adaptive control of a two input – two output system

Proceedings of the 10th Mediterranean Conference on Control and Automation - MED2002 Lisbon, Portugal, July 9-12, 2002.

ADAPTIVE CONTROL OF A TWO INPUT – TWO OUTPUT SYSTEM USING DELTA MODELS V. Bobál, P. Navrátil, P. Dostál Department of Control Theory, Institute of Information Technologies, Tomas Bata University in Zlin, Nám. TGM 275, 762 72 Zlín, Czech Republic Phone: +420 67 754 3217, Fax: +420 67 7543333 e-mail: [email protected] Keywords: adaptive control, multivariable control, delta model, decoupling, real time control

Abstract This paper presents the design of an adaptive controller for a two input – two output (TITO) system using delta models. This controller has been verified by simulation and real time control of a non-linear laboratory model CE108 - coupled drives apparatus. The recursive least squares method is used in identification part of this controller. The synthesis is based on a polynomial approach. Decoupling, where the compensator is placed ahead of the system, suppresses the interactions between control loops. The results of the simulation and the real-time experiments are also given.

1 Introduction Many technological processes require that several variables relating to one system are controlled simultaneously. Each input may influence all system outputs. The design of a controller able to cope with such a system must be quite sophisticated. There are many different methods of controlling multivariable systems. Several of these use decentralized PID controllers [10], others apply single input-single-output (SISO) methods extended to cover multiple inputs [6]. Here decoupling methods are used to transform the

multivariable system into a series of independent SISO loops [8], [14], [15], [16]. This paper is organized as follows: in Section 2 is defined a δ - model and it described its identification; Section 3 presents the controlled model; Section 4 describes how feedback control without decoupling is designed; Section 5 describes two decoupling methods; Section 6 describes the system identification method; Section 7 gives the simulation results; Section 8 contains the experimental results; finally, Section 9 concludes the paper.

2 Delta models and their identification If G(s) is the transfer function of a continuous-time dynamic system (s is a complex variable), then the following expression for the discrete transfer function with the zero - order holder is valid G( z) =

z − 1  −1 G (s ) Z L  z s  

(1)

This step transfer function (1) is a rational polynomial function with variable z. The simple model structure, easy recursive identification using measurable data, suitability for the synthesis of the discrete control loop as well as for the description and expression of different types of stochastic process, including disturbance modelling, are all advantages of the z – transform function. The step z - transfer functions have some disadvantages when the sampling period decreases:

identification algorithm to this model. To parameters estimates of the δ - model, the recursive least squares method (RLSM) with directional forgetting is applied (Kulhavý [7], Bobál et al. [1].

•

the Z - transformation parameters do not converge as the sampling period decreases to the Laplace – transformation continuous parameters from which they were derived,

•

very small sampling periods yield very small A useful model to apply this method of numbers from the transfer function numerator, identification is the regression (ARX) model which the poles transfer function approach the is often expressed in its compact form unstable domain as the sampling period y (k ) = Θ T (k )Φ (k − 1) + n(k ) (7) decreases.

•

The disadvantages of the discrete models can be where Θ T (k ) is the vector of the parameter and avoided by introducing a more suitable discrete Θ T (k − 1) is the regression vector ( y(k) is the process model. The δ - model, where operator δ converges output variable, u(k) is the controller output with decreased sampling period T0 to a differential variable and n(k) is the non-measurable random operator p component). For the identification a stochastic δ second order model with following discrete lim δ = p (2) T0 → 0 equation is used is best suited to this purposes. yδ (k ) = −α 1 yδ (k − 1) − α 2 y δ (k − 2) + (8) Middleton and Goodwin [11] or Feuer and + β 1u δ (k − 1) + β 2 uδ (k − 2) + n(k ) Goodwin [5] published one of these approaches to the design of these new discrete δ - models. If the where new variable γ is introduced then it is possible to y (k ) − 2 y (k − 1) + y (k − 2) yδ ( k ) = ; prove (Mukhopadhyay et al. [12]), that equality T02 z −1 y (k − 1) − y (k − 2) (3) γ = yδ (k − 1) = ; αT0 z + (1 − α )T0 T0 holds for interval 0 ≤ α ≤ 1 . By substituting α in yδ (k − 2) = y (k − 2); (9) equation (3) we obtain an infinite number of new δ u (k − 1) − u (k − 2) uδ (k − 1) = ; - models. There are several well-known δ - models T0 in applied use uδ (k − 2) = u (k − 2) z −1 for a = 0 γ = forward δ - model (4) From equation (7), (8) and (9) it is obvious that the T0 vector of parameters has the form z −1 for a = 1 γ = backward δ - model (5) ΘδT (k ) = [α 1 ,α 2 , β 1 , β 2 ] (10) zT0 and the regression vector is 2 z −1 Tustin δ - model (6) for a = 0.5 γ = y (k − 1) − y (k − 2 ) T0 z + 1 , − y (k − 2), ΦδT (k − 1) = [− T0 (11) This paper will only be concerned with the forward u (k − 1) − u (k − 2 ) δ - model (4). The δ - models will be used in , u (k − 2 )] T0 process modelling for adaptive control based on the self - tuning controller (STC). The main idea of an Then in the identification part of the designed STC is based on a recursive identification controller algorithms the regression (ARX) model procedure and selected control synthesis. For this of the following form reason it is necessary to apply suitable recursive

