State Estimation based Inverse Dynamic Controller for Hybrid system using Artificial Neural Network

State Estimation based Inverse Dynamic Controller for Hybrid system
using Artificial Neural Network
Xiao Laui,
Research Scholar,
City University of Hong Kong, China.
Rui-Lain Chua,
Professor,
City University of Hong Kong, China.
Abstract:
Authors have proposed State Estimation Based Inverse Dynamics Controller (SEBIDC), which utilizes an Artificial
Neural Network (ANN) based state estimation scheme for nonlinear autonomous hybrid systems which are subjected to
state disturbances and measurement noises that are stochastic in nature. A salient feature of the proposed scheme is
that it offers better state estimates and hence a better control of non-measurable state variables with a non linear
approach in correcting the a priori estimates by avoiding statistical linearization involved in existing approaches based
on derivative free estimation methods. Simulation results guarantees significant reduction in Integral Square Error
(ISE) and standard deviation (σ) of error, between the controlled variable and set point and control signal computation
time when compared with best existing related work based on Unscented Kalman Filter (UKF) and Ensemble Kalman
Filter (EnKF). Detailed analysis of the experimental results on real plant under different operating conditions such as
servo and regulatory operations, initial condition mismatch, and different types of faults in the system, confirms
robustness of proposed approach in these conditions and support the simulation results obtained. The main advantage
of the proposed controller is that the control signal computation time is very much less than the selected sampling time
of the process, so direct control of the plant is possible with this approach.
Key words: Artificial Neural Network, Hybrid Dynamic Systems, State Estimation, Inverse Dynamics Controller.
1. Introduction
Systems in which it may be required to model inherent process discontinuities, where the continuous behavior is drastically
changed, or use actuators and sensors which are often fundamentally discontinuous, or use discrete events that can be a useful
abstraction to model various mode switching used in the specification and control of the basically continuous process, require
hybrid systems modeling and control approach ([1], [2], [3] and [4]). Kalman filter (KF) [5] and extended Kalman filter (EKF)
[6] are used as a state estimator in conventional state observers. For linear systems as uncertainties in state and measurement
equations can be modeled as Gaussian white noise processes Kalman filter can generate optimal estimates of state. Extended
Kalman filter (EKF), which is a natural extension of the linear filter to the nonlinear domain through analytical linearization,
can be used in state estimation of nonlinear systems. In this approach, the estimated states are obtained by using Taylor series
expansion of the nonlinear state transition operator. But, it requires analytical computation of Jacobians at each time step,
which is considered to be computationally demanding for complex nonlinear systems. Also it is required that nonlinear
function vector appearing in state dynamics and output dynamics should be smooth and at least once differentiable. However,
dynamical models of Hybrid systems involve discontinuities due to switching of the discrete state values. As Jacobians cannot
be computed for non-smooth functions, EKF cannot be realized in Hybrid systems [7]. In [8] a moving horizon based state
estimation approach has been reported for hybrid system estimation. But, the use of fixed arrival cost used in the moving
horizon estimator formulation result in sub-optimal state estimates. The authors in [9] have proposed UKF as an alternative to
EKF so that the main shortcoming of EKF when used for highly nonlinear system is eliminated. [10] has proposed a derivative
free non linear filtering technique for nonlinear hybrid systems. As far as the control element is concerned, [11] had proposed a
robust model predictive control (RMPC) scheme for a class of hybrid system such as piece wise affine system to ensure simple
and fast suboptimal solution for the control problem with reduced computation time. Similarly, the nonlinear model predictive
control (NMPC) in [7] and [12], and fault tolerant model predictive control in [13], for hybrid systems, used UKF approach in
its state estimation part. All the state estimation schemes for hybrid systems in literature involve analytical or statistical
linearization [14] and preclude their use from systems which require more accurate state estimates. Introduction of Artificial
Neural Network (ANN) in state estimation and control of different systems considerably improves the performance which is
The International journal of analytical and experimental modal analysis
Volume XII, Issue III, March/2020
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very clear from the works reported in [15], [16], [17] and [18]. But, in the case of [15], if a nonlinear scheme is used in
correction of a priori estimates, more accurate state estimates can be obtained in hybrid systems also and this has been
implemented in this work. The distinguished feature of the proposed approach is that it uses nonlinear ANN to correct the a
priori estimates and gives better estimate than its counterparts. Once trained for sufficient variety of input data, including ill-
conditioned system data, the ANN-based state estimator provides accurate estimates of the system states. Further, an ANN
based Controller (ANNC), using this estimator, is developed for controlling the non-measurable states of the hybrid system.
