Bilinear Compensed Generalized Predictive Control: An Adaptive

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based in MPC (Model Predictive Control) that is a class of algorithms widely used in academy and in chemical industry. The most part of industrial processes are ...
Bilinear Compensed Generalized Predictive Control: An Adaptive Approach Adhemar de Barros Fontes1, André Laurindo Maitelli2 and Anderson Luiz de Oliveira Cavalcanti3 1

Electrical Engineering Departament Federal University of Bahia, Brazil e-mail: [email protected]

2,3

Computation and Automation Engineering Departament Federal University of the Rio Grande do Norte, Brazil e-mails: [email protected] , [email protected]

Abstract The present paper presents an adaptive implementation of bilinear compensed generalized predictive control. The bilinear compensated model adds a compensation term in bilinear model wich minimize the prediction error variance generated by its linearization. Since the relation between the prediction error and the control signal is nonlinear, and the compensation term model is linear, a different model may be estimated to each different level of control signal. Thus, in the present approach, is proposed the adaptive calculation of compensation term. The estimation of compensation term is obtained by recursive least squares algorithm (RLS). Keywords – bilinear systems; estimation; identification; model predictive control and nonlinear systems.

1

Introduction

Predictive control is based in a control law obtained by the prediction of output signal of the process. There are several algorithms of predictive control, however this paper is based in MPC (Model Predictive Control) that is a class of algorithms widely used in academy and in chemical industry. The most part of industrial processes are nonlinear, so a more realistic representation becomes necessary. In this case, a single linear model often can not represents exactly its dynamic behavior [3]. The difficults in use nonlinear models in MPC algorithms are: few models available (there are few methods to identificate them); the computational complexity of resolution of predictive control algorithms based in

nonlinear models; the lack of theoretical results to prove the robustness of these controllers [1]. Bilinear models are nonlinear models, however are more simple than general nonlinear models and represent better the behavior of processes than linear models. The choice of bilinear models to represent the dynamic of the processes is because they are mathematicaly more treatable than general nonlinear models and represent better nonlinear dynamics than linear models. Predictives laws shown in this paper are an optimization problem and in general a quadratic cost function is choiced to be minimized. When is used optimization techniques in bilinear models, it becomes to a nonlinear optimization problem, and an explicit equation for the control input cannot be obtained [6]. To solve this problem, some nonlinear programming thechniques are used, however the present approach uses linearization thechniques, what becomes to timestep quasilinear NARIMAX model. This model is used in BGPC (Bilinear Generalized Predictive Control) [5]. In approach [5], due to the used model, a prediction error exists, wich increases with the prediction horizon, degrading the performance of that controller. A compensation term is proposed in [4] to minimize the error prediction variance, however in that case, a constant offline compensation term is calculated. The present approach shows the compensation term calculated on-line due the nonlinear dynamic of the prediction error.

2

NARMAX Time-Step Quasi-Linear Model

NARMAX (Nonlinear, Auto-Regressive, Moving Average with exogenous Input) time-step quasi-linear model consists of a linearization of NARIMAX bilinear model. Considering the following bilinear model:

A(q 1 ) y k q  d B (q 1 )u (k  1) na

m

i 1

j 1

(1) 1

 ¦ ¦ d ( i  d ), j u ( k  j  i  1) y (k  i ) 

C (q ) e( k ) ' (q 1 )

where the polynomials A(q 1 ) , B (q 1 ) and C (q 1 ) are defined by the following form: P(q 1 ) with

a0

c0

p 0  p1 q 1    p np q  np

1 e b0 z 1 ;

' 1  q 1 ;

d i, j

(2) 0

for

i d 0 ; u (k ) , y (k ) and e(k )  ƒ are the sequences of

3

Bilinear Compensed Generalized Predictive Controller

Bilinear Compensed Generalized Predictive Control is based in NARIMAX time-step quasi-linear compensed model. This model introduces in NARIMAX time-step quasi-linear model a compensation term that minimize the prediction error generated by linearization.The prediction error must be minimized because it degrades the controller performance. The prediction error is given by:

input, output and white noise, respectively.

H i (k )

The linearization consists of rewrite the bilinear model showed above as follow: m § · ¨¨ a i  d ( i  d ), j u (k  i  j  1¸¸ y (k  i )  i 1 © j 1 ¹ d 1 (3) q B(q )u (k  1)  C (q 1 )e(k ) na

y (k )



¦

¦

defining a~i (u )

m § · ¨¨ a i  ¦ d ( i  d ), j u (k  i  j  1) ¸¸ j 1 © ¹

~ A(q 1 , u ) 1  a~1 (u )q 1    a~na (u )q  na

(5)

Observe that, in each instant k the last values of u(k) are known, allowing so that the ãi(u) are determinates and considered constant until the following instant. The NARMAX time-step quasi-linear is given by: ~ A(q 1 , u ) y (k )

q  d B(q 1 )u (k  1)  C (q 1 )e(k )