yδ (k ) = ΘδT (k )Φδ (k − 1) + n(k )

(12) The matrices of the discrete model are γ 2 + α1γ + α 2 α 3γ + α 4  A(γ ) =   γ 2 + α 7γ + α 8   α 5γ + α 6

is used.

The recursive least squares method is utilized for calculating of the parameter estimates Θˆ δ (k ) and  β γ + β 2 β 3γ + β 4  B (γ ) =  1 (15) adaptation is supported by directional forgetting  + + β γ β β γ β 5 6 7 8   [4]. The value of the directional forgetting factor ϕ(k) basically depends on the level of conformity and the differential equations of the model are achieved between the model and the real behaviour y1δ (k ) = −α 1 y1δ (k − 1) − α 2 y1δ (k − 2) − of the system. − α 3 y 2δ (k − 1) − α 4 y 2δ (k − 2) + β 1u1δ (k − 1) + A self – tuning SISO (single input – single output)

PID controller based on the recursive identification of the δ - model and of a modified Ziegler – Nichols criterion has been designed in [4] and some modifications of the pole placement controllers in [2].

+ β 2 u1δ (k − 2) + β 3u 2δ (k − 1) + α 4 u 2δ (k − 2)

y 2δ (k ) = −α 5 y1δ (k − 1) − α 6 y1δ (k − 2) −

− α 7 y 2δ (k − 1) − α 8 y 2δ (k − 2) + β 5 u1δ (k − 1) + (16)

+ β 6 u1δ (k − 2) + β 7 u 2δ (k − 1) + β 8 u 2δ (k − 2 )

3 Description of TITO system The internal structure of the system is shown in 4 Design of feedback MIMO system Fig. 1. w + u1

G 11 y1

e

P-1Q

F-1

u

A-1B

y

−

G 12

Fig. 2. Block diagram of the closed loop system

G 21 y2 u2

G 22

Fig. 1. A two input – two output system – the “P” structure The transfer matrix of the system is G G =  11 G 21

G12  G 22 

(13)

In the same way as the controlled system, the transfer matrix of the controller takes the form of matrix fraction G (γ ) = P −1 (γ )Q (γ ) = Q1 (γ )P1−1 (γ )

(16)

The matrix of an integrator for permanent zero control error is γ 0  F (γ ) =   0 γ 

(17)

The control law apparent in the block diagram It is possible to assume that the system is described (variable γ will be omitted from some operations by the matrix fraction for the sake of simplification) has the form −1 −1 G (γ ) = A (γ )B (γ ) = B1 (γ )A1 (γ ) (14) U = F −1Q1 P1-1 E (18) Where polynomial matrices A∈Rmm[γ], B∈Rmn[γ] It is possible to derive the following equation for are the left indivisible decomposition of matrix the system output G(γ) and matrices A1∈Rmm[γ], B1∈Rmn[γ] are the Y = A −1 BF −1 P −1QE = A −1 BF −1 P −1Q (W − Y ) (19) right indivisible decomposition.

which can be modified to give Y = P1 ( AFP1 + BQ1 ) BQ1 P1−1W −1

(20)

composed of a string of single input – single output methods.

