Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
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Mathematical Approaches in Advanced Control Theories

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Research Article | Open Access

Volume 2012 |Article ID 514504 | 22 pages | https://doi.org/10.1155/2012/514504

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

Academic Editor: Xianxia Zhang
Received28 Mar 2012
Accepted09 May 2012
Published18 Jul 2012

Abstract

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 πΌπ‘‘Μ‚π‘œ 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 (Ξ©,β„±,{ℱ𝑑}𝑑β‰₯0,𝑃) a complete probability space with a filtration {ℱ𝑑}𝑑β‰₯0 satisfying the usual conditions (i.e., it is right continuous and β„±0 contains all 𝑝-null sets). Let |π‘₯| stand for the usual Euclidean norm for a vector π‘₯, and let β€–π‘₯𝑑‖ stand for the supremum of vector π‘₯ over time period [𝑑0,𝑑], that is, β€–π‘₯𝑑‖=sup𝑑0≀𝑠≀𝑑|π‘₯(𝑠)|. The superscript 𝑇 will denote transpose and we refer to Tr(β‹…) 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: 𝑑π‘₯=𝑓(π‘₯,𝑒,𝑑,π‘Ÿ(𝑑))𝑑𝑑+𝑔(π‘₯,𝑒,𝑑,π‘Ÿ(𝑑))π‘‘πœ”(𝑑),(2.1) where π‘₯βˆˆβ„π‘›, π‘’βˆˆβ„π‘š are state vector and input vector of the system, respectively. π‘Ÿ(𝑑), 𝑑β‰₯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 πœ”(𝑑)={πœ”1,πœ”2,…,πœ”π‘™} is 𝑙-dimensional independent Wiener process defined on the probability space, with covariance matrix 𝐸{π‘‘πœ”π‘‘πœ”π‘‡}=Ξ₯(𝑑)Ξ₯𝑇(𝑑)𝑑𝑑, where Ξ₯(𝑑) is an unknown bounded matrix-value function. Furthermore, we assume that the Wiener noise πœ”(𝑑) is independent of the Markov chain π‘Ÿ(𝑑). The functions π‘“βˆΆβ„π‘›+π‘šΓ—β„+×𝑆→ℝ𝑛 and π‘”βˆΆβ„π‘›+π‘šΓ—β„+×𝑆→ℝ𝑛×𝑙 are locally Lipschitz in (π‘₯,𝑒,π‘Ÿ(𝑑)=π‘˜)βˆˆβ„π‘›+π‘šΓ—π‘† for all 𝑑β‰₯0; namely, for any β„Ž>0, there is a constant πΎβ„Žβ‰₯0 such that ||𝑓π‘₯1,𝑒1ξ€Έξ€·π‘₯,𝑑,π‘˜βˆ’π‘“2,𝑒2ξ€Έ||∨||𝑔π‘₯,𝑑,π‘˜1,𝑒1ξ€Έξ€·π‘₯,𝑑,π‘˜βˆ’π‘”2,𝑒2ξ€Έ||,𝑑,π‘˜β‰€πΎβ„Žξ€·||π‘₯1βˆ’π‘₯2||+||𝑒1βˆ’π‘’2||ξ€Έβˆ€ξ€·π‘₯(2.2)1,𝑒1ξ€Έ,ξ€·π‘₯,𝑑,π‘˜2,𝑒2ξ€Έ,𝑑,π‘˜βˆˆβ„π‘›+π‘šΓ—β„+||π‘₯×𝑆,1||∨||π‘₯2||∨||𝑒1||∨||𝑒2||β‰€β„Ž.(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 π‘Ÿ(𝑑) with regime transition rate matrix Ξ =[πœ‹π‘˜π‘—]𝑁×𝑁, the entries πœ‹π‘˜π‘—,π‘˜,𝑗=1,2,…,𝑁 are interpreted as transition rates such that ξ‚»πœ‹π‘ƒ(π‘Ÿ(𝑑+𝑑𝑑)=π‘—βˆ£π‘Ÿ(𝑑)=π‘˜)=π‘˜π‘—π‘‘π‘‘+π‘œ(𝑑𝑑)ifπ‘˜β‰ π‘—,1+πœ‹π‘˜π‘—π‘‘π‘‘+π‘œ(𝑑𝑑)ifπ‘˜=𝑗,(2.4) where 𝑑𝑑>0 and π‘œ(𝑑𝑑) satisfies lim𝑑𝑑→0(π‘œ(𝑑𝑑)/𝑑𝑑)=0. Here πœ‹π‘˜π‘—>0(π‘˜β‰ π‘—) is the transition rate from regime π‘˜ to regime 𝑗. Notice that the total probability axiom imposes πœ‹π‘˜π‘˜ negative and 𝑁𝑗=1πœ‹π‘˜π‘—=0,βˆ€π‘˜βˆˆπ‘†.(2.5) For each regime transition rate matrix Ξ , there exists a unique stationary distribution 𝜁=(𝜁1,𝜁2,…,πœπ‘) such that [14] Ξ β‹…πœ=0,π‘ξ“π‘˜=1πœπ‘˜=1,πœπ‘˜>0,βˆ€π‘˜βˆˆπ‘†.(2.6) Let 𝐢2,1(ℝ𝑛×ℝ+×𝑆) denote the family of all functions 𝐹(π‘₯,𝑑,π‘˜) on ℝ𝑛×ℝ+×𝑆 which are continuously twice differentiable in π‘₯ and once in 𝑑. Furthermore, we give the stochastic differentiable equation of 𝐹(π‘₯,𝑑,π‘˜) as 𝑑𝐹(π‘₯,𝑑,π‘˜)=πœ•πΉ(π‘₯,𝑑,π‘˜)πœ•π‘‘π‘‘π‘‘+πœ•πΉ(π‘₯,𝑑,π‘˜)+1πœ•π‘₯𝑓(π‘₯,𝑒,𝑑,π‘˜)𝑑𝑑2ξ‚ΈΞ₯Trπ‘‡π‘”π‘‡πœ•(π‘₯,𝑒,𝑑,π‘˜)2𝐹(π‘₯,𝑑,π‘˜)πœ•π‘₯2ξ‚Ή+𝑔(π‘₯,𝑒,𝑑,π‘˜)Ξ₯𝑑𝑑𝑁𝑗=1πœ‹π‘˜π‘—πΉ(π‘₯,𝑑,𝑗)𝑑𝑑+πœ•πΉ(π‘₯,𝑑,π‘˜)+πœ•π‘₯𝑔(π‘₯,𝑒,𝑑,π‘˜)π‘‘πœ”(𝑑)𝑁𝑗=1[𝐹](π‘₯,𝑑,𝑗)βˆ’πΉ(π‘₯,𝑑,π‘˜)𝑑𝑀𝑗(𝑑),(2.7) where 𝑀(𝑑)=(𝑀1(𝑑),𝑀2(𝑑),…,𝑀𝑁(𝑑)) is a martingale process.

