Design and Implementation of Model Reference Adaptive Controller using Coefficient Diagram Method for a nonlinear process

IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE)
e-ISSN: 2278-1676,p-ISSN: 2320-3331, Volume 6, Issue 4 (Jul. - Aug. 2013), PP 70-76
www.iosrjournals.org
www.iosrjournals.org 70 | Page
Design and Implementation of Model Reference Adaptive Controller
using Coefficient Diagram Method for a nonlinear process
R.Satheesh Babu1
, S.Abraham Lincon2
1
(PG Scholar, Electronics & Instrumentation Engg, Annamalai University, India)
2
(Professor, Electronics & Instrumentation Engg, Annamalai University, India)
Abstract: In this work an adaptive control along with robust control have been developed to address the problem
of system performance in the face of system uncertainty in control-system design without excessive reliance on
system models. The direct MRAC control scheme with unknowns is designed with control law that is to be
combined with an adaptive law. The adaptive laws are developed using the SPR-Lyapunov design approach and
are driven by the estimation error for continuous-time systems guaranteeing asymptotic stability of the system
states. CDM is a polynomial approach which was developed and introduced for a good transient response of the
control systems. An MRAC is constructed in two-degree of- freedom (2DOF) control structure and the adaptation
gains of controller parameters are found through CDM. Spherical tank level process is used for validating the
design procedure of CDM-MRAC in real time and the performance of controller is compared with linear PI
controller. Simulation is carried out using Mat Lab Simulink software.
Keywords - Adaptive control, SPR-Lyapunov DMRAC, CDM-MRAC.
I. INTRODUCTION
The purpose of feedback control is to achieve desirable system performance in the face of system
uncertainty and system disturbances. Although system identification can reduce uncertainty to some extent,
residual modelling discrepancies always remain. Controllers must therefore be robust to achieve desired
disturbance rejection and/or tracking performance requirements in the presence of such modelling uncertainty. To
this end, adaptive control along with robust control have been developed to address the problem of system
performance. Adaptive controllers directly or indirectly adjust feedback gains to maintain closed-loop stability and
improve performance in the face of system errors. Specifically, indirect adaptive controllers utilize parameter
update laws to estimate unknown system parameters and adjust feedback gains to account for system variation,
while direct adaptive controllers directly adapt the controller gains in response to system variations. Even though
adaptive control algorithms have been developed in the literature for both continuous-time and discrete-time
systems, the majority of the discrete-time results are based on recursive least squares and least mean squares
algorithms with primary focus on state convergence. Alternatively, Lyapunov-based adaptive controllers have been
developed for continuous-time systems guaranteeing asymptotic stability of the system states. In this work, the
design and implementation of MRAC using Coefficient diagram method is presented to improve standard designs
in adaptive control schemes.
Section 2 describes the design schemes of MRAC for SISO plants, and the development of MRAC for the
same plant. Section 3 is devoted to the basics of Coefficient Diagram Method design and development of MRAC
using CDM for SISO plants. Section 4 presents the simulation results and comparison of performance of
controllers. The conclusion is presented in section 5.
II. MRAC FOR SISO PLANTS
Consider the SISO, LTI plant described by the vector differential equation
Where and have the appropriate dimensions. The transfer function of the plant is
given by
with expressed in the form
where are monic polynomials and is a constant referred to as the high frequency gain. The reference
model, selected by the designer to describe the desired characteristics of the plant, is described by the differential
equation

Design and Implementation of Model Reference Adaptive Controller using Coefficient Diagram Method
where for some integer and r is the reference input which is assumed to be a uniformly
bounded and piecewise continuous function of time. The transfer function of the reference model given by
Expressed in the same form as (3), i.e.,
Where are monic polynomials and is a constant.
The MRAC objective is to determine the plant input so that all signals are bounded and the plant output
tracks the reference model output as close as possible for any given reference input r(t) of the class defined
above. If is chosen so that the closed-loop transfer function from r to has stable poles and is equal to ,
the transfer function of the reference model. Such a transfer function matching guarantees that for any reference
input signal r(t), the plant output converges to exponentially fast. This leads to the closed-loop transfer
function
This control law, however, is feasible only when is Hurwitz. Otherwise, (6) may involve zero-pole
cancellations outside , which will lead to unbounded internal states associated with non-zero initial conditions.
Consider the feedback control law shown in fig 1.
Where
are constant parameters to be designed and Ʌ(s) is an arbitrary monic Hurwitz
polynomial of degree that contains as a factor, i.e.,
Which implies that is monic, Hurwitz and degree .
The controller parameter vector is to be chosen so that the transfer function from r to is equal to .
Fig 1 Structure of the MRAC scheme
The I/O properties of the closed-loop plant shown in fig 1 are described by the transfer function equation
Where
If the controller parameters are selected to meet the control objective so that the closed-loop
poles are stable and the closed-loop transfer function i.e.,

