Adaptive State-Feedback Stabilization for High-Order Stochastic Nonlinear Systems Driven by Noise of Unknown Covariance

Abstract

This paper further considers more general high-order stochastic nonlinear system driven by noise of unknown covariance and its adaptive state-feedback stabilization problem. A smooth state-feedback controller is designed to guarantee that the origin of the closed-loop system is globally stable in probability.

1. Introduction

In this paper, we consider the following high-order stochastic nonlinear system:𝑑𝑥₁=𝑥^𝑝₂𝑑𝑡+𝑓₁𝑥₁𝑑𝑡+𝑔₁𝑥₁Σ𝑑𝜔,𝑑𝑥₂=𝑥^𝑝₃𝑑𝑡+𝑓₂𝑥₂𝑑𝑡+𝑔₂𝑥₂⋮Σ𝑑𝜔,𝑑𝑥_𝑛=𝑢^𝑝𝑑𝑡+𝑓_𝑛𝑥_𝑛𝑑𝑡+𝑔_𝑛𝑥_𝑛Σ𝑑𝜔, (1.1) where 𝑥=(𝑥₁,…,𝑥_𝑛)∈𝑅^𝑛, 𝑢∈𝑅, are the state and control input, respectively. 𝑥_𝑖=(𝑥₁,…,𝑥_𝑖), 𝑖=1,…,𝑛. 𝑝≥1 is odd integer. 𝑤 is an 𝑟-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. Σ∶𝑅₊→𝑅^𝑟×𝑟 is an unknown bounded nonnegative definite Borel measurable matrix function and ΣΣ^𝑇 denotes the infinitesimal covariance function of the driving noise Σ𝑑𝜔. 𝑓_𝑖∶𝑅^𝑖→𝑅 and 𝑔_𝑖∶𝑅^𝑖→𝑅^𝑟, 𝑖=1,…,𝑛, are assumed to be smooth with 𝑓_𝑖(0)=0 and 𝑔_𝑖(0)=0.

When 𝑝=1, system (1.1) reduced to the well-known normal form whose study on feedback control problem has achieved great development in recent years. According to the difference of selected Lyapunov functions, the existing literature on controller design can be mainly divided into two types. One type is based on quadratic Lyapunov functions which are multiplied by different weighting functions, see, for example, [1–5] and the references therein. Another essential improvement belongs to Krstić and Deng. By introducing the quartic Lyapunov function, [6, 7] presented asymptotical stabilization control in the large under the assumption that the nonlinearities equal to zero at the equilibrium point of the open-loop system. Subsequently, for several classes of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions, by combining Krstić and Deng’s method with stochastic small-gain theorem [8], and with dynamic signal and changing supply function [9, 10], different adaptive output-feedback control schemes are studied.

When 𝑝>1, some intrinsic features of (1.1), such as that its Jacobian linearization is neither controllable nor feedback linearizable, lead to the existing design tools hardly applicable to this kind of systems. Motivated by the fruitful deterministic results in [11, 12] and the related papers and based on stochastic stability theory in [13–15], and so forth, [16] firstly considered, this class of systems with stochastic inverse dynamics. Subsequently, [17–21] considered respectively, the state-feedback stabilization problem for more general systems with different structures. [22, 23] solved the output-feedback stabilization, and [24] addressed the inverse optimal stabilization.

All the papers mentioned above, however, only consider the case of ΣΣ^𝑇=𝐼. In this paper, we will further consider more general high-order stochastic nonlinear system driven by noise of unknown covariance and its adaptive state-feedback stabilization problem. A smooth state-feedback controller is designed to guarantee that the origin of the closed-loop system is globally stable in probability. A simulation example verifies the effectiveness of the control scheme.

The paper is organized as follows. Section 2 provides some preliminary results. Section 3 gives the state-feedback controller design and stability analysis, following a simulation example in Section 4. In Section 5, we conclude the paper.

2. Preliminary Results

The following notations definitions and lemmas are to be used throughout the paper.

𝑅₊ stands for the set of all nonnegative real numbers, 𝑅^𝑛 is the 𝑛-dimensional Euclidean space, 𝑅^𝑛×𝑚 is the space of real 𝑛×𝑚-matrixes. 𝒞² denotes the family of all the functions with continuous second partial derivatives. |𝑥| is the usual Euclidean norm of a vector 𝑥. ‖𝑋‖=(Tr{𝑋𝑋^𝑇})^1/2, where Tr{𝑋} is its trace when 𝑋 is a square matrix and 𝑋^𝑇 denotes the transpose of 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 class 𝒦 for each fixed 𝑡 and decrease to zero as 𝑡→∞ for each fixed 𝑠. To simplify the procedure, we sometimes denote 𝜒(𝑡) by 𝜒 for any variable 𝜒(𝑡).