Take the expectation in (2.7), so that the the infinitesimal generator produces [2, 15] ℒ𝐹(π‘₯,𝑑,π‘˜)=πœ•πΉ(π‘₯,𝑑,π‘˜)+πœ•π‘‘πœ•πΉ(π‘₯,𝑑,π‘˜)πœ•π‘₯𝑓(π‘₯,𝑒,𝑑,π‘˜)+𝑁𝑗=1πœ‹π‘˜π‘—+1𝐹(π‘₯,𝑑,𝑗)2ξ‚ΈΞ₯Trπ‘‡π‘”π‘‡πœ•(π‘₯,𝑒,𝑑,π‘˜)2𝐹(π‘₯,𝑑,π‘˜)πœ•π‘₯2ξ‚Ή.𝑔(π‘₯,𝑒,𝑑,π‘˜)Ξ₯(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œ‹π‘˜π‘— and martingale process 𝑀(𝑑), which are both caused by the Markov chain π‘Ÿ(𝑑). 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 πœ–>0, there exist 𝒦ℒ function 𝛽(β‹…,β‹…), π’¦βˆž function 𝛾(β‹…), and a constant 𝑑𝑐β‰₯0 such that 𝑃||π‘₯||ξ€·||π‘₯(𝑑,π‘˜)<𝛽0||‖‖𝑒,𝑑+𝛾𝑑‖‖(π‘˜)+𝑑𝑐β‰₯1βˆ’πœ–βˆ€π‘‘β‰₯0,π‘˜βˆˆπ‘†,π‘₯0βˆˆβ„π‘›β§΅{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 π‘₯(𝑑,π‘˜) and control signal 𝑒𝑑(π‘˜). For nonjump system, system state and control signal contain only continuous time 𝑑 with π‘˜β‰‘1. While jump systems concern with both continuous time 𝑑 and discrete regime π‘˜. For different regime π‘˜, control signal 𝑒𝑑(π‘˜) will differ with different sample taken even at the same time 𝑑, 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 π‘˜β‰‘1, and the definition of JISpS will degenerate to ISpS.

Consider the jump interconnected dynamic system described in Figure 1: 𝑑π‘₯1=𝑓1ξ€·π‘₯1,π‘₯2,Ξ1ξ€Έ(π‘Ÿ(𝑑)),π‘Ÿ(𝑑)𝑑𝑑+𝑔1ξ€·π‘₯1,π‘₯2,Ξ1ξ€Έ(π‘Ÿ(𝑑)),π‘Ÿ(𝑑)π‘‘π‘Šπ‘‘1,𝑑π‘₯2=𝑓2ξ€·π‘₯1,π‘₯2,Ξ2(ξ€Έπ‘Ÿ(𝑑)),π‘Ÿ(𝑑)𝑑𝑑+𝑔2ξ€·π‘₯1,π‘₯2,Ξ2(ξ€Έπ‘Ÿ(𝑑)),π‘Ÿ(𝑑)π‘‘π‘Šπ‘‘2,(2.10) where π‘₯=(π‘₯𝑇1,π‘₯𝑇2)π‘‡βˆˆβ„π‘›1+𝑛2 is the state of system, Ξžπ‘–(π‘Ÿ(𝑑)),𝑖=1,2 denotes exterior disturbance and/or interior uncertainty. π‘Šπ‘‘π‘– 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 π‘₯1-system and π‘₯2-system are JISpS in probability with (Ξ1(π‘˜),π‘₯2(𝑑,π‘˜)) as input and π‘₯1(𝑑,π‘˜) as state and (Ξ2(π‘˜),π‘₯1(𝑑,π‘˜)) as input and π‘₯2(𝑑,π‘˜) as state, respectively; that is, for any given πœ–1,πœ–2>0, 𝑃||π‘₯1||(𝑑,π‘˜)<𝛽1ξ€·||π‘₯1||ξ€Έ(0,π‘˜),𝑑+𝛾1ξ€·β€–β€–π‘₯2β€–β€–ξ€Έ(𝑑,π‘˜)+𝛾𝑀1ξ€·β€–β€–Ξž1𝑑‖‖(π‘˜)+𝑑1ξ€Ύβ‰₯1βˆ’πœ–1,𝑃||π‘₯2(||𝑑,π‘˜)<𝛽2ξ€·||π‘₯2(||ξ€Έ0,π‘˜),𝑑+𝛾2ξ€·β€–β€–π‘₯1(‖‖𝑑,π‘˜)+𝛾𝑀2ξ€·β€–β€–Ξž2𝑑(β€–β€–ξ€Έπ‘˜)+𝑑2ξ€Ύβ‰₯1βˆ’πœ–2,(2.11) hold with 𝛽𝑖(β‹…,β‹…) being 𝒦ℒ function, 𝛾𝑖 and 𝛾𝑀𝑖 being π’¦βˆž 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 ξ€·1+𝜌1𝛾1βˆ˜ξ€·1+𝜌2𝛾2(𝑠)≀𝑠,βˆ€π‘ β‰₯𝑠0,(2.12) the interconnected system is JISpS in probability with Ξ(π‘˜)=(Ξ1(π‘˜),Ξ2(π‘˜)) as input and π‘₯=(π‘₯1,π‘₯2) as state; that is, for any given πœ–>0, there exist a 𝒦ℒ function 𝛽𝑐(β‹…,β‹…), a π’¦βˆž function 𝛾𝑀(β‹…), and a parameter 𝑑𝑐β‰₯0 such that 𝑃||π‘₯||(𝑑,π‘˜)<𝛽𝑐||π‘₯0||ξ€Έ,𝑑+π›Ύπ‘€ξ€·β€–β€–Ξžπ‘‘β€–β€–ξ€Έ(π‘˜)+𝑑𝑐β‰₯1βˆ’πœ–.(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 π‘‘πœ‰=π‘ž(𝑦,πœ‰,𝑑,π‘Ÿ(𝑑))𝑑𝑑,𝑑π‘₯𝑖=π‘₯𝑖+1𝑑𝑑+π‘“π‘‡π‘–ξ€·π‘‹π‘–ξ€Έπœƒ,𝑑,π‘Ÿ(𝑑)βˆ—π‘‘π‘‘+Δ𝑖(𝑋,πœ‰,𝑑,π‘Ÿ(𝑑))𝑑𝑑+𝑔𝑇𝑖𝑋𝑖,𝑑,π‘Ÿ(𝑑)π‘‘πœ”,𝑑π‘₯𝑛=𝑒𝑑𝑑+𝑓𝑇𝑛(𝑋,𝑑,π‘Ÿ(𝑑))πœƒβˆ—π‘‘π‘‘+Δ𝑛(𝑋,πœ‰,𝑑,π‘Ÿ(𝑑))𝑑𝑑+𝑔𝑇𝑛(𝑋,𝑑,π‘Ÿ(𝑑))π‘‘πœ”π‘–=1,2,…,π‘›βˆ’1,𝑦=π‘₯1,(3.1) where 𝑋𝑖=(π‘₯1,π‘₯2,…,π‘₯𝑖)π‘‡βˆˆβ„π‘–(π‘‹βˆˆβ„π‘›) is state vector, π‘’βˆˆβ„ is system input signal, πœ‰βˆˆβ„π‘›0 is unmeasured state vector, and 𝑦 is output signal. πœƒβˆ—βˆˆβ„π‘0 is a vector of unknown adaptive parameters. The Markov chain π‘Ÿ(𝑑)βˆˆπ‘† and Wiener noise πœ” are as defined in Section 2. π‘“π‘–βˆΆβ„π‘–Γ—β„+×𝑆→ℝ𝑝0, π‘”π‘–βˆΆβ„π‘–Γ—β„+×𝑆→ℝ𝑙 are vector-valued smooth functions, and Δ𝑖(𝑋,πœ‰,𝑑,π‘Ÿ(𝑑)) denotes the unmodeled dynamic uncertainty which could vary with different regime π‘Ÿ(𝑑) taken. Both 𝑓𝑖, 𝑔𝑖 and Δ𝑖 are locally Lipschitz as in Section 2.

