppress the vibration of a piezoelectric tube actuator [11]. LQG was also combined with proportional integral (PI) controller to nullify steady state error [12]. Grid voltage disturbance was tackled with the help of frequency-adaptive multi-resonator augmented LQG current controller [13]. Two separately designed PID controllers were brought together with resonant controller to suppress the current harmonic components [14]. Data driven controller was augmented with d-step ahead prediction optimal controller and PID controller to control a pulp neutralization process by compensating unmodelled dynamics [15]. To tune the gains of the PID controller, different improved strategies were adopted rather than the Ziegler-Nichols method [10] in different literatures. Variable gains of PID controller were designed for a linear model of unmanned aerial vehicle (UAV) [16]. Relay feedback tuning method was also utilized to design the gains of the PID controller [17]. However, the performances of the classical controllers were unsatisfactory to tackle systems with uncertainties, disturbances, time varying parameters, nonlinearities and delays [18]. To overcome such drawbacks, modern control theories draw attention. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 R. Maiti et al., Hybrid L1 Adaptive Control, Studies in Systems, Decision and Control 422, https://doi.org/10.1007/978-3-030-97102-1_1 3
- 45. 4 1 Introduction 1.2 Overview of Modern Control Methodologies Proper estimations of time varying uncertainties and disturbances are required to eliminate these quantities, which can be obtained through adaptive controllers [19, 20]. To eliminate the drawbacks of conventional adaptive controller, a newly invented adaptive controller, viz., L1 adaptive controller comes into place and increases its applications in different fields. The parameters of different controllers are tuned employing stochastic optimization and metaheuristics techniques to obtain optimized performance of the system under control. In different literatures, fuzzy logic systems are utilized to approximate nonlinearities [21], to produce controller by approxi- mating uncertainties [22], disturbances [23], unmodelled dynamics [24], etc. Liter- ature surveys of these modern control methodologies are provided in the following sections. 1.2.1 Stochastic Optimization and Metaheuristics Techniques Arbitrarily chosen parameter values of controllers, within the ranges obtained from the stability condition, do not guarantee optimal performance. Optimization tech- niques acquire great attention to choose the optimal parameter values of controller during last two decades. Different optimization techniques, viz., genetic algorithm (GA) [25, 26], particle swarm optimization (PSO) [27–29], ant colony optimization (ACO) [30, 31], firefly (FF) algorithm [32–34], cuckoo search (CS) [35, 36], artificial bee colony (ABC) [37, 38], harmony search algorithm (HS) [39, 40] were used in different literatures to obtain optimized performance of the controllers. It is observed from the literatures that, the computational cost reduces considerably in case of harmony search algorithm compared to other optimization techniques [39]. Involve- ment of lesser number of internal parameters and their easy selections increase the use of HS algorithm in different optimization problems, viz., pipe network designing [41], vehicle routing [42], structural designing [43], etc. Different parameter modi- fications of the HS algorithm provides the easy enhancement of exploration and exploitation capability of it. The advantages of HS algorithm over other optimization process increase its utilization gradually as elaborated follows. 1.2.1.1 Harmony Search Algorithm HS resembles with the group of musicians playing pitches with the aim to develop an aesthetic harmony all together [40]. Being beneficial than other optimization techniques, the utilization of HS algorithm increases gradually. The HS algorithm was successfully utilized in low impact development and management of urban storm water systems to avoid flooding risk and assure environmental interventions [44]. The HS algorithm performed better than random sample consensus algorithm to detect
- 46. 1.2 Overview of Modern Control Methodologies 5 vanishingpointpresentattheextendedlinesofroadlanesforself-drivingautonomous vehicle [45]. In power system engineering, HS was employed in different literatures [46–50]. A distributed-generation (DG) system was controlled through proportional integral (PI) controller whose gains were tuned by means of HS algorithm [51]. The parameters of fuzzy controller were tuned by utilizing HS for air heater system with transport-delay [52]. HS was also applied in medical fields to RNA secondary structure prediction [53, 54], medical physics [55], hearing AIDS [56] etc. It is of great importance to balance between exploration and exploitation capa- bility of HS algorithm [57]. Good exploration phenomenon indicates finding more search spaces which avoids local minima trapping problem. Good exploitation signi- fies proper convergence of the optimization process [58]. Different modifications of basic HS algorithm were manifested by designing its parameter values in different ways to obtain optimal balance between diversification and intensification of the optimization technique [59]. Dynamic pitch adjustment rate (PAR) [60], dynamic bandwidth (bw) selection formulas [60, 61] were developed in different literatures. Gaussian distribution was utilized to formulate the PAR with the aim to improvise exploration at first and then exploitation at the end of the harmony improvisation process of HS [62]. The HS algorithm was hybridized with hill climbing and global best particle swarm optimization to improve local exploitation and global conver- gence respectively which was employed to design university course timetable [63]. Basic HS algorithm was also combined with wavelet mutation and new harmony was produced based on roulette wheel mechanism to improve convergence speed, accuracy and robustness for economic load dispatch (ELD) problem [64]. Differen- tial mutation in pitch adjustment rate was incorporated to solve job shop scheduling problem [65]. Position update formula of PSO and mutation phenomenon of GA replaced the harmony memory considering rate (HMCR) and pitch adjustment rate (PAR) parameters of HS respectively in [66]. A detail review of harmony search algorithm, its developments and applications can be found in [67] and the references therein. To improvise the exploitation capability of basic HS algorithm, local search based phenomenon was incorporated in it [68–70]. Literatures show that, HS algorithm and its different modifications can be used efficiently to solve many problems. It can be used to tune the parameter values of the controller to provide optimize performance of the system. However, these modified HS may not find the proper system matrix values to model unmodelled dynamics, nonlinearities, delays, etc. This phenomenon can be addressed properly employing fuzzy logic. Literature reviews of fuzzy logic systems to model nonlinear systems, unmodelled dynamics, delays, etc., are elaborated in the next section. 1.2.2 Fuzzy Logic Systems To tackle nonlinearities, delays, unmodelled dynamics with the uncertainties and disturbances, fuzzy logic based approaches gain much attention [71]. Uncertainties and disturbances can be nullified through adaptive controllers but they fail to provide