The proposed controller (ANNC) provides better performance and has following advantages over existing schemes.
 Apart from analytical and statistical linearization in the correction part of EKF and UKF based controllers respectively;
proposed controller uses a nonlinear correction approach using ANN to correct the a priori estimates and hence offers a
better state estimates by avoiding the linearization. The correction part of ANNC is completely parameter independent,
and thereby gives better state estimates even under there is mismatch in the parameters.
 ANN has built in noise rejection capability which makes the ANNC scheme robust in performance.
Comparative analysis of the proposed approach with best relating work based on UKF (statistical linearization approach) and
EnKF (Particle Filter) is made in terms of ISE in state estimate on the same benchmark model and it reveals that the proposed
approach is able to reduce the error in state estimates. Detailed performance evaluation of the proposed approach under servo –
regulatory operations and plant model parameter mismatch were conducted. Detailed analysis of the experimental results on
the real plant under different operating conditions such as initial condition mismatch, and different types of faults in the system
confirms efficacy of proposed approach. Here, an inverse dynamics controller is utilized for controlling the non-measurable
states of the system so that the computational burden is very much reduced when compared with the model predictive control
scheme implemented in [7], [12] and [13] without compromising the performance. Also the constraints handling capability for
this scheme is also achieved with this approach by introducing upper and lower limiting functions at the output of both
estimator and controller. The rest of the paper is organized as follows. In section 2, description of the ANN state estimation
algorithm developed for hybrid systems is given. The Section 3 explains the SEBIDC scheme. Simulation results and detailed
performance analysis of the proposed scheme and with comparison to the best related work is given in section 4. Experimental
results and its analysis are presented in Section 5. Finally, section 6 summarizes the paper.
2. ANN based Hybrid State Estimation
In this scheme, an ANN based correction is developed. As in the case of EKF and UKF, ANN based state estimation is also
recursive in nature. Even though it has the same framework of Kalman filter based state estimator, it is designed for
eliminating the analytical and statistical linearization [14] used in the case of EKF and UKF. This structure is suggested
because recurrent type of ANN is better for the complex dynamic system [18]. The schematic diagram of proposed ANNC is
as given in Fig. 1.
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Fig.1. Schematic representation of proposed ANNC
The detailed nonlinear auto regressive with exogenous input (NARX) structure used for the considered problem is given in Fig.
2 and the other NARX parameters used for this study are provided in Table 1.
The current output can be predicted as a function of present and past inputs and past outputs as given below, in which Y and X
represent the outputs and inputs of the network respectively and KNN is a nonlinear ANN function.
