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Journal of Applied Mathematics
Volume 2012, Article ID 514504, 22 pages
Research Article

Robust Adaptive Switching Control for Markovian Jump Nonlinear Systems via Backstepping Technique

1Department of Automation, University of Science and Technology of China, Hefei, Anhui 230027, China
2Center of Information Science Experiment and Education, University of Science and Technology of China, China

Received 28 March 2012; Accepted 9 May 2012

Academic Editor: Xianxia Zhang

Copyright © 2012 Jin Zhu et al. This is an open access article distributed under the Creative 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 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 𝐼𝑡̂𝑜 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,,{𝑡}𝑡0,𝑃) a complete probability space with a filtration {𝑡}𝑡0 satisfying the usual conditions (i.e., it is right continuous and 0 contains all 𝑝-null sets). Let |𝑥| stand for the usual Euclidean norm for a vector 𝑥, and let 𝑥𝑡 stand for the supremum of vector 𝑥 over time period [𝑡0,𝑡], that is, 𝑥𝑡=sup𝑡0𝑠𝑡|𝑥(𝑠)|. The superscript 𝑇 will denote transpose and we refer to Tr() as the trace for matrix. In addition, we use 𝐿2(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(𝑡))𝑑𝑡+𝑔(𝑥,𝑢,𝑡,𝑟(𝑡))𝑑𝜔(𝑡),(2.1) where 𝑥𝑛, 𝑢𝑚 are state vector and input vector of the system, respectively. 𝑟(𝑡), 𝑡0 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)={𝜔1,𝜔2,,𝜔𝑙} is 𝑙-dimensional independent Wiener process defined on the probability space, with covariance matrix 𝐸{𝑑𝜔𝑑𝜔𝑇}=Υ(𝑡)Υ𝑇(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 𝑓𝑛+𝑚×+×𝑆𝑛 and 𝑔𝑛+𝑚×+×𝑆𝑛×𝑙 are locally Lipschitz in (𝑥,𝑢,𝑟(𝑡)=𝑘)𝑛+𝑚×𝑆 for all 𝑡0; namely, for any >0, there is a constant 𝐾0 such that ||𝑓𝑥1,𝑢1𝑥,𝑡,𝑘𝑓2,𝑢2||||𝑔𝑥,𝑡,𝑘1,𝑢1𝑥,𝑡,𝑘𝑔2,𝑢2||,𝑡,𝑘𝐾||𝑥1𝑥2||+||𝑢1𝑢2||𝑥(2.2)1,𝑢1,𝑥,𝑡,𝑘2,𝑢2,𝑡,𝑘𝑛+𝑚×+||𝑥×𝑆,1||||𝑥2||||𝑢1||||𝑢2||.(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=[𝜋𝑘𝑗]𝑁×𝑁, the entries 𝜋𝑘𝑗,𝑘,𝑗=1,2,,𝑁 are interpreted as transition rates such that 𝜋𝑃(𝑟(𝑡+𝑑𝑡)=𝑗𝑟(𝑡)=𝑘)=𝑘𝑗𝑑𝑡+𝑜(𝑑𝑡)if𝑘𝑗,1+𝜋𝑘𝑗𝑑𝑡+𝑜(𝑑𝑡)if𝑘=𝑗,(2.4) where 𝑑𝑡>0 and 𝑜(𝑑𝑡) satisfies lim𝑑𝑡0(𝑜(𝑑𝑡)/𝑑𝑡)=0. Here 𝜋𝑘𝑗>0(𝑘𝑗) is the transition rate from regime 𝑘 to regime 𝑗. Notice that the total probability axiom imposes 𝜋𝑘𝑘 negative and 𝑁𝑗=1𝜋𝑘𝑗=0,𝑘𝑆.(2.5) For each regime transition rate matrix Π, there exists a unique stationary distribution 𝜁=(𝜁1,𝜁2,,𝜁𝑁) such that [14] Π𝜁=0,𝑁𝑘=1𝜁𝑘=1,𝜁𝑘>0,𝑘𝑆.(2.6) Let 𝐶2,1(𝑛×+×𝑆) denote the family of all functions 𝐹(𝑥,𝑡,𝑘) on 𝑛×+×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 𝑑𝐹(𝑥,𝑡,𝑘)=𝜕𝐹(𝑥,𝑡,𝑘)𝜕𝑡𝑑𝑡+𝜕𝐹(𝑥,𝑡,𝑘)+1𝜕𝑥𝑓(𝑥,𝑢,𝑡,𝑘)𝑑𝑡2ΥTr𝑇𝑔𝑇𝜕(𝑥,𝑢,𝑡,𝑘)2𝐹(𝑥,𝑡,𝑘)𝜕𝑥2+𝑔(𝑥,𝑢,𝑡,𝑘)Υ𝑑𝑡𝑁𝑗=1𝜋𝑘𝑗𝐹(𝑥,𝑡,𝑗)𝑑𝑡+𝜕𝐹(𝑥,𝑡,𝑘)+𝜕𝑥𝑔(𝑥,𝑢,𝑡,𝑘)𝑑𝜔(𝑡)𝑁𝑗=1[𝐹](𝑥,𝑡,𝑗)𝐹(𝑥,𝑡,𝑘)𝑑𝑀𝑗(𝑡),(2.7) where 𝑀(𝑡)=(𝑀1(𝑡),𝑀2(𝑡),,𝑀𝑁(𝑡)) is a martingale process.

Take the expectation in (2.7), so that the the infinitesimal generator produces [2, 15] 𝐹(𝑥,𝑡,𝑘)=𝜕𝐹(𝑥,𝑡,𝑘)+𝜕𝑡𝜕𝐹(𝑥,𝑡,𝑘)𝜕𝑥𝑓(𝑥,𝑢,𝑡,𝑘)+𝑁𝑗=1𝜋𝑘𝑗+1𝐹(𝑥,𝑡,𝑗)2ΥTr𝑇𝑔𝑇𝜕(𝑥,𝑢,𝑡,𝑘)2𝐹(𝑥,𝑡,𝑘)𝜕𝑥2.𝑔(𝑥,𝑢,𝑡,𝑘)Υ(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 𝜋𝑘𝑗 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 𝜖>0, there exist 𝒦 function 𝛽(,), 𝒦 function 𝛾(), and a constant 𝑑𝑐0 such that 𝑃||𝑥||||𝑥(𝑡,𝑘)<𝛽0||𝑢,𝑡+𝛾𝑡(𝑘)+𝑑𝑐1𝜖𝑡0,𝑘𝑆,𝑥0𝑛{0}.(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], and the difference between JISpS in probability and ISpS in probability lies in the expressions of system state 𝑥(𝑡,𝑘) and control signal 𝑢𝑡(𝑘). For nonjump system, system state and control signal contain only continuous time 𝑡 with 𝑘1. While jump systems concern with both continuous time 𝑡 and discrete regime 𝑘. For different regime 𝑘, control signal 𝑢𝑡(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. Based on this, the corresponding stability is called Jump ISpS, and it is an extension of ISpS. Let 𝑘1, and the definition of JISpS will degenerate to ISpS.

