Output-Feedback and Inverse Optimal Control of a Class of Stochastic Nonlinear Systems with More General Growth Conditions

Abstract

This paper investigates the problem of output-feedback stabilization for a class of stochastic nonlinear systems in which the nonlinear terms depend on unmeasurable states besides measurable output. We extend linear growth conditions to power growth conditions and reduce the control effort. By using backstepping technique, choosing a high-gain parameter, an output-feedback controller is designed to ensure the closed-loop system to be globally asymptotically stable in probability, and the inverse optimal stabilization in probability is achieved. The efficiency of the output-feedback controller is demonstrated by a simulation example.

1. Introduction

The design of output-feedback controller for stochastic nonlinear systems has achieved remarkable research development, because output feedback control is more suitable for practical engineering systems; for example, see [1–12] and references therein. In recent years such research hotspot has mainly focused on a class of special nonlinear stochastic systems in which the nonlinear vector terms depend on the unmeasurable states besides the measurable output; for example, see [13–17] and references therein. The work of [13] discussed the output-feedback controller design by introducing a stability concept named globally asymptotically stable in probability. Based on the purpose of reducing the amount of control, [14] considered the output-feedback stabilization problem.

However, in [13–17], the nonlinear vector terms satisfy the linear growth conditions strictly, which greatly narrows the scope of application of the research results. Naturally, one may ask about an interesting and challenging problem: can we further relax the linear growth conditions? To our knowledge, the existing research results on this problem are as in [18–20]. In [18], authors discussed the output-feedback stabilization problem by introducing a rescaling transformation under more relaxed growth conditions. On the basis of [18], the work of [19] and [20] further considered the output-feedback controller design problem for high-order stochastic nonlinear systems. However, for [18–20], the observer gain 𝐾 is usually larger than 1, and the choice can lead to a controller design which needs larger control effort. So another challenging problem is proposed that is whether the assumption 𝐾>1 can be removed.

In this paper, we investigate the output-feedback stabilization problem for a class of stochastic nonlinear systems satisfying power growth conditions. Inspired by [13, 14], we find the maximum value interval of observer gain for the desired controller by using backstepping technique. For this interval, the designed output-feedback controller ensures that the equilibrium at the origin is globally asymptotically stable in probability and the inverse optimal stabilization in probability is achieved. The main contributions of this paper are characterized as follows. (i) We extend the linear growth conditions to the power growth conditions. (ii) The assumption of 𝐾>1 in [18–20] is removed so that we can get less control effort.

The paper is organized as follows. Section 2 provides some preliminary results. In Section 3, the problem to be investigated is presented. In Sections 4 and 5, an output-feedback controller is designed and analysed. In Section 6, the inverse optimal stabilization in probability is achieved. Section 7 provides a simulation example. Section 8 concludes this paper.

2. Notations and Preliminary Results

Throughout this paper, the following notations will be used. 𝑅 denotes the set of all real numbers; 𝑅₊ denotes the set of all nonnegative real numbers; 𝑅^𝑛 denotes the real 𝑛-dimensional space; 𝑅^𝑛×𝑟 denotes the real 𝑛×𝑟 matrix space; Tr(⋅) denotes the trace for square matrix 𝑋; |𝑋| denotes the Euclidean norm of a vector 𝑋, and ‖𝑋‖ is the Frobenius norm of matrix 𝑋 defined by ‖𝑋‖=(∑^𝑛_𝑖=1∑^𝑟_𝑗=1𝑥²_𝑖𝑗)^1/2; for a given vector 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑥_[𝑖] denotes (𝑥₁,…,𝑥_𝑖)^𝑇; 𝐶^𝑖 denotes the set of all function with continuous 𝑖th partial derivatives; 𝐶^2,1(𝑅^𝑛×𝑅₊,𝑅₊) is the family of all nonnegative functions 𝑉(𝑥,𝑡) on 𝑅^𝑛×𝑅₊, which are 𝐶² in 𝑥 and 𝐶¹ in 𝑡; 𝐾 denotes the set of all functions: 𝑅⁺→𝑅⁺, which are continuous, strictly increasing, and vanish at zero; 𝐾_∞ denotes 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.