Y (k) = KNN{X (k), X (k-m), Y (k-1)… Y (k-n)} (1)
Table 1
ANN Parameters
Parameter Value
ANN Structure NARX
No. of hidden layers 1
Hidden Layer neurons 5
Hidden layer activation function ‘tan sigmoid’
Output layer activation function ‘purelin’
No of epochs 100
No of exogenous inputs 4
No. of delayed inputs 0
No of outputs 3
No. of feedback output delays 2
Training method Back propagation
Training function Levenberg–Marquardt
Process Noise, ( )w t
Set point
Input, ( )u t
Measurement Noise, ( )v t
Output, ( )y t
Prediction
Model
ANN
Correction
Function
ITD
OTD
SEBIDC
Innovation,
( 1)k kˆ( 1)x k k
xk/k-1
ˆ( )x k
( )u k
( )y k
Process
SampleHold
ˆ( 1)y k k
ˆ( )x k
Estimator
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Performance Function Mean Square Error
A sequence of current and past input vectors (X (k), X (k-1), X (k-m)) are obtained by passing X (k) through an input time
delay unit, ITD (0: m). Similarly output time delay unit, OTD (1: n) provides a sequence of past output vectors (Y (k-1), Y (k-
n)). For the considered problem, the input and the output are T
1 2 3
ˆ ˆ ˆX( ) [ ( 1), ( 1), ( 1), ( 1)]    k h k k h k k h k k k k
T
1 2 3
ˆ ˆ ˆY( ) [ ( ), ( ), ( )]k h k h k h k respectively.
Similar to Kalman filter based state estimators and its nonlinear extensions, the proper value for the initial state vector is
assumed for the prediction model. The input and output measurements are made from the process and the input measurement
are presented to the state prediction model (F) along with the assumed initial state vector in order to compute the time updated
values for states.
ˆ ˆ( 1) F( ( 1), ( ))  x k k x k u k (2)
With, ˆ ˆ( 1) (0) [ (0)]  x k x E x , the assumed initial value of state vector.
The a priori state estimates, ˆ( 1)x k k can be given to the output model (H) so that a priori estimates of the output, ˆ( 1)y k k
can be obtained as
ˆ ˆ( 1) H ( 1)    y k k x k k (3)
The innovation between plant output ( )y k and a priori output estimate ˆ( 1)y k k is calculated as
ˆ( 1) ( ) ( 1)   k k y k y k k (4)
In the correction step of the algorithm, the a priori state estimates will be corrected using this innovation with the help of the
ANN to obtain a posteriori estimates of state vector ˆ( )x k k .
 NNˆ ˆ ˆ( ) K ITD( ( 1), ( 1)),OTD( ( ))  x k x k k k k x k (5)
These estimated states are fed back to the controller for calculating the new input signal to the plant.
Fig.2. NARX structure for the three-tank hybrid system
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3. State Estimation Based Inverse Dynamics Controller (SEBIDC)
In hybrid systems, for the controller design, its mode switching property has also to be taken in to account. For such processes,
model based control schemes are proposed in the literature ([12], [13], [19], [20], [21] and [22]) for obtaining the satisfactory
control of the output variables. In this work a model based control namely inverse dynamics controller (IDC) have been
implemented for controlling the non-measurable states of Hybrid System. The objective of this section is to review the non-
linear dynamic control technique that can be applied to develop a non-measurable level control system that is valid over the
entire operating region of the hybrid three-tank system which is described below.
Consider the nonlinear system of the form,
( ) ( ) x A x B x u (6)
y Cx (7)
Where, A(x) = (n × 1) vector, B(x) = (n × m) matrix, C = (m × 1) vector.
Using the inverse dynamics of (6) and (7), input vector u can be represented as a function of v and x,
Let it be 1F ( , )u v x
Where, F1 is a nonlinear function and v is the input to the inverse system.
Implementation of the inverse dynamics controller of (6) and (7) is done by repeatedly differentiating the measurement
function until the input variable u appears.
Differentiating (7)
y Cx (8)
[ ( ) ( ). ] y C A x B x u (9)
* *
( ) ( ). y A x B x u (10)
Where, A*
(x) =C.A(x) and B*
(x) =B.A(x)
So the control law u can be written as per [16] as
1
* *
( )[ ( )]

 u B x v A x (11)
A sufficient condition for the existence of an inverse system model to (6) and (7) is that B*
in (11) be non-singular. If this is the
case, then the inverse system model takes the form,
( ) ( )[ ( ) ( ) ]   x A x B x F x G x v (12)
Where,
1
* *
( ) ( ) ( )

F x B x A x and
1
*
( ) ( )

G x B x
( ) ( )[ ( ) ( ) ]   x A x B x F x G x v (13)
The input to the inverse system is v = y – yref
As the hybrid three-tank system, considered under this study can be directly represented in the form of (6) and (7), applying
this procedure will yield the control law as
 1 1 1sp=A C h - + + +1in 1 1 3 5F h Q Q Q (14)
 2 2 2sp=A C h - + + +2in 2 2 4 7F h Q Q Q (15)
Since h1, h2, Q1, Q2, Q3, Q4, Q5, and Q7, are non-measureable for the considered problem, the estimated values can be used so
that the controller becomes an estimator based inverse dynamics controller.
