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Hindawi Publishing Corporation

Journal of Applied Mathematics

Volume 2012, Article ID 514504, 22 pages

doi:10.1155/2012/514504

Research Article

Robust Adaptive Switching Control

for Markovian Jump Nonlinear Systems via

Backstepping Technique

Jin Zhu,

Hongsheng Xi,

Qiang Ling,

and Wanqing Xie

Department of Automation, University of Science and Technology of China, Hefei, Anhui 230027, China

Center of Information Science Experiment and Education, University of Science and Technology of China,

China

Correspondence should be addressed to Jin Zhu, jinzhu@ustc.edu.cn

Received 28 March 2012; Accepted 9 May 2012

Academic Editor: Xianxia Zhang

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

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.

1. Introduction

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 1, 2,

ﬁltering 3, 4 and controller design 5, 6, and so forth. But studies are far from complete

2 Journal of Applied Mathematics

because researchers are facing big challenges while dealing with the nonlinear case of such

complicated systems.

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 7,

satellite attitude 8, and electrohydraulic servosystem 9. However, such models should

be modiﬁed since stochastic structure variations exist in these practical systems, and this

speciﬁc nonlinear system has been extended to jump case. For Markovian jump nonlinear

systems of strict-feedback form, 10, 11 investigated stabilization and tracking problems for

such MJNSs, respectively. And 12 studied the robust controller design for such systems

with unmodeled dynamics. However, for the MJNSs suﬀering aforementioned factors in this

paragraph, research work has not been performed yet.

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 12,

several practical limitations are considered which include the following: the uncertainties

are with unmodeled dynamics, and the upper bound of dynamics is not necessarily

known. Meanwhile the statistics information of Wiener noise is unknown. Also the adaptive

parameter is introduced to the controller design whose advantage has been described in 13.

The control strategy consists of several steps: ﬁrstly, by applying generalized Ito formula, the

stochastic diﬀerential equation for MJNS is deduced and the concept of JISpS jump input-

to-state practical stability is deﬁned. Then with backstepping technology and small-gain

theorem, robust adaptive switching controller is designed for such strict-feedback system.

Also the upper bound of the uncertainties can be estimated. Finally according to the stochastic

Lyapunov criteria, it is shown that all signals of the closed-loop system are globally uniformly

bounded in probability. Moreover, system states can converge to an attractive region whose

radius can be made as small as possible with appropriate control parameters chosen.

The rest of this paper is organized as follows. Section 2 begins with some mathematical

notions including diﬀerential equation for MJNS, and we introduce the notion of JISpS and

stochastic Lyapunov stability criterion. Section 3 presents the problem description, and a

robust adaptive switching controller is given based on backstepping technique and stochastic

small-gain theorem. In Section 4, stochastic Lyapunov criteria are applied for the stability

analysis. Numerical examples are given to illustrate the validity of this design in Section 5.

Finally, a brief conclusion is drawn in Section 6.

2. Mathematical Notions

2.1. Stochastic Differential Equation of MJNS

Throughout the paper, unless otherwise speciﬁed, we denote by Ω, F, {F

}

t≥0

,P a complete

probability space with a ﬁltration {F

}

t≥0

satisfying the usual conditions i.e., it is right

Journal of Applied Mathematics 3

continuous and F

contains all p-null sets.Let|x| stand for the usual Euclidean norm for

avectorx,andletx

 stand for the supremum of vector x over time period t

,t,thatis,

x

  sup

≤s≤t

|xs|. The superscript T will denote transpose and we refer to Tr· as the

trace for matrix. In addition, we use L

P to denote the space of Lebesgue square integrable

vector.

Take into account the following Markovian jump nonlinear system:

dx  f



x, u, t, r





dt  g



x, u, t, r





dω





, 2.1

where x ∈ R

, u ∈ R

are state vector and input vector of the system, respectively.

rt, t ≥ 0 is named system regime, a right-continuous Markov chain on the probability

space taking values in ﬁnite state space S  {1, 2,...,N}. And ωt{ω

,ω

,...,ω

} is l-

dimensional independent Wiener process deﬁned on the probability space, with covariance

matrix E{dωdω

} ΥtΥ

tdt, where Υt is an unknown bounded matrix-value function.

Furthermore, we assume that the Wiener noise ωt is independent of the Markov chain rt.

The functions f : R

nm

× R



× S → R

and g : R

nm

× R



× S → R

n×l

are locally Lipschitz in

x, u, rtk ∈ R

nm

× S for all t ≥ 0; namely, for any h>0, there is a constant K

≥ 0 such

that





,t,k



− f



,t,k



∨





,t,k



− g



,t,k



≤ K



− x



− u



2.2

∀



,t,k





,t,k



∈ R

nm

× R



× S,

∨

≤ h. 2.3

It is known by 2 that with 2.3 standing, MJNS 2.1 has a unique solution.

