Adaptive Output Feedback Control for a Class of Stochastic Nonlinear Systems with SiISS Inverse Dynamics

Abstract

The adaptive stabilization scheme based on tuning function for stochastic nonlinear systems with stochastic integral input-to-state stability (SiISS) inverse dynamics is investigated. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is constructively designed. It is shown that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.

1. Introduction

Global stabilization control design of stochastic nonlinear systems is one of the most important topics in nonlinear control theory, which has received and is increasingly receiving a great deal of attention; see, for example, [1–37] and the references therein. For a class of stochastic nonlinear systems with stochastic inverse dynamics, much progress has been made in the design of the global stabilization controller [12, 13, 15, 16, 24, 25, 31]. However, all these controllers are only robust against stochastic inverse dynamics with stringent stability margin. To weaken the stringent condition on stochastic inverse dynamics, a natural idea is to benefit from input-to-state stability (ISS) in [38] and integral input-to-state stability (iISS) in [39] which are now recognized as the central unifying concepts in feedback design and stability analysis of deterministic nonlinear systems. Tsinias in [21], Tang and Basar in [19] first proposed the concept of stochastic input-to-state stability (SISS) independently. Further in-depth study on SISS and its applications are presented in [9–11, 18, 22]. Motivated by these aforementioned important results, [34] showed that SISS condition can be weakened to stochastic integral input-to-state stability (SiISS) and developed a unifying output feedback framework for global regulation.

Nonlinear small-gain theorem plays an important role in the controller design and stability analysis of deterministic nonlinear systems in [40, 41]. While for stochastic nonlinear systems, there are fewer results on the small-gain theorem. [25] firstly established a gain-function-based stochastic nonlinear small-gain theorem for ISpS in probability. In some succeeding research work, [9, 11, 35] presented some Lyapunov-based small-gain type conditions on SISS and SiISS, respectively. [4] further discusses the relationship of small-gain type conditions on SiISS and studies the problem of GAS in probability via output feedback.

In this paper, inspired by [4], a more general class of stochastic nonlinear systems with uncertain parameters and stochastic integral input-to-state stability (SiISS) inverse dynamics is investigated. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is proposed to guarantee that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.

The paper is organized as follows. Section 2 begins with the mathematical preliminaries. Section 3 presents statement of the problem. The design of adaptive output feedback controller is given in Section 4. The corresponding analysis is given in Section 5. Section 6 concludes the paper.

2. Mathematical Preliminaries

The following notations are used throughout the paper. ℝ₊ stands for the set of all nonnegative real numbers, ℝ^𝑛 is the 𝑛-dimensional Euclidean space, and ℝ^𝑛×𝑚 is the space of real 𝑛×𝑚-matrices. For 𝑥=(𝑥₁,…,𝑥_𝑛), one defines 𝑥_𝑖=(𝑥₁,…,𝑥_𝑖), 𝑖=1,…,𝑛−1. 𝐶^𝑖 denotes the family of all the functions with continuous 𝑖th partial derivatives. 𝐿¹(ℝ₊;ℝ₊) is the family of all functions 𝑙∶ℝ₊→ℝ₊ such that ∫^∞₀𝑙(𝑡)𝑑𝑡<∞. For a given vector or matrix 𝑋, 𝑋^𝑇 denotes its transpose, Tr{𝑋} denotes its trace when 𝑋 is square. |𝑋| denotes the Euclidean norm of a vector 𝑋, and ‖𝑋‖=(Tr{𝑋𝑋^𝑇})^1/2 for a matrix 𝑋. 𝒦 denotes the set of all functions: ℝ₊→ℝ₊, which are continuous, strictly increasing, and vanishing at zero; 𝒦_∞ is the set of all functions which are of class 𝒦 and unbounded; 𝒦ℒ denotes the set of all functions 𝛽(𝑠,𝑡): ℝ₊×ℝ₊→ℝ₊, which are of 𝒦 for each fixed 𝑡 and decrease to zero as 𝑡→∞ for each fixed 𝑠.

Consider the stochastic nonlinear delay-free system𝑑𝑥=𝑓(𝑥,𝑡)𝑑𝑡+𝑔(𝑥,𝑡)𝑑𝑤,∀𝑥(0)=𝑥₀∈ℝ^𝑛, (2.1) where 𝑥∈ℝ^𝑛, 𝑤 is an 𝑚-dimensional standard Wiener process defined in a complete probability space {Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃} with Ω being a sample space, ℱ being a 𝜎-field, {ℱ_𝑡}_𝑡≥0 being a filtration, and 𝑃 being the probability measure. Borel measurable functions 𝑓: ℝ^𝑛×ℝ₊→ℝ^𝑛 and 𝑔: ℝ^𝑛×ℝ₊→ℝ^𝑛×𝑚 are piecewise continuous in 𝑡 and locally bounded and locally Lipschitz continuous in 𝑥 uniformly in 𝑡∈ℝ₊. Let ℒ𝑉(𝑥) denote infinitesimal generator of function 𝑉∈𝐶² along stochastic system (2.1) with the definition of(𝑥)=(𝑥)ℒ𝑉𝜕𝑉𝑓()+1𝜕𝑥𝑥,𝑡2Tr𝑔^𝑇()𝜕𝑥,𝑡²𝑉(𝑥)𝜕𝑥²𝑔(𝑥,𝑡). (2.2)

Definition 2.1 (see [34]). Stochastic process {𝜉(𝑡)}_𝑡≥𝑡0 is said to be bounded almost surely if sup_𝑡≥𝑡0|𝜉(𝑡)|<∞ a.s.

Lemma 2.2 (Stochastic LaSalle Theorem [14]). For system (2.1), if there exist functions 𝑉∈𝐶², 𝛼₁,𝛼₂∈𝒦_∞, 𝑙∈𝐿¹(ℝ₊;ℝ₊), and a continuous nonnegative function 𝑊∶ℝ^𝑛→ℝ such that for all 𝑥∈ℝ^𝑛, 𝑡≥0, 𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂(|𝑥|),ℒ𝑉(𝑥)≤−𝑊(𝑥)+𝑙(𝑡), (2.3) then for each 𝑥₀∈ℝ^𝑛,(i)system (2.1) has a unique strong solution on [0,∞), and solution 𝑥(𝑡) is bounded almost surely;(ii)when 𝑓(0,𝑡)≡0, 𝑔(0,𝑡)≡0, 𝑙(𝑡)≡0, the equilibrium 𝑥=0 is globally stable in probability.