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 1 1 1sp
ˆ ˆ ˆ ˆ=A C h - + + +1in 1 1 3 5F h Q Q Q (16)
 2 2 2sp
ˆ ˆ ˆ ˆ=A C h - + + +2in 2 2 4 7F h Q Q Q (17)
Where, C1 and C2 are controller tuning parameter, and its values are varied from 0 < C1,C2 < 1 on separate runs and the
integral square error between the controlled variable and the set point, is noted. The values of C1 and C2, which give the
minimum ISE, are selected as the tuning parameters. The h1sp and h2sp are the corresponding desired values of water levels.
4. Simulation Results and Performance Analysis
The schematic representation of hybrid three-tank system is given in Fig.3.The benchmark system used in [7] is used here with
same levels in three tanks (h1, h2 and h3) as continuous states and z1 and z2 variables as discrete states for evaluating the
performance the controller in comparison to best related work.
Fig.3. Schematic representation of the benchmark hybrid three tank system
Detailed modeling of the hybrid three-tank system is given as appendix. The algorithm has been implemented in MATLAB.
The detailed discussions about the results obtained in the simulation are given in the following subsections.
4.1 Performance in Servo Operation
Servo operation of the closed loop system when a change in set point occurs was conducted by introducing a step change with
magnitude 0.04 m at 100th
sampling instant. The results are given in Fig.4. Comparison with the best existing related work is
shown in Table 2. Comparison of the proposed approach to UKF based approach based on ISE shows that the new approach is
better as ISE reduced from 3.3278 to 0.5533 in level, h1 and from 3.7203 to 0.4568 in level, h2. Also, the average computation
time per iteration reduced from 60.65 seconds to 0.0777 seconds. Evolution of true and estimated states of hybrid three-tank
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system with ANNC (Servo operation) is shown in Fig.5. Evolution of true and estimated values of discrete variables of hybrid
three-tank system with ANNC (Servo Operation) is shown in Fig.6.
Fig.4. Servo response of hybrid three tank system with ANNC (a) Level in Tank 1, (b) Level in tank 2 (c) Manipulating variables
Fig.5. True and estimated states of hybrid three tank system with ANNC (Servo operation) (a) Level in Tank 1, (b) Level in Tank 2, (c) Level in tank 3
50 100 150 200 250 300
.2
.4
Level(h
1
)
(a)
Setpoint CV(Proposed)
50 100 150 200 250 300
0.2
0.4
Level(h
2
)
(b)
Setpoint CV(proposed)
50 100 150 200 250 300
0
0.5
1
(c)
Sampling Instants
Manipulating
Variables
Fin1 Fin2
50 100 150 200 250 300
0.2
0.4
Level(h1)
(a)
True Estimated
50 100 150 200 250 300
0.2
0.4
Level(h2)
(b)
True Estimated
50 100 150 200 250 300
0.2
0.4
0.6
Level(h
3
)
(c)
Sampling Instants
True Estimated
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Fig.6. Evolution of true and estimated values of discrete variables of hybrid three tank system with ANNC (Servo Operation)
4.2 Performance in Regulatory Operation
The results of the performance of the estimator in regulatory operation are given in Fig.7. Comparison with the best existing
related work is shown in Table 3. Comparison of proposed approach to UKF based approach based on standard deviation
shows that the new approach is better as standard deviation has reduced from 0.0130 to 0.0127 in level h2 and from 0.0484 to
0.0065 in level h3 with very close standard deviation in level h1. It may be noted that the maximum standard deviation in the
case of nonmeasured state variables is 0.0217 in the proposed approach and 0.0484 in the UKF based NMPC(45% reduction
with proposed approach).