- 47. 6 1 Introduction satisfactoryperformanceincaseofnonlinearities,delays,unmodelleddynamics,etc., present in the system simultaneously. Universal approximator fuzzy logic system [72] was used extensively to tackle nonlinearities [73–77] during last few decades. Fuzzy logic system can be utilized to approximate system models with nonlinearities, unmodelled dynamics, uncertainties [78, 79] as well as to design different types of controllers [80–84]. Approximation of nonlinear system and developing controller for that system should be synchronized. With this aim, fuzzy parallel distributed compensation (PDC) type controller was introduced [85, 86] and utilized in many applications [87–90]. The nonlinear system can be approximated by fuzzy parallel distributed compensation logic [91–94]. The nonlinear system is considered as the fuzzy blending of finite number of linear systems. Then, for each linearized zone, linear state feedback controller is designed and fuzzy augmented to produce overall control action [95–97]. This is known as PDC control law which is nonlinear in nature and can handle nonlinear systems efficiently [98, 99]. At first, fuzzy PDC logic is employed to design the system and then same premise part is utilized to generate PDC control law. Fuzzy PDC control law was applied in different cases, viz., atti- tude control of spacecraft [99], speed control of surface-mounted permanent magnet synchronous motor [100], wind energy system control [101], maximum power point tracking control of solar photovoltaic electricity generation system [102], nonlinear vehicle dynamics control [103], etc. Yet, the performance of PDC controller degrades when uncertainties and disturbances appear in the system [101]. Fuzzy controller can be efficiently hybridized with other controllers to tackle different quantities consequently. To tackle uncertainties and disturbances, a special type of adaptive controller, viz., L1 adaptive controller gains much attention during few years. The literature reviews of the L1 adaptive controller are provided in the following section. 1.2.3 L1 Adaptive Control Methodologies Systems with time varying uncertainties, disturbances can be controlled through adaptive controllers [104–106]. Conventional adaptive controller utilizes low adap- tation gain to retain the robustness of the system [107]. However, utilization of low adaptation gain possesses sluggish transient performance [108]. To overcome this drawback in 2006, L1 adaptive controller was proposed with high adaptation gain to incur quick transient performance and a low-pass filter is attached in the control channel to retain robust performance [109–111]. L1 adaptive controller comprises of predictor, adaptation laws, state feedback control action, L1 control law and low-pass filter. Predictor predicts the unknown constant, time varying uncertainties and time varying disturbances present in the system and their values are adapted following the adaptation laws. From these adaptive estimates, L1 adaptive control law is formulated which contains high frequency component. These high frequencies, may yield insta- bility of the system, are nullified by a low-pass filter attached after the control block. The filtered output is augmented with state feedback control signal and provided to
- 48. 1.2 Overview of Modern Control Methodologies 7 the system under control. Designing of these components and selection of param- eters from L1 norm condition were elaborated in the seminal papers of L1 adap- tive controller, published in 2006 by Chengyu Cao and Naira Hovakimyan [107, 108]. Despite of utilizing high adaptation gain, the guaranteed transient performance was analyzed in [109–111]. The exponential decay of non-zero trajectory initial- ization error, transient and steady state system performances with respect to the reference system were also analyzed by Cao and Hovakimyan [112]. Stability anal- ysis of L1 adaptive controller was performed in digital mode [113], linear matrix inequality based approach [114, 115], etc. L1 adaptive controller can efficiently handles systems with unknown time varying parameters [112], unmatched uncer- tainties [116, 117], unmatched disturbances [118], unmodelled dynamics [119], etc. Nonlinear systems with uncertainties and disturbances were also controlled through L1 adaptive controller in different literatures [120–123]. Though, consideration of uncertainties, disturbances, nonlinearities, unmodelled dynamics, delays, etc., simul- taneously is quite difficult and missing in the literatures, as far as author’s belief and knowledge goes. Advantages of L1 adaptive controller made it so prevalent that, its applications grow very fast and spread to different fields. L1 adaptive controller was successfully employed to control different types of flight systems, viz., aircraft [124, 125], wing- rock [126], X-wing tail-sitter micro aerial vehicle (MAV) [127], unmanned aerial vehicle (UAV) [128, 129], fighter aircraft [130], autopilot [131, 132]. Pitch break uncertainty as well as actuator failure of an unmanned military tailless unstable aircraft were controlled through L1 adaptive controller and compared with model reference adaptive controller (MRAC) to show the effectiveness of L1 adaptive controller [133]. L1 adaptive controller was used to control magnetic torque coil in attitude control system of a Pico-scale satellite test bed [134]. Depth and pitch angle of a multi-input multi-output (MIMO) submarine system was controlled through this controller [135]. Wind turbine was controlled by regulating the speed of a generator for maximum power point tracking through the L1 adaptive controller [136]. Precise controlling phenomenon of L1 adaptive controller extended its application in medical field for delivering anaesthesia to the patient during surgery [137]. To enhance the performance of L1 adaptive controller, it is augmented with other strategies which are elaborated in the subsequent section. 1.3 State of the Art of Hybrid L1 Adaptive Control Methodologies Practical systems contain nonlinearities, time varying structured and unstructured uncertainties, disturbances, unknown constants, cross-couplings, delays, etc. To tackle all of these quantities simultaneously, different controllers are hybridized together. L1 adaptive controller provides fast transient performance as well as retains
- 49. 8 1 Introduction high robustness to tackle uncertainties and disturbances. Arbitrary parameter selec- tion, within the ranges of L1 norm condition, does not guarantee optimal perfor- mance of it, although ensures the stability of the closed loop system. Hence, proper balance between quick transient performance and high robustness should be main- tained by designing the components and selecting the parameter values of L1 adap- tive controller appropriately. Different strategies were applied to design its different components. A greedy randomized optimization with multi-criteria was utilized to design the filter of L1 adaptive controller [138]. Fuzzy logic was employed to design the pre-filter gain of L1 adaptive controller where the filter structure was kept fixed [139]. Adaptive estimates of uncertainties, in designing L1 adaptive controller, were also approximated utilizing fuzzy logic system for electropneumatic actuator [140] and wind energy conversion systems [141]. Being beneficial in controlling systems withtimevaryinguncertaintiesanddisturbanceswhichpresentinpracticalsystem,L1 adaptive controller was combined with other control strategies in many cases. Base- line dynamic inversion controller was augmented with L1 adaptive controller to elim- inate time-varying uncertainty of a combination of Novlit-3 micro aerial vehicle and a novel non-orthogonal X shaped wing layout with a single propeller system [142]. Dynamic inversion based L1 adaptive controller was implemented for missile [143], generic transport model [144], etc. L1 adaptive controller was augmented with differ- ential proportional-integral (PI) baseline controller to tackle longitudinal dynamics of a F16 aircraft [145]. L1 adaptive back-stepping controller was designed for unmanned aerial vehicles (UAVs) with position kinematics and dynamics present in strict feed- back form [146]. PID controller was appended with L1 adaptive controller to obtain proper tracking by eliminating the time lag for depth control of an under-actuated underwater vehicle [147]. Though in different literatures, different components of the L1 adaptive controller were designed through optimization technique or utilizing fuzzy logic but the entire controller structure design was absent there, as far as author’s knowledge goes. In case of designing L1 adaptive controller for systems consisting of uncertainties, disturbances and nonlinearities, separate consideration of nonlinearities were also absent in the literatures. After modelling the system and developing controller for that system, it is impor- tant to analyze the stability conditions of it. There are different ways to investigate the stability conditions of stochastic optimization and metaheuristics techniques, fuzzy logic controllers, etc., whose literature surveys are provided in the subsequent section. 1.4 Overview of Stability Conditions It is essential to investigate the exploration (/intensification) and exploitation (/diver- sification) capabilities in case of any optimization process. Good balance between exploration and exploitation shows that, the optimization process searches huge spaces with the aim to converge at the end of the process without diverging from the