Our design purpose is to find a switching controller 𝑒 of the form 𝑒(π‘₯,𝑑,π‘˜),π‘˜βˆˆπ‘† 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 πœ‰ subsystem with input 𝑦 is JISpS in probability; namely, for any given πœ–>0, there exist 𝒦ℒ function 𝛽(β‹…,β‹…), π’¦βˆž function 𝛾(β‹…), and a constant 𝑑𝑐β‰₯0 such that 𝑃||πœ‰||ξ€·||πœ‰(𝑑,π‘˜)<𝛽0||ξ€Έ(,𝑑+𝛾‖𝑦‖)+𝑑𝑐β‰₯1βˆ’πœ–βˆ€π‘‘β‰₯0,π‘˜βˆˆπ‘†,πœ‰0βˆˆβ„π‘›0⧡{0}.(3.2) (A2) For each 𝑖=1,2,…,𝑛, π‘˜βˆˆπ‘†, there exists an unknown bounded positive constant π‘βˆ—π‘– such that ||Δ𝑖||(𝑋,πœ‰,𝑑,π‘˜)β‰€π‘βˆ—π‘–πœ™π‘–1𝑋𝑖,π‘˜+π‘βˆ—π‘–πœ™π‘–2ξ€·||πœ‰||ξ€Έ,,π‘˜(3.3) where πœ™π‘–1(β‹…,π‘˜), πœ™π‘–2(β‹…,π‘˜) are known nonnegative smooth functions for any given π‘˜βˆˆπ‘†. Notice that π‘βˆ—π‘– is not unique since any π‘βˆ—π‘–>π‘βˆ—π‘– satisfies inequality (3.3). To avoid confusion, we define π‘βˆ—π‘– 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 π‘₯,π‘¦βˆˆβ„π‘›, the following inequality holds π‘₯π‘‡πœ–π‘¦β‰€π‘π‘|π‘₯|𝑝+1π‘žπœ–π‘ž||𝑦||π‘ž,(3.4) where πœ–>0 and the constants 𝑝>1, π‘ž>1 satisfy (π‘βˆ’1)(π‘žβˆ’1)=1.

Lemma 3.2 (martingale representation [17]). Let 𝐡(𝑑)=[𝐡1(𝑑),𝐡2(𝑑),…,𝐡𝑁(𝑑)] be N-dimensional standard Wiener noise. Supposing 𝑀(𝑑) is an ℱ𝑁𝑑-martingale (with respect to P) and that 𝑀(𝑑)∈𝐿2(𝑃) for all 𝑑β‰₯0, then there exists a stochastic process Ξ¨(𝑑)∈𝐿2(𝑃), such that 𝑑𝑀(𝑑)=Ξ¨(𝑑)⋅𝑑𝐡(𝑑).(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 πœƒβˆ—, π‘βˆ—π‘– needs to be estimated. Denote the estimation of adaptive parameter πœƒβˆ— with πœƒ and the estimation of upper bound of uncertainty π‘βˆ—π‘– with 𝑝𝑖. Perform a new transformation as 𝑧𝑖=π‘₯𝑖(π‘˜)βˆ’π›Όπ‘–βˆ’1ξ€·π‘‹π‘–βˆ’1,𝑑,πœƒ,𝑝𝑖,π‘˜βˆ€π‘–=1,2,…,𝑛,π‘˜βˆˆπ‘†.(3.6) For simplicity, we just denote π›Όπ‘–βˆ’1(π‘‹π‘–βˆ’1,𝑑,πœƒ,𝑝𝑖,π‘˜), 𝑓𝑖(𝑋𝑖,𝑑,π‘˜), 𝑔𝑖(𝑋𝑖,𝑑,π‘˜), Δ𝑖(𝑋,πœ‰,𝑑,π‘˜), π‘ž(𝑦,πœ‰,𝑑,π‘˜) by π›Όπ‘–βˆ’1(π‘˜), 𝑓𝑖(π‘˜), 𝑔𝑖(π‘˜), Δ𝑖(π‘˜), π‘ž(π‘˜), respectively, where 𝛼0(π‘˜)=0, 𝛼𝑛(π‘˜)=𝑒(π‘˜), for all π‘˜βˆˆπ‘†, and the new coordinate is 𝑍(π‘˜)=(𝑧1(π‘˜),𝑧2(π‘˜),…,𝑧𝑛(π‘˜)).

According to stochastic differential equation (2.7), one has 𝑑𝑧𝑖=𝑑π‘₯π‘–βˆ’π‘‘π›Όπ‘–βˆ’1=ξ€Ίπ‘₯(π‘˜)𝑖+1+𝑓𝑇𝑖(π‘˜)πœƒβˆ—+Δ𝑖(π‘˜)π‘‘π‘‘βˆ’πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘‘π‘‘π‘‘βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗π‘₯𝑗+1+𝑓𝑇𝑗(π‘˜)πœƒβˆ—+Ξ”π‘—ξ€»βˆ’(π‘˜)π‘‘π‘‘πœ•π›Όπ‘–βˆ’1(π‘˜)Μ‡πœ•πœƒπœƒπ‘‘π‘‘βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘π‘–Μ‡π‘π‘–1π‘‘π‘‘βˆ’2π‘–βˆ’1𝑝,π‘ž=1πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žπ‘”π‘‡π‘(π‘˜)Ξ₯Ξ₯π‘‡π‘”π‘ž(π‘˜)π‘‘π‘‘βˆ’π‘ξ“π‘—=1πœ‹π‘˜π‘—π›Όπ‘–βˆ’1+𝑔(𝑗)𝑑𝑑𝑇𝑖(π‘˜)βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗𝑔𝑇𝑗(π‘˜)π‘‘πœ”+𝑁𝑗=1ξ€Ίπ›Όπ‘–βˆ’1(π‘˜)βˆ’π›Όπ‘–βˆ’1ξ€»(𝑗)𝑑𝑀𝑗=𝑧(𝑑)𝑖+1+𝛼𝑖(π‘˜)+πœπ‘‡π‘–(π‘˜)πœƒβˆ—+Λ𝑖(π‘˜)π‘‘π‘‘βˆ’πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘‘π‘‘π‘‘βˆ’πœ•π›Όπ‘–βˆ’1(π‘˜)Μ‡πœ•πœƒπœƒπ‘‘π‘‘βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘π‘–Μ‡π‘π‘–βˆ’π‘‘π‘‘π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗π‘₯𝑗+11π‘‘π‘‘βˆ’2π‘–βˆ’1𝑝,π‘ž=1πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žπ‘”π‘‡π‘(π‘˜)Ξ₯Ξ₯π‘‡π‘”π‘ž(π‘˜)π‘‘π‘‘βˆ’π‘ξ“π‘—=1πœ‹π‘˜π‘—π›Όπ‘–βˆ’1(𝑗)𝑑𝑑+πœŒπ‘‡π‘–(π‘˜)π‘‘πœ”+Γ𝑖(π‘˜)𝑑𝑀(𝑑).(3.7) Here we define Λ𝑖(π‘˜)β‰œΞ”π‘–(π‘˜)βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘—Ξ”π‘—πœ(π‘˜),𝑖(π‘˜)β‰œπ‘“π‘–(π‘˜)βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘—π‘“π‘—πœŒ(π‘˜),𝑖(π‘˜)β‰œπ‘”π‘–(π‘˜)βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗𝑔𝑗Γ(π‘˜),𝑖𝛼(π‘˜)β‰œπ‘–βˆ’1(π‘˜)βˆ’π›Όπ‘–βˆ’1(1),π›Όπ‘–βˆ’1(π‘˜)βˆ’π›Όπ‘–βˆ’1(2),…,π›Όπ‘–βˆ’1(π‘˜)βˆ’π›Όπ‘–βˆ’1ξ€».(𝑁)(3.8) From assumption (A2), one gets that there exists nonnegative smooth function πœ™π‘–1, πœ™π‘–2 satisfying ||Λ𝑖||(π‘˜)β‰€π‘βˆ—π‘–πœ™π‘–1𝑋𝑖,π‘˜+π‘βˆ—π‘–πœ™π‘–2ξ€·||πœ‰||ξ€Έ,π‘˜.(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(𝑑), according to Lemma 3.2, there exist a function Ξ¨(𝑑)∈𝐿2(𝑃) and an 𝑁-dimensional standard Wiener noise 𝐡(𝑑) satisfying 𝑑𝑀(𝑑)=Ξ¨(𝑑)𝑑𝐡(𝑑), where 𝐸[Ξ¨(𝑑)Ξ¨(𝑑)𝑇]=πœ“(𝑑)πœ“(𝑑)𝑇≀𝑄<∞ and 𝑄 is a positive bounded constant. Therefore we have 𝑑𝑧𝑖=𝑧𝑖+1+𝛼𝑖(π‘˜)+πœπ‘‡π‘–(π‘˜)πœƒβˆ—+Λ𝑖(π‘˜)βˆ’πœ•π›Όπ‘–βˆ’1(π‘˜)βˆ’πœ•π‘‘πœ•π›Όπ‘–βˆ’1(π‘˜)Μ‡πœ•πœƒπœƒβˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘π‘–Μ‡π‘π‘–βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗π‘₯𝑗+1βˆ’12π‘–βˆ’1𝑝,π‘ž=1πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žπ‘”π‘‡π‘(π‘˜)Ξ₯Ξ₯π‘‡π‘”π‘ž(π‘˜)βˆ’π‘ξ“π‘—=1πœ‹π‘˜π‘—π›Όπ‘–βˆ’1ξƒ°(𝑗)𝑑𝑑+πœŒπ‘‡π‘–(π‘˜)π‘‘πœ”+Γ𝑖(π‘˜)Ξ¨(𝑑)𝑑𝐡(𝑑).(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 𝐡(𝑑).

