CONSTRAINED NONLINEAR MODEL-BASED PREDICTIVE CONTROL USING ARX-PLUS VOLTER
FAOUZI M’SAHLI (
( I ) Ecole Nationale d’hgtnieurs de Gabes, universitt de Sfax, Route de Medenine 6029, Gabes, Tunisie. e-mail: Belgacem.Agoubi. @.enig.rnu.tn
ABSTRACT In this paper a nonlinear adaptive constrained Model Predictive Control scheme based on models identified from input-output data is proposed. We consider single input - single output (SISO) nonlinear systems described by ARX-plus volterra models. The proposed control action is obtained by solving a fourth order nonlinear programming problem on-line subject to linear constraints on the input signal. The adaptive nonlinear control strategy is obtained by augmenting the non adaptive controller with an indirect parameter estimation scheme which accounts for unknown andor slowly time-varying parameters. Simulation case study is used to demonstrate the practical utility of the proposed control scheme and to evaluate its performance.
1. INTRODUCTION Model Predictive Control (MPC) involves the computation of a set of manipulated variable moves to minimize an objective function over a prediction horizon subject to constraints on states, manipulated and controlled variables. Garcia et al [I] and Clarke [2] describe the main advances in model based predictive control. They indicate that this control is emerging as one of the most popular and effective control techniques in the process industries. This is due to the fact that many attributes commonly found in many practical industrial control design may be incorporated in model based control such as future reference trajectory, measurable or predictible disturbances and constraints. However, most MPC applications are based on the phylosophy that the systems to be controlled can be described by linear models. This also constitues an important deficiency, because such simple models are often inadequate if a more realistic approximation of the usually complex, nonlinear processes is desired. Futhermore, most of the industrial processes are nonlinear in nature, yet not many of these processes adequately described by models derived from tirst principales. Thus, it becomes imperative to model them from available input-output data. A very lage variety of nonlinear input-output models have been proposed for use in MPC strategies for the control of nonlinear systems. Yeo and williams [ 3 ] used a bilinear models in an MPC scheme for simulated control of bilinear process model descriptions. No formal approach for the identification of their model is presented, and the solution of the resulting optimization problem still requires nonlinear programming techniques. Hamandez E. [4],Harnandez and Arkun [5] proposed a nonlinear MPC scheme based on polynomial autoregressive moving-average ( A R M A ) models obtained from input-output data. The performance of the control scheme was investigated with state estimation and several disturbance 0-7803-4778-1/98 $10.00 0 1998 IEEE 3764
FAOUZI BOUANI(l1, ABDELKADER EL KAMEL(3) RIDHA BEN ABDENNOUR(~),MEKKI KSOURI(~) (2) Ecole Nationale d’Ingtnieurs de Tunis, universitt Tunis 11, Bevedere, 1002, Tunis, Tunisie. e-mail:
[email protected] (3) Ecole Centrale de Lille, LAIL, B.P.:48, 59651 villeneuve d’Ascq Cedex, France. e-mail:
[email protected]
models for a variety of case studies. Second-order volterra models were used in the control of nonlinear processes using the Internal Model Control (IMC) framework [6]. Kimbauer and Jorji [7] implemented an ARX-plus volterra model in an MPC scheme and performed simulations for the control of a distillation column. In this paper, an adaptive and modified version of the MPC strategy proposed by Kimbauer and Jorji is developed and numericly evaluated. The new indirect adaptive nonlinear controller is designed by combining the predictive controller with an indirect parameter estimation. Several MPC methods have been employed to solve the constrained control problem for nonlinear systems. While for linear plants the MPC problems are usually reduced to a standard quadratic program (QP)[8,9], application of the MPC concept to nonlinear systems involves a high order nonlinear program (NLP). However, this nonlinear program is much more difficult to solve than the QP problem. In order to reduce the complexity of NLP, an alternative predictive control methods have been developed for the control of a class of nonlinear systems. The basic idea for these alternative methods is to combine two different control schemes: feedback linearization and standard linear MPC. The basic difficulty with this approach is caused by the fact that the origmal optimization problem for the nonlinear system subject to linear constraints on the input has been transformed into an optimization problem for a linear system subject to nonlinear constraints and state dependent [10,1 I]. This second problem is not necessarily easier to solve. In this work, we consider the original nonlinear MPC problem and we propose to solve it using an ellipsoidal cutting-plane algorithm. 2. STATEMENT OF THE PROBLEM 2.1. The plant model We consider a nonlinear system that can be approximate by the toilowing nonlinear discrete-time autoregressive plus second-order volterra model “Y
nu
y( k ) = yo + E a, y(k - i 1+ E b,u( k - 1 ) ,=I
I=I
+ T i b , ,u(k-i)u(k-j)+&(k)
(1)
1=11=1
where y o is a bias term that appears in the general form of the second-order volterra model y( k ) E 3 is the output, u(k) E ’% is the input, a , , b , ,b,, are the parameters of the autoregressive plus volterra model, nu and ny are the number of lags on the input and the output, respectively E ( k ) E % contains all terms up to second-order. The third term on the nght-hand side of (I), which represents the second-order volterra term, have a tnangular form representation. One
advantage of usirlg the autoregressive plus volterra model is that the one-ahead prediction problem can be formulated as a linear regression, greatly simplifying the identification of the parameters from input-output data. 2.2. Control probllem formulation In order to satisfy the model analysis objectives and the control law computation, it is necessary to build a state space realisation of the autoregressive plus volterra model (1). The nonminimal state !;pace representation used in [3] was adopted in this paper. The states are defined by: xl'(k)= u ( k - I )
x;'"(k)=y(k)
x i t ( k ) = u(k -nu)
x;$+i ( k ) = y( k - ny )
(8)
i ( k + l / k ) = F k ( k / k),u(k).G(k)]
1
0 ... 0 O 00
(7)
$(t)=[E[:i]
Then, an Weighted Recursive Prediction Error Method (WRPEM) is used to obtain the values of 6 ( k ) as time progresses. The appropriate structure of the WRPEM for the system described in (3) is given by:
Given the identification of the state variables in (2), we can write the realisation (1) as:
1:
sequence of future control increments computed by the optimisation problem at time k; Au(k+j)=O for j> Nu. A stepwise regression method was used off-line to select the model structure of the ARX-plus volterra model [12]. in the rest of this section, we shall describe a recursive identification algorithm which allow us to estimate the parameters and the state simultaneously. Thus, we construct a parameter vector by augmenting the vector of model parameters, 0, with an estimator gain, G:
i ( k / k-l)=H[i(k/
k)]
&k) = &k- I ) + P ( k - I)cp(k)i(k/k- 1) r 1-1
P ( k ) = P ( k - I)-P(k-l)cp(k)cp'(k)P(k-l)
(3)
y(k)=[O
[$;;;]
0 .. 0 I 0 .. 01
In this paper wf: will consider a nonlinear model-based predictive controller (NLMBPC) scheme based on the ARXplus volterra model (1). To do this, a state estimation procedure can be incorporated into the NLMBPC strategy using the followed realisation: a ( k + l / k ) = F G ( k I k),u(k)] (4)
i(k/ k)=i(k/k-I)+
