The problems of reachable set estimation and state-feedback controller design are investigated for singular Markovian jump systems with bounded input disturbances. Based on the Lyapunov approach, several new sufficient conditions on state reachable set and output reachable set are derived to ensure the existence of ellipsoids that bound the system states and output, respectively. Moreover, a state-feedback controller is also designed based on the estimated reachable set. The derived sufficient conditions are expressed in terms of linear matrix inequalities. The effectiveness of the proposed results is illustrated by numerical examples.
The research on singular systems has attracted significant attention in the past years due to the fact that singular systems can better describe a larger class of physical systems such as robotic systems, electric circuits, and mechanical systems. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems [
Reachable set is one of the important techniques for parameter estimation or state estimation problems [
In this paper, we consider the problems of reachable set estimation and synthesis of singular Markovian jump systems. By using the Lyapunov approach, the estimation conditions on state reachable set and output reachable set are derived, respectively. Moreover, the desired state-feedback controller is designed based on the estimated reachable set.
Consider the following singular Markovian jump system:
For notational simplicity, in the sequel, for each possible
In this paper we are interested in determining ellipsoids that contain, respectively, the state reachable set and output reachable set. In the reachable set analysis, it is required that systems should be asymptotically stable. When this requirement is not met, we will further design a state-feedback controller such that the reachable set of the closed-loop system is contained in the smallest ellipsoid.
The state reachable set of the free system in (
Since
The following definition and lemma are also useful in deriving the main results.
(I) The free system is said to be regular if
(II) The free system is said to be impulse-free if
For any matrices
Let
In this subsection, we will focus our attention on determining a ball which contains the state reachable set of the free system.
If there exist nonsingular matrices
We first prove the regularity and nonimpulsiveness of the free system. Let
Next, we will show the state reachable set of free system is mean-square bounded within the set
Since
By (
It should be noted that inequality (
If there exist symmetric positive definite matrices
Let
In order to make the ellipsoid
If there exist symmetric positive definite matrices
By Theorem
The output reachable set is also expected to be as small as possible. To achieve this goal, we first solve LMI (
In this section, we turn our attention to the state-feedback control problem. Our goal here is to find a state-feedback controller, which not only stabilizes the closed-loop system, but also makes the ellipsoid bound on the reachable set of closed-loop system as small as possible.
Now, consider the state-feedback controller
Consider singular Markov jump system (
Denote
Using Lemma
By Schur complement, the previous matrix inequality becomes
Since
Pre- and postmultiplying (
Next, we show that the reachable set of the closed-loop system (
Noting that
By (
In order to make the ellipsoid
In this section, two numerical simulation examples are given to show the effectiveness of the main results derived above.
Consider the free system in (
In this example, we choose
By applying Theorem
For simulation we assume that
The switching between two modes.
The state reachable set
The output reachable set
Consider system (
By Theorem
For the purpose of the simulation, we assume the initial condition
The switching between two modes.
The state reachable set
This paper has dealt with the problems of reachable set estimation and state-feedback controller design for singular Markovian jump systems. New sufficient conditions for the state reachable set estimation and output reachable set estimation have been, respectively, derived in terms of linear matrix inequalities. Based on the estimated reachable set, the state-feedback controller has also been designed. Numerical examples and simulation results have been provided to demonstrate the effectiveness of the proposed methods.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by Applied Basic Research Project of Science and Technology Department of Sichuan Province (no. 2017JY0336), a project supported by Scientific Research Fund of Sichuan Provincial Education Department (no. 16ZA0146), the National Natural Science Foundation of China (no. 11501474), the Longshan Academic Talents Research Support Program of the Southwest University of Science and Technology (no. 17LZX537 and no. 17LZXY11), and the Doctoral Research Foundation of Southwest University of Science and Technology (no. 13zx7141).