Elsevier

Automatica

Volume 36, Issue 4, April 2000, Pages 563-571
Automatica

Brief Paper
Constrained robust predictive controller for uncertain processes modeled by orthonormal series functions☆

https://doi.org/10.1016/S0005-1098(99)00179-XGet rights and content

Abstract

The present work focuses on robust predictive control (RPC) of uncertain processes and proposes a new approach based on orthonormal series function modeling. In such unstructured modeling, the output signal is described as a weighted sum of orthonormal functions that uses approximative information about the time constant of the process. Due to an efficient uncertainty representation, this kind of modeling is advantageous in the RPC context, even for constrained systems and processes with integral action. The stability of the closed-loop system is guaranteed by the setting of sufficient conditions for the selection of the controller prediction horizon. Simulation results are presented to illustrate the performance of this new RPC algorithm.

Introduction

Model based predictive controllers (MBPC) are, by definition, based on predicting the behavior of the process to be controlled. The principle of MBPC consists in calculating the control input by the minimization of a cost function over a future time horizon under certain process constraints (Clarke, 1994), and the closed-loop performance depends on the choice of an appropriate model for prediction and of several tuning parameters. In most cases, a single linear model has been adopted to describe the behavior of the process, but this involves only an approximation; hence great uncertainty in the value of the process parameters may result. Therefore, it is very important to consider the effect of these uncertain parameters on both the optimality and the stability of the closed-loop system during the design stage.

Predictive controllers that explicitly consider the uncertainty in the process model for calculating the control law are called robust predictive controllers (RPC). When structured uncertainties are taken into account to calculate a set of output predictions and then to optimise the control law, that results in the solution of a min–max optimisation problem (Campo & Morari, 1987; Prett & Garcia, 1989; Allwright & Papavasiliou, 1992; Zheng & Morari, 1993). These authors proposed to model the process by means of its impulse response with uncertain coefficients and used this model to calculate the set of possible values for output prediction. Nevertheless, the use of impulse response modeling requires dealing with a large number of uncertain parameters, thus augmenting the overall computational efforts of the control law and the complexity of the robust identification, i.e., the identification of the parameters of the uncertainty set.

Other types of models for RPC have been also proposed, including the controlled auto regressive integrated moving average (CARIMA) model. In this approach (Gutierrez & Camacho, 1995), the number of uncertain parameters is low, but the use of a CARIMA model requires a priori knowledge about process order and time delay; moreover, the uncertainty must not be in the auto regressive part of the model, or the control law will no longer be optimal.

This paper presents a robust predictive controller based on orthonormal series function modeling (RPC-OSF). The robust approach presented here has the following characteristics (Oliveira, Favier, Dumont & Amaral, 1996; Oliveira, 1997): (i) Ensuring the convexity of the cost function in relation to uncertain parameters permits the consideration of a finite number of constraints during the solution of the optimization problem, as well as guaranteeing the optimality of the solution. (ii) Orthonormal series functions-based models are capable of representing a stable system with fewer parameters than impulse response models, thus reducing the computational efforts of the control law and simplifying the robust identification problem. (iii) The stability of closed-loop system is guaranteed by setting sufficient conditions for the selection of controller parameters. (iv) Processes including integral action and input signal constraints can also be considered.

This paper is organized as follows. Section 2 presents the orthonormal series function model (OSF-Model) for uncertain systems, whereas Section 3 presents robust predictive control based on this kind of modeling (RPC-OSF), considering the SISO constrained case; the use of Euclidean and infinite norms in the cost function is described. Section 4 establishes some stability results, while Section 5 illustrates the performance of the algorithm by means of an example. The final section of the paper presents the conclusions.

Section snippets

Uncertain system modeling using orthonormal series functions

An open-loop stable linear system can be represented with arbitrary accuracy by a series of orthonormal functions, as follows (Ninness & Gustafsson, 1995; Wahlberg & Makila, 1996):y(k)=∑i=1nci(εi)Φi(q−1)u(k),where y(k) is the output signal, u(k) is the input signal and Φi(q−1), i=1,…,n, is a set of transfer functions, in the backward shift operator q−1, associated with the basis of orthonormal functions used in the series expansion. The uncertain parameters ci(εi) characterize the process

Robust predictive control

A predictive controller is defined by using a model to compute the predicted process output and a cost function that describes the closed-loop performance of the system; the cost function is then minimized in relation to the future control signals. Finally, the first of these control signals is applied to the process (receding horizon strategy).