The results of the performance of the controller in regulatory operation are given in Fig.8 and evolution of true and estimated
values of discrete variables of hybrid three tank system with ANNC (Regulatory Operation; Disturbance by varying the valve
position of fifth hand valve) in Fig.9. Comparison with the best existing related work is shown in Table 4. Comparison of
proposed approach to UKF based approach based on ISE shows that the new approach is better as ISE has reduced from
3.5869 to 0.0587 in level h1 and from 1.5212 to 0.0366 in level h2. Also, the average computation time per iteration has
reduced from 59.05 seconds to 0.0739 seconds.
100 200 300
-2
-1
0
1
2
Sampling Instants
z
1
(True)
100 200 300
-2
-1
0
1
2
Sampling Instants
z
2
(True)
100 200 300
-2
-1
0
1
2
Sampling Instants
z
1
(Estimate)
100 200 300
-2
-1
0
1
2
Sampling Instants
z
2
(True)
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Fig.7. Evolution of true and estimated states of hybrid three tank system with ANNC (a) Level in Tank 1, (b) Level in Tank 2, (c) Level in tank 3
Fig .8. Regulatory response of hybrid three tank system with ANNC (a) disturbance, (b) Level in Tank 1, (c) Level in tank 2 (d) Manipulating variables
20 40 60 80 100 120 140
0.2
0.3
0.4
Level(h
1
)
(a)
True Estimated
20 40 60 80 100 120 140
0.2
0.3
0.4
Level(h
2
)
(b)
True Estimated
20 40 60 80 100 120 140
0.2
0.3
0.4
Level(h3
)
(c)
Sampling Instants
True Estimated
20 40 60 80 100 120 140
0.2
0.3
0.4
Level(h
1
)
(b)
Setpoint C V(Proposed)
20 40 60 80 100 120 140
0.2
0.3
0.4
Level(h
2
)
(c)
Setpoint CV (Proposed)
20 40 60 80 100 120 140
0
0.5
1
(d)
Sampling Instants
Manipulating
Variables
Fin1
Fin2
20 40 60 80 100 120 140
0
0.5
1
Disturbance
(a)
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Fig.9. Evolution of true and estimated values of discrete variables of hybrid three tank system with ANNC (Regulatory Operation)
In order to obtain better insight of the ability of the proposed controller to achieve decoupling and offset-free control action,
hypothetical situation, in which state and measurement noise are not present, is simulated. Response given in Fig. 10 reveals
that the effect in level of tank 2, due to the disturbance in tank 1 is very much less compared with that given in [7]. As in the
case of UKF based NMPC in [7], proposed method also giving a slight offset.
Table 3
Regulatory Control Problem: Estimator performance Comparison
Controller σE(h1) σE(h2) σE(h3)
Proposed 0.0217 0.0127 0.0065
UKF based NMPC[7] 0.0213 0.0130 0.0484
UKF based NMPC[7] 0.0242 0.0141 0.0488
Table 4
Regulatory Control Problem: Controller Performance Comparison
Controller ISE(h1) ISE(h2) Avg. Computation time per iteration (S)
Proposed 0.0587 0.0366 0.0739
UKF based NMPC[7] 3.5869 1.5212 59.05
EnKF based NMPC[7] 3.3928 1.4011 206.44
50 100 150
-2
-1
0
1
2
Sampling Instants
z
1
(True)
50 100 150
-2
-1
0
1
2
Sampling Instants
z
2
(True)
50 100 150
-2
-1
0
1
2
Sampling Instants
z
1
(Estimate)
50 100 150
-2
-1
0
1
2
Sampling Instants
z
2
(True)
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Fig .10. Regulatory response of hybrid three tank system with ANNC (without state and measurement noise) (a) disturbance, (b) Level in Tank 1, (c) Level in
tank 2 (d) Manipulating variables
4.3 Plant Model Parameter Mismatch
The performance of the controller in case of plant model parameter mismatch is considered and the performance is given in
Fig. 11. From Table.5, it can be seen that the ISE is improved to 0.0276 from 0.9231 for level h1 and to 0.0228 from 1.0579 for
level h2 when compared to best existing related work based on UKF.