In the following, we cite two small-gain type conditions on SiISS in [35].

Consider the following stochastic nonlinear system𝑑𝑥=𝑓(𝑥,𝑣,𝑡)𝑑𝑡+𝑔(𝑥,𝑣,𝑡)𝑑𝑤,∀𝑥(0)=𝑥₀∈ℝ^𝑛, (2.4) where 𝑥∈ℝ^𝑛 is the state, 𝑣∈ℝ^𝑚 is the input, and 𝑤 is an 𝑟-dimensional standard Wiener process defined as in (2.1). Borel measurable functions 𝑓: ℝ^𝑛×ℝ^𝑚×ℝ₊→ℝ^𝑛 and 𝑔: ℝ^𝑛×ℝ^𝑚×ℝ₊→ℝ^𝑛×𝑟 are locally bounded and locally Lipschitz continuous with respect to (𝑥,𝑣) uniformly in 𝑡∈ℝ₊.

Definition 2.3 (see [34]). System (2.4) is said to be stochastic integral input-to-state stable (SiISS) using Lyapunov function if there exist functions 𝑉∈𝐶²(ℝ^𝑛;ℝ), 𝛼₁,𝛼₂,𝛾∈𝒦_∞, and a merely positive definite continuous function 𝛼 such that 𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂(|𝑥|),ℒ𝑉(𝑥)≤−𝛼(|𝑥|)+𝛾(|𝑣|). (2.5) The function 𝑉 satisfying (2.5) is said to be a SiISS-Lyapunov function, and (𝛼,𝛾) in (2.5) is called the SiISS supply rate of system (2.4).

Lemma 2.4 (see [35]). For system (2.4) satisfying (2.5), if there exists a positive definite continuous function 𝛼 such that limsup_+𝑠→0𝛼(𝑠)/𝛼(𝑠)<∞, limsup_𝑠→∞𝛼(𝑠)/𝛼(𝑠)<∞, then there exists a function 𝛾∈𝒦_∞ such that (𝛼,𝛾) is a new SiISS supply rate of system (2.4). Moreover, if limsup_𝑠→0+𝛾(𝑠)/𝑠^𝑚<∞, then limsup_𝑠→0+𝛾(𝑠)/𝑠^𝑚<∞, where 𝑚 is any positive integer.

The following lemma shows that the condition at infinity can be removed if more prior information on stochastic system is known.

Assumption H. For functions 𝑔, 𝑉, 𝛼 given in (2.4), (2.5) with liminf_𝑠→∞𝛼(𝑠)=∞, there exist known smooth positive definite functions 𝜙₁, 𝜙₂ such that ‖𝑔(𝑥,𝑣,𝑡)‖≤𝜙₁(|𝑥|), |𝜕𝑉(𝑥)/𝜕𝑥|≤𝜙₂(|𝑥|) and limsup_𝑠→0+𝜙²₁(𝑠)𝜙²₂(𝑠)/𝛼(𝑠)<∞.

Lemma 2.5 (see [35]). For system (2.4) satisfying (2.5) and Assumption H, if there exists a positive definite function 𝛼 such that limsup_𝑠→0+𝛼(𝑠)𝛼(𝑠)<∞,^∞₀𝑒^{−∫𝑠0(1/𝜁(𝛼1−1(𝜏)))𝑑𝜏}𝜉𝛼₁⁻¹(𝑠)^′𝑑𝑠<∞, (2.6) where 𝜉(⋅)≥0, 𝜁(⋅)>0 are smooth increasing functions with 𝜉(𝑠)𝛼(𝑠)≥2𝛼(𝑠), 𝜁(𝑠)𝛼(𝑠)≥𝜙²₁(𝑠)𝜙²₂(𝑠) for any 𝑠≥0, then there exists a function 𝛾∈𝒦_∞ such that (𝛼,𝛾) is a new SiISS supply rate of system (2.4). Moreover, if limsup_𝑠→0+𝛾(𝑠)/𝑠^𝑚<∞, then limsup_𝑠→0+𝛾(𝑠)/𝑠^𝑚<∞, where 𝑚 is any positive integer.

Lemma 2.6. Let 𝑥,𝑦 be real variables, then for any positive integers 𝑚, 𝑛 and continuous function 𝑎(⋅)≥0, 𝑎(⋅)𝑥^𝑚𝑦^𝑛≤𝑏|𝑥|^𝑚+𝑛+(𝑛/(𝑚+𝑛))((𝑚+𝑛)/𝑚)^−𝑚/𝑛𝑎^{(𝑚+𝑛)/𝑛}(⋅)𝑏^𝑚/𝑛|𝑦|^𝑚+𝑛, where 𝑏>0 is any real number.

3. Problem Statement

In this paper, we consider a class of stochastic nonlinear systems described by𝑑𝜂=𝜑₀𝜂,𝑥₁𝑑𝑡+𝜓₀𝜂,𝑥₁𝑑𝑤,𝑑𝑥₁=𝑥₂𝑑𝑡+𝜑₁𝜂,𝑥𝑑𝑡+𝜓₁⋮𝜂,𝑥𝑑𝑤,𝑑𝑥_𝑛−1=𝑥_𝑛𝑑𝑡+𝜑_𝑛−1𝜂,𝑥𝑑𝑡+𝜓_𝑛−1𝜂,𝑥𝑑𝑤,𝑑𝑥_𝑛=𝑢𝑑𝑡+𝜑_𝑛𝜂,𝑥𝑑𝑡+𝜓_𝑛𝜂,𝑥𝑑𝑤,𝑦=𝑥₁, (3.1) where (𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛−1, 𝑢,𝑦∈ℝ represent the unmeasurable state, the control input, and the measurable output, respectively. 𝜂∈ℝ^𝑛0 is referred to as the stochastic inverse dynamics. The initial value (𝜂^𝑇(0),𝑥₁(0),…,𝑥_𝑛(0)) can be chosen arbitrarily. 𝑤 is an 𝑚-dimensional standard Wiener process defined as in (2.1). Uncertain functions 𝜑_𝑖∶ℝ^𝑛0×ℝ^𝑛→ℝ, 𝜓_𝑖∶ℝ^𝑛0×ℝ^𝑛→ℝ^𝑚, 1≤𝑖≤𝑛, are smooth functions. It is assumed that 𝜑₀ and 𝜓₀ are locally Lipschitz continuous functions.