- 50. 1.4 Overview of Stability Conditions 9 stability region. Owing to the easy implementation and lower computational time of HS, in this book, the literatures of stability analysis of HS algorithm are provided. It is important to analyze mathematically the exploration and exploitation capabilities of the HS algorithm which was performed in [148]. The authors of [148] showed that, the population variance of the HS algorithm increased gradually with the itera- tion which exhibited its better exploration capability. They also proved the spectral radius convergence of the iterative matrix of the HS algorithm. This demonstrated the better exploitation capability of the HS algorithm [148]. Stability analysis of different modified HS algorithm was also provided in different literatures [149]. Mathematical analysis of the exploration and exploitation capabilities of the local search based HS (lbest HS) algorithm was absent in the literatures to the best of author’s understanding. The stochastic optimization and metaheuristics techniques were utilized to provide the optimal parameter setting of the controllers. Therefore, it is important to analyze the stability condition of the optimization techniques along with the controller which was also not present in the literatures as per as author’s knowledge goes. In another case, for nonlinear system, a set of stability conditions have to be derived which are difficult from a single Lyapunov function. In case of fuzzy PDC logic, multiple number of fuzzy rules are formulated and combined to model the system and controller [88, 150]. In case of such large number of fuzzy rules, use of quadratic Lyapunov function [151, 152] introduces conservative conditions [153, 154]. To reduce conservatism, different approaches were made to formulate the Lyapunov functions in different literatures, viz., piecewise Lyapunov function [153, 155–158], polynomial Lyapunov function [159, 160], multiple Lyapunov function [161–165]. Based on this multiple Lyapunov function approach, fuzzy Lyapunov function was introduced [166]. Fuzzy Lyapunov function is smooth in nature which is advantageous over piecewise Lyapunov function. This fuzzy Lyapunov function is developed by augmenting finite number of zone-wise Lyapunov functions where in each zone, positive definite matrix is present [166, 167]. The same premise part utilized to design the nonlinear system and controller is employed to formulate the fuzzy Lyapunov function [168–171]. From these literature surveys of the existing controllers, optimization techniques, fuzzy logic systems, etc., and their stability analysis, the motivations of designing the control methodologies presented in this book arise which are as follows. 1.5 Aims and Scopes of the Book 1.5.1 Aims System with uncertainties and disturbances can be tackled through adaptive controllers. Though, conventional adaptive controllers provide sluggish transient performance to retain robustness. To overcome such problem, in 2006, L1 adaptive
- 51. 10 1 Introduction controller originates with high adaptation gain to achieve quick transient perfor- mance and a low-pass filter to retain the robust performance [109]. Literature shows that, the components of L1 adaptive controller should be chosen sensibly within the L1 norm bound to obtain fast transient performance with high robustness [111]. Different components of L1 adaptive controller were designed utilizing different optimization techniques in different literatures [138, 139]. Nevertheless, designing the overall controller optimally is of great importance to achieve the accurate param- eter setting and optimal control performance of L1 adaptive controller. Again the literatures show that, HS and its different modifications can be efficiently utilized to choose the parameters of any controller. This motivates to design local best HS (lbest HS) based L1 adaptive controller and to analyze overall stability condition [172]. On the other hand, though nonlinear systems with uncertainties and disturbances are controlled through L1 adaptive controller but tackling nonlinearities separately was absent in the literatures [120–123]. Decentralized L1 adaptive controller was designed for large scale nonlinear system with unknown interconnection and unmod- elled dynamics with the help of decentralized passive identifiers [120]. Non-affine nonlinear systems, consist of unmeasured states, were controlled by implementing piece-wise continuous adaptive laws of L1 adaptive controller [121]. Nonlinear time varying reference system was also considered to control nonlinear system where an upper bound had to be considered additionally with complicated calculation which increased the conservatism [173, 174]. To deal with nonlinearities, in particular, fuzzy parallel distributed compensation (PDC) logic gained much attention in last two decades [175]. Motivated with these ideas, the L1 adaptive controller can be hybridized with fuzzy PDC controller to handle system with nonlinearities, unmod- elled dynamics, delays, uncertainties and disturbances all in conjunction. Stability of the overall system can be analyzed utilizing fuzzy Lyapunov function due to its less conservatism. 1.5.2 Scopes Motivatedfromtheaforementionedconcerns,inthisbook,theadvantagesofdifferent methodologies are hybridized to tackle different quantities simultaneously with the aim to produce superior control performances for the practical systems. Owing to the fast transient performance as well as high robustness, in this book, a special type of adaptive controller, viz., L1 adaptive controller is utilized. All the components of L1 adaptive controller are designed by tuning the parameters utilizing stochastic optimization and metaheuristics techniques to obtain overall optimal performance. Considering the advantages of HS algorithm and to enhance the exploration and exploitation capabilities, a variant of HS algorithm, viz., metaheuristic lbest HS is utilized in this book. The overall stability analysis of the concurrent lbest HS based L1 (lbest HS-L1) adaptive controller is also performed. On the other hand, in many literaturesL1 adaptivecontrollerwasemployedtotacklenonlinearsystemwithuncer- tainties and disturbances where nonlinearity was not considered in a separate way. To
- 52. 1.5 Aims and Scopes of the Book 11 tackle uncertainties, disturbances as well as nonlinearities, delays, etc., in this book, fuzzy logic is utilized to design PDC controller augmented L1 adaptive controller. This method is termed as fuzzy parallel distributed compensation type L1 (fuzzy PDC-L1) adaptive controller. The stability of the system with controller is investi- gated by means of fuzzy Lyapunov function. The proposals and the contributions of this book are elaborated as follows. The layout of the book proposals are depicted in Fig. 1.1. In this book, to obtain fast transient performance by eliminating uncertainties and disturbances present in the system, L1 adaptive controller with fast adaptation and high robustness is employed. To obtain optimal performance from L1 adaptive controller, its parameters are tuned utilizing optimization techniques. Being aware of easy implementation and good balance between exploration and exploitation capability, a novel variant of HS algo- rithm, viz., local best harmony search (lbest HS) algorithm is utilized. The values of unknown constant, time varying uncertainties and time varying disturbances present in the system are at first obtained utilizing lbest HS algorithm. Then, their values are adapted concurrently following L1 adaptation laws to fine tune those values. In local search based HS, the total harmony memory is divided into some groups and from eachgrouplocalbestsolutionisfoundout.Fromthoselocalbestsolutions,globalbest solution is obtained. In this local best search phenomenon, the exploitation capability of the HS algorithm increases enormously. The exploration and exploitation capa- bilities of the whole optimization technique augmented controller are investigated. It is proved analytically that, the population variance of the lbest HS-L1 adaptive controller increases gradually with iteration which shows its better exploration capa- bility [172]. With good exploration capability, high exploitation capability of the optimization process is also required. The exploitation capability of the designed method is assured by means of spectral radius convergence phenomenon [172]. Again to handle nonlinearities present in the system along with uncertainties and disturbances, in this book, fuzzy parallel distributed compensation (PDC) logic Fig. 1.1 Layout of the book proposal