Consider the jump interconnected dynamic system described in Figure 1: 𝑑𝑥1=𝑓1𝑥1,𝑥2,Ξ1(𝑟(𝑡)),𝑟(𝑡)𝑑𝑡+𝑔1𝑥1,𝑥2,Ξ1(𝑟(𝑡)),𝑟(𝑡)𝑑𝑊𝑡1,𝑑𝑥2=𝑓2𝑥1,𝑥2,Ξ2(𝑟(𝑡)),𝑟(𝑡)𝑑𝑡+𝑔2𝑥1,𝑥2,Ξ2(𝑟(𝑡)),𝑟(𝑡)𝑑𝑊𝑡2,(2.10) where 𝑥=(𝑥𝑇1,𝑥𝑇2)𝑇𝑛1+𝑛2 is the state of system, Ξ𝑖(𝑟(𝑡)),𝑖=1,2 denotes exterior disturbance and/or interior uncertainty. 𝑊𝑡𝑖 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].

Figure 1: Interconnected feedback system.

Lemma 2.4 (stochastic small-gain theorem). Suppose that both the 𝑥1-system and 𝑥2-system are JISpS in probability with (Ξ1(𝑘),𝑥2(𝑡,𝑘)) as input and 𝑥1(𝑡,𝑘) as state and (Ξ2(𝑘),𝑥1(𝑡,𝑘)) as input and 𝑥2(𝑡,𝑘) as state, respectively; that is, for any given 𝜖1,𝜖2>0, 𝑃||𝑥1||(𝑡,𝑘)<𝛽1||𝑥1||(0,𝑘),𝑡+𝛾1𝑥2(𝑡,𝑘)+𝛾𝑤1Ξ1𝑡(𝑘)+𝑑11𝜖1,𝑃||𝑥2(||𝑡,𝑘)<𝛽2||𝑥2(||0,𝑘),𝑡+𝛾2𝑥1(𝑡,𝑘)+𝛾𝑤2Ξ2𝑡(𝑘)+𝑑21𝜖2,(2.11) hold with 𝛽𝑖(,) being 𝒦 function, 𝛾𝑖 and 𝛾𝑤𝑖 being 𝒦 functions, and 𝑑𝑖 being nonnegative constants, 𝑖=1,2.
If there exist nonnegative parameters 𝜌1, 𝜌2, 𝑠0 such that nonlinear gain functions 𝛾1, 𝛾2 satisfy 1+𝜌1𝛾11+𝜌2𝛾2(𝑠)𝑠,𝑠𝑠0,(2.12) the interconnected system is JISpS in probability with Ξ(𝑘)=(Ξ1(𝑘),Ξ2(𝑘)) as input and 𝑥=(𝑥1,𝑥2) as state; that is, for any given 𝜖>0, there exist a 𝒦 function 𝛽𝑐(,), a 𝒦 function 𝛾𝑤(), and a parameter 𝑑𝑐0 such that 𝑃||𝑥||(𝑡,𝑘)<𝛽𝑐||𝑥0||,𝑡+𝛾𝑤Ξ𝑡(𝑘)+𝑑𝑐1𝜖.(2.13)

Remark 2.5. 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 [16]. The reason is that in Lemma 3.1 we only take into account the interconnection relationship between synthetical system and its subsystems, despite the fact that subsystems are of jump or nonjumpform. If both subsystems are nonjump and ISpS in probability, respectively, the synthetical system is ISpS in probability. By contraries, if both subsystems are jump and JISpS in probability, respectively, the synthetical system is JISpS in probability correspondingly.

3. Problem Description and Controller Design

3.1. Problem Description

Consider the following Markovian jump nonlinear systems with dynamic uncertainty and noise described by 𝑑𝜉=𝑞(𝑦,𝜉,𝑡,𝑟(𝑡))𝑑𝑡,𝑑𝑥𝑖=𝑥𝑖+1𝑑𝑡+𝑓𝑇𝑖𝑋𝑖𝜃,𝑡,𝑟(𝑡)𝑑𝑡+Δ𝑖(𝑋,𝜉,𝑡,𝑟(𝑡))𝑑𝑡+𝑔𝑇𝑖𝑋𝑖,𝑡,𝑟(𝑡)𝑑𝜔,𝑑𝑥𝑛=𝑢𝑑𝑡+𝑓𝑇𝑛(𝑋,𝑡,𝑟(𝑡))𝜃𝑑𝑡+Δ𝑛(𝑋,𝜉,𝑡,𝑟(𝑡))𝑑𝑡+𝑔𝑇𝑛(𝑋,𝑡,𝑟(𝑡))𝑑𝜔𝑖=1,2,,𝑛1,𝑦=𝑥1,(3.1) where 𝑋𝑖=(𝑥1,𝑥2,,𝑥𝑖)𝑇𝑖(𝑋𝑛) is state vector, 𝑢 is system input signal, 𝜉𝑛0 is unmeasured state vector, and 𝑦 is output signal. 𝜃𝑝0 is a vector of unknown adaptive parameters. The Markov chain 𝑟(𝑡)𝑆 and Wiener noise 𝜔 are as defined in Section 2. 𝑓𝑖𝑖×+×𝑆𝑝0, 𝑔𝑖𝑖×+×𝑆𝑙 are vector-valued smooth functions, and Δ𝑖(𝑋,𝜉,𝑡,𝑟(𝑡)) denotes the unmodeled dynamic uncertainty which could vary with different regime 𝑟(𝑡) taken. Both 𝑓𝑖, 𝑔𝑖 and Δ𝑖 are locally Lipschitz as in Section 2.