Fig.11. Response of hybrid three tank system with ANNC (Plant-Model mismatch ) (a) Level in Tank 1, (b) Level in Tank 2, (c) Level in tank 3
Table 5
Plant Model Parameter Mismatch: Controller Performance Comparison
Controller ISE(h1) ISE(h2)
Proposed 0.0276 0.0228
UKF based NMPC [7] 0.9231 1.0579
EnKF based NMPC[7] 0.8647 0.9688
20 40 60 80 100 120 140
0
0.5
1
Disturbance
(a)
20 40 60 80 100 120 140
0.2
0.3
0.4
Level(h
1
)
(b)
20 40 60 80 100 120 140
0.2
0.3
0.4Level(h
2
)
(c)
20 40 60 80 100 120 140
0
0.5
1
(d)
Sampling Instants
Manipulating
Variables
Fin1 Fin2
20 40 60 80 100
0.2
0.3
0.4
Level(h
1
)
(a)
20 40 60 80 100
0.2
0.3
0.4
Level(h
2
)
(b)
20 40 60 80 100
0
0.5
1
(c)
Sampling Instants
Manipulating
Variables
Fin1 Fin2
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5. Experimental Results and Performance Analysis
Real-time experimental validations were carried out on the experimental setup. In addition to the experimental setup, other
tools used, which were for the real time implementation are the software Lab VIEW and the NI DAQ (USB6251). In the real
system, the performance of the controller in regulatory operation and servo operation based on ISE and average computation
time per iteration is shown in Fig. 12, Table 6, Fig. 13, and Table 7. In Table 8 and Fig. 14 response of the system in initial
condition mismatch is shown. The response of the system in +10% and -10% plant model parameter mismatch is given in
Tables 9 and 10 and Figures 15 and 16 respectively. Results of hand valve faults which can occur in real time application are
given in Table 11, Fig. 17, Table 12 and Fig. 18. The real time experimental results support the simulation results on
performance.
Table 6
Regulatory Control Problem: Controller Performance Comparison
Avg. Computation
time per iteration (S)
Proposed 0.0200 0.0193 0.1038
Table 7
Servo Control Problem: Controller Performance Comparison
Avg. Computation
Proposed 0.0172 0.0144 0.1152
Table 8
Initial Condition Mismatch: Controller Performance Comparison
Avg. Computation
Proposed 0.0214 0.0592 0.1017
Table 9
Plant Model Parameter Mismatch (+10%): Controller Performance Comparison
Avg. Computation
Proposed 0.0045 0.0069 0.2112
Table 10
Plant Model Parameter Mismatch (-10%): Controller Performance Comparison
Avg. Computation
Proposed 0.0122 0.0131 0.1037
Table 11
Hand Valve Faults -Leakage: Controller Performance Comparison
Avg. Computation
Proposed 0.0600 0.0077 0.0768
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Table 12
Hand Valve Faults -Clogging: Controller Performance Comparison
Avg. Computation
Proposed 0.0082 0.0078 0.0943
Fig.12. Regulatory response of hybrid three tank system with ANNC (a) Level in Tank 1, (b) Level in tank 2
Fig.13. Servo response of hybrid three tank system with ANNC (a) Level in Tank 1, (b) Level in tank 2
50 100 150 200 250 300 350 400
0.24
0.27
0.3
0.33
Level(h
1
)
(a)
CV1
(Proposed) SETPOINT1
100 200 300 400
0.24
0.27
0.3
0.33
Level(h
2
)
(b)
Sampling Instants
CV2
50 100 150 200
0.23
0.26
0.29
0.32
Level(h
1
)
(a)
50 100 150 200
0.23
0.26
0.29
0.32
Level(h
2
)
(b)
Sampling Instants
CV2
CV1
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Fig.14. Closed response of hybrid three tank system with ANNC (Initial Condition Mismatch ) (a) Level in Tank 1, (b) Level in Tank 2