The research purpose of this paper is to design an adaptive output feedback controller for system (3.1) by using stochastic LaSalle theorem and small-gain type conditions on SiISS, in such a way that, for all initial conditions, the solutions of the closed-loop system are bounded almost surely and the closed-loop systems are globally stable in probability. To achieve the control purpose, we need the following assumptions.

Assumption 3.1. For each 1≤𝑖≤𝑛, there exist the unknown constant 𝜃_𝑖>0, the known nonnegative smooth functions 𝜑_𝑖0, 𝜑_𝑖1, 𝜓_𝑖0 and 𝜓_𝑖1 with 𝜑_𝑖𝑗(0)=0, 𝜓_𝑖𝑗(0)=0, 𝑗=0,1, such that |𝜑_𝑖(𝜂,𝑥)|≤𝜃_𝑖(𝜑_𝑖0(|𝜂|)+𝜑_𝑖1(𝑥₁)), |𝜓_𝑖(𝜂,𝑥)|≤𝜃_𝑖(𝜓_𝑖0(|𝜂|)+𝜓_𝑖1(𝑥₁)).

For Assumption 3.1, there exist smooth functions 𝜑_𝑖1 and 𝜓_𝑖1 satisfying𝜑_𝑖1𝑥₁=𝑥₁𝜑_𝑖1𝑥₁,𝜓_𝑖1𝑥₁=𝑥₁𝜓_𝑖1𝑥₁, (3.2) which will be frequently used in the subsequent sections.

Assumption 3.2. For the 𝜂-subsystem, there exists an SiISS-Lyapunov function 𝑉₀(𝜂). Namely, 𝑉₀ satisfies 𝛼(|𝜂|)≤𝑉₀(𝜂)≤𝛼(|𝜂|), ℒ𝑉₀(𝜂)≤−𝛼(|𝜂|)+𝛾(|𝑥₁|), where 𝛼,𝛼,𝛾 are class 𝒦_∞ functions, and 𝛼 is merely a continuous positive definite function.

4. Design of an Adaptive Output Feedback Controller

4.1. Reduced-Order Observer Design

Introduce the following reduced-order observer:𝑥̇_𝑖=𝑥_𝑖+1+𝑘_𝑖+1𝑦−𝑘_𝑖𝑥₁+𝑘₁𝑦̇,1≤𝑖≤𝑛−2,𝑥_𝑛−1=𝑢−𝑘_𝑛−1𝑥₁+𝑘₁𝑦, (4.1) where 𝑘=(𝑘₁,…,𝑘_𝑛−1)^𝑇 is chosen such that 𝐴₀=^{𝐼−𝑘𝑛−2}_0…0 is asymptotically stable. Define the error variable𝑒_𝑖=1𝜃^∗𝑥_𝑖+1−𝑥_𝑖−𝑘_𝑖𝑥₁,1≤𝑖≤𝑛−1,𝜃^∗=max1,𝜃₁,…,𝜃_𝑛. (4.2) By (3.1), (4.1), and (4.2), one gets 𝑑𝑒_𝑖=𝑒_𝑖+1−𝑘_𝑖𝑒₁1𝑑𝑡+𝜃^∗𝜑_𝑖+1𝜂,𝑥−𝑘_𝑖𝜑₁1𝜂,𝑥𝑑𝑡+𝜃^∗𝜓_𝑖+1𝜂,𝑥−𝑘_𝑖𝜓₁𝜂,𝑥𝑑𝑤, (4.3) whose compact form is 𝑑𝑒=𝐴₀𝑒+Φ𝜂,𝑥𝑑𝑡+Ψ𝜂,𝑥𝑑𝑤, (4.4) where𝑒=𝑒₁,…,𝑒_𝑛−1^𝑇,Φ=1𝜂,𝑥𝜃^∗𝜑₂𝜂,𝑥−𝑘₁𝜑₁𝜂,𝑥,𝜑₃𝜂,𝑥−𝑘₂𝜑₁𝜂,𝑥,…,𝜑_𝑛𝜂,𝑥−𝑘_𝑛−1𝜑₁𝜂,𝑥^𝑇,Ψ=1𝜂,𝑥𝜃^∗𝜓₂𝜂,𝑥−𝑘₁𝜓₁𝜂,𝑥,𝜓₃𝜂,𝑥−𝑘₂𝜓₁𝜂,𝑥,…,𝜓_𝑛𝜂,𝑥−𝑘_𝑛−1𝜓₁𝜂,𝑥^𝑇. (4.5)

4.2. The Design of Adaptive Backstepping Controller

From (3.1), (4.1), (4.2), and (4.4), the interconnected system is represented as𝑑𝜂=𝜑₀𝜂,𝑦𝑑𝑡+𝜓₀𝐴𝜂,𝑦𝑑𝑤,𝑑𝑒=₀𝑒+Φ𝜂,𝑥𝑑𝑡+Ψ𝜂,𝑥𝑑𝑤,𝑑𝑦=𝑥₁+𝑘₁𝑦+𝜃^∗𝑒₁+𝜑₁𝜂,𝑥𝑑𝑡+𝜓₁𝜂,𝑥𝑑𝑤,𝑑𝑥₁=𝑥₂+𝑘₂𝑦−𝑘₁𝑥₁+𝑘₁𝑦𝑑𝑡,𝑑𝑥₂=𝑥₃+𝑘₃𝑦−𝑘₂𝑥₁+𝑘₁𝑦⋮𝑑𝑡,𝑑𝑥_𝑛−1=𝑢−𝑘_𝑛−1𝑥₁+𝑘₁𝑦𝑑𝑡. (4.6) Next, we will develop an adaptive backstepping controller by using the backstepping method. Firstly, a coordinate transformation is introduced𝑧₁=𝑦,𝑧_𝑖+1=𝑥_𝑖−𝛼_𝑖𝑦,𝑥₁,…,𝑥_𝑖−1,𝜃,𝑖=1,…,𝑛−1. (4.7)