Our design purpose is to find a switching controller 𝑢 of the form 𝑢(𝑥,𝑡,𝑘),𝑘𝑆 such that the closed-loop jump system could be JISpS in probability and the system output 𝑦 could be within an attractive region around the equilibrium point. In this paper, the following assumptions are made for MJNS (3.1). (A1) The 𝜉 subsystem with input 𝑦 is JISpS in probability; namely, for any given 𝜖>0, there exist 𝒦 function 𝛽(,), 𝒦 function 𝛾(), and a constant 𝑑𝑐0 such that 𝑃||𝜉||||𝜉(𝑡,𝑘)<𝛽0||(,𝑡+𝛾𝑦)+𝑑𝑐1𝜖𝑡0,𝑘𝑆,𝜉0𝑛0{0}.(3.2) (A2) For each 𝑖=1,2,,𝑛, 𝑘𝑆, there exists an unknown bounded positive constant 𝑝𝑖 such that ||Δ𝑖||(𝑋,𝜉,𝑡,𝑘)𝑝𝑖𝜙𝑖1𝑋𝑖,𝑘+𝑝𝑖𝜙𝑖2||𝜉||,,𝑘(3.3) where 𝜙𝑖1(,𝑘), 𝜙𝑖2(,𝑘) are known nonnegative smooth functions for any given 𝑘𝑆. Notice that 𝑝𝑖 is not unique since any 𝑝𝑖>𝑝𝑖 satisfies inequality (3.3). To avoid confusion, we define 𝑝𝑖 the smallest nonnegative constant such that inequality (3.3) is satisfied.

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

Lemma 3.1 (Young’s inequality [12]). For any two vectors 𝑥,𝑦𝑛, the following inequality holds 𝑥𝑇𝜖𝑦𝑝𝑝|𝑥|𝑝+1𝑞𝜖𝑞||𝑦||𝑞,(3.4) where 𝜖>0 and the constants 𝑝>1, 𝑞>1 satisfy (𝑝1)(𝑞1)=1.

Lemma 3.2 (martingale representation [17]). Let 𝐵(𝑡)=[𝐵1(𝑡),𝐵2(𝑡),,𝐵𝑁(𝑡)] be N-dimensional standard Wiener noise. Supposing 𝑀(𝑡) is an 𝑁𝑡-martingale (with respect to P) and that 𝑀(𝑡)𝐿2(𝑃) for all 𝑡0, then there exists a stochastic process Ψ(𝑡)𝐿2(𝑃), such that 𝑑𝑀(𝑡)=Ψ(𝑡)𝑑𝐵(𝑡).(3.5)

3.2. Controller Design

Now we seek for the switching controller for MJNS (3.1) so that the closed-loop system could be JISpS in probability, where the parameter 𝜃, 𝑝𝑖 needs to be estimated. Denote the estimation of adaptive parameter 𝜃 with 𝜃 and the estimation of upper bound of uncertainty 𝑝𝑖 with 𝑝𝑖. Perform a new transformation as 𝑧𝑖=𝑥𝑖(𝑘)𝛼𝑖1𝑋𝑖1,𝑡,𝜃,𝑝𝑖,𝑘𝑖=1,2,,𝑛,𝑘𝑆.(3.6) For simplicity, we just denote 𝛼𝑖1(𝑋𝑖1,𝑡,𝜃,𝑝𝑖,𝑘), 𝑓𝑖(𝑋𝑖,𝑡,𝑘), 𝑔𝑖(𝑋𝑖,𝑡,𝑘), Δ𝑖(𝑋,𝜉,𝑡,𝑘), 𝑞(𝑦,𝜉,𝑡,𝑘) by 𝛼𝑖1(𝑘), 𝑓𝑖(𝑘), 𝑔𝑖(𝑘), Δ𝑖(𝑘), 𝑞(𝑘), respectively, where 𝛼0(𝑘)=0, 𝛼𝑛(𝑘)=𝑢(𝑘), for all 𝑘𝑆, and the new coordinate is 𝑍(𝑘)=(𝑧1(𝑘),𝑧2(𝑘),,𝑧𝑛(𝑘)).

According to stochastic differential equation (2.7), one has 𝑑𝑧𝑖=𝑑𝑥𝑖𝑑𝛼𝑖1=𝑥(𝑘)𝑖+1+𝑓𝑇𝑖(𝑘)𝜃+Δ𝑖(𝑘)𝑑𝑡𝜕𝛼𝑖1(𝑘)𝜕𝑡𝑑𝑡𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑥𝑗+1+𝑓𝑇𝑗(𝑘)𝜃+Δ𝑗(𝑘)𝑑𝑡𝜕𝛼𝑖1(𝑘)̇𝜕𝜃𝜃𝑑𝑡𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑝𝑖̇𝑝𝑖1𝑑𝑡2𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞𝑔𝑇𝑝(𝑘)ΥΥ𝑇𝑔𝑞(𝑘)𝑑𝑡𝑁𝑗=1𝜋𝑘𝑗𝛼𝑖1+𝑔(𝑗)𝑑𝑡𝑇𝑖(𝑘)𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑔𝑇𝑗(𝑘)𝑑𝜔+𝑁𝑗=1𝛼𝑖1(𝑘)𝛼𝑖1(𝑗)𝑑𝑀𝑗=𝑧(𝑡)𝑖+1+𝛼𝑖(𝑘)+𝜏𝑇𝑖(𝑘)𝜃+Λ𝑖(𝑘)𝑑𝑡𝜕𝛼𝑖1(𝑘)𝜕𝑡𝑑𝑡𝜕𝛼𝑖1(𝑘)̇𝜕𝜃𝜃𝑑𝑡𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑝𝑖̇𝑝𝑖𝑑𝑡𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑥𝑗+11𝑑𝑡2𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞𝑔𝑇𝑝(𝑘)ΥΥ𝑇𝑔𝑞(𝑘)𝑑𝑡𝑁𝑗=1𝜋𝑘𝑗𝛼𝑖1(𝑗)𝑑𝑡+𝜌𝑇𝑖(𝑘)𝑑𝜔+Γ𝑖(𝑘)𝑑𝑀(𝑡).(3.7) Here we define Λ𝑖(𝑘)Δ𝑖(𝑘)𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗Δ𝑗𝜏(𝑘),𝑖(𝑘)𝑓𝑖(𝑘)𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑓𝑗𝜌(𝑘),𝑖(𝑘)𝑔𝑖(𝑘)𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑔𝑗Γ(𝑘),𝑖𝛼(𝑘)𝑖1(𝑘)𝛼𝑖1(1),𝛼𝑖1(𝑘)𝛼𝑖1(2),,𝛼𝑖1(𝑘)𝛼𝑖1.(𝑁)(3.8) From assumption (A2), one gets that there exists nonnegative smooth function 𝜙𝑖1, 𝜙𝑖2 satisfying ||Λ𝑖||(𝑘)𝑝𝑖𝜙𝑖1𝑋𝑖,𝑘+𝑝𝑖𝜙𝑖2||𝜉||,𝑘.(3.9) The inequality (3.9) could easily be deduced by using Lemma 3.1.