Consider the nonlinear stochastic system 𝑑𝑥=𝑓(𝑥)𝑑𝑡+𝑔(𝑥)𝑑𝜔, (2.1) where 𝑥∈𝑅^𝑛 is the state, 𝑤 is an 𝑟-dimensional independent Wiener process with incremental covariance ΣΣ^𝑇𝑑𝑡, that is, 𝐸{𝑑𝜔𝑑𝜔^𝑇}=ΣΣ^𝑇𝑑𝑡, where Σ is a bounded function taking values in the set of nonnegative definite matrices, 𝑓∶𝑅^𝑛→𝑅^𝑛 and 𝑔∶𝑅^𝑛→𝑅^𝑛×𝑟 are locally Lipschitz functions.

Definition 2.1 (see [13]). For any given 𝑉(𝑥)∈𝒞² associated with stochastic system (2.1), the differential operator ℒ is defined as (𝑥)≜(𝑥)ℒ𝑉𝜕𝑉𝑓(𝑥)+1𝜕𝑥2Tr𝑔^𝑇(𝑥)𝜕²𝑉(𝑥)𝜕𝑥²𝑔(𝑥). (2.2)

Definition 2.2 (see [25]). For the stochastic system (2.1) with 𝑓(0)=0, 𝑔(0)=0, the equilibrium 𝑥(𝑡)=0 is globally asymptotically stable (GAS) in probability if for any 𝜉>0, there exists a class 𝒦ℒ function 𝛽(⋅,⋅) such that 𝑃|𝑥(𝑡)|<𝛽𝑥₀,𝑡≥1−𝜉,𝑡≥0,∀𝑥₀∈𝑅^𝑛⧵{0}. (2.3)

Lemma 2.3 (see [14]). Consider the stochastic system (2.1). If there exist a 𝒞² function 𝑉(𝑥), class 𝒦_∞ function 𝛼₁ and 𝛼₂, constants 𝑐₁>0 and 𝑐₂≥0, and a nonnegative function 𝑊(𝑥) such that for all 𝑥∈𝑅^𝑛, 𝑡≥0𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂(|𝑥|),ℒ𝑉≤−𝑐₁𝑊(𝑥)+𝑐₂, (2.4) then,(a)there exists an almost surely unique solution on [0,∞) for each 𝑥₀∈𝑅^𝑛,(b)when 𝑐₂=0, 𝑓(0)=0, 𝑔(0)=0, and 𝑊(𝑥) is continuous, then the equilibrium 𝑥=0 is globally stable in probability and the solution x(𝑡) satisfies 𝑃{lim_𝑡→∞𝑊(𝑥(𝑡))=0}=1.

Lemma 2.4 (see [12]). Let 𝑥,𝑦 be real variables, for any positive integers 𝑚,𝑛, positive real number 𝑏 and nonnegative continuous function 𝑎(⋅), then 𝑎(⋅)𝑥^𝑚𝑦^𝑛≤b|𝑥|^𝑚+𝑛+𝑛𝑚+𝑛𝑚+𝑛𝑚^−𝑚/𝑛(𝑎(⋅))^{(𝑚+𝑛)/𝑛}𝑏^−𝑚/𝑛𝑦^𝑚+𝑛, (2.5) when 𝑎(⋅)=1,𝑏=(𝑚/(𝑚+𝑛))𝑑, 𝑑 is a positive constant, then the above inequality is 𝑥^𝑚𝑦^𝑛≤𝑚𝑑|𝑥|𝑚+𝑛^𝑚+𝑛+𝑛𝑑𝑚+𝑛^−𝑚/𝑛𝑦^𝑚+𝑛. (2.6)

Lemma 2.5 (see [12]). Let 𝑥,𝑦 and 𝑧_𝑖,  𝑖=1,…,𝑝, be real variables and let 𝑙₁(⋅) and 𝑙₂(⋅) be smooth mappings. Then for any positive integers 𝑚, 𝑛 and real number 𝑁>0, there exist two nonnegative smooth functions ℎ₁(⋅) and ℎ₂(⋅) such that the following inequalities hold:(i)|𝑥^𝑚[(𝑦+𝑥𝑙₁(𝑥))^𝑛−(𝑥𝑙₁(𝑥))^𝑛]|≤|𝑥|^𝑚+𝑛/𝑁+|𝑦|^𝑚+𝑛ℎ₁(𝑥,𝑦), (ii)|𝑦^𝑚(𝑧^𝑛₁+⋯+𝑧^𝑛_𝑝+𝑦^𝑛)𝑙₂(𝑧₁,…,𝑧_𝑝,𝑦)|≤(1/𝑁)∑^𝑝_𝑘=1|𝑧_𝑘|^𝑚+𝑛+|𝑦|^𝑚+𝑛ℎ₂(𝑧₁,…,𝑧_𝑝,𝑦).

Lemma 2.6 (see [12]). Let 𝑥₁,…,𝑥_𝑛, 𝑝, be positive real variables, then 𝑥₁+⋯+𝑥_𝑛^𝑝≤max𝑛^𝑝−1,1𝑥^𝑝₁+⋯+𝑥^𝑝_𝑛. (2.7)

3. Controller Design and Stability Analysis

The objective of this paper is to design a smooth state-feedback controller for system (1.1), such that the solution of the closed-loop system is GAS in probability. To achieve the control objective, we need the following assumption.