Choose the quartic Lyapunov function as 1𝑉(π‘˜)=4𝑛𝑖=1𝑧4𝑖+1Μƒπœƒ2π›Ύπ‘‡Μƒπœƒ+𝑛𝑖=112πœŽπ‘–Μƒπ‘2𝑖,(3.11) where 𝛾>0, πœŽπ‘–>0 are constants. Μƒπœƒ=πœƒβˆ—βˆ’πœƒ and ̃𝑝𝑖=π‘π‘€π‘–βˆ’π‘π‘– are parameter estimation errors, where π‘π‘€π‘–β‰œmax{π‘βˆ—π‘–,𝑝0𝑖} and 𝑝0𝑖 are given positive constants.

In the view of (3.10) and (3.11), the infinitesimal generator of 𝑉 satisfies ℒ𝑉(π‘˜)=𝑛𝑖=1𝑧3𝑖𝑧𝑖+1+𝛼𝑖(π‘˜)+πœπ‘‡π‘–(π‘˜)πœƒβˆ—+Λ𝑖(π‘˜)βˆ’πœ•π›Όπ‘–βˆ’1(π‘˜)βˆ’πœ•π‘‘πœ•π›Όπ‘–βˆ’1(π‘˜)Μ‡πœ•πœƒπœƒβˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘π‘–Μ‡π‘π‘–βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗π‘₯𝑗+1βˆ’12π‘–βˆ’1𝑝,π‘ž=1πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žπ‘”π‘‡π‘(π‘˜)Ξ₯Ξ₯π‘‡π‘”π‘ž(π‘˜)βˆ’π‘ξ“π‘—=1πœ‹π‘˜π‘—π›Όπ‘–βˆ’1ξƒ°+3(𝑗)2𝑛𝑖=1𝑧2π‘–πœŒπ‘‡π‘–(π‘˜)Ξ₯Ξ₯π‘‡πœŒπ‘–3(π‘˜)+2𝑛𝑖=1𝑧2𝑖Γ𝑖(π‘˜)πœ“πœ“π‘‡Ξ“π‘‡π‘–1(π‘˜)βˆ’π›ΎΜƒπœƒπ‘‡Μ‡πœƒβˆ’π‘›ξ“π‘–=11πœŽπ‘–Μƒπ‘π‘–Μ‡π‘π‘–+𝑁𝑗=1πœ‹π‘˜π‘—β‰€π‘‰(𝑗)𝑛𝑖=1𝑧3𝑖34𝛿𝑖4/3+14𝛿4π‘–βˆ’1ξƒͺ𝑧𝑖+𝛼𝑖(π‘˜)+πœπ‘‡π‘–(π‘˜)πœƒβˆ’πœ•π›Όπ‘–βˆ’1(π‘˜)βˆ’πœ•π‘‘πœ•π›Όπ‘–βˆ’1(π‘˜)Μ‡πœ•πœƒπœƒβˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘π‘–Μ‡π‘π‘–βˆ’π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗π‘₯𝑗+1+πœ†π‘§3π‘–π‘–βˆ’1𝑝,π‘ž=1ξƒ¬πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žξƒ­2𝑔𝑇𝑝(π‘˜)π‘”π‘žξ€»(π‘˜)2+πœ‡1π‘§π‘–ξ€ΊπœŒπ‘‡π‘–(π‘˜)πœŒπ‘–ξ€»(π‘˜)2+πœ‡2𝑧𝑖Γ𝑖(π‘˜)Γ𝑇𝑖(π‘˜)2βˆ’π‘ξ“π‘—=1πœ‹π‘˜π‘—π›Όπ‘–βˆ’1ξƒ°+ξ‚Έ(𝑗)(π‘›βˆ’1)𝑛(2π‘›βˆ’1)+96πœ†9𝑛16πœ‡1ξ‚Ή||Ξ₯||4+9𝑛16πœ‡2𝑄2βˆ’Μƒπœƒπ‘‡ξƒ¬1π›ΎΜ‡πœƒβˆ’π‘›ξ“π‘–=1𝑧3π‘–πœπ‘–ξƒ­βˆ’(π‘˜)𝑛𝑖=1ξ‚Έ1πœŽπ‘–Μƒπ‘π‘–Μ‡π‘π‘–βˆ’π‘§3𝑖Λ𝑖+(π‘˜)𝑁𝑗=1πœ‹π‘˜π‘—π‘‰(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: 𝑛𝑖=1𝑧3𝑖𝑧𝑖+1≀34π‘›βˆ’1𝑖=1𝛿𝑖4/3𝑧4𝑖+14π‘›βˆ’1𝑖=11𝛿4𝑖𝑧4𝑖+1=𝑛𝑖=134𝛿𝑖4/3+14𝛿4π‘–βˆ’1ξƒͺ𝑧4π‘–βˆ’12𝑛𝑖=1𝑧3π‘–π‘–βˆ’1𝑝,π‘ž=1πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žπ‘”π‘‡p(π‘˜)Ξ₯Ξ₯π‘‡π‘”π‘žβ‰€(π‘˜)𝑛𝑖=1πœ†π‘§6π‘–π‘–βˆ’1𝑝,π‘ž=1ξƒ¬πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žξƒ­2𝑔𝑇𝑝(π‘˜)𝑔𝑝(π‘˜)π‘”π‘‡π‘ž(π‘˜)π‘”π‘ž(π‘˜)+𝑛𝑖=1π‘–βˆ’1𝑝,π‘ž=11||16πœ†Ξ₯Ξ₯𝑇||2=𝑛𝑖=1πœ†π‘§6π‘–π‘–βˆ’1𝑝,π‘ž=1ξƒ¬πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žξƒ­2𝑔𝑇𝑝(π‘˜)π‘”π‘žξ€»(π‘˜)2+||Ξ₯Ξ₯𝑇||2396πœ†(π‘›βˆ’1)𝑛(2π‘›βˆ’1),2𝑛𝑖=1𝑧2π‘–πœŒπ‘‡π‘–(π‘˜)Ξ₯Ξ₯π‘‡πœŒπ‘–β‰€(π‘˜)𝑛𝑖=1πœ‡1𝑧4π‘–ξ€ΊπœŒπ‘‡π‘–(π‘˜)πœŒπ‘–ξ€»(π‘˜)2+𝑛𝑖=1916πœ‡1||Ξ₯Ξ₯𝑇||2=𝑛𝑖=1πœ‡1𝑧4π‘–ξ€ΊπœŒπ‘‡π‘–(π‘˜)πœŒπ‘–ξ€»(π‘˜)2+9𝑛16πœ‡1||Ξ₯Ξ₯𝑇||2,32𝑛𝑖=1𝑧2𝑖Γ𝑖(π‘˜)πœ“πœ“π‘‡Ξ“π‘‡π‘–(≀3π‘˜)2𝑛𝑖=1𝑧2𝑖Γ𝑖(π‘˜)𝑄Γ𝑇𝑖≀(π‘˜)𝑛𝑖=1πœ‡2𝑧4𝑖Γ𝑖(π‘˜)Γ𝑇𝑖(π‘˜)2+𝑛𝑖=1916πœ‡2𝑄2=𝑛𝑖=1πœ‡2𝑧4𝑖Γ𝑖(π‘˜)Γ𝑇𝑖(π‘˜)2+9𝑛16πœ‡2𝑄2,(3.13) where 𝛿0=∞, 𝛿𝑛=0 and πœ†>0, πœ‡1>0, πœ‡2>0, 𝛿𝑖>0, 𝑖=1,2,…,𝑛 are design parameters to be chosen.