Considering the transformation 𝑧𝑖 in (3.7) which contains the martingale process 𝑀(𝑡), according to Lemma 3.2, there exist a function Ψ(𝑡)𝐿2(𝑃) and an 𝑁-dimensional standard Wiener noise 𝐵(𝑡) satisfying 𝑑𝑀(𝑡)=Ψ(𝑡)𝑑𝐵(𝑡), where 𝐸[Ψ(𝑡)Ψ(𝑡)𝑇]=𝜓(𝑡)𝜓(𝑡)𝑇𝑄< and 𝑄 is a positive bounded constant. Therefore we have 𝑑𝑧𝑖=𝑧𝑖+1+𝛼𝑖(𝑘)+𝜏𝑇𝑖(𝑘)𝜃+Λ𝑖(𝑘)𝜕𝛼𝑖1(𝑘)𝜕𝑡𝜕𝛼𝑖1(𝑘)̇𝜕𝜃𝜃𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑝𝑖̇𝑝𝑖𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑥𝑗+112𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞𝑔𝑇𝑝(𝑘)ΥΥ𝑇𝑔𝑞(𝑘)𝑁𝑗=1𝜋𝑘𝑗𝛼𝑖1(𝑗)𝑑𝑡+𝜌𝑇𝑖(𝑘)𝑑𝜔+Γ𝑖(𝑘)Ψ(𝑡)𝑑𝐵(𝑡).(3.10) Differential equation of new coordinate 𝑍=(𝑧1,𝑧2,,𝑧𝑛) is deduced by (3.10). The martingale process resulting from Markov process is transformed into Wiener noise by using Martingale representation theorem. To deal with this, quartic Lyapunov function is proposed, and in the controller design, consideration must be taken for the Wiener noise 𝐵(𝑡).

Choose the quartic Lyapunov function as 1𝑉(𝑘)=4𝑛𝑖=1𝑧4𝑖+1̃𝜃2𝛾𝑇̃𝜃+𝑛𝑖=112𝜎𝑖̃𝑝2𝑖,(3.11) where 𝛾>0, 𝜎𝑖>0 are constants. ̃𝜃=𝜃𝜃 and ̃𝑝𝑖=𝑝𝑀𝑖𝑝𝑖 are parameter estimation errors, where 𝑝𝑀𝑖max{𝑝𝑖,𝑝0𝑖} and 𝑝0𝑖 are given positive constants.

In the view of (3.10) and (3.11), the infinitesimal generator of 𝑉 satisfies 𝑉(𝑘)=𝑛𝑖=1𝑧3𝑖𝑧𝑖+1+𝛼𝑖(𝑘)+𝜏𝑇𝑖(𝑘)𝜃+Λ𝑖(𝑘)𝜕𝛼𝑖1(𝑘)𝜕𝑡𝜕𝛼𝑖1(𝑘)̇𝜕𝜃𝜃𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑝𝑖̇𝑝𝑖𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑥𝑗+112𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞𝑔𝑇𝑝(𝑘)ΥΥ𝑇𝑔𝑞(𝑘)𝑁𝑗=1𝜋𝑘𝑗𝛼𝑖1+3(𝑗)2𝑛𝑖=1𝑧2𝑖𝜌𝑇𝑖(𝑘)ΥΥ𝑇𝜌𝑖3(𝑘)+2𝑛𝑖=1𝑧2𝑖Γ𝑖(𝑘)𝜓𝜓𝑇Γ𝑇𝑖1(𝑘)𝛾̃𝜃𝑇̇𝜃𝑛𝑖=11𝜎𝑖̃𝑝𝑖̇𝑝𝑖+𝑁𝑗=1𝜋𝑘𝑗𝑉(𝑗)𝑛𝑖=1𝑧3𝑖34𝛿𝑖4/3+14𝛿4𝑖1𝑧𝑖+𝛼𝑖(𝑘)+𝜏𝑇𝑖(𝑘)𝜃𝜕𝛼𝑖1(𝑘)𝜕𝑡𝜕𝛼𝑖1(𝑘)̇𝜕𝜃𝜃𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑝𝑖̇𝑝𝑖𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑥𝑗+1+𝜆𝑧3𝑖𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞2𝑔𝑇𝑝(𝑘)𝑔𝑞(𝑘)2+𝜇1𝑧𝑖𝜌𝑇𝑖(𝑘)𝜌𝑖(𝑘)2+𝜇2𝑧𝑖Γ𝑖(𝑘)Γ𝑇𝑖(𝑘)2𝑁𝑗=1𝜋𝑘𝑗𝛼𝑖1+(𝑗)(𝑛1)𝑛(2𝑛1)+96𝜆9𝑛16𝜇1||Υ||4+9𝑛16𝜇2𝑄2̃𝜃𝑇1𝛾̇𝜃𝑛𝑖=1𝑧3𝑖𝜏𝑖(𝑘)𝑛𝑖=11𝜎𝑖̃𝑝𝑖̇𝑝𝑖𝑧3𝑖Λ𝑖+(𝑘)𝑁𝑗=1𝜋𝑘𝑗𝑉(𝑗).(3.12) The following inequalities could be deduced by using Young’s inequality and norm inequalities with the help of changing the order of summations or exchanging the indices of the summations: 𝑛𝑖=1𝑧3𝑖𝑧𝑖+134𝑛1𝑖=1𝛿𝑖4/3𝑧4𝑖+14𝑛1𝑖=11𝛿4𝑖𝑧4𝑖+1=𝑛𝑖=134𝛿𝑖4/3+14𝛿4𝑖1𝑧4𝑖12𝑛𝑖=1𝑧3𝑖𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞𝑔𝑇p(𝑘)ΥΥ𝑇𝑔𝑞(𝑘)𝑛𝑖=1𝜆𝑧6𝑖𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞2𝑔𝑇𝑝(𝑘)𝑔𝑝(𝑘)𝑔𝑇𝑞(𝑘)𝑔𝑞(𝑘)+𝑛𝑖=1𝑖1𝑝,𝑞=11||16𝜆ΥΥ𝑇||2=𝑛𝑖=1𝜆𝑧6𝑖𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞2𝑔𝑇𝑝(𝑘)𝑔𝑞(𝑘)2+||ΥΥ𝑇||2396𝜆(𝑛1)𝑛(2𝑛1),2𝑛𝑖=1𝑧2𝑖𝜌𝑇𝑖(𝑘)ΥΥ𝑇𝜌𝑖(𝑘)𝑛𝑖=1𝜇1𝑧4𝑖𝜌𝑇𝑖(𝑘)𝜌𝑖(𝑘)2+𝑛𝑖=1916𝜇1||ΥΥ𝑇||2=𝑛𝑖=1𝜇1𝑧4𝑖𝜌𝑇𝑖(𝑘)𝜌𝑖(𝑘)2+9𝑛16𝜇1||ΥΥ𝑇||2,32𝑛𝑖=1𝑧2𝑖Γ𝑖(𝑘)𝜓𝜓𝑇Γ𝑇𝑖(3𝑘)2𝑛𝑖=1𝑧2𝑖Γ𝑖(𝑘)𝑄Γ𝑇𝑖(𝑘)𝑛𝑖=1𝜇2𝑧4𝑖Γ𝑖(𝑘)Γ𝑇𝑖(𝑘)2+𝑛𝑖=1916𝜇2𝑄2=𝑛𝑖=1𝜇2𝑧4𝑖Γ𝑖(𝑘)Γ𝑇𝑖(𝑘)2+9𝑛16𝜇2𝑄2,(3.13) where 𝛿0=, 𝛿𝑛=0 and 𝜆>0, 𝜇1>0, 𝜇2>0, 𝛿𝑖>0, 𝑖=1,2,,𝑛 are design parameters to be chosen.