- 53. 12 1 Introduction is hybridized with L1 adaptive controller [175]. In this fuzzy PDC-L1 adaptive controller, the nonlinear system is considered as the combination of finite number of linear fuzzy systems. Then, the same fuzzy premise part, utilized to design the system, is employed to design different components of L1 adaptive controller, viz., predictor, L1 adaptation law, L1 control law, PDC control law and low-pass filter. For each linearized zone, fuzzy predictor is developed and combined to produce overall predictor model. Uncertainties, disturbances present in these linearized zones are adapted following zone-wise L1 adaptation laws and are accumulated to develop whole L1 adaptation law. From zone-wise adaptation laws, L1 control signals are developed keeping the zonal nature and then fuzzy blended to produce whole L1 control law. Due to the use of high adaptation gain, the control law of each zone contains high frequency components. To nullify that high frequency from each of the linearized zone, the produced control signals are filtered through low-pass filter dedicated for each of those zones. The state feedback controller of basic L1 adaptive controller is designed utilizing fuzzy PDC logic. The zone-wise developed fuzzy PDC controllers are accumulated to produce overall fuzzy PDC control law. Filtered L1 adaptive control law and fuzzy PDC control signal are augmented and provided to the system under control. All of these components of the L1 adaptive controller are designed separately from the fuzzy adaptation laws, developed from the stability criteria of fuzzy Lyapunov functions. The overall stability of the closed loop system is also investigated utilizing fuzzy Lyapunov function. The transient and steady state stable performances of the system are guaranteed analytically. The designed controllers are employed on four simulation case studies and three experimental case studies to show their effectiveness. In simulation case studies, two single-input single-output (SISO) system, viz., Duffing’s oscillatory system with disturbance and nonlinear spring mass damper system are considered to show the controller performance when disturbances are present in the system. Then, one single-input multi-output (SIMO) system, viz., unstable non-minimum phase 4th order inverted pendulum with cart is controlled and stabilized. A nonlinear, cross- coupled twin rotor multi-input multi-output (MIMO) system (TRMS) is investigated to show the controller performance in case of nonlinearities and cross-couplings. Furthermore, in experimental case studies, speed control of a SISO system, viz., an electrical actuator is performed owing to its large number of applications in practical systems. Angular position control of a MIMO, nonlinear two link robot manipulator (TLRM) with unknown dynamics is evaluated to examine the effectiveness of the designed controllers in tackling system with nonlinearities, unmodelled dynamics, etc. Finally, temperature control of a delay dependent air heater system with distur- bance is performed to show the efficiency of the designed controllers to tackle large delay. All of these systems are controlled through different types of controllers and their performances are compared. At first, two classical controllers, viz., proportional integral derivative (PID) controller, linear quadratic Gaussian (LQG) controller are tested. Then, model reference adaptive controller (MRAC) and basic L1 adaptive controller are imposed on them. After that, a stochastic optimization, viz., PSO based L1 (PSO-L1) adaptive controller; metaheuristic, viz., basic HS based L1 (HS- L1) adaptive controller and metaheuristic lbest HS based L1 (lbest HS-L1) adaptive
- 54. 1.5 Aims and Scopes of the Book 13 controller are examined. Finally, the fuzzy PDC-L1 adaptive controller is employed. Results show that, disturbance rejection phenomenon of basic L1 adaptive controller is better than the classical controllers and MRAC. The lbest HS-L1 adaptive controller provides better transient performance and tracking control than the basic L1 adaptive controller. The fuzzy PDC-L1 adaptive controller outperforms the basic L1 adaptive controller as well as lbest HS-L1 adaptive controller. 1.6 Summary This chapter illustrates the introduction of the book. The journey of classical controllers towards modern control methods are portrayed with the current status of modern control methodologies. To deal with nonlinear system containing uncer- tainties and disturbances, modern controllers, viz., adaptive control, fuzzy control, optimization based control techniques come into place. Then, the motivations of this book are discussed owing to the shortcomings of basic L1 adaptive controller as well as strengths of local best harmony search optimization technique and fuzzy parallel distributed compensation logic. Next the proposals of the book are elaborated with the architecture of the book proposal. In this book, two hybridized L1 adaptive controllers are implemented with mathematical analysis and experimental demon- strations. lbest HS algorithm is hybridized with basic L1 adaptive controller with the aim to obtain optimal parameter setting of L1 adaptive controller. The lbest HS-L1 adaptive controller provides satisfactory transient and steady state performance as well as high robustness. To tackle nonlinearities, delays along with uncertainties and disturbances, fuzzy PDC logic is hybridized with L1 adaptive controller. The fuzzy PDC-L1 adaptive controller provides fast transient performance by means of fuzzy L1 adaptive controller as well as robust and satisfactory steady state tracking performance utilizing fuzzy PDC logic. The performances of the designed control methodologies are evaluated on four simulation case studies and three experimental case studies to show the effectiveness. After that, contributions of the book are summarized. Then, the structure and orientation of the book are elaborated followed by this summary of the present chapter. Next chapter introduces the formulation of basic L1 adaptive controller and its stability analysis. Four simulation case studies are performed to show the effectiveness of basic L1 adaptive controller over PID, LQG and MRAC. References 1. Liu, J., Wang, H., Zhang, Y.: New result on PID controller design of LTI systems via dominant eigenvalue assignment. Automatica 62, 93–97 (2015) 2. Sujitjorn, S., Wiboonjaroen, W.: State-PID feedback for pole placement of LTI systems. Math. Probl. Eng. 2011, 1–20 (2011)
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- 56. “Star,” with more brandy at his elbow, he fell into a drunken stupor by the fire. The whole district was, however, aroused, and the road being searched while he lay in that condition. Three soldiers traced him to the “Star,” and he awoke to find himself covered by their pistols. To them he yielded all his varied wealth, with the exception of a few trifles hidden in his neckcloth, and then staggered up to bed with the troopers at his heels. There they watched him all night. He was, as he says in his last account and confession of a wild career, “a good deal chagrined” at this. How to escape? He thought of a plan. Throwing the few remaining trinkets suddenly in the fire, the soldiers, as he had expected, made a dash to save them, while he pounced upon the pistols. He seized a couple, and, standing at the door, desperately pulled the triggers. The soldiers would probably have been sent to Kingdom Come but for the trifling circumstance that the weapons missed fire. It was a mishap that cost Simms his life, for he was quickly seized and more vigilantly guarded, and, when morning dawned, taken up to London on the road he had so blithely travelled the day before. One more journey followed—to Tyburn, where they hanged him in the following June. The pilgrim of the roads who looks for the “Bull” at Dunstable, or the “Star” at Hockliffe, will not find those signs: the shrines of the saints and the haunts of the highwaymen are alike the food of ravenous Time.