Here we suggest the following adaptive laws [18]: Μ‡ξƒ¬πœƒ=𝛾𝑛𝑖=1𝑧3π‘–πœπ‘–ξ€·(π‘˜)βˆ’π‘Žπœƒβˆ’πœƒ0ξ€Έξƒ­,̇𝑝𝑖=πœŽπ‘–ξ€Ίπ‘§3π‘–πœ›π‘–(π‘˜)βˆ’π‘šπ‘–ξ€·π‘π‘–βˆ’π‘0𝑖.ξ€Έξ€»(3.14) Here π‘Ž>0, πœƒ0βˆˆβ„π‘0, π‘šπ‘–>0, 𝑖=1,2,…,𝑛 are design parameters to be chosen. And define function 𝛽(π‘˜) as πœ›π‘–(π‘˜)=πœ™π‘–1𝑋𝑖𝑧,π‘˜β‹…tanh3π‘–πœ™π‘–1𝑋𝑖,π‘˜πœ€π‘–ξƒ­+𝑧3𝑖𝑧tanh6π‘–πœπ‘–ξƒͺ,𝛽𝑖(π‘˜)=π‘π‘–β‹…πœ›π‘–(π‘˜),(3.15) where πœ€π‘–>0, πœπ‘–>0, 𝑖=1,2,…,𝑛 are control parameters to be chosen, and let the virtual control signal be 𝛼𝑖(π‘˜)=βˆ’π‘π‘–π‘§π‘–βˆ’ξƒ©34𝛿𝑖4/3+14𝛿4π‘–βˆ’1ξƒͺπ‘§π‘–βˆ’πœπ‘‡π‘–(π‘˜)πœƒ+πœ•π›Όπ‘–βˆ’1(π‘˜)+πœ•π‘‘πœ•π›Όπ‘–βˆ’1(π‘˜)Μ‡πœ•πœƒπœƒ+π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘π‘–Μ‡π‘π‘–+π‘–βˆ’1𝑗=1πœ•π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯𝑗π‘₯𝑗+1βˆ’πœ†π‘§3π‘–π‘–βˆ’1𝑝,π‘ž=1ξƒ¬πœ•2π›Όπ‘–βˆ’1(π‘˜)πœ•π‘₯π‘πœ•π‘₯π‘žξƒ­2𝑔𝑇𝑝(π‘˜)π‘”π‘žξ€»(π‘˜)2βˆ’πœ‡1π‘§π‘–ξ€ΊπœŒπ‘‡π‘–(π‘˜)πœŒπ‘–ξ€»(π‘˜)2βˆ’πœ‡2𝑧𝑖Γ𝑖(π‘˜)Γ𝑇𝑖(π‘˜)2+𝑁𝑗=1πœ‹π‘˜π‘—π›Όπ‘–βˆ’1(𝑗)βˆ’π›½π‘–(π‘˜).(3.16) Thus the real control signal 𝑒(π‘˜) satisfies 𝑒(π‘˜)=𝛼𝑛(π‘˜) such that β„’π‘‰β‰€βˆ’π‘›ξ“π‘–=1𝑐𝑖𝑧4π‘–Μƒπœƒξ€·+π‘Žπœƒβˆ’πœƒ0ξ€Έ+𝑛𝑖=1𝑧3𝑖Λ𝑖(π‘˜)βˆ’π‘π‘€π‘–πœ›π‘–ξ€»+(π‘˜)𝑛𝑖=1π‘šπ‘–Μƒπ‘π‘–ξ€·π‘π‘–βˆ’π‘0𝑖+ξ‚Έ(π‘›βˆ’1)𝑛(2π‘›βˆ’1)+96πœ†9𝑛16πœ‡1ξ‚Ή||Ξ₯||4+9𝑛16πœ‡2𝑄2+𝑁𝑗=1πœ‹π‘˜π‘—π‘‰(𝑗).(3.17) Based on assumption (A2) and (3.9), we obtain the following inequality by applying Lemma 3.1: 𝑧3𝑖Λ𝑖(π‘˜)βˆ’π‘π‘€π‘–π‘§3π‘–πœ›π‘–||𝑧(π‘˜)≀3𝑖Λ𝑖||(π‘˜)βˆ’π‘π‘€π‘–π‘§3π‘–πœ™π‘–1𝑋𝑖𝑧,π‘˜β‹…tanh3π‘–πœ™π‘–1𝑋𝑖,π‘˜πœ€π‘–ξƒ­βˆ’π‘π‘€π‘–π‘§6𝑖𝑧tanh6π‘–πœπ‘–ξƒͺ≀||𝑧3𝑖||βˆ—ξ€Ίπ‘βˆ—π‘–πœ™π‘–1𝑋𝑖,π‘˜+π‘βˆ—π‘–πœ™π‘–2ξ€·||πœ‰||,π‘˜ξ€Έξ€»βˆ’π‘π‘€π‘–π‘§3π‘–πœ™π‘–1𝑋𝑖𝑧,π‘˜β‹…tanh3π‘–πœ™π‘–1𝑋𝑖,π‘˜πœ€π‘–ξƒ­βˆ’π‘π‘€π‘–π‘§6𝑖𝑧tanh6π‘–πœπ‘–ξƒͺ≀||𝑧3𝑖||π‘βˆ—π‘–πœ™π‘–1𝑋𝑖,π‘˜βˆ’π‘π‘€π‘–π‘§3π‘–πœ™π‘–1𝑋𝑖𝑧,π‘˜β‹…tanh3π‘–πœ™π‘–1𝑋𝑖,π‘˜πœ€π‘–ξƒ­+𝑝𝑀𝑖||𝑧3𝑖||πœ™π‘–2ξ€·||πœ‰||ξ€Έ,π‘˜βˆ’π‘π‘€π‘–π‘§6𝑖𝑧tanh6π‘–πœiξƒͺ≀𝑝𝑀𝑖||𝑧3π‘–πœ™π‘–1𝑋𝑖||,π‘˜βˆ’π‘§3π‘–πœ™π‘–1𝑋𝑖𝑧,π‘˜β‹…tanh3π‘–πœ™π‘–1𝑋𝑖,π‘˜πœ€π‘–ξƒ­ξƒ­+𝑝𝑀𝑖𝑧6π‘–βˆ’π‘§6𝑖𝑧tanh6π‘–πœπ‘–ξƒͺ+14πœ™2𝑖2ξ€·||πœ‰||ξ€Έξƒ­β‰€πœ€,π‘˜π‘–+πœπ‘–2𝑝𝑀𝑖+𝑝𝑀𝑖4πœ™2𝑖2ξ€·||πœ‰||ξ€Έ.,π‘˜(3.18) In (3.18), the following inequality is applied: ||πœ‚||ξ‚€πœ‚0β‰€βˆ’πœ‚β‹…tanhπœ–ξ‚β‰€12πœ–.(3.19) Notice the fact that π‘ŽΜƒπœƒπ‘‡ξ€·πœƒβˆ’πœƒ0ξ€Έ1=βˆ’2π‘ŽΜƒπœƒπ‘‡Μƒ1πœƒβˆ’2π‘Žξ€·πœƒβˆ’πœƒ0ξ€Έπ‘‡ξ€·πœƒβˆ’πœƒ0ξ€Έ+12π‘Žξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έπ‘‡ξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έ1β‰€βˆ’2π‘ŽΜƒπœƒπ‘‡Μƒ1πœƒ+2π‘Žξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έπ‘‡ξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έ,π‘šπ‘–Μƒπ‘π‘–ξ€·π‘π‘–βˆ’π‘0𝑖1=βˆ’2π‘šπ‘–Μƒπ‘2π‘–βˆ’12π‘šπ‘–ξ€·π‘π‘–βˆ’π‘0𝑖2+12π‘šπ‘–ξ€·π‘π‘€π‘–βˆ’π‘0𝑖21β‰€βˆ’2π‘šπ‘–Μƒπ‘2𝑖+12π‘šπ‘–ξ€·π‘π‘€π‘–βˆ’π‘0𝑖2.(3.20) Submitting (3.18), (3.20) into (3.12), there is ℒ𝑉(π‘˜)β‰€βˆ’π‘›ξ“π‘–=1𝑐𝑖𝑧4π‘–βˆ’12π‘ŽΜƒπœƒπ‘‡Μƒπœƒβˆ’π‘›ξ“π‘–=112π‘šπ‘–Μƒπ‘2𝑖+12π‘Žξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έπ‘‡ξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έ+𝑛𝑖=112π‘šπ‘–ξ€·π‘π‘€π‘–βˆ’π‘0𝑖2+ξ‚Έ(π‘›βˆ’1)𝑛(2π‘›βˆ’1)+96πœ†9𝑛16πœ‡1ξ‚Ή||Ξ₯||4+9𝑛16πœ‡2𝑄2+𝑛𝑖=1πœ€π‘–+πœπ‘–2𝑝𝑀𝑖+𝑛𝑖=1𝑝𝑀𝑖4πœ™2𝑖2ξ€·||πœ‰||ξ€Έ+,π‘˜π‘ξ“π‘—=1πœ‹π‘˜π‘—π‘‰(𝑗)β‰€βˆ’π›Ό1𝑉(π‘˜)+π‘‰πœ‰ξ€·||πœ‰||ξ€Έ,π‘˜+𝑑𝑧+𝑁𝑗=1πœ‹π‘˜π‘—π‘‰(𝑗).