Here we suggest the following adaptive laws [18]: ̇𝜃=𝛾𝑛𝑖=1𝑧3𝑖𝜏𝑖(𝑘)𝑎𝜃𝜃0,̇𝑝𝑖=𝜎𝑖𝑧3𝑖𝜛𝑖(𝑘)𝑚𝑖𝑝𝑖𝑝0𝑖.(3.14) Here 𝑎>0, 𝜃0𝑝0, 𝑚𝑖>0, 𝑖=1,2,,𝑛 are design parameters to be chosen. And define function 𝛽(𝑘) as 𝜛𝑖(𝑘)=𝜙𝑖1𝑋𝑖𝑧,𝑘tanh3𝑖𝜙𝑖1𝑋𝑖,𝑘𝜀𝑖+𝑧3𝑖𝑧tanh6𝑖𝜐𝑖,𝛽𝑖(𝑘)=𝑝𝑖𝜛𝑖(𝑘),(3.15) where 𝜀𝑖>0, 𝜐𝑖>0, 𝑖=1,2,,𝑛 are control parameters to be chosen, and let the virtual control signal be 𝛼𝑖(𝑘)=𝑐𝑖𝑧𝑖34𝛿𝑖4/3+14𝛿4𝑖1𝑧𝑖𝜏𝑇𝑖(𝑘)𝜃+𝜕𝛼𝑖1(𝑘)+𝜕𝑡𝜕𝛼𝑖1(𝑘)̇𝜕𝜃𝜃+𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑝𝑖̇𝑝𝑖+𝑖1𝑗=1𝜕𝛼𝑖1(𝑘)𝜕𝑥𝑗𝑥𝑗+1𝜆𝑧3𝑖𝑖1𝑝,𝑞=1𝜕2𝛼𝑖1(𝑘)𝜕𝑥𝑝𝜕𝑥𝑞2𝑔𝑇𝑝(𝑘)𝑔𝑞(𝑘)2𝜇1𝑧𝑖𝜌𝑇𝑖(𝑘)𝜌𝑖(𝑘)2𝜇2𝑧𝑖Γ𝑖(𝑘)Γ𝑇𝑖(𝑘)2+𝑁𝑗=1𝜋𝑘𝑗𝛼𝑖1(𝑗)𝛽𝑖(𝑘).(3.16) Thus the real control signal 𝑢(𝑘) satisfies 𝑢(𝑘)=𝛼𝑛(𝑘) such that 𝑉𝑛𝑖=1𝑐𝑖𝑧4𝑖̃𝜃+𝑎𝜃𝜃0+𝑛𝑖=1𝑧3𝑖Λ𝑖(𝑘)𝑝𝑀𝑖𝜛𝑖+(𝑘)𝑛𝑖=1𝑚𝑖̃𝑝𝑖𝑝𝑖𝑝0𝑖+(𝑛1)𝑛(2𝑛1)+96𝜆9𝑛16𝜇1||Υ||4+9𝑛16𝜇2𝑄2+𝑁𝑗=1𝜋𝑘𝑗𝑉(𝑗).(3.17) Based on assumption (A2) and (3.9), we obtain the following inequality by applying Lemma 3.1: 𝑧3𝑖Λ𝑖(𝑘)𝑝𝑀𝑖𝑧3𝑖𝜛𝑖||𝑧(𝑘)3𝑖Λ𝑖||(𝑘)𝑝𝑀𝑖𝑧3𝑖𝜙𝑖1𝑋𝑖𝑧,𝑘tanh3𝑖𝜙𝑖1𝑋𝑖,𝑘𝜀𝑖𝑝𝑀𝑖𝑧6𝑖𝑧tanh6𝑖𝜐𝑖||𝑧3𝑖||𝑝𝑖𝜙𝑖1𝑋𝑖,𝑘+𝑝𝑖𝜙𝑖2||𝜉||,𝑘𝑝𝑀𝑖𝑧3𝑖𝜙𝑖1𝑋𝑖𝑧,𝑘tanh3𝑖𝜙𝑖1𝑋𝑖,𝑘𝜀𝑖𝑝𝑀𝑖𝑧6𝑖𝑧tanh6𝑖𝜐𝑖||𝑧3𝑖||𝑝𝑖𝜙𝑖1𝑋𝑖,𝑘𝑝𝑀𝑖𝑧3𝑖𝜙𝑖1𝑋𝑖𝑧,𝑘tanh3𝑖𝜙𝑖1𝑋𝑖,𝑘𝜀𝑖+𝑝𝑀𝑖||𝑧3𝑖||𝜙𝑖2||𝜉||,𝑘𝑝𝑀𝑖𝑧6𝑖𝑧tanh6𝑖𝜐i𝑝𝑀𝑖||𝑧3𝑖𝜙𝑖1𝑋𝑖||,𝑘𝑧3𝑖𝜙𝑖1𝑋𝑖𝑧,𝑘tanh3𝑖𝜙𝑖1𝑋𝑖,𝑘𝜀𝑖+𝑝𝑀𝑖𝑧6𝑖𝑧6𝑖𝑧tanh6𝑖𝜐𝑖+14𝜙2𝑖2||𝜉||𝜀,𝑘𝑖+𝜐𝑖2𝑝𝑀𝑖+𝑝𝑀𝑖4𝜙2𝑖2||𝜉||.,𝑘(3.18) In (3.18), the following inequality is applied: ||𝜂||𝜂0𝜂tanh𝜖12𝜖.(3.19) Notice the fact that 𝑎̃𝜃𝑇𝜃𝜃01=2𝑎̃𝜃𝑇̃1𝜃2𝑎𝜃𝜃0𝑇𝜃𝜃0+12𝑎𝜃𝜃0𝑇𝜃𝜃012𝑎̃𝜃𝑇̃1𝜃+2𝑎𝜃𝜃0𝑇𝜃𝜃0,𝑚𝑖̃𝑝𝑖𝑝𝑖𝑝0𝑖1=2𝑚𝑖̃𝑝2𝑖12𝑚𝑖𝑝𝑖𝑝0𝑖2+12𝑚𝑖𝑝𝑀𝑖𝑝0𝑖212𝑚𝑖̃𝑝2𝑖+12𝑚𝑖𝑝𝑀𝑖𝑝0𝑖2.(3.20) Submitting (3.18), (3.20) into (3.12), there is 𝑉(𝑘)𝑛𝑖=1𝑐𝑖𝑧4𝑖12𝑎̃𝜃𝑇̃𝜃𝑛𝑖=112𝑚𝑖̃𝑝2𝑖+12𝑎𝜃𝜃0𝑇𝜃𝜃0+𝑛𝑖=112𝑚𝑖𝑝𝑀𝑖𝑝0𝑖2+(𝑛1)𝑛(2𝑛1)+96𝜆9𝑛16𝜇1||Υ||4+9𝑛16𝜇2𝑄2+𝑛𝑖=1𝜀𝑖+𝜐𝑖2𝑝𝑀𝑖+𝑛𝑖=1𝑝𝑀𝑖4𝜙2𝑖2||𝜉||+,𝑘𝑁𝑗=1𝜋𝑘𝑗𝑉(𝑗)𝛼1𝑉(𝑘)+𝑉𝜉||𝜉||,𝑘+𝑑𝑧+𝑁𝑗=1𝜋𝑘𝑗𝑉(𝑗).(3.21) Here parameter 𝛼1, 𝑑𝑧 and 𝒦 function 𝑉𝜉(|𝜉|,𝑘) is chosen to satisfy 𝑉𝜉||𝜉||,𝑘𝑛𝑖=1𝑝𝑀𝑖4𝜙2𝑖2||𝜉||,𝑘,𝛼1=min4𝑐𝑖,𝑎𝛾,𝑚𝜎𝑖,𝑑𝑧=12𝑎𝜃𝜃0𝑇𝜃𝜃0+𝑛𝑖=112𝑚𝑖𝑝𝑀𝑖𝑝0𝑖2+(𝑛1)𝑛(2𝑛1)+96𝜆9𝑛16𝜇1||Υ||4+9𝑛16𝜇2𝑄2+𝑛𝑖=1𝜀𝑖+𝜐𝑖2𝑝𝑀𝑖.(3.22)