Step 1. By (4.6) and (4.7), one has

𝑑𝑧₁=𝛼₁+𝑧₂+𝑘₁𝑦+𝜃^∗𝑒₁+𝜑₁𝜂,𝑥𝑑𝑡+𝜓₁𝜂,𝑥𝑑𝑤. (4.8) Since 𝐴₀ is asymptotically stable, there exists a positive definite matrix 𝑃 such that 𝑃𝐴₀+𝐴^𝑇₀𝑃=−𝐼_𝑛−1. Choose𝑉₁𝑒,𝑧₁,𝜃=𝛿2𝑒^𝑇𝑃𝑒²+1𝜃2Γ²+14𝑧⁴₁,𝛿>0,Γ>0, (4.9) where 𝜃=𝜃−𝜃, 𝜃 is the estimate of 𝜃=max{𝜃^∗,𝜃^∗(4/3),𝜃^∗4}. In view of (2.2), (4.4), (4.8), and (4.9), thenℒ𝑉₁=2𝛿𝑒^𝑇𝑃𝑒𝑒^𝑇𝑃𝐴₀+1𝑒+Φ𝜂,𝑥2TrΨ^𝑇𝜂,𝑥4𝛿𝑒^𝑇𝑃^𝑇𝑒^𝑇𝑃+2𝛿𝑒^𝑇Ψ+1𝑃𝑒𝑃𝜂,𝑥Γ𝜃𝜃̇+𝑧³₁𝛼₁+𝑧₂+𝑘₁𝑦+𝜃^∗𝑒₁+𝜑₁+3𝜂,𝑥2𝑧²₁Tr𝜓^𝑇₁𝜂,𝑥𝜓₁𝜂,𝑥=𝛿𝑒^𝑇𝑃𝑒𝑒^𝑇𝑃𝐴₀+𝐴^𝑇₀𝑃𝑒+2𝛿𝑒^𝑇𝑃𝑒𝑒^𝑇Ψ𝑃Φ𝜂,𝑥+Tr^𝑇𝜂,𝑥2𝛿𝑒^𝑇𝑃^𝑇𝑒^𝑇𝑃+𝛿𝑒^𝑇Ψ+1𝑃𝑒𝑃𝜂,𝑥Γ𝜃𝜃̇+𝑧³₁𝛼₁+𝑧₂+𝑘₁𝑦+𝜃^∗𝑒₁+𝜑₁+3𝜂,𝑥2𝑧²₁Tr𝜓^𝑇₁𝜂,𝑥𝜓₁𝜂,𝑥≤−𝛿𝜆_min(𝑃)|𝑒|⁴+2𝛿‖𝑃‖²|𝑒|³Φ𝜂,𝑥+3𝛿‖𝑃‖²|𝑒|²Ψ𝜂,𝑥²+1Γ𝜃𝜃̇+𝑧³₁𝛼₁+𝑧₂+𝑘₁𝑦+𝜃^∗𝑒₁+𝜑₁+3𝜂,𝑥2𝑧²₁Tr𝜓^𝑇₁𝜂,𝑥𝜓₁𝜂,𝑥. (4.10) Applying Assumption 3.1, (3.2), (4.3), (4.7), and Lemma 2.6, it follows that2𝛿‖𝑃‖²|𝑒|³Φ𝜂,𝑥≤𝑎₀₁|𝑒|⁴+𝑎₀₁Φ𝜂,𝑥⁴,3𝛿‖𝑃‖²|𝑒|²Ψ𝜂,𝑥²≤𝑎₀₂|𝑒|⁴+𝑎₀₂Ψ(𝜂,𝑥)⁴,𝑧³₁𝑧₂≤𝑎₁₀𝑧⁴₁+𝑎₁₀𝑧⁴₂,𝜃^∗𝑧³₁𝑒₁≤𝑎₁₁𝑒⁴₁+𝜃^∗(4/3)𝛾₁₁𝑧₁𝑧⁴₁≤𝑎₁₁𝑒⁴₁+𝜃𝛾₁₁𝑧₁𝑧⁴₁,𝑧³₁𝜑₁𝜂,𝑥≤𝑧₁³𝜃^∗𝜑₁₀𝜂+𝜑₁₁𝑧₁≤𝑎₁₂𝜑⁴₁₀𝜂+𝜃𝛾₁₂𝑧₁𝑧⁴₁,32𝑧²₁Tr𝜓^𝑇₁𝜂,𝑥𝜓₁𝜂,𝑥≤3𝑧²₁𝜃^∗2𝜓²₁₀𝜂+𝜓²₁₁𝑧₁≤𝑎₁₃𝜓⁴₁₀𝜂+𝜃𝛾₁₃𝑧₁𝑧⁴₁, (4.11)where 𝑎₀₁, 𝑎₀₂, 𝛾₁₁, 𝛾₁₂, and 𝛾₁₃ are smooth functions, 𝑎₀₁,𝑎₀₂,𝑎₁₀,𝑎₁₀,𝑎₁₁,𝑎₁₂,𝑎₁₃>0 are constants.

Using (𝑎₁+⋯+𝑎_𝑛)²≤(𝑥+𝑎)^𝑛=𝑛∑^𝑛_𝑖=1𝑎²_𝑖, (𝑎+𝑏)⁴≤8𝑎⁴+8𝑏⁴, 𝑦=𝑥₁, Assumption 3.1, (3.2), and (4.5), one getsΦ𝜂,𝑥⁴≤64(𝑛−1)𝑘⁴₁+⋯+𝑘⁴_𝑛−1𝜑⁴₁₀+𝑦⁴𝜑⁴₁₁+𝜑⁴₂₀+⋯+𝜑⁴_𝑛0+𝑦⁴𝜑⁴₂₁+⋯+𝜑⁴_𝑛1,Ψ𝜂,𝑥⁴≤64(𝑛−1)𝑘⁴₁+⋯+𝑘⁴_𝑛−1𝜓⁴₁₀+𝑦⁴𝜓⁴₁₁+𝜓⁴₂₀+⋯+𝜓⁴_𝑛0+𝑦⁴𝜓⁴₂₁+⋯+𝜓⁴_𝑛1. (4.12) Substituting (4.11)-(4.12) into (4.10) and using 𝑧₁=𝑦 lead toℒ𝑉₁≤−𝑎₀₀|𝑒|⁴+𝑎₁₁𝑒⁴₁+Δ₁𝜂+𝑎₁₀𝑧⁴₂+𝑧³₁𝛼₁+𝑘₁𝑧₁+Δ₀₀𝑧₁𝑧₁+𝑎₁₀𝑧₁+𝜃𝛾₁₁𝑧₁+𝛾₁₂𝑧₁+𝛾₁₃𝑧₁𝑧⁴₁+1Γ𝜃𝜃̇, (4.13)