Assumption 3.1. There are nonnegative smooth functions 𝑓_𝑖1, 𝑔_𝑖1,  𝑖=1,…,𝑛, such that 𝑓_𝑖𝑥_𝑖≤^𝑖_𝑗=1𝑥_𝑗^𝑝𝑓_𝑖1𝑥_𝑖,𝑔_𝑖𝑥_𝑖≤^𝑖_𝑗=1𝑥_𝑗^𝑝𝑔_𝑖1𝑥_𝑖. (3.1)

3.1. Controller Design

Now, we give the controller design procedure by using the backstepping method.

First, we introduce the following coordinate change: 𝑧₁=𝑥₁,𝑧_𝑖=𝑥_𝑖−𝛼_𝑖−1𝑥_𝑖−1,𝜃,𝑖=2,…,𝑛, (3.2) where 𝛼_𝑖−1(𝑥_𝑖−1,𝜃), 𝑖=2,…,𝑛, are smooth virtual controllers which will be designed later, 𝜃 is the estimation of 𝜃, and 𝜃≜max_𝑡≥0ΣΣ^{𝑇(𝑝+3)/2},ΣΣ^{𝑇(𝑝+3)/3},ΣΣ^𝑇. (3.3) Then, by Itô’s differentiation rule, one has𝑑𝑧_𝑖=𝑑𝑥_𝑖−𝛼_𝑖−1𝑥_𝑖−1,𝜃=𝑥^𝑝_𝑖+1+𝐹_𝑖𝑥_𝑖−^𝑖−1_𝑙=1𝜕𝛼_𝑖−1𝜕𝑥_𝑙𝑥^𝑝_𝑙+1−12^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖−1𝜕𝑥_𝑘𝜕𝑥_𝑚𝑔_𝑘𝑥_𝑘ΣΣ^𝑇𝑔^𝑇_𝑚𝑥_𝑚−𝜕𝛼_𝑖−1𝜕𝜃𝜃̇𝑑𝑡+𝐺_𝑖𝑥_𝑖Σ𝑑𝜔, (3.4) where 𝐹_𝑖𝑥_𝑖=𝑓_𝑖𝑥_𝑖−^𝑖−1_𝑙=1𝜕𝛼_𝑖−1𝜕𝑥_𝑙𝑓_𝑙𝑥_𝑙,𝐺_𝑖𝑥_𝑖=𝑔_𝑖𝑥_𝑖−^𝑖−1_𝑙=1𝜕𝛼_𝑖−1𝜕𝑥_𝑙𝑔_𝑙𝑥_𝑙. (3.5)

Step 1. Define the first Lyapunov function 𝑉₁𝑧₁,𝜃=14𝑧⁴₁+1𝜃2Γ², (3.6) where Γ is a positive constant, 𝜃=𝜃−𝜃 is the parameter estimation error. By (3.2)–(3.4) and Assumption 3.1, there exist nonnegative smooth functions 𝜇₁₁ and 𝜇₁₅ such that ℒ𝑉₁=𝑧³₁𝑥^𝑝₂+𝑧³₁𝑓₁𝑥₁+32𝑧²₁𝑔₁𝑥₁ΣΣ^𝑇𝑔^𝑇₁𝑥₁−𝜃Γ𝜃̇≤𝑧³₁𝑥^𝑝₂−𝛼^𝑝₁+𝑧³₁𝛼^𝑝₁+𝑧₁^𝑝+3𝜇₁₁𝑧₁+𝑧₁^𝑝+3𝜇₁₅𝑧₁𝜃𝜃−Γ𝜃̇≤𝑧³₁𝑥^𝑝₂−𝛼^𝑝₁+𝑧³₁𝛼^𝑝₁+𝑧₁^𝑝+3𝜇₁₁𝑧₁+𝑧₁^𝑝+3𝜇₁₅𝑧₁1+𝜃²−𝜃Γ𝜃̇−Γ𝑧₁^𝑝+3𝜇₁₅𝑧₁. (3.7) Choosing the first smooth virtual controller 𝛼₁𝑥₁,𝜃=−𝑧₁𝛽₁𝑧₁,𝜃,𝛽₁𝑧₁,𝜃=𝑐₁₁+𝜇₁₁(𝑧₁)+𝜇₁₅(𝑧₁)1+𝜃^21/𝑝, (3.8) and the tuning function 𝜏₁𝑧₁=Γ𝑧₁^𝑝+3𝜇₁₅𝑧₁, (3.9) one has ℒ𝑉₁≤−𝑐₁₁𝑧₁^𝑝+3+𝑧³₁𝑥^𝑝₂−𝛼^𝑝₁−𝜃Γ𝜃̇−𝜏₁, (3.10) where 𝑐₁₁>0 is a design parameter.