(9)

where H i (k ) is the prediction error referring to horizon i and calculated in instant k , y (k  i ) is the prediction istep-ahead of bilinear system output and yˆ (k  i ) is the prediction i-step-ahead of quasi-linear model. In [4] the model proposed to represents the prediction error is a linear moving average model, thus is used the recursive least squares algorithm to estimate its parameters. The model is:

(4)

and

y (k  i )  yˆ (k  i )

H i (k )

Li (q 1 )u (k )

(10)

where Li (q 1 ) is the compensation term referring to i horizon. The polynomial that represents the compensation term is: Li (q 1 )

l 0, i  l1, i q 1  l 2 ,i q 2    l nl ,i q  nl

(11)

The polynomial shown in (10) is a dynamic compensation term such that the quasilinear model static gain either the same that compensated quasilinear model. In this way it is possible to conclude that:

(6) nl

In [5] is proposed NARMAX time-step quasi-linear model, however, NARIMAX model is most suitable to guarantee null steady-state error to a step change in set-point. NARIMAX time-step quasi-linear model introduces an integration operator as follow: A (q 1 , u) y(k )

q d B(q 1 )'u(k  1)  C(q 1 )e(k )

¦l

(12)

(7)

with: A (q 1 , u )

0 i

Note that the order of Li (q 1 ) can be determined by the Akaike criterion. Thus, considering the polynomial´s degree A (q 1 , u ) is (na  1) , the compensation term model proposed by [4] is: Li (q 1 )

~ 'A(q 1 , u )

j ,i

j 0

(8)

l 0 ,i  l1, i q  i  l 2 ,i q  ( i 1)    l ( na 1 i ),i q  ( na 1 i ) (13)

In this sense, NARIMAX compensated timestep quasilinear i-step ahead is: A (q 1 , u ) y (k  i )

q  d [ B(q 1 )

 Li (q 1 )]'u (k  i  1)  C (q 1 )e(k  i )

(14)

doing:

and:

Bi (q 1 )

B(q 1 )  Li (q 1 )

(15) u

BGPC controller is based in minimization of the following cost function: NY

J

¦[ yˆ(k  i)  r(k  i)] ¦O(i)['u(k  i 1)]

i N1

2

(16) Free response is calculated by prediction equation.

i 1

where yˆ (k  i ) is the predition i-step ahead of NARIMAX timestep quasilinear model output; r (k  i ) is the trajectory

The minimization of cost function is obtained in analitical form: wJ wu

of future reference; O (i ) are weighting factors sequence of the control signal; NY is the prediction horizon; NU is the control horizon; N1 is the minimum prediction horizon; It is important to remember that due the prediction error, the prediciton above is a sub-optimal prediction and not an optimal prediction. In [2], the weighting factors sequence of the control signal is considered constant. The cost function shown in (16) may be rewrite of matrix form: J

( y  r ) T ( y  r )  Ou T u

( H (u )u  y l  r ) T ( H (u )u  y l  r )  Ou T u

Forced response is given by H (u )u , where H calculated by predictor equation as following:

(18) is

yˆ (k  i ) H i (q 1 , u )'u (k  d  i  1)  F (q 1 , u ) y (k ) (19) Predictor equation is calculated by diophantine equation [4]: F j (q 1 , u )

1 A (q 1 , u )

E j (q 1 , u )  q  j

Ei (q 1 , u )

ei , 0  ei ,1 q 1    e j , j 1 q  ( j 1)

Fi (q 1 , u )

f i , 0  f i ,1 q 1    f j ,na q ( na )

A (q 1 , u )

(20)

Ei (q 1 , u ) Bi (q 1 )

( H (u ) T H (u )  OI ) 1 H (u ) T (r  y l )

(25)

GPC controller is based in recending prediction horizon that calculates a sequence of control signal and uses only the first element of vector u shown in (26). Thus, the current control signal is calculated by expression: 'u (k )

K (r  y l )

(26)

where K is the frist row of matrix :

( H (u ) T H (u )  OI ) 1 H (u ) T

4

(27)

Bilinear Compensated Adaptive Generalized Predictive Control

4.1 Analisys of the Compensation Term There is a nonlinear bahavior between the input signal and the prediction error. However it is intended in this research to use a linear, moving average model to represent the compensation term. The parameters of compensation term are estimated using RLS algorithm. It was expected that differents parameters would be estimated for differents input signals. For example, considering the following second order billinear system: y (k )

0.2 y (k  1)  0.1 y (k  2)  0.4u (k  1) 

0.8 y (k  1)u (k  1)  0.6 y (k  2)u (k  2) 

e( k ) '

The compensation term to horizon 1 and input signal u 0.3 is: 0.3432  0.0858q 1  0.2575q 2

Other estimation may be calculated to horizon 1 and input signal u 0.4 resulting in: (21)

and: H i ( q 1 , u )

u

L1 (q 1 )

where:

(24)

0

Calculating (25), it is obtained the flollowing control law:

(17)

The complete response of the system may be rewrite as an overlapping of free response and forced response. Free response of a system in instant k is the response of the system for a null variation of the input signal after instant k . Forced response considers futures variations of input sinal with null initial condition. Rewriting (17) considering free response and forced response it is obtained: J

(23)

Above expressions are considered to C (q 1 ) 1 .