- 57. XIX Who shall say certainly how or when the phrase “Downright Dunstahle” first arose, or what it originally meant. Not the present historian, who merely sucks his wisdom from local legends as he goes. And when it happens, as not infrequently is the case, they have no agreement, but lead the questing toiler after truth into culs-de-sac of falsehoods and blind-alleys and mazes of contradictions, the labour were surely as profitless as the mediæval search for the Philosopher’s Stone. Briefly, then, “Downright Dunstable” is a figurative expression for either or both of two things: a state of helpless intoxication, or for that kind of candid speech often called “brutal frankness.” At any rate, it is ill questing at Dunstable for light on the subject, and it is quite within the usual run of things to find the old saying unknown nowadays in the place that gave it birth. There is no evidence in the long broad street of Dunstable town of the age and ancient importance of the place. It looks entirely modern, and the Priory church is hidden away on the right. When— in the course of a century or so—the young limes, sycamores, and chestnuts, planted on either side of that main thoroughfare, have grown to maturity, the view coming into Dunstable from London will be a noble one. At present it is merely neat and cheerful. No mention is made of Dunstable in Doomsday Book. When that work was compiled, the old Roman station of Durocobrivæ, occupied in turn by the Saxons and burnt to the ground by marauding Danes, lay in a heap of blackened ruins, the only living creatures in the neighbourhood the fierce robbers who lay wait for travellers at this ancient crossing of the Watling and the Icknield Streets. If any of the surveyors who took notes for the making of Doomsday Book were so rash as to come here for that purpose, certainly they must have perished in the doing. At that period and until the beginning of
- 58. Henry I.’s reign the road was bordered by dense woodlands, affording a safe hiding-place for malefactors, chief among whom, according to an absurd monkish legend, purporting to account for the place-name, was a robber named Dun. The ruined town and the impenetrable thickets were known, they said, as “Dun’s Stable.” The first step towards reclaiming the road and the ruins from anarchy and violence was the clearing of these woods. This was followed by the building of a house—probably a hunting-lodge—for the King, and the founding of the once powerful and stately Priory of Dunstable, portions of whose noble church remain to day as the parish church of the town. To the Augustine priors the town and its market rights were given, and the place, new-risen from its ashes, throve under the combined patronage of Church and State. Whatever the religious merits of those old monks may have been, certainly they were business men, stock-raisers, and wool-growers of the first order. Their flocks and herds covered those downs that remain much the same now as eight hundred years ago, and their Dunstable wool was prized as the best in the kingdom. But these business-like monks were not altogether loved by the townsfolk, who resented the taxes laid upon them by the Church, all-powerful here in those days. It seemed to men unjust that fat priors and their crew should command the best of both worlds: should wield the keys of heaven and take heavy toll of goods in the market. The townsfolk, indeed, in 1229 made a bold stand, and protesting that they “would sooner go to hell than be taxed,” vainly attempted to form a new settlement outside the town. The sole results were that they were taxed rather more heavily than before, and ecclesiastically cursed. To detail here the grandeur and the pride of that great Priory would be to halt too long on the way. All who had, in those ancient times, any business along this great road were entertained by the Prior. The common herd in those early days were entertained at the guest house, a building facing the main road, on a site now occupied by a house called “The Priory.” King John in 1202 had given his hunting-lodge to the Priory, and from that time onward Kings and Queens were lodged in the Priory itself. Here rested—the next halting-place from Stony Stratford the body of Queen Eleanor on the way to Westminster, in 1290; and one of the long series of Eleanor crosses remained in the market-place until 1643, when it was destroyed by the Parliamentary troops. In the Lady Chapel (long since swept away) of the Priory
- 59. Church, Cranmer promulgated the divorce of Henry VIII. and Katherine of Arragon. Two years later, the Priory itself was dissolved. At first it seemed likely that Dunstable would be made the seat of a Bishop and the great church erected to the dignity of a Cathedral, but the project came to nothing, and the sole remaining portions of the old buildings are the nave and the west front. Presbytery, choir, transepts, lady chapel, and aisles were torn down. The aisles and east end of the church are modern, the nave a majestic example of Norman architecture, and the west front a curiously picturesque mass of Transitional Norman, Early English, and Perpendicular, worthy the dexterous pencil of a Prout. DUNSTABLE PRIORY CHURCH. The spoliation of the Priory Church was a long but thorough process. Many of its carved stones are worked into houses and walls in and around the town, but it was left for modern times to complete the vandalism; when, for example, great numbers of decorative pillars and capitals were discovered, some put to use to form an “ornamental rockery” in a neighbouring garden, the remaining cartloads taken to a secluded spot in the downs and buried; when the stone coffin of a prior was sold for use as a horse-trough and afterwards broken up for road-metal; when a rector could find it possible to destroy a holy-water stoup, the old font could be thrown away, and the pulpit sold to a publican for the decoration of a tea-
- 60. garden. Among other objects that have disappeared in modern times is the life-size effigy of St. Fredemund, the sole remaining portion of his shrine. Fredemund was a son of King Offa. His body had been brought hither in ancient times, on the way to Canterbury, but was, by some miraculous interposition, prevented from leaving Dunstable. No miracle saved his statue. The ancient sanctus bell of the church, inscribed “Ave Maria, gracia plena,” hangs on the wall of the modern town-hall. “Dunstable,” says Ogilby, writing in 1675, “is full of Inns for Accommodation, and noted for good Larks.” This would seem to hint at an unwonted sprightliness in the hostelries and town of Dunstable, were it not that larks bore but one signification in Ogilby’s day. Slang had not then stepped in to give the word a double meaning. Of the notable old inns of Dunstable the “Sugarloaf” remains, roomy and staid, reprobating unseemliness. Larks, like Dunstable wool in still older days, and straw-plaiting in more recent times, no longer render the town notable. Straw-plaiting and hat- making are, it is true, yet carried on, but the industry is a depressed one. A greater feature, perhaps, is seen in the extensive printing works established here in recent years by the great London firm of Waterlow Sons.