Step i (2≤𝑖≤𝑛). For notational coherence, denote 𝑢=𝑥_𝑛+1. Assuming that at step 𝑖−1, one has ℒ𝑉_𝑖−1≤−^𝑖−1_𝑗=1𝑐_{𝑗,𝑖−1}𝑧_𝑗^𝑝+3−𝜃Γ+^𝑖−1_𝑗=2𝑧³_𝑗𝜕𝛼_𝑗−1𝜕𝜃𝜃̇−𝜏_𝑖−1+𝑧³_𝑖−1𝑥^𝑝_𝑖−𝛼^𝑝_𝑖−1, (3.11) where 𝜏_𝑖−1=𝜏_𝑖−2+Γ𝑧^𝑝+3_𝑖−1(𝜇_𝑖−1,4+𝜇_𝑖−1,5). In the sequel, we will prove that (3.11) still holds for the 𝑖th Lyapunov function 𝑉_𝑖(𝑧_𝑖,𝜃)=𝑉_𝑖−1(𝑧_𝑖−1,𝜃)+(1/4)𝑧⁴_𝑖. By (3.4) and (3.11), one has ℒ𝑉_𝑖≤ℒ𝑉_𝑖−1+𝑧³_𝑖𝑥^𝑝_𝑖+1+𝐹_𝑖𝑥_𝑖−^𝑖−1_𝑙=1𝜕𝛼_𝑖−1𝜕𝑥_𝑙𝑥^𝑝_𝑙+1−12^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖−1𝜕𝑥_𝑘𝜕𝑥_𝑚𝑔_𝑘𝑥_𝑘ΣΣ^𝑇𝑔^𝑇_𝑚𝑥_𝑚−𝑧³_𝑖𝜕𝛼_𝑖−1𝜕𝜃𝜃̇+32𝑧²_𝑖TrΣ^𝑇𝐺^𝑇_𝑖𝑥_𝑖𝐺_𝑖𝑥_𝑖Σ≤−^𝑖−1_𝑗=1𝑐_{𝑗,𝑖−1}𝑧_𝑗^𝑝+3−𝜃Γ+^𝑖−1_𝑗=2𝑧³_𝑗𝜕𝛼_𝑗−1𝜕𝜃𝜃̇−𝜏_𝑖−1+𝑧³_𝑖−1𝑥^𝑝_𝑖−𝛼^𝑝_𝑖−1+𝑧³_𝑖𝑥^𝑝_𝑖+1+𝐹_𝑖𝑥_𝑖−^𝑖−1_𝑙=1𝜕𝛼_𝑖−1𝜕𝑥_𝑙𝑥^𝑝_𝑙+1−12^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖−1𝜕𝑥_𝑘𝜕𝑥_𝑚𝑔_𝑘𝑥_𝑘ΣΣ^𝑇𝑔^𝑇_𝑚𝑥_𝑚−𝑧³_𝑖𝜕𝛼_𝑖−1𝜕𝜃𝜃̇+32𝑧²_𝑖TrΣ^𝑇𝐺^𝑇_𝑖𝑥_𝑖𝐺_𝑖𝑥_𝑖Σ+𝑧³_𝑖𝜕𝛼_𝑖−1𝜕𝜃𝜏_𝑖−1−𝑧³_𝑖𝜕𝛼_𝑖−1𝜕𝜃𝜏_𝑖−1. (3.12) To proceed further, an estimate for each term in the right-hand side of (3.12) is needed. Using Itô’s differentiation rule, Lemmas 2.4–2.6, (3.2), and (3.3), it follows that 𝑧³_𝑖−1𝑥^𝑝_𝑖−𝛼^𝑝_𝑖−1=𝑧³_𝑖−1𝑧_𝑖+𝛼_𝑖−1^𝑝−𝛼^𝑝_𝑖−1≤𝜉_𝑖1𝑧^𝑝+3_𝑖−1+𝜇_𝑖1𝑧_𝑖,𝜃𝑧_𝑖^𝑝+3,𝑧³_𝑖𝐹_𝑖𝑥_𝑖≤𝑧_𝑖^3𝑖_𝑗=1𝑧_𝑗^𝑝𝜌_𝑖2𝑗𝑧_𝑖,𝜃≤^𝑖−1_𝑗=1𝜉_𝑖2𝑗𝑧_𝑗^𝑝+3+𝜇_𝑖2𝑧_𝑖,𝜃𝑧_𝑖^𝑝+3,−𝑧³_𝑖^𝑖−1_𝑙=1𝜕𝛼_𝑖−1𝜕𝑥_𝑙𝑥^𝑝_𝑙+1≤𝑧_𝑖^3𝑖−1_𝑗=1𝑧_𝑗^𝑝𝜌_𝑖3𝑗𝑧_𝑖,𝜃≤^𝑖−1_𝑗=1𝜉_𝑖3𝑗𝑧_𝑗^𝑝+3+𝜇_𝑖3𝑧_