4. Stochastic Stability Analysis

Theorem 4.1. Considering the MJNS (3.1) with Assumptions (A2)  standing, the 𝑋-subsystem is JISpS in probability with the adaptive laws (3.14) and switching control law (3.16) adopted; meanwhile all solutions of closed-loop 𝑋-subsystem are ultimately bounded.

Proof. Considering the MJNS (3.1) with Lyapunov function (3.11), the following equations hold according to [10]: 𝐸𝑉(𝑟(𝑡))=𝑁𝑙=1𝐸𝑉(𝑙)𝜁𝑙,𝐸𝑉(𝑟(𝑡))=𝑁𝑙=1𝐸(𝑉(𝑙))𝜁𝑙.(4.1) Thus (3.21) can be written as 𝐸𝑉(𝑟(𝑡))=𝑁𝑙=1𝐸(𝑉(𝑙))𝜁𝑙𝑁𝑙=1𝐸𝛼1𝜁𝑙𝑉(𝑙)+𝜁𝑙𝑉𝜉||𝜉||,𝑙+𝜁𝑙𝑑𝑧+𝜁𝑙𝑁𝑗=1𝜋𝑙𝑗𝑉(𝑗)=𝛼1𝑁𝑙=1𝜁𝑙𝐸𝑉(𝑙)+𝐸𝑁𝑙=1𝜁𝑙𝑁𝑗=1𝜋𝑙𝑗𝑉+(𝑗)𝑁𝑙=1𝜁𝑙𝐸𝑉𝜉||𝜉||,𝑙+𝑑𝑧||||𝛼𝐸𝑉(𝑟(𝑡))+𝜒𝜉(𝑡)+𝑑𝑧,(4.2) where positive scalar 𝛼 is given as 𝛼𝛼1max𝑙,𝑗𝑆𝜁𝑙𝜁𝑗max𝑗𝑆𝑁𝑙=1𝜋𝑙𝑗𝜒||||𝜉(𝑡)𝑁𝑙=1𝜁𝑙𝐸𝑉𝜉||𝜉||.,𝑙(4.3) It is easily seen that 𝜒(|𝜉(𝑡)|) is a 𝒦 function with 𝑟(𝑡) given, and appropriate control parameter 𝑐𝑖, 𝑙𝛾, 𝑚𝜎𝑖 can be chosen to satisfy 𝛼>0.
For each integer 1, define a stopping time as 𝜏||𝑧||=inf𝑡0(𝑡)(4.4) Obviously, 𝜏 almost surely as . Noticing that 0<|𝑧(𝑡)| if 0𝑡<𝜏, we can apply the generalized 𝐼𝑡̂𝑜 formula to derive that for any 𝑡0, 𝐸𝑒𝛼(𝑡𝜏)𝑉𝑟𝑡𝜏𝑧=𝑉0+,0,𝑟(0)𝑡𝜏0𝑒𝛼𝑠[]𝛼𝐸𝑉(𝑟(𝑠))+𝐸𝑉(𝑟(𝑠))𝑑𝑠𝑉(𝑟(0))+𝑡𝜏0𝑒𝛼𝑠||||𝛼𝐸𝑉(𝑟(𝑠))𝛼𝐸𝑉(𝑟(𝑠))+𝜒𝜉(𝑡)+𝑑𝑧𝑑𝑠=𝑉(𝑟(0))+𝑡𝜏0𝑒𝛼𝑠𝜒||||𝜉(𝑡)+𝑑𝑧𝑑𝑠.(4.5) Let , apply Fatou’s lemma to (4.5), and we have 𝐸𝑒𝛼𝑡𝑉(𝑟(𝑡))𝑉(𝑟(0))+𝐸𝑡0𝑒𝛼𝑠𝜒||||𝜉(𝑡)+𝑑𝑧𝑑𝑠.(4.6) By using mean value theorem for integration, there is 𝐸𝑒𝛼𝑡𝑉(𝑥(𝑡),𝑡,𝑘)=𝑒𝛼𝑡𝐸𝑉(𝑟(𝑡))𝑉(𝑟(0))+sup0𝑠𝑡𝜒||𝜉||(𝑠)+𝑑𝑧𝑡0𝑒𝛼𝑠𝑑𝑠.(4.7) According to the property of 𝒦 function, the following inequality is deduced: 𝑒𝛼𝑡𝜒𝐸𝑉(𝑟(𝑡))𝑉(𝑟(0))+sup0𝑠𝑡||||𝜉(𝑠)+𝑑𝑧𝑡0𝑒𝛼𝑠𝑑𝑠=𝑉(𝑟(0))+𝜒(𝜉(𝑡))+𝑑𝑧1𝛼𝑒𝛼𝑡1𝑉(𝑟(0))+𝜒(𝜉(𝑡))+𝑑𝑧1𝛼𝑒𝛼𝑡.(4.8) According to (3.11), one gives 𝑒𝑁𝛼𝑡𝑗=114𝐸𝑧4𝑖𝑒𝛼𝑡𝐸𝑉(𝑟(𝑡))𝐸𝑉(𝑟(0))+𝜒(𝜉(𝑡))+𝑑𝑧1𝛼𝑒𝛼𝑡.(4.9) Consequently, 𝑁𝑖=1𝐸𝑧4𝑖4𝑒𝛼𝑡4𝑉(𝑟(0))+𝛼𝜒(𝜉(𝑡))+𝑑𝑧.(4.10) Defining 𝒦 function 𝛽(,), 𝒦 function 𝛾(), and nonnegative number 𝑑𝑐 as: 𝛽||𝑧0||,𝑡=4𝑒𝜆𝑡4𝑉(𝑟(0)),𝛾(𝜉(𝑡))=𝛼𝜒(𝜉(𝑡)),𝑑𝑐=4𝛼𝑑𝑧.(4.11) and applying Chebyshev’s inequality, we have that the 𝑋-subsystem of MJNS (3.1) is JISpS in probability.
The proof is completed.

