This paper investigates the stability of a class of stochastic nonlinear systems with Markovian switching via output-feedback. Based on the backstepping design method and homogeneous domination technique, an output-feedback controller is constructed to guarantee that the closed-loop system has a unique solution and is almost surely asymptotically stable. The efficiency of the output-feedback controller is demonstrated by a simulation example.

There are lots of real systems, such as hierarchical control of manufacturing systems, financial engineering, and wireless communications systems, whose structure and parameters may change abruptly. Further, if the occurrence of these events is governed by a Markov chain, these systems are called Markovian jump systems. As one branch of modern control theory, the study of Markovian jump systems has aroused lots of attention with fruitful results achieved for linear case, such as the controllability and observability [

Considering that the system states are incompletely measurable, the problem of output-feedback control is more important and challenging than that of the state-feedback control in practical applications. Reference [

Compared with the existing results, the contributions of this paper are as follows:

The results in [

Since the drift terms and diffusions terms are all Markovian switching, how to design an effective observer to deal with the unmeasurable states and how to design a control to guarantee that the closed-loop system has a unique solution and is almost surely asymptotically stable are nontrivial work.

The remaining part of this paper is organized as follows. Section

The following notations will be used throughout this paper.

Consider the stochastic differential equations with Markovian switching:

For

A stochastic process

For any

Let

Consider the following stochastic nonlinear systems:

We need the following assumption.

There are constants

Condition

Condition

When

By introducing the coordinates

The design of output-feedback controller for system (

For simplicity, we assume

Choose

Choose

We introduce the following transformation:

By choosing the Lyapunov function

Substituting (

Since

The following design procedure proceeds in the similar way as

The construction of

Hence, (

By introducing the dilation weight

For system (

Now, we state the main results in this paper.

If Assumption

For every

For any

By the definition of

From (

In view of Assumption

Similar to (

From (

From the definition of

For any

By Lemma

With (

The unique features of the approaches proposed in this paper include the following:

This paper is the first result about the output-feedback control of stochastic nonlinear systems with Markovian switching and uncontrollable linearizations.

Since the drift terms and diffusions terms are all Markovian switching, a homogeneous domination approach is developed in this paper, which can effectively deal with the Markovian switching and uncontrollable linearizations simultaneously.

Consider the following system with two modes. The Markov process

The system is described by

Here, without detailed arguments, we only state the final results as follows:

In the simulation, one chooses the initial values

The responses of closed-loop system (

Runs of Markov process

This paper investigates the output-feedback stabilization of stochastic nonlinear systems with Markovian switching for the first time. By using the backstepping design method and homogeneous domination technique, an output-feedback controller is constructed to guarantee that the closed-loop system is almost surely asymptotically stable.

There are some related problems for further consideration, for example, how to generalize the results in this paper to more general stochastic nonlinear systems with plant and parameter uncertainties.

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

This work is supported by the National Natural Science Foundation of China (nos. 61174097 and 61573179) and by Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (no. BS2013DX001).

_{2}-control of Markovian jump linear systems via convex analysis