𝑖,𝜃𝑧_𝑖^𝑝+3,−12𝑧³_𝑖^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖−1𝜕𝑥_𝑘𝜕𝑥_𝑚𝑔_𝑘𝑥_𝑘ΣΣ^𝑇𝑔^𝑇_𝑚𝑥_𝑚≤𝑧_𝑖^3𝑖−1_𝑗=1𝑧_𝑗^2𝑝𝜌_𝑖4𝑗𝑧_𝑖,𝜃ΣΣ^𝑇≤^𝑖−1_𝑗=1𝜉_𝑖4𝑗𝑧_𝑗^𝑝+3+𝜇_𝑖4𝑧_𝑖,𝜃𝑧_𝑖^𝑝+33𝜃,2𝑧²_𝑖TrΣ^𝑇𝐺^𝑇_𝑖𝑥_𝑖𝐺_𝑖𝑥_𝑖Σ≤𝑧²_𝑖^𝑖_𝑗=1𝑧_𝑗^𝑝+1𝜌_𝑖5𝑗𝑧_𝑖,𝜃ΣΣ^𝑇≤^𝑖−1_𝑗=1𝜉_𝑖5𝑗𝑧_𝑗^𝑝+3+𝜇_𝑖5𝑧_𝑖,𝜃𝑧_𝑖^𝑝+3𝜃,−𝑧³_𝑖𝜕𝛼_𝑖−1𝜕𝜃𝜏_𝑖−1≤𝑧_𝑖^3𝑖−1_𝑗=1𝑧_𝑗^𝑝𝜌_𝑖6𝑗𝑧_𝑖,𝜃≤^𝑖−1_𝑗=1𝜉_𝑖6𝑗𝑧_𝑗^𝑝+3+𝜇_𝑖6𝑧_𝑖,𝜃𝑧_𝑖^𝑝+3, (3.13) where 𝜉_𝑖1, 𝜉_𝑖2𝑗, 𝜉_𝑖3𝑗, 𝜉_𝑖4𝑗, 𝜉_𝑖5𝑗, 𝜉_𝑖6𝑗, 𝑗=1,…,𝑖−1, are positive constants and 𝜇_𝑖1, 𝜇_𝑖2, 𝜇_𝑖3, 𝜇_𝑖4, 𝜇_𝑖5, 𝜇_𝑖6, 𝜌_𝑖2𝑗, 𝜌_𝑖3𝑗, 𝜌_𝑖4𝑗, 𝜌_𝑖5𝑗, 𝜌_𝑖6𝑗, 𝑗=1,…,𝑖, are nonnegative smooth functions. Substituting (3.13) into (3.12), one has ℒ𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_{𝑗,𝑖−1}𝑧_𝑗^𝑝+3+𝜉_𝑖1𝑧^𝑝+3_𝑖−1−𝜃Γ+^𝑖−1_𝑗=2𝑧³_𝑗𝜕𝛼_𝑗−1𝜕𝜃𝜃̇−𝜏_𝑖−1+𝑧³_𝑖𝛼^𝑝_𝑖+𝑧_𝑖^𝑝+3𝜇_𝑖1+𝜇_𝑖2+𝜇_𝑖3+𝜇_𝑖6+𝜇_𝑖4+𝜇_𝑖51+𝜃²+𝜃Γ𝜇_𝑖4+𝜇_𝑖5Γ𝑧_𝑖^𝑝+3+^𝑖−1_𝑗=1𝜉_𝑖2𝑗+𝜉_𝑖3𝑗+𝜉_𝑖4𝑗+𝜉_𝑖5𝑗+𝜉_𝑖6𝑗𝑧_𝑗^𝑝+3+𝑧³_𝑖𝑥^𝑝_𝑖+1−𝛼^𝑝_𝑖. (3.14) Choosing the 𝑖th smooth virtual controller 𝛼_𝑖𝛼_𝑖𝑥_𝑖,𝜃=−𝑧_𝑖𝛽_𝑖𝑧_𝑖,𝜃,𝛽_𝑖𝑧_𝑖,𝜃=𝑐_𝑖𝑖+𝜇_𝑖1+𝜇_𝑖2+𝜇_𝑖3+𝜇_𝑖6+𝜇_𝑖4+𝜇_𝑖51+𝜃²+^𝑖_𝑗=2𝑧³_𝑗𝜕𝛼_𝑗−1𝜕𝜃Γ^1/𝑝, (3.15) and tuning function 𝜏_𝑖𝜏_𝑖𝑧_𝑖=𝜏_𝑖−1𝑧_𝑖−1+Γ𝑧_𝑖^𝑝+3𝜇_𝑖4+𝜇_𝑖5, (3.16) and substituting (3.15) and (3.16) into (3.14), it follows that ℒ𝑉_𝑖𝑧_𝑖,𝜃≤−^𝑖_𝑗=1𝑐_𝑗𝑖𝑧_𝑗^𝑝+3−𝜃Γ+^𝑖_𝑗=2𝑧³_𝑗𝜕𝛼_𝑗−1𝜕𝜃𝜃̇−𝜏_𝑖+𝑧³_𝑖𝑥^𝑝_𝑖+1−𝛼^𝑝_𝑖, (3.17) where 𝑐_𝑗𝑖=𝑐_𝑗𝑗−𝜉_𝑗+1,1−∑⁶_𝑘=2𝜉_𝑖𝑘𝑗, 𝑗=1,…,𝑖−1.

