On the multimodel stabilizing control of uncertain systems. Part I

Multimodel control and validity analysis

Z.Kardous, N.Ben Hadj Braiek

LECAP/EPT, Tunisia

P.Borne, A.El Kamel

LAGIS/EC Lille, France

 

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 some domain. This principle is adopted by several researchers interested by this field and initiated the development of other approaches such as the Takagi-Sugeno fuzzy approach or the polytopic approach. Based on the same principle, these approaches present similarities in their overall structure but are totally different in some other points such as the modelling procedure or the model validity evaluation.

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 based on Kharitonov's theorem. 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 this 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.

We have considered the problem of multimodel control of complex systems described by discrete-time uncertain models. The multimodel base consists of linear local or generic models. The control law was chosen to be of state-feedback type, and Luenberger type observer was used for state estimation. Then multimodel controller-observer parameters were deduced by a validity-weighted fusion. To evaluate the models' validities, we have presented new types of residues using state estimation and/or horizon.

The multimodel controller-observer designed for the transmission system by pole-assignment strategy has given satisfying responses and has proven the enhancement ensured thanks to the new developed residues.

Quadratic stability study will be carried out in the second section of the present paper entitled "Stabilization study and applications". Sufficient conditions of quadratic stabilization will be proven for the designed multimodel controller and observer, and a new method will be detailed to compute stabilizing control and observation parameters.