On the multimodel stabilizing control of uncertain systems. Part II

Stabilization study and applications

Z.Kardous, N.B.-H.Braiek

LECAP/EPT, Tunisia

P.Borne, A.El Kamel

LAGIS/EC Lille, France

 

The present paper rises the problematic of the stabilizing control of complex uncertain processes in the multimodel approach. It is organized in two parts, that are published in two issues (Int.J."PNAES", No.26, No.27). The first part concerns the generation of the model base and the design of multimodel control and observation systems. The control law is chosen to be of state-feedback type, and the multimodel observer is based on elementary Luenberger observers. The global controller and observer are deduced by a validity-weighted fusion. To evaluate the models' validities, a residue approach is adopted. In this field, new types of residues are developed based on state estimation and/or moving horizon. The stabilization study of the proposed multimodel systems is held in the second part of the paper. Two quadratic criteria are presented: the first concerns state feedback multimodel control system and the second is related to multimodel control system with multimodel observer. A new stabilizing control strategy is then proposed. It consists in computing the control and the observer parameters directly by the resolution of LMI-constraints systems derived from the stability criteria. A complex process is considered in each part to illustrate and discuss the performances and the efficiency of the proposed approaches.

 

The multimodel approach is a control strategy especially adapted to complex and/or uncertain systems. It allows overcoming the drawbacks of nonlinear, non-stationary and uncertain models that may be encountered in the control design task. In fact, it suggests characterizing the process dynamical behavior by means of simpler models each one valid in a given domain.

The idea of multiplying the number of models is interesting since it ensures simple and precise description of the complex process. However, when this number rises, it may entail computational difficulties at the control design step. Therefore it seems necessary to minimize the number of models in the base while ensuring a satisfying description. When local modelling leads to large number of models, or when it is impossible or insufficient because of uncertainties, the multimodel approach offers systematic methods of generic modelling. This strategy provides a model base limited to four or five models.

Thanks to the multimodel description, we can design nonlinear control systems from elementary linear control laws, each of which associated to one model. The global control is therefore deduced by a fusion weighted by the models' validities. These coefficients evaluate the trust degree on the models. They have to be conveniently computed since they may remarkably affect the performance of the global control. Several approaches were developed to evaluate models' validity such as fuzzy, probabilistic and geometric strategies. However, all these strategies require a priori data which are not always available in the case of uncertain systems. Thus a straightforward method is necessary to estimate the models' pertinence using only a posteriori data such as the residue approach. This latter consists in computing the validities online using, for instance, errors between the process responses and the models outputs.

In the first part of the paper, after presenting the modelling strategy in the multimodel approach, we suggest a design procedure of multimodel controller and multimodel observer. The control law is chosen to be of state-feedback type, and the multimodel observer is based on elementary Luenberger observers. The global state-feedback gain and observation gain are deduced by fusion of the elementary gains associated with the models in the base. As a first alternative, we suggest to compute the controller and the observer parameters for each model using a pole-assignment strategy. Another method will be proposed in the second part of this paper, based on the stabilization criteria. To evaluate the models' validities, we adopt the residue approach. In this field, to enhance the performances of the proposed control law, we suggest new types of residues based on state estimation and/or moving horizon. The stability and stabilization study of such multimodel systems will be held in the second part of this paper. In order to assess the performances of the proposed approaches, we consider a flexible transmission system as an illustrative example.

However, control systems can not be useful unless they ensure the stabilization conditions; hence process control task is usually connected to the stability problem. In the case of multimodel control, the stability analysis appears to be particularly hard compared with classic control strategies. The main contribution of this second part consists in the stability and stabilization study of discrete-time multimodel systems. We consider multimodel controller of state-feedback type, and for unreachable state processes, we present multimodel observer based on elementary Luenberger observers.

In second part, we present two quadratic criteria of stability concerning respectively multimodel control systems with and without multimodel observer. Based on these criteria, we suggest a new approach for stabilizing control design. It consists in computing the control parameters or the control and the observation parameters directly from the stability LMI-constraints systems. In order to illustrate the effectiveness of the suggested criteria and to assess the performances of the proposed approach, we consider an inverted pendulum system, which is a strongly nonlinear process.