Integrated fault detection and control for two‐dimensional Markovian jump systems

Abstract

Summary This paper is concerned with the problem of integrated fault detection and control for a class of two‐dimensional (2D) discrete‐time Markovian jump systems. The mathematical model of 2D Markovian jump systems is established upon the well‐known Roesser model, and a faults detection filter/controller is proposed to detect faults and meet some control specifications simultaneously. In this strategy, it takes into account both the fault detection objective and the control objective simultaneously through certain performance levels. The integrated design problem is then formulated as a multi‐objective optimization problem, which is nonconvex in essence. Furthermore, a two‐step algorithm is developed to solve this nonconvex problem. Sufficient conditions for existence of the desired fault detection filter/controller are established in terms of LMIs. A numerical example is used to demonstrate the effectiveness of the proposed method.

Asynchronous observer‐based sliding mode control for 2‐D Markov jump Roesser systems

“…For simplicity, we replace V 2 (s t 1 ,t 2 , 𝛾 t 1 ,t 2 ) with V 2 and based on the sliding surface (10), one has…”

“…In general, 2-D systems are formulated mathematically by partial differential or difference equations associated with more than one complex/real variables in continuous-or discrete-time domains. [6][7][8][9][10] Consequently, the fundamental differences with the one-dimensional (1-D) systems immediately arises, and the traditional control theory for 1-D systems cannot be directly extended to 2-D systems. Therefore, many researchers have devoted to the analysis and synthesis of 2-D systems and some attractive results can be found in References 11 and 12.…”

“…Next, we will synthesize an asynchronous SMC law to drive the system trajectories onto the predefined sliding surface (10) in a finite time. (1), the sliding surface designed as (10), where G 𝜇 are feasible solutions of (15), and the SMC law is synthesized as below, then, the state trajectories of the 2-D system can be driven onto the predefined sliding surface in a finite time:…”

“…Theorem 1. Consider the 2-D Markovian jump system (1), the sliding surface is designed as (10), given a scalar 𝛼, if there exist positive definite symmetric matrices Q i = Q h i ⊕ Q v i and general matrices X, M 𝜇 , i ∈ ⟨N⟩, 𝜇 ∈ ⟨S⟩, such that the following inequalities hold:…”

“…Compared with traditional 1‐D systems updating in a single direction along the time scale, the main characteristic in 2‐D systems is that its states evolve in two directions simultaneously and affect each other. In general, 2‐D systems are formulated mathematically by partial differential or difference equations associated with more than one complex/real variables in continuous‐ or discrete‐time domains 6‐10 . Consequently, the fundamental differences with the one‐dimensional (1‐D) systems immediately arises, and the traditional control theory for 1‐D systems cannot be directly extended to 2‐D systems.…”