- 61. XX From Dunstable the road enters a deep chalk cutting through the Downs—similar to, but not so great a work as, the chalky gash through Butser Hill, on the Portsmouth Road. In this mile-length of cutting the traveller stews on still summer days, blinded by the chalky glare; or, when it blows great autumnal guns and snow-laden winter gales, whistling and roaring through this exposed gullet with the sound of a railway train, freezes to his very marrow. Before this cutting was made, and the “spoil” from it used in the making of the great embankment that carries the road above the deep succeeding valley, this was a precipitous ascent and descent, and a cruel tax upon horses. Looking backwards, the embankment is impressive, even in these days of great engineering feats, and proves to the eye how vigorously the question of road reform was being grappled with just before the introduction of railways. From this point the famous Dunstable Downs are well seen, rising in bold terraces and swelling hills from the hollow, and receding in fold upon fold of treeless wastes where the prehistoric Icknield Way runs and the stone implements and flint arrows dropped by primitive man for lack of reliable pockets, are found.
- 62. DUNSTABLE DOWNS. The neolithic ancestor seems to have been particularly fond of these windy hillsides, and has left a great earthwork on them, ten acres in extent. Maiden Bower they call it nowadays—as grotesquely unsuitable a corruption of the original “Maghdune-burh” as may well be imagined. Its wind-swept terraces, distinctly seen from this embankment, scarce give the idea of a boudoir. Neolithic man was fond of these hillsides in a purely negative way. He would have preferred the warmer valleys, only in those remote times they were filled with dense and almost impenetrable forests, and abounded in the fiercest and wildest of wild animals, that came at night and preyed upon his family circle when the camp-fires burnt low. And when those wild creatures were not to be dreaded, there were always hostile tribes prowling in the thickets. So, on all counts, the Downs were safest. Where that remote ancestor built his bee-hive huts and banded together with his fellows to raise a fortified post, others— Britons, Romans, and Saxons—came and added more and taller earthworks, so that the tallest of them are sixteen feet high even now. Shortly after leaving the embankment behind, a sign-post marks a lane to the left, leading to Tilsworth, a dejected village, looking as though agricultural depression had hit it hard. A deserted schoolhouse, by the church, is falling to pieces. Just within the churchyard is a headstone, standing remotely apart from the others. Its isolation invites scrutiny; an attention rewarded by this epitaph:— THIS STONE WAS ERECTED BY SUBSCRIPTION
- 63. TO THE MEMORY OF A FEMALE UNKNOWN FOUND MURDER’D IN BLACKGROVE WOOD AUG. 15th 1821 Oh pause my friends and drop the silent tear Attend and learn why I was buried here; Perchance some distant earth had hid my clay If I’d outliv’d the sad, the fatal day: To you unknown, my case not understood; From whence I came, or why in Blackgrove Wood. This truth’s too clear; and nearly all that’s known— I there was murder’d, and the villain’s flown. May God, whose piercing eye pursues his flight, Pardon the crime, but bring the deed to light. That the deed was “brought to light” is obvious enough, but that is not what the author of those lines meant. The perpetrator of the deed was never discovered. Blackgrove Wood, a dark mass in a little hollow, is easily seen from the road. In another two miles Hockliffe is reached.
- 64. XXI “A dirty way leads you to Hockley, alias Hockley-in-the-Hole,” said Ogilby, in 1675; and it seems to have gradually become worse during the next few years, for Celia Fiennes, confiding her adventures to her diary, about 1695, tells of “seven mile over a sad road. Called Hockley in ye Hole, as full of deep slows in ye winter it must be Empasable.” It received, in fact, all the surface-water draining from Dunstable Downs to the south and Brickhills to the north. It is not, however, until he has left Hockliffe behind and started to climb out of it that, the amateur of roads discovers how deeply in a hole Hockliffe is, for it is approached from the Dunstable side by a level stretch that dims the memory of the downs, and makes all those old tales of sloughs appear like fantastic inventions. It is at this time perhaps the most perfectly preserved example of Telford’s road-making. Surface, cross-drains, ditches, and hedges are maintained in as good condition as when first made. And why so more than in other places? For this very reason; that it is in a hole, and if not properly drained, would again become as “empasable” as it was over two hundred years ago. Hockliffe, originally a very small village, grew to great importance in coaching times, for here is the junction of the Holyhead and Manchester and Liverpool roads, both in those times of the greatest vogue and highest importance. An after-glow of those radiant glories of the road is seen in the long street. Hockliffe was in Pennant’s time, when coaching had grown enormously in importance, “a long range of houses, mostly inns.” It is so now, with the difference that the houses mostly have been inns, and are so no longer. In his day he observed “the English rage for novelty” to be “strongly tempted by one sagacious publican, who informs us, on his sign, of newspapers being to be seen at his house every day in the week.”
- 65. THE “WHITE HORSE,” HOCKLIFFE. At which of the two principal inns, the “White Hart” or the “White Horse,” this enterprising publican carried on business he does not tell us. Perhaps it was the “White Horse”; now certainly one of the most interesting of inns, and then the chiefest in Hockliffe. Before its hospitable door the “Holyhead Mail,” the Shrewsbury “Greyhound,” the Manchester “Telegraph,” the Liverpool “Royal Umpire,” and many another drew up, together with some of the many “Tally-Hoes” that spread a fierce rivalry down the road. It was probably at Hockliffe and at the hospitable door of the “White Horse,” that the “Birmingham Tally-Ho” conveying Tom Brown to Rugby drew up at dawn “at the end of the fourth stage.” We need not look for exact coaching data in that story; else, among other things, we might cavil at the description of it as a “little” roadside inn. A bright fire gleaming through the red curtains of the bar window gave promise of good refreshment, and so while the horses were changed, the guard took Tom in to give him “a drop of something to keep the cold out,” or rather to drive it out, for poor Tom’s feet were already so cold that they might have been in the next world, for all he could feel of them, and the guard had to pick him off the coach-top and set him in the road. “Early purl” set that right, and warmed the cockles of his heart. There is no nonsense of the plate-glass and electric-bell kind about the “White Horse.” If the old coachmen were to come back, and the passengers they drove, they would find the old house much the same —the stables docked perhaps of some of their old extent and a trifle ruinous, and the house in these less palmy days crying out for some
- 66. fresh paint and a few minor repairs; but still the same well- remembered place. Even the windows in the gables, blocked up over a century ago to escape Mr. Pitt’s window-tax, have not been re- opened. There are low-browed old rooms at the inn, with a cosy kitchen that is as much parlour; with undisguised oaken beams running overhead, rich in pendant hams that by due hanging have acquired artistic old-masterish tones, like mellow Morlands and rich Gainsboroughs. There is a capacious hearth, there are settles to sit easily in, and warming pans that have warmed many a bed for old- time travellers; and there are memories, too, for them that care to summon them. Will they come? Yes, I warrant you. They are memories chiefly of moving accidents by flood and fell, for Hockliffe has had more than its due share of coaching accidents. They happened chiefly on the hills a mile out, where Battlesden Park skirts the road, and where, although Telford did some embanking of the hollows and cutting of the crests, they remain formidable to this day. Battlesden became an ominous name in those days, and the “White Horse” and many another Hockliffe inn very like hospitals. The year 1835 was an especially disastrous one. In May, the “Hope” Halifax coach, on the way to London, was being driven down hill at a furious pace, when the horses became unmanageable, and the coach, overloaded with luggage piled up on the roof, after reeling in several directions, fell on the off side. All the passengers were injured more or less severely. The next happening was when the Shrewsbury “Greyhound,” coming towards London, was overturned at a point almost opposite Battlesden House. Again most of the passengers were seriously injured, and the coachman had a leg broken. Two of the horses suffered similar injuries. This accident was caused by the near-side wheeler kicking over the pole and thus upsetting the coach while it was running at high speed down hill. Of course, when the great Christmas snowstorm of 1836 blocked nearly all the roads in England, Hockliffe was a very special place for drifts, and the Birmingham, Manchester, Holyhead, Chester and Holyhead, and Halifax mails were all snowed up. An attempt made to drag the Chester mail out resulted in the fore-axle giving way and the coach being abandoned. The boys went forward on horseback. The Holyhead mail, with the Irish bags, was more fortunate. When the horses suddenly floundered up to their necks in the snow, the coachman dived off headlong, and was nearly suffocated; but with
