This paper is concerned with model predictive control (MPC) problem for continuous-time Markov Jump Systems (MJSs) with incomplete transition rates and singular character. Sufficient conditions for the existence of a model predictive controller, which could optimize a quadratic cost function and guarantee that the system is piecewise regular, impulse-free, and mean square stable, are given in two cases at each sampling time. Since the MPC strategy is aggregated into continuous-time singular MJSs, a discrete-time controller is employed to deal with a continuous-time plant and the cost function not only refers to the singularity but also considers the sampling period. Moreover, the feasibility of the MPC scheme and the mean square admissibility of the closed-loop system are deeply discussed by using the invariant ellipsoid. Finally, a numerical example is given to illustrate the main results.

As a class of stochastic hybrid systems, Markov Jump Systems (MJSs) are suitable to describe many practical systems whose structures and parameters suffer from random abrupt changes [

In recent years, an ever increasing number of scholars have paid much attention in the study of singular MJSs [

On another research frontline, model predictive control (MPC), which is also named as receding horizon control, has received tremendous attention in practical applications [

Almost all of the above-mentioned studies are based on the assumption that the transition probabilities are completely known. However, in practice, obtaining the complete knowledge of transition probabilities is difficult and costly. Until very recently, some efforts start to focus on MJSs with incomplete transition probabilities. Zhang and Boukas firstly proposed an analysis and design method for MJSs with partly unknown transition probability, which does not require the information of unknown elements [

It is worth noting that MPC for MJSs are usually based on discrete-time version. However, in the real world, we take the controlled objects as continuous-time models. To the best of our knowledge, few results about continuous-time singular MJSs are addressed due to the difficulty in stability analysis and controller design, especially further considering incomplete transition rates. In this paper, a new MPC method is developed for continuous-time singular MJSs with incomplete transitions rates. Our main contributions include the following three aspects.

Consider a continuous-time singular MJS with uncertainties described as follows:

For the sake of convenience,

Moreover, in this paper, the transition rates are considered to be partly available; namely, some elements in matrix

In this paper, we denote

In addition, if

In the case that

MPC for (

About the singular MJS (

The continuous-time singular MJSs (

regular if

impulse-free if

mean square stable if there exists a scalar

mean square admissible if it is regular, impulse-free, and mean square stable.

Given a continuous-time dynamical system

Before giving the main results of the singular MJS (

Given matrices

Given symmetric and positive definite matrices

If a mode-dependent symmetric matrix

In this paper, at each sampling time

In this section, we will design MPC controller for the singular MJSs with incomplete transition rates by presenting sufficient conditions, which could be efficiently solved by LMI toolbox.

Consider the singular MJS (

Define a quadratic function

Considering Lemma

Applying the control law (

Certainly, MPC for singular MJSs without incomplete transition descriptions and the normal MJS with incomplete transition descriptions can be viewed as two special cases of singular MJS. Then, we have the following corollaries.

Consider the singular MJS (

When

The mode-dependent state feedback gain

Previous section gives sufficient conditions, which could guarantee the existence of model predictive controller. In this section, we mainly study the mean square stability of the closed-loop singular MJS (

The solution of the optimization problem in Theorem

At sampling time

The solution of the optimization problem in Theorem

The feedback predictive control gain computed from Theorem

Firstly, we show that the singular MJS (

Based on singular value decomposition, system (

Assuming the optimal solution of optimization problem (

Let

From (

from which we can see that

What is more, we consider the mean square stability of the singular MJS (

From (

Note that

Therefore, according to the property of being regular and impulse-free, the closed-loop singular MJS is mean square admissible.

To illustrate the efficiency of the proposed MPC scheme for continuous-time MJSs, a numerical example is presented in the following.

Consider a system with the form of (

The TRM is given as follows:

The simulation step is taken as 150 time units and each unit length is taken as 0.01. We get the jump mode (Figure

Jump mode.

State responses under MPC.

State responses under normal state feedback control.

Control input under MPC.

Control input under normal state feedback control.

Obviously, the unstable system (

In this paper, MPC strategy is presented for continuous-time singular MJSs with incomplete transition rates. The controller design problem is formulated as LMI optimization algorithm. The predictive control strategy is proved to be feasible at every sampling time and it also can guarantee the mean square admissibility of the closed-loop system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This project was jointly supported by NSFC (61203126), NSFC (61374047), BK2012111, and 111 Project (B12018).

_{∞}control of a class of discrete-time Markov jump linear systems with piecewise-constant TPs subject to average dwell time switching

_{∞}control of systems with parameter uncertainty