^{1}

^{1}

^{1}

^{2}

^{1}

^{2}

This paper investigates robust adaptive switching controller design for Markovian jump nonlinear systems with unmodeled dynamics and Wiener noise. The concerned system is of strict-feedback form, and the statistics information of noise is unknown due to practical limitation. With the ordinary input-to-state stability (ISS) extended to jump case, stochastic Lyapunov stability criterion is proposed. By using backstepping technique and stochastic small-gain theorem, a switching controller is designed such that stochastic stability is ensured. Also system states will converge to an attractive region whose radius can be made as small as possible with appropriate control parameters chosen. A simulation example illustrates the validity of this method.

The establishment of modern control theory is contributed by state space analysis method which was introduced by Kalman in 1960s. This method, describing the changes of internal system states accurately through setting up the relationship of internal system variables and external system variables in time domain, has become the most important tool in system analysis. However, there remain many complex systems whose states are driven by not only continuous time but also a series of discrete events. Such systems are named hybrid systems whose dynamics vary with abrupt event occurring. Further, if the occurring of these events is governed by a Markov chain, the hybrid 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, for example, stability analysis [

The difficulties may result from several aspects for the study of Markovian jump nonlinear systems (MJNSs). First of all, controller design largely relies on the specific model of systems, and it is almost impossible to find out one general controller which can stabilize all nonlinear systems despite of their forms. Secondly Markovian jump systems are applied to model systems suffering sudden changes of working environment or system dynamics. For this reason, practical jump systems are usually accompanied by uncertainties, and it is hard to describe these uncertainties with precise mathematical model. Finally, noise disturbance is an important factor to be considered. More often that not, the statistics information of noise is unknown when taking into account the complexity of working environment. Among the achievements of MJNSs, the format of nonlinear systems should be firstly taken into account. As one specific model, the nonlinear system of strict-feedback form is well studied due to its powerful modelling ability of many practical systems, for example, power converter [

Motivated by this, this paper focuses on robust adaptive controller design for a class of MJNSs with uncertainties and Wiener noise. Compared with the existing result in [

The rest of this paper is organized as follows. Section

Throughout the paper, unless otherwise specified, we denote by

Take into account the following Markovian jump nonlinear system:

Considering the right-continuous Markov chain

Take the expectation in (

Equation (

MJNS (

The definition of ISpS (input-to-state practically stable) in probability for nonjump stochastic system is put forward by Wu et al. [

Consider the jump interconnected dynamic system described in Figure

Interconnected feedback system.

Suppose that both the

If there exist nonnegative parameters

The previously mentioned stochastic small-gain theorem for jump systems is an extension of nonjump case. This extension can be achieved without any mathematical difficulties, and the proof process is the same as in [

Consider the following Markovian jump nonlinear systems with dynamic uncertainty and noise described by

Our design purpose is to find a switching controller

The

For each

For the design of switching controller, we introduce the following lemmas.

For any two vectors

Let

Now we seek for the switching controller for MJNS (

According to stochastic differential equation (

Considering the transformation

Choose the quartic Lyapunov function as

In the view of (

Here we suggest the following adaptive laws [

Considering the MJNS (

Considering the MJNS (

For each integer

The proof is completed.

Considering the MJNS (

From Assumption (A1), the

With loss of generality, in this section we consider a two-order Markovian jump nonlinear system with regime transition space

Here let noise covariance be

The system regime is

The system regime is

Regime transition

System output

System state

Switching controller

Adaptive parameter

Parameter

Parameter

Now we choose different control parameters as

Regime transition

System output

System state

Switching controller

Adaptive parameter

Parameter

Parameter

Comparing the results from two simulations, all the signals of closed-loop system are globally uniformly ultimately bounded, and the system output can be regulated to a neighborhood near the equilibrium point despite different jump samples. As could be seen from the figures, larger values of

Much research work has been performed towards the study of nonlinear system by using small-gain theorem [

In this paper, the robust adaptive switching controller design for a class of Markovian jump nonlinear system is studied. Such MJNSs, suffering from unmodeled dynamics and noise of unknown covariance, are of the strict feedback form. With the extension of input-to-state stability (ISpS) to jump case as well as the small-gain theorem, stochastic Lyapunov stability criterion is put forward. By using backstepping technique, a switching controller is designed which ensures the jump nonlinear system to be jump ISpS in probability. Moreover the upper bound of uncertainties can be estimated, and system output will converge to an attractive region around the equilibrium point, whose radius can be made as small as possible with appropriate control parameters chosen. Numerical examples are given to show the effectiveness of the proposed design.

This work is supported by the National Natural Science Foundation of China under Grants 60904021 and the Fundamental Research Funds for the Central Universities under Grants WK2100060004.