(3.21) Here parameter 𝛼1, 𝑑𝑧 and π’¦βˆž function π‘‰πœ‰(|πœ‰|,π‘˜) is chosen to satisfy π‘‰πœ‰ξ€·||πœ‰||ξ€Έβ‰₯,π‘˜π‘›ξ“π‘–=1𝑝𝑀𝑖4πœ™2𝑖2ξ€·||πœ‰||ξ€Έ,π‘˜,𝛼1ξ€½=min4𝑐𝑖,π‘Žβ‹…π›Ύ,π‘šβ‹…πœŽπ‘–ξ€Ύ,𝑑𝑧=12π‘Žξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έπ‘‡ξ€·πœƒβˆ—βˆ’πœƒ0ξ€Έ+𝑛𝑖=112π‘šπ‘–ξ€·π‘π‘€π‘–βˆ’π‘0𝑖2+ξ‚Έ(π‘›βˆ’1)𝑛(2π‘›βˆ’1)+96πœ†9𝑛16πœ‡1ξ‚Ή||Ξ₯||4+9𝑛16πœ‡2𝑄2+𝑛𝑖=1πœ€π‘–+πœπ‘–2𝑝𝑀𝑖.(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(π‘Ÿ(𝑑))=𝑁𝑙=1𝐸𝑉(𝑙)πœπ‘™,𝐸ℒ𝑉(π‘Ÿ(𝑑))=𝑁𝑙=1𝐸(ℒ𝑉(𝑙))πœπ‘™.(4.1) Thus (3.21) can be written as 𝐸ℒ𝑉(π‘Ÿ(𝑑))=𝑁𝑙=1𝐸(ℒ𝑉(𝑙))πœπ‘™β‰€π‘ξ“π‘™=1πΈξƒ―βˆ’π›Ό1πœπ‘™π‘‰(𝑙)+πœπ‘™π‘‰πœ‰ξ€·||πœ‰||ξ€Έ,𝑙+πœπ‘™π‘‘π‘§+πœπ‘™π‘ξ“π‘—=1πœ‹π‘™π‘—ξƒ°π‘‰(𝑗)=βˆ’π›Ό1𝑁𝑙=1πœπ‘™ξƒ―πΈπ‘‰(𝑙)+𝐸𝑁𝑙=1πœπ‘™π‘ξ“π‘—=1πœ‹π‘™π‘—π‘‰ξƒ°+(𝑗)𝑁𝑙=1πœπ‘™πΈπ‘‰πœ‰ξ€·||πœ‰||ξ€Έ,𝑙+𝑑𝑧||||ξ€Έβ‰€βˆ’π›ΌπΈπ‘‰(π‘Ÿ(𝑑))+πœ’πœ‰(𝑑)+𝑑𝑧,(4.2) where positive scalar 𝛼 is given as π›Όβ‰œπ›Ό1βˆ’max𝑙,π‘—βˆˆπ‘†ξ‚»πœπ‘™πœπ‘—ξ‚Όβˆ—maxπ‘—βˆˆπ‘†ξƒ―π‘ξ“π‘™=1πœ‹π‘™π‘—ξƒ°πœ’ξ€·||||ξ€Έβ‰œπœ‰(𝑑)𝑁𝑙=1πœπ‘™πΈπ‘‰πœ‰ξ€·||πœ‰||ξ€Έ.,𝑙(4.3) It is easily seen that πœ’(|πœ‰(𝑑)|) is a π’¦βˆž function with π‘Ÿ(𝑑) given, and appropriate control parameter 𝑐𝑖, 𝑙⋅𝛾, π‘šβ‹…πœŽπ‘– can be chosen to satisfy 𝛼>0.
For each integer β„Žβ‰₯1, define a stopping time as πœβ„Žξ€½||𝑧||ξ€Ύ=inf𝑑β‰₯0∢(𝑑)β‰₯β„Ž(4.4) Obviously, πœβ„Žβ†’βˆž almost surely as β„Žβ†’βˆž. Noticing that 0<|𝑧(𝑑)|β‰€β„Ž if 0≀𝑑<πœβ„Ž, we can apply the generalized πΌπ‘‘Μ‚π‘œ formula to derive that for any 𝑑β‰₯0, 𝐸𝑒𝛼⋅(π‘‘βˆ§πœβ„Ž)π‘‰ξ€·π‘Ÿξ€·π‘‘βˆ§πœβ„Žξ€·π‘§ξ€Έξ€Έξ€»=𝑉0ξ€Έ+ξ€œ,0,π‘Ÿ(0)π‘‘βˆ§πœβ„Ž0𝑒𝛼𝑠[]ξ€œπ›ΌπΈπ‘‰(π‘Ÿ(𝑠))+𝐸ℒ𝑉(π‘Ÿ(𝑠))𝑑𝑠≀𝑉(π‘Ÿ(0))+π‘‘βˆ§πœβ„Ž0𝑒𝛼𝑠||||𝛼𝐸𝑉(π‘Ÿ(𝑠))βˆ’π›ΌπΈπ‘‰(π‘Ÿ(𝑠))+πœ’πœ‰(𝑑)+π‘‘π‘§ξ€»ξ€œπ‘‘π‘ =𝑉(π‘Ÿ(0))+π‘‘βˆ§πœβ„Ž0π‘’π›Όπ‘ ξ€Ίπœ’ξ€·||||ξ€Έπœ‰(𝑑)+𝑑𝑧𝑑𝑠.(4.5) Let β„Žβ†’βˆž, apply Fatou’s lemma to (4.5), and we have πΈξ€Ίπ‘’π›Όπ‘‘ξ€»ξ€œπ‘‰(π‘Ÿ(𝑑))≀𝑉(π‘Ÿ(0))+𝐸𝑑0π‘’π›Όπ‘ ξ€Ίπœ’ξ€·||||ξ€Έπœ‰(𝑑)+𝑑𝑧𝑑𝑠.(4.6) By using mean value theorem for integration, there is 𝐸𝑒𝛼𝑑𝑉(π‘₯(𝑑),𝑑,π‘˜)=𝑒𝛼𝑑𝐸𝑉(π‘Ÿ(𝑑))≀𝑉(π‘Ÿ(0))+sup0β‰€π‘ β‰€π‘‘ξ€Ίπœ’ξ€·||πœ‰||ξ€Έ(𝑠)+π‘‘π‘§ξ€»β‹…ξ€œπ‘‘0𝑒𝛼𝑠𝑑𝑠.(4.7) According to the property of π’¦βˆž function, the following inequality is deduced: π‘’π›Όπ‘‘ξ‚»πœ’ξ‚΅πΈπ‘‰(π‘Ÿ(𝑑))≀𝑉(π‘Ÿ(0))+sup0≀𝑠≀𝑑||||ξ‚Άπœ‰(𝑠)+π‘‘π‘§ξ‚Όβ‹…ξ€œπ‘‘0𝑒𝛼𝑠𝑑𝑠=𝑉(π‘Ÿ(0))+πœ’(β€–πœ‰(𝑑)β€–)+𝑑𝑧⋅1π›Όξ‚ξ€·π‘’π›Όπ‘‘ξ€Έξ€Ίβˆ’1≀𝑉(π‘Ÿ(0))+πœ’(β€–πœ‰(𝑑)β€–)+𝑑𝑧⋅1𝛼𝑒𝛼𝑑.(4.8) According to (3.11), one gives 𝑒𝑁𝛼𝑑𝑗=114𝐸𝑧4𝑖≀𝑒𝛼𝑑𝐸𝑉(π‘Ÿ(𝑑))≀𝐸𝑉(π‘Ÿ(0))+πœ’(β€–πœ‰(𝑑)β€–)+𝑑𝑧⋅1𝛼𝑒𝛼𝑑.(4.9) Consequently, 𝑁𝑖=1𝐸𝑧4𝑖≀4π‘’βˆ’π›Όπ‘‘4𝑉(π‘Ÿ(0))+π›Όξ€½πœ’(β€–πœ‰(𝑑)β€–)+𝑑𝑧.(4.10) Defining 𝒦ℒ function 𝛽(β‹…,β‹…), π’¦βˆž function 𝛾(β‹…), and nonnegative number 𝑑𝑐 as: 𝛽||𝑧0||ξ€Έ,𝑑=4π‘’βˆ’πœ†π‘‘β€–4𝑉(π‘Ÿ(0)),𝛾(β€–πœ‰(𝑑))=π›Όβ€–πœ’(β€–πœ‰(𝑑)),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 πœ‰ 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 πœ–>0, there exists 𝑇>0 and 𝛿>0 such that if 𝑑>𝑇, the output of jump system 𝑦 satisfies 𝑃||𝑦||ξ€Ύ(𝑑)<𝛿β‰₯1βˆ’πœ–.(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: ξ€·π‘₯π‘‘πœ‰=π‘ž1ξ€Έ,πœ‰,𝑑,π‘Ÿ(𝑑)𝑑𝑑,𝑑π‘₯1=π‘₯2𝑑𝑑+𝑓1ξ€·π‘₯1ξ€Έπœƒ,𝑑,π‘Ÿ(𝑑)βˆ—π‘‘π‘‘+Ξ”1(𝑋,πœ‰,𝑑,π‘Ÿ(𝑑))𝑑𝑑+π‘₯11/3π‘‘πœ”,𝑑π‘₯2=𝑒𝑑𝑑+𝑓2(𝑋,𝑑,π‘Ÿ(𝑑))πœƒβˆ—π‘‘π‘‘+Ξ”2(𝑋,πœ‰,𝑑,π‘Ÿ(𝑑))𝑑𝑑,𝑦=π‘₯1,(5.1) where the transition rate matrix is ξ€ΊΞ =πœ‹11πœ‹12πœ‹21πœ‹22ξ€»=ξ€Ίβˆ’223βˆ’3ξ€» with stationary distribution 𝜁1=𝜁2=0.5.