Hence at step 𝑛, the smooth adaptive state-feedback controller𝑢=𝛼_𝑛𝑥_𝑛,𝜃=−𝑧_𝑛𝛽_𝑛𝑧_𝑛,𝜃,𝜃̇=𝜏_𝑛𝑧_𝑛,𝛽_𝑛𝑧_𝑛,𝜃=𝑐_𝑛𝑛+𝜇_𝑛1+𝜇_𝑛2+𝜇_𝑛3+𝜇_𝑛6+𝜇_𝑛4+𝜇_𝑛51+𝜃²+^𝑛_𝑗=2𝑧³_𝑗𝜕𝛼_𝑗−1𝜕𝜃Γ^1/𝑝,𝜏_𝑛𝑧_𝑛=Γ𝑧₁^𝑝+3𝜇₁₅𝑧₁+^𝑛_𝑗=2Γ𝑧_𝑗^𝑝+3𝜇_𝑗4+𝜇_𝑗5, (3.18) such that the 𝑛th Lyapunov function𝑉_𝑛𝑧_𝑛,𝜃=14^𝑛_𝑗=1𝑧⁴_𝑗+1𝜃2Γ² (3.19) satisfiesℒ𝑉_𝑛≤−^𝑛_𝑗=1𝑐_𝑗𝑛𝑧_𝑗^𝑝+3, (3.20) where 𝜇_𝑛𝑙, 𝑙=1,…,6, are nonnegative smooth functions, 𝑐_𝑗𝑛, 𝑗=1,…,𝑛, are constants, and 𝑐_𝑗𝑛=𝑐_𝑗𝑗−𝜉_𝑗+1,1−⁶_𝑘=2𝜉_𝑛𝑘𝑗,𝑗=1,…,𝑛−1. (3.21)