Theorem 4.2. Considering the MJNS (3.1) with Assumptions (A1), (A2) holding, the interconnected Markovian jump system is JISpS in probability with adaptive laws (3.14) and switching control law (3.16) adopted; meanwhile all solutions of closed-loop system are ultimately bounded. Furthermore, the system output could be regulated to an arbitrarily small neighborhood of the equilibrium point in probability within finite time.

Proof. From Assumption (A1), the 𝜉 subsystem is JISpS in probability. And it has been shown in Theorem 4.1 that the 𝑋 subsystem is JISpS in probability. Similar to the proof in [12], we have that the entire MJNS (3.1) is JISpS in probability; that is, for any given 𝜖>0, there exists 𝑇>0 and 𝛿>0 such that if 𝑡>𝑇, the output of jump system 𝑦 satisfies 𝑃||𝑦||(𝑡)<𝛿1𝜖.(4.12) Meanwhile 𝛿 can be made as small as possible by appropriate control parameters chosen.

5. Simulation

With loss of generality, in this section we consider a two-order Markovian jump nonlinear system with regime transition space 𝑆={1,2}, and the system with unmodeled dynamics and noise is as follows: 𝑥𝑑𝜉=𝑞1,𝜉,𝑡,𝑟(𝑡)𝑑𝑡,𝑑𝑥1=𝑥2𝑑𝑡+𝑓1𝑥1𝜃,𝑡,𝑟(𝑡)𝑑𝑡+Δ1(𝑋,𝜉,𝑡,𝑟(𝑡))𝑑𝑡+𝑥11/3𝑑𝜔,𝑑𝑥2=𝑢𝑑𝑡+𝑓2(𝑋,𝑡,𝑟(𝑡))𝜃𝑑𝑡+Δ2(𝑋,𝜉,𝑡,𝑟(𝑡))𝑑𝑡,𝑦=𝑥1,(5.1) where the transition rate matrix is Π=𝜋11𝜋12𝜋21𝜋22=2233 with stationary distribution 𝜁1=𝜁2=0.5.

Here let noise covariance be 𝐸{𝑑𝜔𝑑𝜔𝑇}=1 and system dynamics for each mode as 𝑞𝑥1,𝜉,𝑡,1=0.5𝜉+0.3𝑥1𝑥,𝑞1,𝜉,𝑡,2=0.4𝜉+0.3𝑥1𝑓cos𝑡,1𝑥1,𝑡,1=𝑥21,𝑓1𝑥1,𝑡,2=𝑥1cos𝑥1,Δ1(𝑋,𝜉,𝑡,1)=0.5𝜉+0.4𝑥1sin2𝑡,Δ1(𝑋,𝜉,𝑡,2)=𝑥1𝑓𝜉,2(𝑋,𝑡,1)=𝑥1sin𝑥2+𝑥2,𝑓2(𝑋,𝑡,2)=𝑥1+2𝑥2,Δ2(𝑋,𝜉,𝑡,1)=0.4𝜉sin𝑡+0.3𝑥1.Δ2(𝑋,𝜉,𝑡,2)=𝑥1||𝜉||1/2.(5.2) From Assumption (A2), we have Δ1(𝑋,𝜉,𝑡,1)𝑝1||𝜉||+𝑝1||𝑥1||,Δ1(𝑋,𝜉,𝑡,2)𝑝1||𝜉||2+𝑝1||𝑥1||2,Δ2(𝑋,𝜉,𝑡,1)𝑝2||𝜉||+𝑝2||𝑥1||,Δ2(𝑋,𝜉,𝑡,2)𝑝2||𝜉||+𝑝2||𝑥1||2,(5.3) where 𝑝10.5 and 𝑝20.5 and the 𝜉 subsystem satisfies 𝑉04(𝜉,𝑡,𝑘)||𝜉||102+𝜒0||𝑥1||+𝑑0,(5.4) where 𝑉0=(1/2)𝜉2, 𝜒0(|𝑥1|)=0.15|𝑥1|2, 𝑑0=0.125, and it can be checked which satisfies the stochastic small-gain theorem. Thus the control law is taken as follows (here 𝛿1=1 ).