Equation (9) satisfied for all . Because the degree of the denominator of is and
that of is , for the matching equation to hold, an additional zero-pole
cancellations must occur in . Now because is Hurwitz by assumption and is
designed to be Hurwitz, it follows that all the zeros of are stable and therefore any zero-pole cancellation can
only occur in . Choosing
and using the matching equation (9) becomes
or
Equating the coefficients of the powers of s on both sides of (12), it can be expressed in terms of the
algebraic equation
Where , S is an matrix that depends on the coefficients of
and , and p is an vector with the coefficients of . The existence of to satisfy (12)
and, therefore, (13) will very much depend on the properties of the matrix S. For example, if , more than one
will satisfy (13), whereas if and S is nonsingular, equation(12) will have only one solution.
III. BASICS OF COEFFICIENT DIAGRAM METHOD (CDM)
To Some mathematical relations extensively used in CDM will be introduced hereafter. The characteristic
polynomial is given in the following form
The stability index , the equivalent time constant τ, and stability limit are defined as follows.
From these equations the following relations are derived.
Then characteristic polynomial will be expressed by and as follows.
The equivalent time constant of the ith
order and the stability index of the jth
order are defined as follows.
Thus τ can be considered the equivalent time constant of the 0-th order and is considered as the stability index of
the 1st
order. The stability index of the 2nd
order is a good measure of stability and is shown below,
When the performance specifications are given, they must be modified to the design specifications. In
CDM, the design specifications are the equivalent time constant τ and the stability indices for the higher order
terms. The stability indices for the lower order terms are already specified. Usually the rise time, the settling time,

the overshoot, and the peak time are used for the time response specification. However from the CDM design point
of view, only the settling time ts is meaningful, because it gives upper bound of τ, where ts = 2.5 3 τ. The
frequency response specifications are used for the high frequency attenuation characteristics and the low frequency
disturbance rejection characteristics. Using equation (8) the denominator polynomial of Gc(s) gives the
characteristic polynomial of the closed loop system and with the equations (19) and (20) the controller parameters
are found.
IV. Simulation Results
The designed MRAC is implemented in real time also compared with a PI controller. The set point is
given in terms of percentage of level. 20% of level is given as nominal operating value, after 14000(s) the set point
has been changed to 30%. The load is applied at the valve in outlet, for 10 liter/min change in outflow. For the
sampling time, 1 sec is selected.
Table 1. Plant parameters
Process variables Nominal operating conditions
Level(h) 1 m
Flow rate, (Fin) 0.2215 m2
/sec
Radius of the tank (r) 1 m
Constant of the outlet valve(cs) 0.05 m2
Outlet valve stem position(xs) 1
Gravitational acceleration (g) 9.8 ms-1
Maximum level 2 m
Table 1 provides the description of Spherical tank parameters where table 2 shows the PI controller
parameters for various linearized plant models as different operating levels 10%, 50% and 66% of tank level.
Table 2. PI controller parameters concerning plant models
Linearized models Transfer function models PI Controller parameters
Kc Ki
Model 1 0.825 0.002
Model 2 1.385 0.003
Model 3 2.29 0.005
Fig 2. Plant response with PI controller Fig 3. Plant response with CDM-MRAC
Fig 2 shows the closed loop response of plant with PI controller. Notice that the response has large rise
time and settling time. Fig 3 shows that the response of plant with MRAC. Notice that the rise time has increased as
the reference model has high rise time also the settling time.