where 𝑎₀₀=𝛿𝜆_min(𝑃)−𝑎₀₁−𝑎₀₂,Δ₁𝜂=64(𝑛−1)𝑎₀₁𝑘⁴₁+⋯+𝑘⁴_𝑛−1𝜑⁴₁₀𝜂+𝜑⁴₂₀𝜂+⋯+𝜑⁴_𝑛0𝜂+64(𝑛−1)𝑎₀₂𝑘⁴₁+⋯+𝑘⁴_𝑛−1𝜓⁴₁₀𝜂+𝜓⁴₂₀𝜂+⋯+𝜓⁴_𝑛0𝜂+𝑎₁₂𝜑⁴₁₀𝜂+𝑎₁₃𝜓⁴₁₀𝜂,Δ₀₀𝑧₁=64(𝑛−1)𝑎₀₁𝑘⁴₁+⋯+𝑘⁴_𝑛−1𝜑⁴₁₁𝑧₁+𝜑⁴₂₁𝑧₁+⋯+𝜑⁴_𝑛1𝑧₁+64(𝑛−1)𝑎₀₂𝑘⁴₁+⋯+𝑘⁴_𝑛−1𝜓⁴₁₁𝑧₁+𝜓⁴₂₁𝑧₁+⋯+𝜓⁴_𝑛1𝑧₁. (4.14) Choosing the virtual control 𝛼₁(⋅) and the tuning function 𝜏₁(⋅)𝛼₁𝑦,𝜃=−𝑧₁𝑐₁+𝜈₁𝑧₁+𝑘₁+Δ₀₀𝑧₁+𝑎₁₀+𝜃𝛾₁₁𝑧₁+𝛾₁₂𝑧₁+𝛾₁₃𝑧₁,𝜏₁𝑦=Γ𝛾₁₁𝑧₁+𝛾₁₂𝑧₁+𝛾₁₃𝑧₁𝑧⁴₁, (4.15) one getsℒ𝑉₁≤−𝑐₁𝑧⁴₁−𝜈₁𝑧₁𝑧⁴₁−𝑎₀₀|𝑒|⁴+𝑎₁₁𝑒⁴₁+Δ₁𝜂+𝑎₁₀𝑧⁴₂+1Γ𝜃𝜃̇−𝜏₁, (4.16) where 𝑐₁>0 is design parameter and 𝜈₁(𝑧₁)>0 is a smooth function to be chosen later.

Step i [𝑖=2,…,𝑛]
For notation coherence, denote 𝑢=𝛼_𝑛, 𝑧_𝑛+1=0. At this step, we can obtain a property similar to (4.16), which is presented by the following lemma.

Lemma 4.1. For the 𝑖th Lyapunov function 𝑉_𝑖(𝑒,𝑧_𝑖,𝜃)=(𝛿/2)(𝑒^𝑇𝑃𝑒)²+(1/2Γ)𝜃²+∑^𝑖_𝑗=1(𝑧⁴_𝑗/4), there are the virtual control law 𝛼_𝑖(𝑦,𝑥₁,…,𝑥_𝑖−1,𝜃) and the tuning function 𝜏_𝑖 with the form 𝛼_𝑖𝑦,𝑥₁,…,𝑥_𝑖−1,𝜃=−Ω_𝑖−𝑧_𝑖𝑐_𝑖+𝑎_𝑖−1,0+𝑎_𝑖0+𝜃𝛾_𝑖1+𝛾_𝑖2+𝛾_𝑖3+𝛾_𝑖4+Γ𝛾_𝑖1+𝛾_𝑖2+𝛾_𝑖3+𝛾_𝑖4^𝑖−1_𝑗=1𝜕𝛼_𝑗𝜕𝜃𝑧³_𝑗+1+𝜕𝛼_𝑖−1𝜕𝜃𝜏_𝑖,𝜏_𝑖𝑦,𝑥₁,…,𝑥_𝑖−1,𝜃=𝜏_𝑖−1+Γ𝛾_𝑖1+𝛾_𝑖2+𝛾_𝑖3+𝛾_𝑖4𝑧⁴_𝑖, (4.17) such that ℒ𝑉_𝑖≤−^𝑖_𝑗=1𝑐_𝑗𝑧⁴_𝑗+^𝑖_𝑗=2𝑏_𝑗𝑧⁴₁−𝜈₁𝑧₁𝑧⁴₁−𝑎₀₀|𝑒|⁴+^𝑖_𝑗=1𝑎_𝑗1𝑒⁴₁+Δ_𝑖𝜂+𝑎_𝑖0𝑧⁴_𝑖+1+1Γ𝜃−^𝑖−1_𝑗=1𝜕𝛼_𝑗𝜕𝜃𝑧³_𝑗+1𝜃̇−𝜏_𝑖, (4.18) where 𝑐_𝑖>0 is the designed parameters, 𝑎_𝑖0, 𝑎_𝑖−1,0, 𝑎_𝑖1, 𝑏_𝑖 are some positive constants, 𝛾_𝑖1, 𝛾_𝑖2, 𝛾_𝑖3, 𝛾_𝑖4 are smooth nonnegative functions, whose choices are given in the proof.

Proof. See the appendix.