Case 1. The system regime is 𝑘=1: 𝛼1𝑐(1)=1+34𝑥1𝑥21𝜃𝜇1𝑥17/3𝑝1𝑥1𝑥tanh41𝜖1𝑥31𝑥tanh61𝑣𝑖𝛼2𝑐(1)=2+14𝑧2(1)𝑥1sin𝑥2𝑢1𝑧32(1)𝑥2114𝑧32𝑐(1)1+34+2𝑥1+34𝑥21𝑥41×𝑥21+𝑥2+𝜋11𝛼1(1)+𝜋12𝛼1(2)𝜇2𝑧2𝛼(1)1(1)𝛼1(2)4𝜏2(1)𝜃𝜏1̇𝜃(1)𝑝1𝑥tanh41𝜖13𝑥21𝑥tanh61𝑣𝑖4𝑝1𝑥41sech2𝑥41𝜖1𝑥81sech2𝑥61𝑣𝑖𝑝2𝜛1𝑧(2),2(1)=𝑥2𝛼1(̇1),𝜃=𝛾2𝑖=1𝑧3𝑖𝜏𝑖(1)𝑎𝜃𝜃0,̇𝑝1=𝜎1𝑥31𝜛𝑖(1)𝑚1𝑝1𝑝01,̇𝑝2=𝜎2𝑧32(1)𝜛2(1)𝑚2𝑝2𝑝02.(5.5)

Case 2. The system regime is 𝑘=2: 𝛼1𝑐(2)=1+34𝑥1𝑥1sin𝑥1𝑥2𝑝1𝑥1𝑥tanh41𝜖1𝑥31𝑥tanh61𝑣𝑖,𝛼2𝑐(2)=2+14𝑧2(2)𝜇1𝑧32(2)𝑥4114𝑧32𝑐(2)1+114+34𝑥21+𝑥61𝑥1+𝑥2+𝜋21𝛼1(1)+𝜋22𝛼1(2)𝜇2𝑧2𝛼(2)1(1)𝛼1(2)4𝜏2(2)𝜃𝜏1̇𝜃(2)𝑝1𝑥tanh41𝜖13𝑥21𝑥tanh61𝑣𝑖4𝑝1𝑥41sech2𝑥41𝜖1𝑥81sech2𝑥61𝑣𝑖𝑝2𝜛2𝑧(2),2(2)=𝑥2𝛼1(̇2),𝜃=𝛾2𝑖=1𝑧3𝑖𝜏𝑖(2)𝑎𝜃𝜃0,̇𝑝1=𝜎1𝑥31𝜛𝑖(2)𝑚1𝑝1𝑝01,̇𝑝2=𝜎2𝑧32(2)𝜛2(2)𝑚2𝑝2𝑝02.(5.6) In computation, we set the initial value to be 𝑥1=1.6, 𝑥2=2.7, 𝜃=0, 𝑝1=𝑝2=0 let parameter 𝜃0=1, 𝛾=1, 𝑎=1, 𝑝0=0.7, 𝜖𝑖=𝑣𝑖=0.5, 𝑚𝑖=1, 𝜇1=𝜇2=1 and the time step to be 0.05 s. For comparison, two groups of different control parameters are given. First we take the parameter with values 𝑐1=𝑐2=0.7, 𝜎1=𝜎2=2, and the simulation results are as follows. Figure 2 shows the regime transition of the jump system, Figure 3 shows the system output 𝑦 which is defined as the system state 𝑥1, and Figure 4 shows system state 𝑥2. Figure 5 shows the corresponding switching controller 𝑢; finally Figure 6 shows the trajectory of adaptive parameter 𝜃 and Figure 7; Figure 8 shows the trajectory of parameter 𝑝1, 𝑝2, respectively.

Figure 2: Regime transition 𝑟(𝑡).
Figure 3: System output 𝑦.
Figure 4: System state 𝑥2.
Figure 5: Switching controller 𝑢.
Figure 6: Adaptive parameter 𝜃.
Figure 7: Parameter 𝑝1.
Figure 8: Parameter 𝑝2.

Now we choose different control parameters as 𝑐1=𝑐2=2, 𝜎1=𝜎2=5 and repeat the simulation. The simulation results are as follows. Figure 9 shows the regime transition of the jump system, Figure 10 shows the system output 𝑦 which is defined as the system state 𝑥1 and, Figure 11 shows system state 𝑥2, and Figure 12 shows the corresponding switching controller 𝑢; the trajectory of adaptive parameter 𝜃 is shown in Figures 13 and 14; Figure 15 shows the trajectory of parameter 𝑝1, 𝑝2, respectively.

Figure 9: Regime transition 𝑟(𝑡).
Figure 10: System output 𝑦.
Figure 11: System state 𝑥2.
Figure 12: Switching controller 𝑢.
Figure 13: Adaptive parameter 𝜃.
Figure 14: Parameter 𝑝1.
Figure 15: Parameter 𝑝2.

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 𝑐1, 𝑐2, 𝜎1, 𝜎2 help to increase the convergence speed of system states. This reason is that the increase of these parameters increases the value of 𝛼, which determines the system states convergence speed. Also adaptive parameters 𝜃 and 𝑝1, 𝑝2 approach convergence faster with the increasing of aforementioned parameters.

Remark 5.1. Much research work has been performed towards the study of nonlinear system by using small-gain theorem [16, 19]. In contrast to their contributions, this paper considers a more general form than nonjump systems. The controller 𝑢(𝑘) varies with different regime 𝑟(𝑡)=𝑘 taken, and it differs in two aspects (see (3.16)): the coupling of regimes 𝜋𝑘𝑗𝛼𝑖1(𝑗) and 𝜇2𝑧𝑖[Γ𝑖(𝑘)Γ𝑇𝑖(𝑘)]2, which are both caused by the Markovian jumps. The switching controller will degenerate to an ordinary one if 𝑟(𝑡)1. This controller design method can also be applied for the nonjump nonlinear system.

6. Conclusion

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.


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