Algebraic State Estimation for a
Class of Switched Linear Systems
Yang TIAN1, Chengcheng SONG1, Wilfrid PERRUQUETTI2
1 Sino-French Joint Laboratory of Automation and Signal Processing, Automation School of Nanjing University of Science and Technology (NUST)
Nanjing, 210094, China firstname.lastname@example.org
2 LAGIS (CNRS, UMR 8219), Ecole Centrale de Lille (ECL),
Villeneuve d’Ascq, 59650, France & Equipe Projet Non-A,
INRIA Lille-Nord Europe
Abstract: In this paper, an algebraic state estimation method for a class of switched linear systems is derived. This approach is based on algebraic tools and distribution theory. Firstly, the unknown switching instant and the active mode are identified on-line, and then the process of state estimation is given by an explicit algebraic formula, rather than by an auxiliary dynamic system, which can be implemented formally and estimated very fast in computer. Numerical example and Simulations illustrate the efficiency of the proposed techniques.
Keywords: Algebraic approach, State estimation, Switching instants identification, Switched linear systems.
CITE THIS PAPER AS:
Yang TIAN, Chengcheng SONG, Wilfrid PERRUQUETTI, Algebraic State Estimation for a Class of Switched Linear Systems, Studies in Informatics and Control, ISSN 1220-1766, vol. 23 (2), pp. 207-214, 2014. https://doi.org/10.24846/v23i2y201409
Hybrid dynamical systems (HDS), in which continuous dynamics and discrete events coexist and interact between each other, can be used to model a large number of practical systems. Switched systems as higher–level abstractions of HDS, obtained by neglecting the details of the discrete behaviour. A family of dynamical linear subsystems and a switching law, which orchestrates the switching between them, can compose a switched linear system (see  for surveys).
In the recent years, there has been an increasing interest in the control problems of switched linear systems due to their significance from both a theoretical and practical point of view. Important results for switched systems have been achieved for problems of stability analysis , stabilization [3-5], tracking design  or controllability [7, 8].
Observability and state estimation is a very challenging problem for such systems since both the active mode and the continuous state have to be estimated during a finite time interval. The notion of state estimation for switched systems was firstly introduced in . Observability notions for some classes of hybrid systems such as switched linear systems has been discussed and characterized in recent works such as [1,10,11].The problem is to recover from available measurements the state of the system and/or the switching signal, and eventually the switching time. Different observation and identification methods have been performed during the last years [12-20]: state estimation for nonlinear switched system using Petri Net , for linear switched systems with unknown inputs , or an original and effective sampled and delayed output observer design which is based on hybrid switching systems  etc. Usually, the hybrid observer consists of two parts: an index estimator of the current active sub-model and a continuous observer that estimates asymptotically in most cases, the continuous state of the hybrid system.
The aim of this paper is to estimate the switching instants, active mode and the continuous state of a class of switched linear systems with the knowledge of the first active mode. The possibility to have finite time estimate for this kind of systems is clearly important. The approach considered here takes root in recent works developed in  for parameter identification of linear time-invariant systems. This method is based on algebraic tools (differential algebra, module theory and operational calculus) and results in finite time estimates given by explicit algebraic formula that can be implemented in a straightforward manner using standard tools from computational mathematics. Those results have been extended to the problems of closed-loop parametric estimation for continuous-time linear systems in , state estimation of linear systems in  or with time-varying parameters in , fault diagnosis in , nonlinear systems with unknown inputs in  or nonlinear feedback control in , switched systems estimation with Zeno phenomenon in . This approach was also applied in  for the estimation of the index corresponding to the current active subsystem, and the state variable of this subsystem. Based on the result in [30, 31], finite time identification of the switching instants and the active mode are firstly studied, and the switching time estimation is given by an explicit formula, as a function of the integral of the output, in order to attenuate the influence of measurement noises. Then, combining our results of state estimation for linear time invariant systems by algebraic approach , we give the main approach of current active mode estimation and the continuous state estimation in real time.
This paper is organized as follows: Section 2 gives the problem statement and the mathematical formulation. The main result is derived in Section 3 & Section 4. First, the switching instant identification of one commutation between two modes is analyzed. Then, the result is extended to the case of commutations among an arbitrary number of modes. In Section 4, with the estimated switching instants sequence and the algebraic state estimator for each mode (), the state estimation of the autonomous switched linear systems is achieved. In section 5, simulation results that illustrate the proposed approach are provided. Finally, the last section is devoted to main conclusions and future works.
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