3.2. Stability Analysis

Theorem 3.2. If Assumption 3.1 holds for the high-order stochastic nonlinear system (1.1), under the state-feedback controller (3.18), then(i)the closed-loop system consisting of (1.1), (3.2), (3.8), (3.9), (3.15), (3.16), and (3.18) has an almost surely unique solution on [0,∞) for each (𝑥₀,𝜃(0))∈𝑅^𝑛+1,(ii)the origin of the closed-loop system is globally stable in probability,(iii)𝑃{lim_𝑡→∞|𝑥(𝑡)|=0}=1 and 𝑃{lim_𝑡→∞𝜃(𝑡) exists and is finite}=1.

Proof. It is easy to verify that 𝑉_𝑛(𝑧_𝑛,𝜃) is 𝒞² on 𝑧_𝑛 and 𝜃. For 𝑗=1,…,𝑛−1, choose the design parameter 𝑐_𝑗𝑗>𝜉_𝑗+1,1+∑⁶_𝑘=2𝜉_𝑛𝑘𝑗, 𝑐_𝑛𝑛>0, then by (3.21), 𝑐_𝑗𝑛>0, 𝑗=1,…,𝑛−1. Since 𝑉_𝑛(𝑧_𝑛,𝜃) is continuous, positive, and radially bounded, by (3.20), (3.21), and Lemma  4.3 in [25], there exist two class 𝒦_∞ functions 𝛼₁ and 𝛼₂ such that 𝛼₁(|𝑥|,|𝜃|)≤𝑉_𝑛(𝑧_𝑛,𝜃)≤𝛼₂(|𝑥|,|𝜃|). Hence, the condition of Lemma 2.3 holds.

By Lemma 2.3, it follows that conclusion (i), (ii) hold, and 𝑃{lim_𝑡→∞|𝑧(𝑡)|=0}=1. In view of 𝛼_𝑖(0,𝜃)=0 and 𝑥_𝑖=𝑧_𝑖+𝛼_𝑖−1(𝑥_𝑖−1,𝜃), one has 𝑃{lim_𝑡→∞|𝑥(𝑡)|=0}=1. By (3.20) and the definition of 𝑉_𝑛(𝑧_𝑛,𝜃) in (3.19), it holds that 𝜃(𝑡) converges a.s. to a finite limit 𝜃_∞ as 𝑡→∞, therefore 𝜃(𝑡) converges a.s. to a finite limit as 𝑡→∞.

4. A Simulation Example

Consider a two-order nonlinear stochastic system 𝑑𝑥₁=𝑥³₂𝑑𝑡+𝑓₁𝑥₁𝑑𝑡+𝑥³₁Σ𝑑𝜔,𝑑𝑥₂=𝑢³𝑑𝑡+𝑓₂𝑥₂𝑑𝑡+𝑥³₂Σ𝑑𝜔, (4.1) where 𝑓₁(𝑥₁)=𝑥³₁, 𝑓₂(𝑥₂)=𝑥₁𝑥²₂. By Lemma 2.4, one gets |𝑓₁(𝑥₁)|≤|𝑥₁|³, |𝑔₁(𝑥₁)|≤|𝑥₁|³, |𝑓₂(𝑥₂)|≤(1/3)|𝑥₁|³+(2/3)|𝑥₂|³, 𝑔₂(𝑥₂)≤|𝑥₂|³. We choose 𝑓₁₁(𝑥₁)=1, 𝑔₁₁(𝑥₁)=1, 𝑓₂₁(𝑥₂)=2/3, 𝑔₂₁(𝑥₂)=1, Assumption 3.1 is satisfied.

