Pth moment regional stability/stabilization and generalized pole assignment of linear stochastic systems: Based on the generalized ‐representation method

Abstract

This paper investigates pth moment regional stability, stabilization, and generalized pole assignment of stochastic systems, which are most useful in many specific systems and applications. With the help of the generalized -representation method, the necessary and sufficient conditions of pth moment regional stability and stabilization of stochastic systems are addressed.Meanwhile, the generalized pole assignment of stochastic systems is also considered. Finally, two examples are presented to show the effectiveness of the proposed methods. K E Y W O R D Spth moment -stability, generalized -representation, generalized pole assignment, power vector 1 3234 /journal/rnc Int J Robust Nonlinear Control. 2020;30:3234-3249.ZHANG et al. 3235control system. Frequently used pole assignment regions are generally including negative semi-plane, sector, disk, vertical stripe, and the intersection of them. Regional pole assignment of the linear deterministic system has been investigated in various aspects, such as robustness bounds of pole clustering, 12 robust pole placement, 13,14 and H 2 or H ∞ performance with pole placement constraints. 15,16 Recently, the -representation has been explicitly presented in Reference 17. As the most advantage of this method, many classical results of linear systems could be directly applied to the stochastic systems. [17][18][19][20][21][22] For example, Reference 17 proposed the concept of -representation method, and discussed the problem of generalized -stability/stabilization for linear stochastic systems. The stabilization with state-multiplicative noise was investigated, and the infinite horizon stochastic H 2 ∕H ∞ control with state-dependent and control-dependent noise were studied in Reference 20. Moment stability theorems for the systems with the stability properties were considered in Reference 22. However, it should be pointed out that the -representation method will be no longer valid for the corresponding cases in the meaning of pth moment stability/stabilization although it effectively solves the mean square stability and other related issues for the stochastic linear systems.Motivated by the pole placement, the -representation method and Itô formula, we will generalize the state matrix of the linear stochastic system to the extended state matrix by using the generalized -representation technique. The quadratic terms of the state vector in -representation could be extend to pth terms using the lexicographical order of the monomials. Then, the problems of pth moment regional stability, stabilization, and the generalized pole assignment of linear stochastic systems could be handled effectively as the means of the deterministic system. The main advantages of the proposed approach can be summarized as follows.First, the linear stochastic systems can be transformed into deterministic systems with the state matrix n⋅p and the state vector x {p} by the generalized -representation method, which is the generalization of -representation in Reference 17. Then, the problems of p...

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