When the model parameters are uncertain, with known uncertainty bounds, a robust predictive control (RPC) law can be derived by minimizing the maximum

Stability results

In this section, the stability of the closed-loop system is analyzed. Sufficient conditions for this stability are derived by setting a minimal value for the choice of the final prediction horizon Ny in the unconstrained RPC-OSF case with infinite norm cost function. Let us define Jk as the optimal value of the cost function:Jk≜minΔumaxε∈Ωmaxj=N1,…,Ny|ŷ(k+j|k,ε)−w|,the set-point w being assumed to be constant.

The set of all the output predictions ŷ(k+j|k,ε), for ε∈Ω, is defined asŶ(k+j|k,Ω)≜{

Simulation results

In this section, a simulation example is presented to illustrate the behavior of the proposed robust predictive controller. The process is represented by the following transfer function, with 0≤κ≤1 (Campo & Morari, 1987):G(s)=110s+1+κ110s+1−2s+12s+1−110s+1.The discrete orthonormal series model for this process is defined as follows, with a sampling time of 1 s. The pole p is set at 0.9 to approximate the dynamics of the process when κ=0 (nominal model) and the number n of functions is set at 8.

Conclusion

This paper has presented a new approach based on orthonormal series function modeling for Robust Predictive Control of uncertain processes. The proposed strategy consists in minimizing the maximum of the cost function for all the models within the family of models described by the uncertain parameters, i.e., it consists in solving a min–max optimization problem. Such a predictive controller uses the information about the uncertainty in the process model and can also deal with constraints in the

Gustavo Henrique da Costa Oliveira was born in Rio de Janeiro, Brazil, in 1967. He received the engineering diploma from Federal University of Juiz de Fora, Brazil, in 1989, the MS degree from State University of Campinas (UNICAMP), in 1991, and the doctorate degree from University of Nice-Sophia-Antipolis/Laboratory 13S, in 1997. Currently, he is an associate professor of the Laboratory of System and Automation of the Pontifical Catholic University of Parana, Brazil. His research interests are

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Gustavo Henrique da Costa Oliveira was born in Rio de Janeiro, Brazil, in 1967. He received the engineering diploma from Federal University of Juiz de Fora, Brazil, in 1989, the MS degree from State University of Campinas (UNICAMP), in 1991, and the doctorate degree from University of Nice-Sophia-Antipolis/Laboratory 13S, in 1997. Currently, he is an associate professor of the Laboratory of System and Automation of the Pontifical Catholic University of Parana, Brazil. His research interests are in predictive control, system identification using orthonormal basis functions and mobile robots.

Gérard Favier was born in Avignon, France, in 1949. He obtained the engineering diploma from ENSCM (1973), Besançon, and ENSAE (1974), Toulouse, and received the Engineering Doctorate (1977) and State Doctorate (1981) degrees at Nice University. He is Research Director at the CNRS, and he has the position of director of the 13S laboratory, since 1995. His main research activities are in system identification, predictive control, channel equalization for telecommunications and data fusion for radar tracking.

Wagner Caradori do Amaral was born in Campinas, SP, Brazil on March 25, 1952. He received the BS, MS and doctorate degrees in Electrical Engineering from State University of Campinas, UNICAMP, in 1974, 1976 and 1981, respectively. Since 1991 he has been a Full Professor at the Department of Computer Engineering and Industrial Automation at UNICAMP. Presently, he is the Director of the School of Electrical Engineering at UNICAMP. His research interests are in the areas of modeling, identification and adaptive control.

Guy A. Dumont received his Diplôme d'lngénieur from ENSAM, Paris, France in 1973 and his Ph.D., Electrical Engineering from McGill University Motreal in 1977. After working for Tioxide Europe in France, in 1979 he joined Paprican first in Motreal and then in Vancouver. In 1989, he joined the Department of Electrical and Computer Engineering at the University of British Columbia to hold the Paprican/NSERC Chair in Process Control. There, he founded and leads the Process Control Group at the UBC Pulp and Paper Centre. For the last 25 years, he has been involved in the development of advanced process control methodologies for the process industries, in particular the pulp and paper industry. His current research interests are in adaptive control, control loop performance monitoring, predictive control and control of sheet making processes.

☆

This paper was presented at IFAC Workshop on ACSP, Glasgow/UK, 1998. This paper was recommended for publication in revised form by Associate Editor P.-O. Gutman under the direction of Editor T. Basar.

1

Supported by CNPq/Brazil.

2

Supported by Ministère des Affaires Étrangères/France.

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