Therefore, at the end of the recursive procedure, the controller can be constructed as𝑢=𝛼_𝑛𝑦,𝑥₁,…,𝑥_𝑛−1,𝜃,𝜃̇=𝜏_𝑛𝑦,𝑥₁,…,𝑥_𝑛−1,𝜃. (4.19) Choosing parameters 𝛿, 𝑎₀₁, 𝑎₀₂, 𝑎₁₁, …, 𝑎_𝑛1, 𝑐₂,…,𝑐_𝑛 to satisfy𝑎₀=𝛿𝜆_min(𝑃)−𝑎₀₁−𝑎₀₂−^𝑛_𝑖=1𝑎_𝑖1>0,𝑐₂,…,𝑐_𝑛>0, (4.20) by (4.16) and (4.18), one hasℒ𝑉_𝑛≤−^𝑛_𝑖=1𝑐_𝑖𝑧⁴_𝑖+^𝑛_𝑗=2𝑏_𝑗𝑧⁴₁−𝜈₁𝑧₁𝑧⁴₁−𝑎₀|𝑒|⁴+Δ_𝑛𝜂, (4.21) where𝑉_𝑛𝜃=𝛿𝑒,𝑧,2𝑒^𝑇𝑃𝑒²+1𝜃2Γ²+14^𝑛_𝑖=1𝑧⁴_𝑖,Δ_𝑛𝜂=64(𝑛−1)𝑎₀₁𝑘⁴₁+…+𝑘⁴_𝑛−1𝜑⁴₁₀𝜂+𝜑⁴₂₀𝜂+…+𝜑⁴_𝑛0𝜂+64(𝑛−1)𝑎₀₂𝑘⁴₁+…+𝑘⁴_𝑛−1𝜓⁴₁₀𝜂+𝜓⁴₂₀𝜂+…+𝜓⁴_𝑛0𝜂+^𝑛_𝑖=1𝑎_𝑖2𝜑⁴₁₀𝜂+𝑎_𝑖3𝜓⁴₁₀𝜂+^𝑛_𝑗=2𝑎_𝑗4𝜓⁴₁₀𝜂. (4.22)

5. Stability Analysis

We state the main theorems in this paper. This section is divided into two parts.

5.1. Case I

Theorem 5.1. Assume that Assumptions 3.1 and 3.2 hold with the following properties: limsup_𝑠→0+Δ_𝑛𝜂𝛼(s)<∞,limsup_𝑠→∞Δ_𝑛𝜂𝛼(𝑠)<∞. (5.1) If limsup_𝑠→0+𝛾(𝑠)/𝑠⁴<∞ in Assumption 3.2, by appropriately choosing the positive smooth function 𝜈₁(⋅) in (4.15) and the parameters 𝛿, 𝑎₀₁, 𝑎₀₂, 𝑎₁₁, …, 𝑎_𝑛1, 𝑐₂, …, 𝑐_𝑛 to satisfy (4.20), then(i)the closed-loop system consisting of (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), and (4.19) has a unique and almost surely bounded strong solution on [0,∞);(ii)for each initial value (𝜂(0),𝑥(0),𝑥(0),𝜃(0)), the equilibrium (𝜂,𝑥)=(0,0) is globally stable in probability, where 𝑥=(𝑥₁,…,𝑥_𝑛−1).

Proof. For any constant 𝜖>0, by (5.1), one has limsup_𝑠→0+(1+𝜖)Δ_𝑛𝜂𝛼(𝑠)<∞,limsup_𝑠→∞(1+𝜖)Δ_𝑛𝜂𝛼(𝑠)<∞. (5.2) For the 𝜂-subsystem, by Lemma 2.4, there exists a positive and radially unbounded Lyapunov function 𝑉₀(𝜂)∈𝐶² and 𝛾=𝜌𝛾 such that ℒ𝑉₀𝜂≤−(1+𝜖)Δ_𝑛𝜂+𝛾𝑧₁,limsup_𝑠→0+𝛾(𝑠)𝑠⁴<∞, (5.3)(5.4) where 𝜌 is a positive constant satisfying (1+𝜖)Δ_𝑛(|𝜂|)≤𝜌𝛼(𝑠) for all 𝑠≥0. Choosing the following Lyapunov function for the entire closed-loop system 𝑉𝜂,𝑒,𝑧,𝜃=𝑉₀𝜂+𝑉_𝑛𝑒,𝑧,𝜃 (5.5) and combining (4.21) and (5.3), one obtains ℒ𝑉𝜂,𝑒,𝑧,𝜃≤−^𝑛_𝑖=1𝑐_𝑖𝑧⁴_𝑖−𝜈₁𝑧₁−^𝑛_𝑗=2𝑏_𝑗𝑧⁴₁−𝑎₀|𝑒|⁴−𝜖Δ_𝑛𝜂+𝛾𝑧₁. (5.6) By (5.4), there always exists a smooth function 𝜈₁(⋅) to satisfy the following two inequalities: 𝑧⁴₁𝜈₁𝑧₁−^𝑛_𝑗=2𝑏_𝑗≥𝛾𝑧₁,𝜈₁𝑧₁≥^𝑛_𝑗=2𝑏_𝑗. (5.7) Substituting (5.7) into (5.6) leads to ℒ𝑉𝜂,𝑒,𝑧,𝜃≤−^𝑛_𝑖=1𝑐_𝑖𝑧⁴_𝑖−𝑎₀|𝑒|⁴−𝜖Δ_𝑛𝜂≜−𝑊𝜂,𝑒,𝑧. (5.8) Noting that 𝜑_𝑖0(⋅) and 𝜓_𝑖0(⋅) are nonnegative smooth functions and using (4.22), it follows that Δ_𝑛(⋅) and 𝑊(⋅) are continuous nonnegative functions. By (5.5), (5.8), and Lemma 2.2, one concludes that all the solutions of the closed-loop system are bounded almost surely, the equilibrium (𝜂,𝑒,𝑧)=(0,0,0) is globally stable in probability. By (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), (4.19), and the almost sure boundedness of all the signals, it is not difficult to recursively prove that the equilibrium (𝜂,𝑥)=(0,0) is globally stable in probability.

5.2. Case II

If more information about 𝛼 in Assumption 3.2 is known, that is, liminf_𝑠→∞𝛼(𝑠)=∞, further results under the weaker conditions is given as follows.

Assumption 5.2. For functions 𝜓₀ and 𝑉₀ given by (4.22) and Assumption 3.2, there exist known smooth nonnegative functions 𝜓₁ and 𝜓₂ with 𝜓₁(0)=𝜓₂(0)=0, such that ‖𝜓₀(𝜂,𝑥₁)‖≤𝜓₁(|𝜂|) and |𝜕𝑉₀(𝜂)/𝜕𝜂|≤𝜓₂(|𝜂|).