Considering the right-continuous Markov chain rt with regime transition rate matrix

Ππ



N×N

, the entries π

,k,j  1, 2,...,N are interpreted as transition rates such that





t  dt



 j | r





 k







dt  o





if k

 j,

1  π

dt  o





if k  j,

2.4

where dt > 0andodt satisﬁes lim

dt → 0

odt/dt0. Here π

> 0k

 j is the transition

rate from regime k to regime j. Notice that the total probability axiom imposes π

negative

and



j1

 0, ∀k ∈ S.

2.5

For each regime transition rate matrix Π, there exists a unique stationary distribution ζ 

ζ

,ζ

,...,ζ

 such that 14

Π · ζ  0,



k1

 1,ζ

> 0, ∀k ∈ S.

2.6

4 Journal of Applied Mathematics

Let C

2,1

R

× R



× S denote the family of all functions Fx, t, k on R

× R



× S which are

continuously twice diﬀerentiable in x and once in t. Furthermore, we give the stochastic

diﬀerentiable equation of Fx, t, k as



x, t, k





∂F



x, t, k



∂t

dt 

∂F



x, t, k



∂x



x, u, t, k









x, u, t, k



∂



x, t, k



∂x



x, u, t, k









j1



x, t, j



dt 

∂F



x, t, k



∂x



x, u, t, k



dω









j1





x, t, j



− F



x, t, k









2.7

where MtM

t,M

t,...,M

t is a martingale process.

Take the expectation in 2.7, so that the the inﬁnitesimal generator produces 2, 15



x, t, k





∂F



x, t, k



∂t



∂F



x, t, k



∂x



x, u, t, k







j1



x, t, j









x, u, t, k



∂



x, t, k



∂x



x, u, t, k





2.8

Remark 2.1. Equation 2.7 is the diﬀerential equation of MJNS 2.1 . It is given by 12,and

the similar result is also achieved in 15. Compared with the diﬀerential equation of general

nonjump systems, two parts come forth as diﬀerences: transition rates π

and martingale

process Mt, which are both caused by the Markov chain rt. And we will show in the

following section that the martingale process also has eﬀects on the controller design.

2.2. JISpS and Stochastic Small-Gain Theorem

Deﬁnition 2.2. MJNS 2.1 is JISpS in probability if for any given >0, there exist KL function

β·, ·, K

∞

function γ·, and a constant d

≥ 0 such that





t, k



<β





 γ









 d



≥ 1 −  ∀t ≥ 0,k∈ S, x

∈ R

{

}

. 2.9

Remark 2.3. The deﬁnition of ISpS input-to-state practically stable in probability for

nonjump stochastic system is put forward by Wu et al. 16,andthediﬀerence between JISpS

in probability and ISpS in probability lies in the expressions of system state xt, k and control

signal u

k. For nonjump system, system state and control signal contain only continuous

time t with k ≡ 1. While jump systems concern with both continuous time t and discrete

regime k. For diﬀerent regime k, control signal u

k will diﬀer with diﬀerent sample taken

even at the same time t, and that is the reason why the controller is called a switching one.

Journal of Applied Mathematics 5

(k)

-system

Figure 1: Interconnected feedback system.

Based on t his, the corresponding stability is called Jump ISpS, and it is an extension of ISpS.

Let k ≡ 1, and the deﬁnition of JISpS will degenerate to ISpS.

Consider the jump interconnected dynamic system described in Figure 1:

 f



, Ξ









dt  g



, Ξ









 f



, Ξ









dt  g



, Ξ









2.10

where x x



∈ R

n

is the state of system, Ξ

rt,i  1, 2 denotes exterior

disturbance and/or interior uncertainty. W

is independent Wiener noise with appropriate

dimension, and we introduce the following stochastic nonlinear small-gain theorem as a

lemma, which is an extension of the corresponding result in Wu et al. 16.

Lemma 2.4 stochastic small-gain theorem. Suppose that both the x

-system and x

-system are

JISpS in probability with Ξ

k,x

t, k as input and x

t, k as state and Ξ

k,x

t, k as input

and x

t, k as state, respectively; that is, for any given 

,

> 0,





t, k



<β



0,k



 γ







t, k



 γ









 d



≥ 1 − 





t, k



<β



0,k



 γ







t, k



 γ









 d



≥ 1 − 

2.11

hold with β

·, · being KL function, γ

and γ

being K

∞

functions, and d

being nonnegative

constants, i  1, 2.

If there exist nonnegative parameters ρ

, ρ

, s

such that nonlinear gain functions γ

, γ

satisfy



1  ρ



◦



1  ρ







≤ s, ∀s ≥ s

, 2.12

the interconnected system is JISpS in probability with ΞkΞ

k, Ξ

k as input and x 

x

 as state; that is, for any given >0,thereexistaKL function β

·, ·,aK

∞

function γ

·,

and a parameter d

≥ 0 such that





t, k



<β





 γ









 d



≥ 1 − . 2.13

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