Here let noise covariance be 𝐸{π‘‘πœ”π‘‘πœ”π‘‡}=1 and system dynamics for each mode as π‘žξ€·π‘₯1ξ€Έ,πœ‰,𝑑,1=βˆ’0.5πœ‰+0.3π‘₯1ξ€·π‘₯,π‘ž1ξ€Έ,πœ‰,𝑑,2=βˆ’0.4πœ‰+0.3π‘₯1𝑓cos𝑑,1ξ€·π‘₯1ξ€Έ,𝑑,1=π‘₯21,𝑓1ξ€·π‘₯1ξ€Έ,𝑑,2=βˆ’π‘₯1cosπ‘₯1,Ξ”1(𝑋,πœ‰,𝑑,1)=0.5πœ‰+0.4π‘₯1sin2𝑑,Ξ”1(𝑋,πœ‰,𝑑,2)=π‘₯1π‘“πœ‰,2(𝑋,𝑑,1)=π‘₯1sinπ‘₯2+π‘₯2,𝑓2(𝑋,𝑑,2)=π‘₯1+2π‘₯2,Ξ”2(𝑋,πœ‰,𝑑,1)=0.4πœ‰sin𝑑+0.3π‘₯1.Ξ”2(𝑋,πœ‰,𝑑,2)=π‘₯1||πœ‰||1/2.(5.2) From Assumption (A2), we have Ξ”1(𝑋,πœ‰,𝑑,1)β‰€π‘βˆ—1||πœ‰||+π‘βˆ—1||π‘₯1||,Ξ”1(𝑋,πœ‰,𝑑,2)β‰€π‘βˆ—1||πœ‰||2+π‘βˆ—1||π‘₯1||2,Ξ”2(𝑋,πœ‰,𝑑,1)β‰€π‘βˆ—2||πœ‰||+π‘βˆ—2||π‘₯1||,Ξ”2(𝑋,πœ‰,𝑑,2)β‰€π‘βˆ—2||πœ‰||+π‘βˆ—2||π‘₯1||2,(5.3) where π‘βˆ—1≀0.5 and π‘βˆ—2≀0.5 and the πœ‰ subsystem satisfies ℒ𝑉04(πœ‰,𝑑,π‘˜)β‰€βˆ’||πœ‰||102+πœ’0ξ€·||π‘₯1||ξ€Έ+𝑑0,(5.4) where 𝑉0=(1/2)πœ‰2, πœ’0(|π‘₯1|)=0.15|π‘₯1|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 π‘˜=1: 𝛼1𝑐(1)=βˆ’1+34π‘₯1βˆ’π‘₯21πœƒβˆ’πœ‡1π‘₯17/3βˆ’π‘1π‘₯1π‘₯tanh41πœ–1ξƒͺβˆ’π‘₯31π‘₯tanh61𝑣𝑖ξƒͺ𝛼2𝑐(1)=βˆ’2+14𝑧2(1)βˆ’π‘₯1sinπ‘₯2βˆ’π‘’1𝑧32(1)π‘₯21βˆ’14𝑧32𝑐(1)βˆ’1+34+2π‘₯1+34π‘₯21π‘₯41×π‘₯21+π‘₯2ξ€Έ+πœ‹11𝛼1(1)+πœ‹12𝛼1(2)βˆ’πœ‡2𝑧2𝛼(1)1(1)βˆ’π›Ό1ξ€»(2)4βˆ’πœ2(1)πœƒβˆ’πœ1Μ‡πœƒ(1)βˆ’π‘1π‘₯tanh41πœ–1ξƒͺβˆ’3π‘₯21π‘₯tanh61𝑣𝑖ξƒͺβˆ’4𝑝1π‘₯41sech2π‘₯41πœ–1ξƒͺβˆ’π‘₯81sech2π‘₯61𝑣𝑖ξƒͺβˆ’π‘2πœ›1𝑧(2),2(1)=π‘₯2βˆ’π›Ό1(̇1),πœƒ=𝛾2𝑖=1𝑧3π‘–πœπ‘–ξ€·(1)βˆ’π‘Žπœƒβˆ’πœƒ0ξ€Έξƒ­,̇𝑝1=𝜎1ξ€Ίπ‘₯31πœ›π‘–(1)βˆ’π‘š1𝑝1βˆ’π‘01,̇𝑝2=𝜎2𝑧32(1)πœ›2(1)βˆ’π‘š2𝑝2βˆ’π‘02.ξ€Έξ€»(5.5)