One possibility is the serial insertion of a The closed loop system is stable when the compensator ahead of the system. The aim here is following diophantine equation is satisfied to suppress of undesirable interactions between the input and output variables so that each input affects AF P1 + BQ1 = M (21) only one controlled variable. where M(γ) ∈ Rmm[γ] is a stable diagonal polynomial matrix. γ 4 + m1γ 3 + m2γ 2 +  0   + m γ + m4  (22) M (γ ) =  3  γ 4 + m5γ 3 + m6γ 2 +  0   + m7γ + m8  

w

e

+

−

u R

K

y G

Fig. 3. Closed loop system with compensator

The resulting transfer function H is then given by The roots of this polynomial matrix are the ruling H = KG (24) factor in the behaviour of the closed loop system. They must be inside the circle with the centre at The decoupling conditions are fulfilled when point -1/T0 with the radius 1/T0 if the system is to matrix H is diagonal. be stable. Several well – known compensators are given in The degree of the controller matrix polynomials [8], {9}, [15], [16]. Control algorithms were depends on the internal properness of the closed derived for the model above with two loop. The structure of matrices P1 and Q1 was compensators. These will be referred to as C1 and chosen so that the number of unknown controller C2. Compensator C1 is the inversion of the parameters equals the number of algebraic controlled system. Matrix H is, therefore, a unit equations resulting from the solution of the matrix. diophantine equations using the uncertain coefficients method. γ + p1δ P1 (γ ) =   p 3δ  q γ + q 2δ γ + q 3δ Q1 (γ ) =  1δ 2 q 7δ γ + q 8δ γ + q 9δ

p 2δ  γ + p 4δ 

w +

e

u Q1P1-1

−

F-1

B-1A

y A-1B

controller

q 4δ γ + q 5δ γ + q 6δ   q10δ γ 2 + q11δ γ + q12δ  Fig. 4. Closed loop system with compensator C1 (23) This block diagram leads to an equation for the The solution to the diophantine equation results in a system output, which takes the form set of sixteen algebraic equations with unknown −1 Y = P1 (FP1 + Q1 ) Q1 P1−1W (25) controller parameters. The controller parameters are given by solving these equations. The following equation must be satisfied if the closed loop system is to be stable 2

2

FP1 + Q1 = M (26) 5 Design of decoupling control using compensators The structures of the polynomial matrices of the There are several ways to control multivariable controller were chosen to suit physical demands. systems with internal interactions. Some make use 1 0 P1 (γ ) =  of decentralized PID controllers, whilst others are  0 1 

q Q1 (γ ) =  1δ  0

0  q 2δ 

The controller polynomial matrices are chosen as shown below

(27)

γ 2 + p1δ γ + p 2δ P1 (γ ) =  0 

Consequently, matrix M was chosen to be 0  γ + m1 M (γ ) =  γ + m2   0

(28)

q1δ γ 2 + q 2δ γ + q 3δ Q1 (γ ) =  0 

The controller parameters are the result from the equation (26). The control law can be described by matrix equation and matrix M is −1

FU = B −1 AQ1 P1 E

(29)

Compensator C2 is adjugated matrix B. When C2 was included in the design of the closed loop the model was simplified by considering matrix A as diagonal. The multiplication of matrix B and adjugated matrix B results in diagonal matrix H. The determinants of matrix B represent the diagonal elements. When matrix A is nondiagonal, its inverted form must be placed ahead of the system in order to obtain diagonal matrix H, otherwise it may increase the order of the controller and sophistication of the closed loop system. Although designed for a diagonal matrix, compensator C2 also improves the control process for non – diagonal matrix A in the controlled system. This is demonstrated in the simulation results. w +

e −

Q1P1-1

F-1

u Adj(B)

A-1B

y

 0  γ + p 3 γ + p 4δ  2

 0  q10δ γ + q11δ γ + q12δ  (33) 2

 γ 5 + m1γ 4 + m2γ 3 + 0   2 + m3γ + m4γ + m5   M (γ ) =  γ 5 + m6γ 4 + m7γ 3 + 0   + m8γ 2 + m9γ + m10  

(34)

Solving the diophantine equation defines a set of algebraic equations, which we subsequently use to obtain the unknown controller parameters. The control law is given by the block diagram FU = adj (B )Q1 P1 E −1

(35)

6 Recursive identification The algorithms designed here were incorporated into an adaptive control system with recursive identification. The recursive least squares method proved effective for self-tuning controllers and was used as the basis for this algorithm. The parameter vector are completed as shown below:

controller

Fig. 5. Closed loop system with compensator C2 The equation for the system output as shown in this block diagram takes the form Y = P1 ( AFP1 + B v Q1 )B v Q1 P W −1 1

(30)

where 0  det (B ) B v = Badj (B ) =  det (B )  0

(31)

α α 2 α 3 α 4 T Θδ (k ) =  1 α 5 α 6 α 7 α 8

β1 β5

β2 β6

β3 β7

β4  β 8  (36)