- 67. the aid of the guard and the passengers he was pulled out by the legs, and, a team of cart-horses being requisitioned, the coach itself dragged through. These are examples of the perils His Majesty’s Mails encountered in those times, and of the discomforts endured by the men who carried them for little wage. The Post Office has never been generous to the rank and file of its staff. The secretarial staff, whose business it is to receive complaints and to scientifically fob off the public with tardy promises of enquiries never intended to be made, draw handsome salaries, but those who do the actual work have always been paid something less than they could obtain from other walks of life. The guards in Post Office employment received half a guinea a week salary in the old mail-coach days—as, in fact, a retaining fee—it being estimated by the Department that they could make a good thing of it by the “tips” they would be receiving from passengers. That they did make a good thing of it we know, but the principle was a shabby one for a Government Department to adopt, and really created a kind of indirect taxation. No traveller could refuse to “tip” the guard as well as the coachman, unless very hard-hearted or possessed of a moral courage quite beyond the ordinary. Beyond his half-guinea a week, an annual suit of clothes, and a superannuation allowance of seven shillings a week, a mail guard had no official prospects. Occasionally some crusty passenger, whom the guard, being extra busy with his letters and parcels, had perhaps no time to humour, would refuse to tip, and would write to the Post Office to complain; whereupon the Secretary would indite some humbug of this kind:—
- 68. THE GREAT SNOWSTORM, DEC. 26TH, 1836. THE BIRMINGHAM MAIL FAST IN THE SNOW, WITH LITTLE CHANCE OF A SPEEDY RELEASE: THE GUARD PROCEEDING TO LONDON WITH THE LETTER-BAGS. From a Print after J. Pollard. “Sir,—I have the honour of your letter of the ——, to which I beg leave to observe that neither coachman nor guard should claim anything of ‘vails’ as a right, having ten and sixpence per week each; but the custom too much prevails of giving generally a shilling each at the end of the ground, but as a courtesy, not a right; and it is the absolute order of the office that they shall not use a word beyond solicitation. This is particularly strong in respect of the guard—for, indeed, over the coachman we have not much power; but if he drives less than thirty miles, as your first did, they should think themselves well content with sixpence from each passenger.” In those times sixpence might have been enough, but when, in later days, the coachman or the guard at the end of their respective journeys would come round with the significant remark, “I leaves you here, gentlemen!” he who offered sixpence would have been as daring as one who gave nothing at all. The sixpence would have been returned with a sarcastic courtesy, and a shilling not received with any remarks of gratitude. This custom was known as “kicking the passengers.”
- 69. Very occasionally, and under pressure, the Post Office doled out an extra half-guinea in seasons of extraordinary severity, when passengers were few and tips scarce, and on occasions when the mails were so heavy that the seats generally occupied by passengers were given up to the bags, the guards had an allowance made them. Their zeal under difficulties also received rare and grudging recognition, as when Thomas Sweatman, guard of the Chester mail in the early part of 1795, was awarded half a guinea for his labours at Hockliffe, where, in the middle of the night and up to his waist in water, he helped to put on new traces, travelling to town on his box with his wet clothes freezing to him.
- 70. XXII The red-brick face of the “White Horse” is set off and embellished by a very wealth of elaborate old Renaissance wood-carving that decorates the coach-entrance. It was obviously never intended for its present position, and is said to have come from an old manor-house at Chalgrave, demolished many years ago. Long exposure to the weather and generations of neglect have wrought sad havoc with this old work. A fragment in the kitchen gives the date 1566, and some strips under the archway, with the inscription “John Havil dwiling in cars,” present a mystery not easy to solve. The ominous Battlesden Park, belonging to the Dukes of Bedford, with jealously locked lodge-gates that hinder the harmless tourist from inspecting the church within the demesne, is one of a vast chain of Russell properties stretching for miles across country, from here to Woburn and away to the Great North Road at Wansford. Battlesden is without a tenant, except for those who tenant family vaults and resting-places in the little churchyard: Duncombes within and nobodies in particular without. It was one of these Duncombes of Battlesden—Sir Samuel—who in 1624 introduced Sedan-chairs into England. Weeping marble cherubs on Duncombe monuments, rubbing marble knuckles into marble eyes, testify to grief overpast, but Nature, indifferent as ever, keeps a cheerful face. It here becomes evident that we are on the borders of a stone country, for the little church tower is partly built of that ferruginous sandstone whose rusty red and yellow is for the next thirty miles to become very noticeable. Gaining the summit of Sandhill, a house lying back from the road, on the left, is seen, with traces of a slip-road to it and through its grass-grown stable-yard. It is a noticeable red-brick house, with a steep tiled roof crowned by a weather-vane. Once the “Peacock” inn,
- 71. it has for many years been a private residence. A short distance beyond, past the cross-roads known as Sheep Lane, Bedfordshire is left behind for the county of Buckingham, through which for the next twelve miles, to the end of Stony Stratford, the Holyhead Road takes its way. Buckinghamshire, on the map, is a quaintly shaped county, standing as it were on end, washing its feet in the Thames at Staines, and with its head in the Ouse, in the neighbourhood of Olney. Wags have compared it with a cattle-goad, “because it sticks into Oxon and Herts.” The glimmerings of possible similar verbal atrocities are apparent in the fact that it is also bordered by Beds and Berks. Northants and Middlesex also march with its frontiers. Its name is derived from the Anglo-Saxon word “bucken,” alluding to the beech woods that spread over it, but more particularly in the south, on the densely wooded Chiltern Hills. The Welsh language, innocent of any word for the beech, bears out the statement of Cæsar, that this tree was unknown in Britain at the time of his invasion. Little Brickhill is the first place that Buckinghamshire has to show, and a charming old-world place it is, despite its name, which, together with those of its brothers Great and Bow Brickhills near by, prepares the traveller for—of course—bricks. But the greater number of houses here are stone. It is difficult to imagine this little hillside village an assize town; but so it once was, and the “Sessions House,” a small Tudor building, one of the few in red brick, still stands as a memento of the time when this was the scene of the General Gaol Delivery for the county of Bucks, from 1433 to 1638. The chief reason for this old-time judicial distinction appears in the fact that Aylesbury, the county town, was practically unapproachable during three parts of the year, owing to the infamously bad bye-roads.