We now give the design of state-feedback controller for system (4.1).

Step 1. Define 𝑧₁=𝑥₁, 𝑉₁(𝑧₁,𝜃)=(1/4)𝑧⁴₁+(1/2Γ)𝜃². A smooth virtual controller 𝛼₁𝑥₁,𝜃=−𝑧₁𝛽₁𝑧₁,𝜃,𝛽₁𝑧₁,𝜃=𝑐₁₁+1+𝜇₁₅(𝑧₁)1+𝜃^21/3, (4.2) and the tuning function 𝜏₁𝑧₁=Γ𝑧⁶₁𝜇₁₅𝑧₁ (4.3) yield ℒ𝑉₁(𝑧₁,𝜃)≤−𝑐₁₁𝑧⁶₁+𝑧³₁(𝑥³₂−𝛼³₁)+(𝜃/Γ)(𝜃̇−𝜏₁), where 𝜇₁₅𝑧₁=32𝑧²₁,𝜃(𝑡)=max_𝑡≥0Σ(𝑡)Σ(𝑡)^𝑇3,Σ(𝑡)Σ(𝑡)^𝑇2,Σ(𝑡)Σ(𝑡)^𝑇. (4.4)

Step 2. Defining 𝑧₂=𝑥₂−𝛼₁(𝑥₁,𝜃), 𝑉₂(𝑧₂,𝜃)=𝑉₁(𝑧₁,𝜃)+(1/4)𝑧⁴₂, by (3.12), one has ℒ𝑉₂𝑧₁,𝜃≤−𝑐₁₁𝑧⁶₁+𝑧³₁𝑥³₂−𝛼³₁+𝜃Γ𝜃̇−𝜏₁+𝑧³₂𝑢³+𝐹₂𝑥₂−𝜕𝛼₁𝜕𝑥₁𝑥³₂−12𝜕²𝛼₁𝜕𝑥²₁𝑔₁𝑥₁ΣΣ^𝑇𝑔^𝑇₁𝑥₁−𝑧³₂𝜕𝛼₁𝜕𝜃𝜃̇+32𝑧²₂TrΣ^𝑇𝐺^𝑇₂𝑥₂𝐺₂𝑥₂Σ, (4.5) where 𝐹₂(𝑥₂)=𝑓₂(𝑥₂)−(𝜕𝛼₁/𝜕𝑥₁)𝑓₁(𝑥₁), 𝐺₂(𝑥₂)=𝑔₂(𝑥₂)−(𝜕𝛼₁/𝜕𝑥₁)𝑔₁(𝑥₁). By Lemma 2.4, the definition of 𝑧₂, and (4.2), one can obtain 𝑧³₁𝑥³₂−𝛼³₁≤12𝑑₁₁𝑧⁶₁+12𝑑⁻¹₁₁𝑧⁶₂2+33𝑑₁₂𝑧⁶₁+13𝑑⁻¹₁₂𝛽³₁𝑧⁶₂+56𝑑₁₃𝑧⁶₁+16𝑑⁻¹₁₃𝛽₁¹²𝑧⁶₂=𝜉₂₁𝑧⁶₁+𝜇₂₁𝑧₁,𝜃𝑧⁶₂,𝑧³₂𝐹₂𝑥₂≤2𝑧₂³13𝑧₁³+23𝑧₂³+𝑧₁³𝛽²₁−𝜕𝛼₁𝜕𝑥₁𝑧³₁=𝜉₂₂₁𝑧⁶₁+𝜇₂₂𝑧₂,𝜃,−𝑧³₂𝜕𝛼₁𝜕𝑥₁𝑥³₂≤𝑧₂³𝜕𝛼₁𝜕𝑥₁𝑧³₂−3𝑧²₂𝑧₁𝛽₁+3𝑧₂𝑧₁𝛽₁−𝑧³₁𝛽³₁≤𝜉₂₃₁𝑧⁶₁+𝜇₂₃𝑧₂,𝜃𝑧⁶₂,−12𝑧³₂𝜕²𝛼₁𝜕𝑥²₁𝑔³₁ΣΣ^𝑇𝑔³₁≤𝑧₂³𝜕²𝛼₁𝜕𝑥²₁z⁶₁ΣΣ^𝑇≤𝜉₂₄₁𝑧⁶₁+𝜇₂₄𝑧₂,𝜃𝑧⁶₂𝜃; (4.6) by (4.3), Lemmas 2.4, 2.6, and the definitions of 𝑧₂ and 𝐺₂(𝑥₂), one has −𝑧³₂𝜕𝛼₁𝜕𝜃𝜏₁≤𝑧₂³𝑧₁³32𝜕𝛼₁𝜕𝜃Γ𝑧₁⁵≤𝜉₂₆₁𝑧⁶₁+𝜇₂₆𝑧₂,𝜃𝑧⁶₂,32𝑧²₂TrΣ^𝑇𝐺^𝑇₂𝑥₂𝐺₂𝑥₂Σ≤32𝑧²₂𝑧₂−𝑧₁𝛽₁³−𝜕𝛼₁𝜕𝑥₁𝑧³₁²ΣΣ^𝑇≤𝑧²₂3⋅2⁵𝛽⁶₁+3⋅𝜕𝛼₁𝜕𝑥₁²𝑧⁶₁+2⁵𝑧⁶₂ΣΣ^𝑇≤𝜉₂₅₁𝑧⁶₁+𝜇₂₅𝑧₂,𝜃𝑧⁶₂𝜃, (4.7) where 𝜉₂₁=(1/2)𝑑₁₁+2𝑑₁₂+(5/2)𝑑₁₃, 𝜉₂₂₁=(1/3)𝑑₂₁+𝑑₂₂+𝑑₂₃, 𝜉₂₃₁=(1/2)𝑑₃₁+(1/2)𝑑₃₂, 𝜉₂₄₁=(1/2)𝑑₄₁, 𝜉₂₅₁=(2/3)𝑑₅₁, 𝜉₂₆₁=(1/2)𝑑₆₁, 𝜇₂₁(𝑧₁,𝜃)=(1/2)𝑑⁻¹₁₁+𝑑⁻¹₁₂𝛽³₁(𝑧₁,𝜃)+(1/2)𝑑⁻⁵₁₃𝛽₁¹²(𝑧₁,𝜃), 𝜇₂₂(𝑧₂,𝜃)=(1/3)𝑑⁻¹₂₁+(4/3)+𝑑⁻¹₂₂𝛽⁴₁+𝑑⁻¹₂₃(𝜕𝛼₁/𝜕𝑥₁)², 𝜇₂₃(𝑧₂,𝜃)=(𝜕𝛼₁/𝜕𝑥₁)(1−2𝛽₁^3/2+𝛽⁶₁)+(𝜕𝛼₁/𝜕𝑥₁)²((1/2)𝑑⁻¹₃₁+(1/2)𝑑⁻¹₃₂), 𝜇₂₄(𝑧₂,𝜃)=(1/2)𝑑⁻¹₄₁(𝜕²𝛼₁/𝜕𝑥²₁)²𝑧⁶₁, 𝜇₂₅(𝑧₂,𝜃)=32𝑧²₂+𝑑⁻¹₅₁(128𝛽₁¹⁸+4(𝜕𝛼₁/𝜕𝑥₁)⁶)𝑧⁶₁, 𝜇₂₆(𝑧₂,𝜃)=(9/8)𝑑⁻¹₆₁(𝜕𝛼₁/𝜕𝜃)²Γ²𝑧₁¹⁰, 𝑑₁₁, 𝑑₁₂, 𝑑₁₃, 𝑑₂₁, 𝑑₂₂, 𝑑₂₃, 𝑑₃₁, 𝑑₃₂, 𝑑₄₁, 𝑑₅₁, 𝑑₆₁ are positive constants.
Choosing the smooth adaptive controller 𝑢=−𝑧₂𝛽₂𝑧₂,𝜃,𝜃̇=𝜏₂𝑧₂,𝜏₂𝑧₂=Γ𝑧⁶₁𝜇₁₅𝑧₁+Γ𝑧⁶₂𝜇₂₄𝑧₂,𝜃+𝜇₂₅𝑧₂,𝜃,𝛽₂𝑧₂,𝜃=𝑐₂₂+𝜇₂₁+𝜇₂₂+𝜇₂₃+𝜇₂₆+(𝜇₂₄+𝜇₂₅)1+𝜃²+Γ𝑧³₂𝜕𝛼₁𝜕𝜃^1/3, (4.8) and substituting (4.6)–(4.8) into (4.5), one has ℒ𝑉₂≤−𝑐₁₂𝑧⁶₁−𝑐₂₂𝑧⁶₂, (4.9) where 𝑐₁₂=𝑐₁₁−𝜉₂₁−𝜉₂₂₁−𝜉₂₃₁−𝜉₂₄₁−𝜉₂₅₁−𝜉₂₆₁>0.
In simulation, we choose Σ(𝑡)≡1, the parameters Γ=1, 𝑐₁₁=11, 𝑐₂₂=1, 𝑑₁₁=1, 𝑑₁₂=1, 𝑑₁₃=1, 𝑑₂₁=1, 𝑑₂₂=1, 𝑑₂₃=1, 𝑑₃₁=0.01, 𝑑₃₂=1, 𝑑₄₁=1, 𝑑₅₁=1, 𝑑₆₁=1, the initial values 𝜃(0)=0, 𝑥₁(0)=0, 𝑥₂(0)=−0.5, the sampling period =0.01. Figure 1 verifies the effectiveness of the control scheme.

(a)

(b)

(c)

(d)

Figure 1 

The responses of closed-loop system (4.1)–(4.3), (4.8).

5. Conclusion

In this paper, we further consider more general high-order stochastic nonlinear system driven by noise of unknown covariance and its adaptive state-feedback stabilization problem.

There is a still remaining problem to be investigated: under current investigation, how to design an output feedback controller for system (1.1) with Assumption 3.1?

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 10971256 and 61104222), the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20103705110002), and Natural Science Foundation of Jiangsu Province (BK2011205), Natural Science Foundation of Xuzhou Normal University.