Fig 4. MRAC controller parameter theta1 Fig 5. MRAC controller parameter theta2
Fig 4 and 5 show that the convergence of controller parameters as the tracking error goes asymptotically
zero. And fig 6 shows the comparison of performance between the PI controller and CDM-MRAC.
Fig 6. Comparison of plant responses with CDM-MRAC and PI controller
Table 3. Comparison of controllers performance in time domain
Table 4. Comparison of controllers performance
Controller structure ISE IAE ITAE
CDM-MRAC 5.504e+5 7.699e+4 1.078e+9
PI controller 1.65e+6 1.717e+5 2.442e+9
From fig 6, Table 3 and 4 provides the various performance specifications in time domain between CDM-
MRAC and PI controller. From these, the CDM-MRAC outperforms the PI controller.
Controller structure Settling time (ts) secs %MP Rise time (tr) secs
MRAC using CDM
Servo 1 12500 48 2000
Servo 2 1200 50 500
Load 1 1100 - -
PI controller
Servo 1 12500 - 1200
Servo 2 1200 - 1100
Load 1 1100 - -

V. Hardware Implementation
Fig 7. Closed loop apparatus setup of Spherical tank level process (Real time)
Fig 10. Comparison of Plant responses of CDM-MRAC and PI controller
Fig 8 shows the closed loop response of plant with PI controller and fig 9 gives the closed loop response
of plant with CDM-MRAC scheme. Notice that from Fig 10, it shows that the MRAC provides improved rise time
and settling time and it has 50% of overshoot. The response with PI controller provides less rise time and settling
Fig 8. Spherical tank response of PI
controller in real time
Fig 9.Spherical tank response of CDM-
MRAC in real time

time and 30% of overshoot. And the servo and regulatory responses are good in MRAC scheme. The following
tables 5 and 6 show that the comparison of various performance specifications in time domain.
Table 5. Performance comparison of controllers in real time
Controller structure
Settling time
(ms)
%MP
Rise time
(tr)
MRAC using
CDM
Servo 1 1.2e+5 40 1.6e+4
Servo 2 8e+5 10 1.5e+4
Load 1 1.4e+5 - -
Load 2 4.8e+4 - -
PI controller
Servo 1 NA 32 2.9e+4
Servo 2 1.6e+5 50 1.6e+4
Load 1 NA - -
Table 6. Comparison of controllers performance
Controller structure ISE IAE
CDM-MRAC 4532492 24.83563
PI controller 10532359 25.74069
VI. Conclusion
The model reference control is designed for the stability of a closed loop system in the sense of Lyapunov.
From the closed loop transfer function, the characteristics polynomial has been taken, the CDM is applied on the
characteristics polynomial to find the unknown adaptation gains. The strength of CDM lies in that, for any plant,
minimum phase or non-minimum phase, the simplest and robust controller under practical limitation can be found.
Such controller closely agrees with the controllers which are accepted as good controllers in practical application.
REFERENCES
[1] S. Manabe. Coefficient Diagram Method. Automatic Control in Aerospace, Seoul, Korea. 211-222, 1998.
[2] S. Manabe. A simple proof and the physical interpretation of Routh stability criterion. J. IEEE Japan, 117, (12), 851-854, 1997c.
[3] Petros A. Ioannou and Jing Sun. Robust Adaptive control, 8-12, 105-132, 330-363.
[4] Karl Johan Astrom and Bjorn Wittenmark. Adaptive control, second edition. 210-15.
[5] Pavan K. Vempaty, Ka C. Cheok, and Robert N. K. Loh. Experimental Implementation of Lyapunov based MRAC for Small Biped
Robot Mimicking Human Gait. ISA Transactions- 21(3), 2007.
[6] Muhammad Nasiruddin Mahyuddin and Mohd Rizal Arshad. Performance Evaluation of Direct Model Reference Adaptive Control on a
Coupled-tank Liquid Level System. Elektrika-10(2), 2008.
[7] Serdar Ethem Hamamci Nusret Tan. Design of PI controllers for achieving time and frequency domain specifications simultaneously.
ISA Transactions- 45(4), 2006.

Design and Implementation of Model Reference Adaptive Controller using Coefficient Diagram Method for a nonlinear process

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Design and Implementation of Model Reference Adaptive Controller using Coefficient Diagram Method for a nonlinear process