PD observer parametrization design for descriptor systems
Introduction
Descriptor systems, which are also referred to singular systems, differential algebraic systems, generalized dynamical systems, implicit systems, etc., often appear in electronic circuits, aircraft systems, chemical engineering, economic systems, etc. During the past 30 years, many results have been reported for descriptor systems, see e.g. [1], [2]. The factorization approach has proven to be a powerful tool in multivariable control system analysis and synthesis [3]. Recently, the factorization approach has been introduced to investigate descriptor systems. Specifically, the algebraic framework for studying descriptor systems with the factorization approach has been established [4], [5], [6]. The problems of Bezout factorizations and controller parametrizations have been solved for descriptor systems, see [6], [7], [8], [9], [10].
As dual results of controller parametrizations, the observer parametrizations have been developed effectively to solve the robust observer design and the optimal filtering problems for normal systems [11], [12], [13], [14]. For descriptor systems, the problem of observer parametrization is first studied by Minamide et al. [15], is then discussed by Gao et al. [16] and Gao [17]. Specifically, in the work by Minamide et al. [15], by using descriptor standard form [18], a parametrization of observers is derived. In Ref. [16], a class of reduced-order observers is characterized on the basis of restrictive equivalent transformations. The work in [17] is related to descriptor proportional observers, which is a natural extension of the work [11] for normal systems. However, it is not difficult to find that additional transformations in [15], [16] make the computations involved. Moreover in [15], [17], it is not systematic to find a gain to make the estimation error dynamics both stable and impulse free for descriptor observers.
In this study, a new observer parametrization related to the design of PD descriptor observer is developed. Not any additional transformations are introduced in the derived process. Moreover, the derivative gain and the proportional gain are designed systematically to ensure the estimation error dynamics to be impulse free and stable. This kind of design, therefore, makes the present parameterization convenient and efficient to be performed. The interesting connections between the present parametrization and those given by Minamide et al. [15] and Gao [17] are also established. When disturbances occur in the plant, the estimation error would never be zero. In this case, the estimation error dynamics are also parametrized, which are useful for designing optimal robust observers for descriptor systems. An example is given to illustrate the parametrization procedures.
Section snippets
Preliminaries
Some notations are first given here. denotes the field of real numbers, denotes the -dimensional Euclidean space, denotes the closed, complex right half-plane (or the set of all unstable and finite complex numbers), denotes the set of proper stable, real-rational functions, denotes the class of matrices of size with all elements in , represents the transpose of the matrix, denotes the unity matrix, and 0 denotes scalar zero or a null matrix with compatible size.
New observer parametrization design
In this section, a new observer parametrization is proposed and its interesting connections compared to the known results are shown.
Robust observer design
When disturbances occur, the estimation error would never be zero. In this case, it is desired that the estimation error is kept as small as possible. We now characterize the relation between the estimation error and the disturbance vector . Theorem 3 Given plant (1)–(3) under Assumption 1, the set of all achievable transfer function matrices of the estimation errors in the presence of the disturbance vector is parametrized aswhere , and Proof Without loss
Example
Consider the system given in the form (1)–(3). The corresponding matrices are
According to the well-known criteria [1], one can show easily that this plant comprises impulsive modes, but satisfies Assumption 1. Choose such that is non-singular, and then select such that is a stable polynomial with zeros . The observer parametrization in the form (16) becomes
Conclusions
A new parametrization of all causal observers for descriptor systems has been obtained on the basis of the PD observer design. The proposed parametrization is more convenient and efficient to be performed compared with the previous work. The estimation error dynamics of observers for descriptor systems with respect to unknown input disturbances have also been parametrized. An example is included to illustrate the parametrization procedures. The proposed observer parametrization provides useful
Acknowledgments
This work is supported partially by the Alexander Humboldt Foundation (IV-CHM/1117303 STP). The author would like to thank the referees for their helpful comments.
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