Fig.15. Closed response of hybrid three tank system with ANNC (Plant-Model mismatch +10% ) (a) Level in Tank 1, (b) Level in Tank 2
Fig.16. Closed response of hybrid three tank system with ANNC (Plant-Model mismatch -10% ) (a) Level in Tank 1, (b) Level in Tank 2
50 100 150 200 250 300 350 400
0.26
0.3
0.34
0.38
Level(h
1
)
(a)
CV1
50 100 150 200 250 300 350 400
0.26
0.3
0.34
0.38
Level(h
2
)
(b)
Sampling Instants
CV2
50 100 150 200 250
0.28
.3
0.32
Level(h
1
)
(a)
CV1
50 100 150 200 250
0.28
0.3
0.32
Level(h
2
)
(b)
Sampling Instants
CV2
50 100 150 200 250
0.26
0.28
0.3
0.32
Level(h
1
)
(a)
CV1
50 100 150 200 250
0.26
0.28
0.3
0.32
Level(h
2
)
(b)
Sampling Instants
CV2
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Fig.17. Closed response of hybrid three tank system with ANNC (Handvalve fault-Leakage ) (a) Level in Tank 1, (b) Level in Tank 2
Fig.18. Closed response of hybrid three tank system with ANNC (Handvalve fault-Clogging ) (a) Level in Tank 1, (b) Level in Tank 2
6. Conclusion
An ANN estimation based control scheme which offers better state estimates and hence better control, with a non linear
approach in correcting the a priori estimates by avoiding statistical linearization involved in derivative free estimation is
proposed. On comparing with best exiting related work based on statistical linearization, 85%, 97% and 97% reduction in
Integral Square Error (ISE), between controlled variable and set point, for servo, regulatory and plant model parameter
mismatch operations respectively, and a 45% reduction in standard deviation (σ) of error between true and estimated values of
non-measurable states for regulatory control operation were obtained. In addition to these performance improvements, the
attracting feature of the proposed method is the time required for the computation of control signals and is very much less than
the sampling time of the process which ensures the capability of online implementation as direct control algorithm using the
proposed approach. The experimental studies conducted on a real plant illustrate the robust performance of the proposed
controller in offering better online control of hybrid dynamic system under real time operating constraints.
References
[1] Lennartson, M. Tittus, B. Egardt, and S. Pettersson, “Hybrid systems in process control”, IEEE Control Systems, 16,(5),
(2002), 45-56.
[2] Goebel R., Sanfelice R. G. Teel, A. R., Hybrid Dynamical Systems , IEEE Control Systems Magazine, 29(2), (2009), 28-
93.
100 200 300 400 500
0.26
0.28
Level(h
1
)
(a)
100 200 300 400 500
0.26
0.28
Level(h
2
)
(b)
Sampling Instants
CV2
CV1
50 100 150 200 250
0.25
0.3
0.35
Level(h
1
)
(a)
100 200 300 400 500
0.25
0.3
0.35
Level(h
2
)
(b)
Sampling Instants
CV2
CV1
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[3] Goebel, R; Sanfelice, R, G.and Teel, A.R., Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton
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Controller, In Proc. 17th World Congress of IFAC, (2008), 12510-12515.
[13]Jagadeeshan P., Sachin C. P., Sirish L. S, Design and Implementation Fault Tolerant Model Predictive Control Scheme on
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[14]Lefebvre, T.; Bruyninckx, H.; De Schutter, J., Comment on "A new method for the nonlinear transformation of means and
covariances in filters and estimators", IEEE Transactions on Automatic Control, 47(.8), (2002), 1406-1409.