Lemma 1. The inequality (|𝑥|+|𝑦|)^𝑝≤2^𝑝−1(|𝑥|^𝑝+|𝑦|^𝑝) is established for any 𝑥∈𝑅, 𝑦∈𝑅, 𝑝≥1, 𝑝∈𝑍.

Proof. For an assumption of 𝜶=(𝑎₁,𝑎₂)=(|𝑥|,|𝑦|), 𝜷=(𝑏₁,𝑏₂)=(1,1), inspired by Holder inequality [21], we can get |𝑥|+𝑦≤|𝑥|^𝑝+𝑦^𝑝1/𝑝2^1−1/𝑝 (1) and further get |𝑥|+𝑦^𝑝≤2^𝑝−1|𝑥|^𝑝+𝑦^𝑝. (2) Then the proof is completed.

Consider the following stochastic system: 𝑑𝑥=𝑓(𝑥,𝑢)𝑑𝑡+𝑔(𝑥)𝑑𝜔,∀𝑥₀∈𝑅^𝑛, (3) where 𝑥=(𝑥₁,…,𝑥_𝑛)∈𝑅^𝑛 is the state of the system. 𝑢∈𝑅^𝑚 is the control input of the system. 𝜔 is an 𝑟-dimensional standard Wiener process defined on a probability space {Ω,𝐹,𝑃}. The nonlinear functions 𝑓∶𝑅^𝑚+𝑛→𝑅^𝑛 and 𝑔∶𝑅^𝑛→𝑅^𝑛×𝑟 are locally Lipschitz with 𝑓(0)=0,  𝑔(0)=0. For any given 𝑉∈𝐶²(𝑅^𝑛;𝑅) associated with stochastic system (3), the differential operator 𝐿 is defined as follows: 𝐿𝑉(𝑥)=𝜕𝑉𝑓(𝑥)+1𝜕𝑥2Tr𝑔^𝑇(𝑥)𝜕²𝑉𝜕𝑥²𝑔(𝑥). (4) In order to discuss the stability of stochastic nonlinear systems, we introduce the following stability notion.

Definition 2 (see [22]). For the stochastic nonlinear system (3) with 𝑓(0,𝑢)=0, 𝑔(0)=0, the equilibrium 𝑥(𝑡)=0 of (3) is said to be globally asymptotically stable (GAS) in probability if, for any 𝜀>0, there exists a class 𝐾𝐿 function 𝛽(⋅,⋅) such that 𝑃{|𝑥(𝑡)|<𝛽(|𝑥₀|,𝑡)}≥1−𝜀, ∀𝑡≥0, 𝑥₀∈𝑅^𝑛⧵{0}.

The following lemmas give some sufficient conditions ensuring global asymptotical stability in probability.

Lemma 3 (see [23]). For system (3), if there exist 𝑉(𝑥)∈𝐶², class 𝐾_∞ functions 𝛼₁, 𝛼₂, and a class 𝐾 function 𝛼₃ such that 𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂(|𝑥|), 𝐿𝑉(𝑥)≤−𝛼₃(|𝑥|), then there exists an almost surely unique solution to system (3) on [0,∞), and the equilibrium 𝑥(𝑡)=0 is globally asymptotically stable in probability.

Lemma 4 (see [24]). Consider the following control law: 𝑢=𝛼(𝑥)=−𝑅₂⁻¹𝐿_𝜑2𝑉^𝑇ℓ𝛾𝐿_𝜑2𝑉𝑅₂^−1/2𝐿_𝜑2𝑉𝑅₂^−1/22, (5) where 𝛾(⋅) is a class 𝐾_∞ function, ℓ𝛾(𝑠)=𝑠(𝛾̇)⁻¹(𝑠)−𝛾((𝛾̇)⁻¹(𝑠)), and 𝑅₂(𝑥) is a matrix valued function such that 𝑅₂(𝑥)=𝑅^𝑇₂(𝑥)>0. If the control law (5) to be ensures system (3) globally asymptotically stable in probability, then the control law 𝑢^∗=𝛽^∗(𝑥)𝜃=−2𝑅₂⁻¹𝐿_𝜑2𝑉^𝑇ℓ𝛾𝐿_𝜑2𝑉𝑅₂^−1/2𝐿_𝜑2𝑉𝑅₂^−1/22,𝜃≥2, (6) solves the problem of inverse optimal stabilization in probability for system (3) by minimizing the cost functional 𝐽(𝑢)=𝐸^∞₀𝑙(𝑥)+𝜃²𝛾2𝜃𝑅₂(𝑥)^1/2𝑢𝑑𝜏, (7) where 𝑙(𝑥) is a positive definite radially unbounded function satisfying 𝑙(𝑥)=2𝜃ℓ𝛾𝐿_𝜑2𝑉𝑅₂^−1/2−12Tr𝜑^𝑇₁𝜕²𝑉𝜕𝑥²𝜑₁+𝜃(𝜃−2)ℓ𝛾𝐿_𝜑2𝑉𝑅₂^−1/2. (8)