NU

2

ª 'u (k ) º « 'u ( k  1) » » « » «  » « ¬'u ( k  N  1)¼

(22)

L1 (q 1 )

0.9150  0.2287q 1  0.6866q 2

The parameters of the model estimated to a control signal is not valid to other control signal anymore. RLS algorithm calculates on-line the news parameters referring to the new

input signal according to the set-point. A Pseudo Random Binary Signal, PRBS, is generated and applied in the plant. Expression (10) calculates the prediction error minimized by the compensation term with a sequence of input signal. Figure 1 shows the minimized prediction error calculated with PRBS signal generated with u 0.4 .

compensation term to each variation in control signal. This increase in computational cost is not relevant in controller performance. 4.3 RLS Algorithm Recursive least squares algorithm is used to on-line identification of the compensation term. Based in expression (10):

H i (k ) l 0 ,i u (k )  l1,i u (k  i )  l 2 ,i u (k  i  1)    l ( na 1i ),i u (k  na  1  i)

(28)

Rewriting (28):

H i (k )

xiT, kT i

(29)

where: x iT, k

[u ( k ) 0 u (k  i)  u (k  na  1  i)]

(30)

being 0 a vector of dimension 1 u (i  1) with zero elements and the vector of parameters is:

Ti Figure 1: Prediction error The prediction error calculated by the compensation term estimated to u 0.3 and PRBS generated to u 0.4 had a worse performance than prediction error calculated by compensation term estimated to u 0.4 and PRBS generated to u 0.4 . It justifies the use of adaptive calculation of compensation term. A compensation term estimated to some operation point, will not minimized with same performance the prediction error to other operation point.

[l 0 ,i

l1,i

l 2 ,i  l ( na 1 i ),i ]

(31)

In this way, with above expressions, are used the expressions of RLS algorithm to estimate the parameters of compensation term. 4.4 Results A simulation was implemented in MATLAB. These results are referring to second order bilinear system presented in section 4.1. Figure 3 shows the comparison between output plant using adaptive and not adaptive controller.

4.2 Proposed Structure of the Controller The proposed structure verifies the value of the new control signal generated by optmization of cost function. According to the control signal value generated, a PRBS signal is applied in bilinear model and in quasilinear model to calculate the prediction error to use in recursive least squares algorithm. Figure 2 shows the block diagram to the proposed controller structure.

Figure 3: Comparison of output signal of bilinear system using adaptive and not adaptive controller Figure 4 shows the comparison between control signal generated by adaptive control and not adaptive control.

Figure 2: Block diagram of proposed controller There is a increase in computational cost in relation of non adaptive controller. It is justified by the estimation of

Volterra Models. Automatica: Vol. 31, n.º5, pp. 697714, 1995. [4] Fontes, A. B., Maitelli, A. L. and Salazar, A. O. A New Bilinear Generalized Predictive Control Approach:Algorithm and Results. 15th Triennial World Congress, Barcelona, Spain, IFAC, 2002. [5] Goodhart S. G., Burnham K. J. and James D.J. G. Bilinear Self-tuning Control of a high temperature Heat Treatment Plant. IEEE Control Theory Appl.: Vol. 141, n.º1, January, 1994. [6] Yeo Y. K. and Williams D. C. Bilinear model Predictive control. Ind. Eng. Chem. Res., 26, pp. 22672274, 1987. Figure 4: Control Signal generated by adaptive and not adaptive controller This comparison was generated with same tuning parameters, NU 4 and O 200 . In this, it is observed that the adaptive predictive controller presents a better performance, so much of the point of view of the process response as of the control effort, when compared with non adaptive controller. Both controllers are based in quasilinear compensated model. It is interesting to observe that the less response time obtained with the adaptive predictive controller is significant when the process time constant is large and also that the reduction of the response time was at about of 20%. In relation of the control effort was observed that the final value is the same, however this was anticipated due of the best model utilized in the prediction.

5

Conclusions

The main contribution of this work is the introduction of the adaptive compensation term. This adaptive term is calculated and the results presented, evidencing the better performance of adaptive controller in comparison with non adaptive controller. Although the results presented didn’t show a better expected performance, this can be achieved when a parameter variation of the bilinear system is introduced and also when happen a variation in trajectory reference.

6

References

[1] Camacho E. F. and Bordons C. Model Predictive Control. Springer-Verlag, New York, 1998. [2] Clarke D. W., Mohtadi C. And Tuffs P. S. Generalized Predictive Control – Parts 1 and 2. Automatica: Vol. 21, n.º2, 1987. [3] Doyle III F. J., Ogunnaike B. A. And Pearson Ronal K.. Nonlinear Model-Based Control Using Second-Order