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APPENDIX: PROCESS DESCRIPTION
In the hybrid three-tank system, three tanks of uniform area (A1, A2 and A3) are connected to each other and to sump
through seven hand valves and two variable speed pumps as in figure 3. The seven valve discharge coefficients are represented
by k1 to k7. By conducting simple steady state experiments and necessary calculations, these hand valve coefficients are found,
which are given in table A.1. Now throughout this study the position of these hand valves should not be altered (For the
simulation results, all the parameter values are taken from [7]).
After calculating ki (where, i =1, 2,…,7), the system is modeled using volume balance equations as in (A.1) – (A.3),
in which Qi represent the flows through ith
hand valve and its equations are provided in (A.4) –(A.10). This is used as the
prediction model for calculating a priori state estimates in the estimation algorithms described in section 2.
1
1 1 3 5
1
1
[ ]
A
   
dh
Fin Q Q Q
dt
(A.1)
2
2 2 4 7
2
1
[ ]
A
   
dh
Fin Q Q Q
dt
(A.2)
3
1 2 3 4 6
3
1
[ ]
A
    
dh
Q Q Q Q Q
dt
(A.3)
1 1 1 3 1 3=k ( - ) 2g -Q sign h h h h (A.4)
2 2 3 2 3=k ( - ) 2g -2Q sign h h h h (A.5)
3 1 1 0 3 0k3 2g ( - h ) ( - h ) Q z a h b h (A.6)
4 2 4 2 0 3 0k 2g ( - h ) ( - h ) Q z c h b h (A.7)
15 5 1 d2g( h ) Q k h (A.8)
6 6 3 z=k 2g( h )Q h (A.9)
27 7 2 d=k 2g( h )Q h (A.10)
The system is modeled as hybrid system because the model contains both continuous and discrete states in it. The
flow through the middle inter connecting pipes can be better modeled with the help of discrete variables.
The levels in three tanks (h1, h2 and h3) make the continuous state and the variables z1 and z2 are the discrete state.
Depending on the flows, Q3 and Q4, discrete state variables, z1 and z2 may take the values 0, +1 or -1 as given below.
-1 if Q3 is away from Tank3
z1= 0; if Q3= 0 (A.11)
+1; if Q3 is towards Tank3
Similarly
-1 if Q4 is away from Tank3
z2= 0; if Q4=0 (A.12)
+1; if Q4 is towards Tank3
As Q3 and Q4 are determined by the three levels, it is obvious that the discrete mode switching depends on the
continuous variables; hence the system comes under the definition of AHS. The variables, a, b and c in (A.6) and (A.7) are
ISSN NO:0886-9367
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temporary variables to indicate the levels in three tanks as below or above threshold level. The threshold level, h0 is the height
between middle interconnecting pipes and the tank bottom as marked in figure 3. The height difference between the 5th
, 6th
and
7th
hand valves from the bottom of the corresponding tanks are given by hd1, hz and hd2 respectively. The pumps for delivering
inflows are variable speed pumps which deliver flow ranges from 0 to rated flow based on the input signal value (0-5V) given
to the pump. The experimental setup is given in figure A.1, whose specifications are given in table A.2.
Table A.1
Experimental Values Of Discharge Coefficients Of Hand Valves
Discharge coefficient Value(m2
) Discharge coefficient Value(m2
)
k1 2.6363x10-5
k4 3.4316 x10-5
k2 2.4891 x10-5
k6 2.2538 x10-5
k3 3.7984 x10-5
k5, k7 0
Table A.2
Values Of Different System Parameters
Parameter Value
Tank height 0.60m
Tank over flow height 0.55m
Rated flow rate of pumps 240 lph
Input voltage to pump 0 –5 V
Tank inner diameter 0.15m
Inter connecting pipes inner diameter 0.0125m
Figure A.1: Experimental Setup of autonomous hybrid three – tank system
ISSN NO:0886-9367
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