The data vector is

Φ δT (k ) = [− y1δ (k − 1), − y1δ (k − 2), − y 2δ (k − 1), − y 2δ (k − 2 ), u1δ (k − 1), u1δ (k − 2 ), (37) u 2δ (k − 1), u 2δ (k − 1)]

To achieve stability in the closed loop system the The parameter estimates are actualized using the following diophantine equation must be fulfilled recursive least squares method plus directional AFP1 + B v Q1 = M (32) forgetting. The detailed recursive identification algorithm for TITO system is designed in [3].

adaptive control we tested. The initial parameter estimates were chosen to be

7 Simulation examples

The program system MATLAB - SIMULINK was used to create a program and diagrams to simulate  0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 ΘδT (0) =   and verify the algorithms. Verification by   0 . 5 0 . 6 0 . 7 0 . 8 0 . 5 0 . 6 0 . 7 0 . 8  simulation was carried out on a range of systems (40) with varying dynamics. The control of the model below is given here as our example. The results of simulation are shown in Figs. 7 – 12 2  s + 2s + 0.7 0.2s + 0.4  It is possible to draw several conclusions from the  A(s ) =  simulation results of the experiments on linear  − 0.5s − 0.1 s 2 + 2s + 0.7 static systems. The basic requirement to ensure permanent zero control error was satisfied in all 0.5s + 0.2 0.1s + 0.3 B (s ) =  (38) cases. The criteria on which we judge the quality of  the control process are the overshoot on the  0.5s + 0.1 0.3s + 0.4 controlled values and the speed with which zero Fig. 6 shows the system‘s step responses control error is achieved. According to these Ste p Response criteria the controller incorporating compensator C1 performed the best. However, this controller 0.4 appears to be unsuited to adaptive control due to 0.3 the size of the overshoot and the large numbers of 0.2 process and controller outputs. The controller 0.1 which uses compensator C2 seems to work best in 0 0.8 adaptive control. With regards to decoupling, it is 0.6 clear that controllers with compensators greatly 0.4 reduce interaction. From: U2

To: Y2

A mplitude

To: Y1

From: U1

0.2 0

0

4

8

120

4

8

12

y1

Time (sec.)

w1

Fig. 6. The step response of the system

4 2 0 -2

The right side control matrices are denoted as follows: without compensator - M1, with compensator C1 - M2, and with compensator C2 M3 . γ 4 + 2γ 3 + 1.5γ 2 + 0  + 0.5γ + 0.0625 M1 (γ ) =   γ 4 + 2γ 3 + 1.5γ 2 0  + 0.5γ + 0.0625 

()

M2 γ =

γ  

+ 0.5 0

   +  

0

1500

2000

2500

3000

2

y2 w2

0 -2 -4 -6 30 0

u

500

1000

1500

2000

2500

3000 k

1

20 10 0 -10

  + 0.5

 γ 5 + 2.5γ 4 + 2.5γ 3 + 0   2 + 1.25γ + 3.125γ + 0.0313  M 3 (γ ) =   γ 5 + 2.5γ 4 + 2.5γ 3 +   0 2 + 1.25γ + 3.125γ + 0.0313 

1000

k

0

γ

500

0

50 0

1 000

1 50 0

2 0 00

2 5 00

30 00 k

5 u2

0 -5 -10

(39)

The same initial conditions for system identification were used for all the types of