Lemma 5.3. For Δ_𝑛, 𝛼, 𝛼, 𝜓₁ and 𝜓₂ given by (4.22), Assumptions 3.2 and 5.2, if limsup_𝑠→0+Δ_𝑛(𝑠)+𝜓²₁(𝑠)𝜓²₂(𝑠)𝛼(𝑠)<∞,^∞₀𝑒^{−∫s0(1/𝜁1(𝛼−1(𝜏)))𝑑𝜏}𝜉₁𝛼⁻¹(𝑠)^′𝑑𝑠<∞, (5.9) where 𝜉₁(⋅)≥0 and 𝜁₁(⋅)>0 are smooth increasing functions satisfying 𝜉₁(𝑠)𝛼(𝑠)≥2(1+𝜖)Δ_𝑛(𝑠),𝜁₁(𝑠)𝛼(𝑠)≥𝜓²₁(𝑠)𝜓²₂(𝑠),∀𝑠≥0, (5.10)𝜖 is any positive constant. Then there exists a function 𝛾∈𝒦_∞ such that ((1+𝜖)Δ_𝑛,𝛾) is a new SiISS supply rate of the 𝜂-subsystem in (3.1). Moreover, if 𝛾 in Assumptions 3.2 satisfies limsup_𝑠→0+𝛾(𝑠)/𝑠⁴<∞, then limsup_𝑠→0+𝛾(𝑠)/𝑠⁴<∞.

Since the condition (5.9) is weaker than (5.1), by using Lemma 5.3, we give further results under the weaker condition (5.9).

Theorem 5.4. Suppose that Assumptions 3.1–5.2 and the conditions of Lemma 5.3 hold. If limsup_𝑠→0+𝛾(𝑠)/𝑠⁴<∞ in Assumption 3.2, by appropriately choosing the positive smooth function 𝜈₁(⋅) in (4.15) and the parameters 𝛿, 𝑎₀₁, 𝑎₀₂, 𝑎₁₁, …, 𝑎_𝑛1, 𝑐₂, …, 𝑐_𝑛 to satisfy (4.20), then(1)the closed-loop system consisting of (3.1), (4.1), (4.2), (4.7), (4.15), (4.17), and (4.19) has a unique and almost surely bounded strong solution on [0,∞);(2)for each initial value (𝜂(0),𝑥(0),𝑥(0),𝜃(0)), the equilibrium (𝜂,𝑥)=(0,0) is globally stable in probability.

6. Conclusions

This paper further considers a more general class of stochastic nonlinear systems with uncertain parameters and SiISS inverse dynamics. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is designed to guarantee that all the closed-loop signals are bounded almost surely and the stochastic closed-loop system is globally stable in probability.

There are two remaining problems to be investigated: (1) an essential problem is to find a practical example with explicit physical meaning for system (3.1). A preliminary attempt on high-order stochastic nonlinear system can be found in [28]. (2) How to design an output feedback controller by using this method in this paper for system (3.1) in which the drift and diffusion vector fields depend on the unmeasurable states besides the measurable output?