- 72. LITTLE BRICKHILL. The old “George” inn, that stands directly opposite the Sessions House, is not the only inn at Brickhill against whose name “fuit” must be written. Others, now vanished, were the “White Lion,” now the Post Office, with some delicate decorative carving on its front (the old sign is still preserved upstairs); the “Swan,” the “Shoulder of Mutton,” and the “Waggon.” The class of each one of these old houses may still be traced. The “George” was beyond comparison the chief, and legends still linger of how the old fighting Marquis of Anglesey came up and stayed here as Lord Uxbridge with two legs, and returned after Waterloo as Lord Anglesey with one. They say, too, that the Princess Victoria once halted here the night. In the churchyard, that so steeply overlooks the road at the hither end of the village, you may see stones to the memory of William Ratcliffe, the last host of the “George,” his wife, his relatives, and his servants. He died, aged eighty-two, in 1856; his wife in 1842. Many years before, a servant, Charlotte Osborne, had died, aged thirty-eight; the stone “erected by three sisters, as a tribute of their regard for a faithful servant, and as a testimony to one who anxiously endeavoured to alleviate the sufferings of a beloved and lamented parent upon a dying bed.” Here also is the epitaph of Isaac Webb, “for more than forty years a good and faithful servant to Mr. Ratcliffe of the ‘George Inn,’ during which he gained the esteem of all who knew him.” He died, aged fifty-eight, in 1854.
- 73. YARD OF THE “GEORGE.” The old “George” is now occupied—or partly occupied, for it is a very large house—by a farm bailiff. Just what it and its old coach- yard are like let these sketches tell. Within the church a curious wooden-framed tablet records the death at Little Brickhill of an old-time traveller when journeying from London to Chester. This was “William Bennett, son of the Mayor of Chester. He died March 19th, 1658. But most curious of all is the stone in the churchyard to a certain “True Blue,” who died in 1725, aged fifty-seven. Time has lost all count of “True Blue,” who or what he was, and speculation is futile. If only the vicar who entered his burial in the register had noted some particulars of him, how grateful we should be for the unveiling of this mystery! Those registers have, indeed, no little interest, containing as they do the gruesome records of many criminals executed in the old gaol deliveries, as well as of a woman who was wounded at the battle of Edge Hill and died of her hurts.
- 74. XXIII A long and steep descent into the valley of the Ouse conducts from Little Brickhill into Fenny Stratford, seen in the distance, its roofs glimmering redly amid foliage. The river, a canal, and the low-lying flats illustrate very eloquently the “fenny” adjective in the place- name, and it is in truth a very amphibious, bargee, wharfingery, and mudlarky little town. Agriculture and canal-life mix oddly here. Wharves, the “Navigation” inn, and hunchbacked canal-bridges admit into the town; and the lazy, willow-fringed Ouzel, with pastures and spreading cornfields on either side, bows one out of it at the other end. The arms of Fenny Stratford, to be seen carved above the church door, allude in their wavy lines to its riverain character, but, just as Ipswich and some other ancient ports bear curiously dimidiated arms showing monsters, half lions and half boats, so “Fenny” (as its inhabitants shortly and fondly call it) should bear for arms half a barge and half a plough, conjoined, with, for supporters, a bargee and a ploughman. The church just mentioned is exceedingly ugly, and of the glorified-factory type common at the period when it was built. It owes its present form to Browne Willis, the antiquary, who built it in 1726, and, as an antiquary, ought to have known better. He dedicated it to St. Martin, in memory of his father, who was born in St. Martin’s Lane, and died on St. Martin’s Day. A kindly growth of ivy now screens the greater part of Browne Willis’s egregious architecture. He lies buried beneath the altar, but his memory is kept green by celebration of St. Martin’s Day, November 11th, when the half-dozen small carronades he presented to the town and now known as the “Fenny Poppers,” fire a feu-de-joie, followed by morning service in the church and a dinner in the evening at the “Bull” inn.
- 75. Bletchley and its important railway junction have caused much building here in recent years, and bid fair to presently link up with “Fenny,” just as Wolverton with “Stony.” The distance between the two Stratfords is a little over four miles, the villages of Loughton and Shenley, away from the road, in between, and the main line of the London and North-Western Railway crossing the road on the skew- bridge described in a rapturous railway-guide of 1838 as a “stupendous iron bridge, which has a most noble appearance from below.” At the cross-roads between these two retiring villages stands the “Talbot,” a red-brick coaching inn, mournful in these days and descended to the lower status of a wayside public. It lost its trade at the close of 1838, when the London and Birmingham Railway was completed, but, with other neighbouring inns, did a brisk business at the last, when the line was opened for traffic only as far as “Denbigh Hall,” in the April of that year. The temporary station of that name was situated at the spot where the railway touches the road, at the skew-bridge just passed. Between this point and Rugby, while Stephenson’s contractors were wrestling with the difficulties of the great Roade cutting and the long drawn perils of Kilsby Tunnel, coaches and conveyances of all kinds were run by the railway company, or by William Chaplin, for meeting the trains and conveying passengers the thirty-eight miles across the gap in the rail. From Rugby to Birmingham the railway journey was resumed. “Denbigh Hall” no longer figures in the time-tables, for the idea of a “secondary station,” once proposed to be established here was abandoned. But while the break in the line continued this was a busy place. It is best described in the words of one who saw it then:— “Denbigh Hall, alias hovel, bears much the appearance of a race- course, where tents are in the place of horses—lots of horses, but not much stabling; coachmen, postboys, post-horses, and a grand stand! Here the trains must stop, for the very excellent reason that they can’t go any further. On my arrival I was rather surprised to find all the buildings belonging to the Railway Company of such a temporary description; but this Station will become only a secondary one when the line is opened to Wolverton. There is but one solitary public- house, once rejoicing in the name of the ‘Pig and Whistle,’ but now dignified by the title of ‘Denbigh Hall Inn,’ newly named by Mr. Calcraft, the brewer, who has lately bought the house. Brewers are
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