-15 -20

0

50 0

1 000

1 50 0

2 0 00

2 5 00

30 00 k

Fig. 7. Simulation results: control without compensator

y1 w1

8 Laboratory experiments

4 2 0 -2 0

500

1000

1500

2000

2500

3000 k

y2 w2

2 0 -2 -4 -6 0

500

1000

1500

2000

2500

3000 k

30 u1

20 10 0 -10 0

500

1000

1500

2000

2 500

3000 k

5 u

0

2

-5 -10 -15 -20

0

500

1000

1500

2000

2 500

3000 k

Fig. 8. Simulation results: adaptive control with compensator C1 y1 w1

4 2 0 -2 0

500

1000

1500

2000

2500

3000 k

y2 w2

2 0 -2 -4 -6 0

500

1000

1500

2000

2500

3000 k

30 u1

20 10 0 -10 0

500

1000

1500

2000

2500

3000 k

5 u2

The verification of designed TITO controllers in laboratory conditions operating in real time has laboratory conditions operating in real time has been realized using experimental laboratory model CE 108 - couples drives apparatus. This apparatus, based on experience with authentic industrial control applications, was developed in co-operation with the University of Manchester and made by a British company, TecQuipment Ltd. It allows us to investigate the ever-present difficulty of controlling the tension and speed of material in a continuous process. The process may require the material speed and tension to be controlled to within defined limits. Examples of this occur in the paper-making industry, strip metal and wire manufacture and, indeed, any process where the product is manufactured in a continuous strip. A continuous flexible belt replaces the industrial type material strip. The principle scheme of the model is shown in the Fig. 10. It consists of three pulleys, mounted on a vertical panel so that they form a triangle resting on its base. The two base pulleys are directly mounted on the shafts of two nominally identical servomotors and the apparatus is controlled by manipulating the drive torques to these servomotors. The third pulley, the jockey, is free to rotate and is mounted on a pivoted arm. The jockey pulley assembly, which simulates a material workstation, is equipped with a special sensor and tension measuring equipment. It is the jockey pulley speed and tension which form the principle system outputs. The belt tension is measured indirectly by monitoring the angular deflection of the pivoted tension arm to which the jockey pulley is attached. The controller output variables are the inputs to the servomotors and the process output variables are the tension and speed at the workstation. There are interactions between the control loops.

0 -5 -10 -15 -20

0

500

1000

1500

2000

2500

3000 k

Fig. 9. Simulation results: adaptive control with compensator C2

The task was to apply the methods we designed for the adaptive control of a model representing a nonlinear system with variable parameters which is, therefore, impossible to control deterministically. Adaptive control using recursive identification both with and without the use of compensators was performed. As indicated in the simulation,

compensator C1 was shown to be unsuitable and control broke down. The other two methods gave satisfactory results. The time responses of the control for both cases are shown in Fig.11 and Fig. 12. The figures demonstrate that control with a

compensator reduces interaction. Process output variable y1 is the speed and process output variable y2 is the tension. The variables u1 and u2 are the controller outputs –inputs to the servomotor.

9 Conclusions The adaptive control of a two-variable system based on polynomial theory and using delta models was designed. Decoupling problems were solved by the use of compensators. The designs were simulated and used to control a laboratory model. The simulation results proved that these methods are suitable for the control of linear systems. The control tests on the laboratory model gave satisfactory results despite the fact that the nonlinear dynamics were described by a linear model. 0.5

Fig. 10. Principal scheme of CE 108

y1 w1

0.4 0.3 0.2

0.5 y1 w1

0.1

0.4

0

0.3

50

100

150

200

0.4

0.1 0

0

t [s]

0.2

0

50

100

150

200

y2 w 2

0.2

t [s] y

0.4

0

0.2

-0.2

2

w2

0

50

100

150

200 t [s]

0 1

-0.2

0

50

100

150

200

u

1

t [s]

u1

0.8 0.6

1

0.4

0.8

0.2

0.6

0

0.4

100

150

200 t [s]

150

200

u

2

0.8 0.6 0.4

0.8

0.2

0.6

0

0.4

0

50

100

150

200 t [s ]

0.2 0 0

100

1

50

1 u2

50

t [s ]

0.2 0 0

0

50

100

150

200 t [s]

Fig. 11. Adaptive control of a real model without a compensator

Fig. 12 Adaptive control of a real model using compensator C2

Acknowledgements

7th IFAC Workshop on Adaptation and Learning in Control and Signal Processing ALCOSP 2001, Cernobbio-Como, Italy, pp. 163-168, (2001).

The work has been supported in part by the Grant Agency of the Czech Republic under grants No. 102/02/0204, No. 102/00/0526 and by the Ministry of Education of the Czech Republic under grant [10] W. L. Luyben. “Simple method for tuning SISO controllers in multivariable systems”, No. MSM 281100001. Ing. Eng. Chem. Process Des. Dev., 25, pp. 654 – 660, (1986). [11] R. H. Middleton, G. C. Goodwin. Digital Control and Estimation - A Unified V. Bobál, J. Böhm, R. Prokop. “Practical Approach, Prentice Hall, Englewood Cliffs: aspects of self-tuning controllers”, N. J., 1990. International Journal of Adaptive Control and Signal Processing, 13, pp. 671-690, [12] S. Mukhopadhyay, A. Patra, G. P. Rao. “New (1999). class of discrete-time models for continuostime systems”, International Journal of V. Bobál, P. Dostál, M. Sysel. „Delta Control, 55, pp. 1161-1187, (1992). modification of self-tuning PID controllers”,

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