Case 2. The system regime is π‘˜=2: 𝛼1𝑐(2)=βˆ’1+34π‘₯1βˆ’π‘₯1sinπ‘₯1βˆ’π‘₯2βˆ’π‘1π‘₯1π‘₯tanh41πœ–1ξƒͺβˆ’π‘₯31π‘₯tanh61𝑣𝑖ξƒͺ,𝛼2𝑐(2)=βˆ’2+14𝑧2(2)βˆ’πœ‡1𝑧32(2)π‘₯41βˆ’14𝑧32𝑐(2)βˆ’1+114+34π‘₯21+π‘₯61π‘₯1+π‘₯2ξ€Έ+πœ‹21𝛼1(1)+πœ‹22𝛼1(2)βˆ’πœ‡2𝑧2𝛼(2)1(1)βˆ’π›Ό1ξ€»(2)4βˆ’πœ2(2)πœƒβˆ’πœ1Μ‡πœƒ(2)βˆ’π‘1π‘₯tanh41πœ–1ξƒͺβˆ’3π‘₯21π‘₯tanh61𝑣𝑖ξƒͺβˆ’4𝑝1π‘₯41sech2π‘₯41πœ–1ξƒͺβˆ’π‘₯81sech2π‘₯61𝑣𝑖ξƒͺβˆ’π‘2πœ›2𝑧(2),2(2)=π‘₯2βˆ’π›Ό1(̇2),πœƒ=𝛾2𝑖=1𝑧3π‘–πœπ‘–ξ€·(2)βˆ’π‘Žπœƒβˆ’πœƒ0ξ€Έξƒ­,̇𝑝1=𝜎1ξ€Ίπ‘₯31πœ›π‘–(2)βˆ’π‘š1𝑝1βˆ’π‘01,̇𝑝2=𝜎2𝑧32(2)πœ›2(2)βˆ’π‘š2𝑝2βˆ’π‘02.ξ€Έξ€»(5.6) In computation, we set the initial value to be π‘₯1=1.6, π‘₯2=βˆ’2.7, πœƒ=0, 𝑝1=𝑝2=0 let parameter πœƒ0=1, 𝛾=1, π‘Ž=1, 𝑝0=0.7, πœ–π‘–=𝑣𝑖=0.5, π‘šπ‘–=1, πœ‡1=πœ‡2=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 π‘₯1, and Figure 4 shows system state π‘₯2. Figure 5 shows the corresponding switching controller 𝑒; finally Figure 6 shows the trajectory of adaptive parameter πœƒ and Figure 7; Figure 8 shows the trajectory of parameter 𝑝1, 𝑝2, respectively.

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 π‘₯1 and, Figure 11 shows system state π‘₯2, and Figure 12 shows the corresponding switching controller 𝑒; the trajectory of adaptive parameter πœƒ is shown in Figures 13 and 14; Figure 15 shows the trajectory of parameter 𝑝1, 𝑝2, respectively.

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 πœƒ 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 𝑒(π‘˜) varies with different regime π‘Ÿ(𝑑)=π‘˜ taken, and it differs in two aspects (see (3.16)): the coupling of regimes πœ‹π‘˜π‘—π›Όπ‘–βˆ’1(𝑗) and πœ‡2𝑧𝑖[Γ𝑖(π‘˜)Γ𝑇𝑖(π‘˜)]2, which are both caused by the Markovian jumps. The switching controller will degenerate to an ordinary one if π‘Ÿ(𝑑)≑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.

Acknowledgment

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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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.

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