Appendix

Proof of Lemma 4.1. We prove Lemma 4.1 by induction. Assume that at Step 𝑖−1, there are virtual control laws 𝛼_𝑖−1𝑦,𝑥₁,…,𝑥_𝑖−2,𝜃=−Ω_𝑖−1−𝑧_𝑖−1×𝑐_𝑖−1+𝑎_𝑖−2,0+𝑎_𝑖−1,0+𝑎_𝑖−1,1+𝜃𝛾_𝑖−1,1+𝛾_𝑖−1,2+𝛾_𝑖−1,3+𝛾_𝑖−1,4+Γ𝛾_𝑖−1,1+𝛾_𝑖−1,2+𝛾_𝑖−1,3+𝛾_𝑖−1,4+^𝑖−2_𝑗=1𝜕𝛼_𝑗𝜕𝜃𝑧³_𝑗+1+𝜕𝛼_𝑖−2𝜕𝜃𝜏_𝑖−1,𝜏_𝑖−1𝑦,𝑥₁,…,𝑥_𝑖−2,𝜃=𝜏_𝑖−2+Γ𝛾_𝑖−1,1+𝛾_𝑖−1,2+𝛾_𝑖−1,3+𝛾_𝑖−1,4𝑧⁴_𝑖−1, (A.1) such that 𝑉_𝑖−1(𝑒,𝑧_𝑖−1,𝜃)=(𝛿/2)(𝑒^𝑇𝑃𝑒)²+(1/2Γ)𝜃²+∑^𝑖−1_𝑗=1(𝑧⁴_𝑗/4) satisfies ℒ𝑉_𝑖−1≤−^𝑖−1_𝑗=1𝑐_𝑗𝑧⁴_𝑗+^𝑖−1_𝑗=2𝑏_𝑗𝑧⁴₁−𝜈₁𝑧₁𝑧⁴₁−𝑎₀₀|𝑒|⁴+^𝑖−1_𝑗=1𝑎_𝑗1𝑒⁴₁+Δ_𝑖−1𝜂+𝑎_𝑖−1,0𝑧⁴_𝑖+1Γ𝜃−^𝑖−2_𝑗=1𝜕𝛼_𝑗𝜕𝜃𝑧³_𝑗+1𝜃̇−𝜏_𝑖−1, (A.2) where 𝑐_𝑖−1>0 are the designed parameters, 𝑎_𝑖−1,0, 𝑎_𝑖−1,0, 𝑎_𝑖−1,1, 𝑏_𝑖−1 are some positive constants, and 𝛾_𝑖−1,1, 𝛾_𝑖−1,2, 𝛾_𝑖−1,3, 𝛾_𝑖−1,4 are smooth nonnegative functions.In the sequel, we will prove that Lemma 4.1 still holds for Step 𝑖. Choosing 𝑉_𝑖𝑒,𝑧_𝑖,𝜃=𝑉_𝑖−1𝑒,𝑧_𝑖−1,𝜃+14𝑧⁴_𝑖, (A.3) with the use of (4.6), (4.7), (4.15), and (A.1), the ItÔ differential of 𝑧_𝑖 is given as follows: 𝑑𝑧_𝑖=𝛼_𝑖+𝑧_𝑖+1+Ω_𝑖−𝜕𝛼_𝑖−1𝜃𝜕𝑦^∗𝑒₁−𝜕𝛼_𝑖−1𝜑𝜕𝑦¹𝜂,𝑥−𝜕𝛼_𝑖−1𝜕𝜃𝜃̇−1𝑑𝑡2𝜕²𝛼_𝑖−1𝜕𝑦²𝜓₁𝜂,𝑥𝜓^𝑇₁𝜂,𝑥𝑑𝑡−𝜕𝛼_𝑖−1𝜓𝜕𝑦¹𝜂,𝑥𝑑𝑤, (A.4) where Ω_𝑖=𝑘_𝑖𝑦−𝑘_𝑖−1(𝑥₁+𝑘₁𝑦)  −∑^𝑖−2_𝑗=1(𝜕𝛼_𝑖−1/𝜕𝑥_𝑗)(𝑥_𝑗+1+𝑘_𝑗+1𝑦−𝑘_𝑗(𝑥₁+𝑘₁𝑦))−(𝜕𝛼_𝑖−1/𝜕𝑦)(𝑥₁+𝑘₁𝑦). Using (2.2) and (A.2)–(A.4), we arrive at ℒ𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_𝑗𝑧⁴_𝑗+^𝑖−1_𝑗=2𝑏_𝑗𝑧⁴₁−𝜈₁𝑧₁𝑧⁴₁−𝑎₀₀|𝑒|⁴+^𝑖−1_𝑗=1𝑎_𝑗1𝑒⁴₁+Δ_𝑖−1𝜂+𝑎_𝑖−1,0𝑧⁴_𝑖+1Γ𝜃−^𝑖−2_𝑗=1𝜕𝛼_𝑗𝜕𝜃𝑧³_𝑗+1𝜃̇−𝜏_𝑖−1+𝑧³_𝑖𝛼_𝑖+Ω_𝑖−𝜕𝛼_𝑖−1𝜕𝜃𝜃̇+𝑧_𝑖+1−𝜕𝛼_𝑖−1𝜃𝜕𝑦^∗𝑒₁−𝜕𝛼_𝑖−1𝜑𝜕𝑦¹−1𝜂,𝑥2𝜕²𝛼_𝑖−1𝜕𝑦²𝜓₁𝜂,𝑥𝜓^𝑇₁+3𝜂,𝑥2𝑧²_𝑖Tr𝜕𝛼_𝑖−1𝜕𝑦²𝜓^𝑇₁𝜂,𝑥𝜓₁𝜂,𝑥. (A.5) Now, we estimate the last five terms, respectively, in the right-hand side of (A.5). According to Assumption 3.1, (3.2), (4.7), and Lemma 2.6, there exist positive real numbers 𝑎_𝑖0, 𝑎_𝑖0, 𝑎_𝑖1, 𝑎_𝑖2, 𝑎_𝑖3, 𝑎_𝑖4, 𝑏_𝑖2, 𝑏_𝑖3, 𝑏_𝑖4, smooth nonnegative functions 𝛾_𝑖1, 𝛾_𝑖2, 𝛾_𝑖3, 𝛾_𝑖4 such that 𝑧³_𝑖𝑧_𝑖+1≤𝑎_𝑖0𝑧⁴_𝑖+𝑎_𝑖0𝑧⁴_𝑖+1,−𝑧³_𝑖𝜕𝛼_𝑖−1𝜃𝜕𝑦^∗𝑒₁≤𝑎_𝑖1𝑒⁴₁+𝜃𝛾_𝑖1𝑧_𝑖𝑧⁴_𝑖,−𝑧³_𝑖𝜕𝛼_𝑖−1𝜑𝜕𝑦¹𝜂,𝑥≤𝑧_𝑖³1+𝜕𝛼_𝑖−1𝜕𝑦^21/2𝜑₁₀𝜂+𝜑₁₁𝑧₁≤𝑎_𝑖2𝜑⁴₁₀𝜂+𝑏_𝑖2𝑧⁴₁+𝜃𝛾_𝑖2𝑧_𝑖𝑧⁴_𝑖,−12𝜕²𝛼_𝑖−1𝜕𝑦²𝑧³_𝑖𝜓₁𝜂,𝑥𝜓^𝑇₁𝜂,𝑥≤𝑧_𝑖³𝜕²𝛼_𝑖−1𝜕𝑦²𝜓²₁₀𝜂+𝜓²₁₁𝑧₁≤𝑎_𝑖3𝜓⁴₁₀𝜂+𝑏_𝑖3𝑧⁴₁+𝜃𝛾_𝑖3𝑧_𝑖𝑧⁴_𝑖,32𝑧²_𝑖Tr𝜕𝛼_𝑖−1𝜕𝑦²𝜓^𝑇₁𝜂,𝑥𝜓₁𝜂,𝑥≤3𝑧²_𝑖𝜕𝛼_𝑖−1𝜕𝑦²𝜓²₁₀𝜂+𝜓²₁₁𝑧₁≤𝑎_𝑖4𝜓⁴₁₀𝜂+𝑏_𝑖4𝑧⁴₁+𝜃𝛾_𝑖4𝑧_𝑖𝑧⁴_𝑖. (A.6) Choosing 𝛼_𝑖 and 𝜏_𝑖 as (4.17) and substituting (A.6) into (A.5), (4.18) holds, where 𝑐_𝑖>0 is a design parameter, Δ_𝑖𝜂=Δ_𝑖−1𝜂+𝑎_𝑖2𝜑⁴₁₀𝜂+𝑎_𝑖3𝜓⁴₁₀𝜂+𝑎_𝑖4𝜓⁴₁₀𝜂,𝑏_𝑖=𝑏_𝑖2+𝑏_𝑖3+𝑏_𝑖4. (A.7)

Acknowledgments

This work was supported by National Natural Science Foundation of China (nos. 61104222 and 10971256), Natural Science Foundation of Jiangsu Province (no. BK2011205), and Natural Science Foundation of Jiangsu Normal University (no. 11XLR08).