3. Problem Formulation

Consider the following stochastic nonlinear systems: 𝑑𝑥_𝑖=𝑥_𝑖+1𝑑𝑡+𝜑_𝑖(𝑥)𝑑𝜔,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑢𝑑𝑡+𝜑_𝑛(𝑥)𝑑𝜔,𝑦=𝑥₁, (9) where 𝑥=(𝑥₁,…,𝑥_𝑛)∈𝑅^𝑛, 𝑢∈𝑅, and 𝑦∈𝑅 are the states, the control input, and the measurable output of the system, 𝜔∈𝑅^𝑟 is defined as in (3), and 𝑥₂,…,𝑥_𝑛 are the unmeasurable states. 𝜑_𝑖∶𝑅^𝑛→𝑅^𝑟, 𝑖=1,…,𝑛, are locally Lipschitz with 𝜑_𝑖(0)=0 and satisfy the following power growth conditions.

Assumption 5. For each 1≤𝑖≤𝑛, there exists the known positive constant 𝑑≥0 such that |𝜑_𝑖(𝐱)|≤𝑑(|𝑥₁|^𝑝+|𝑥₂|^𝑝+⋯+|𝑥_𝑖|^𝑝), where 𝑝 is any positive integer.

Remark 6. Assumption 5 can be simplified into linear growth conditions when 𝑝=1. Therefore, linear growth conditions as a special case are included in Assumption 5. This paper extends previous work and gets a new result.

The objective of this paper is to design a smooth output-feedback controller for system (9), such that the closed-loop system is globally asymptotically stable in probability at the origin and achieves the design of the inverse optimal stabilization in probability.

4. Output-Feedback Controller Design

Since 𝑥₂,…,𝑥_𝑛 are unmeasured, the following observer is introduced: 𝑥̇_𝑖=𝑥_𝑖+1+𝐾^𝑖ℎ_𝑖𝑥₁−𝑥₁𝑥̇,𝑖=1,…,𝑛−1,_𝑛=𝑢+𝐾^𝑛ℎ_𝑛𝑥₁−𝑥₁, (10) where 𝑥_𝑖 is the estimated value of 𝑥_𝑖, 𝐾∈𝑅₊ is the observer gain to be determined, and ℎ_𝑖>0,   𝑖=1,…,𝑛, are chosen such that matrix 𝐴=^{−ℎ1⋮𝐼}_{𝑛−1−ℎ𝑛0⋯0} is asymptotically stable; thus there exists a positive definite matrix 𝑃 satisfying 𝐴^𝑇𝑃+𝑃𝐴=−𝐼. Let 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, where 𝑥_𝑖=(𝑥_𝑖−𝑥_𝑖)/𝐾^𝑖−1 for each 𝑖=1,…,𝑛. By (9) and (10), we can get error system 𝑑𝑥=𝐾𝐴𝑥𝑑𝑡+Φ(𝑥)𝑑𝜔, (11) where Φ(𝑥)=(𝜑₁(𝑥),𝜑₂(𝑥)/𝐾,…,𝜑_𝑛(𝑥)/𝐾^𝑛−1)^𝑇.

Now we give the backstepping controller design procedure.

Step  0. Choosing the zeroth Lyapunov function 𝑉₀(𝑥)=(𝑛+1)𝑥^𝑇𝑃𝑥, applying 2𝑎𝑏≤2(𝑎²+𝑏²), (𝑎+𝑏)²≤2(𝑎²+𝑏²), ∑^𝑛_𝑖=1|𝑎_𝑖|²≤(∑^𝑛_𝑖=1|𝑎_𝑖|)², (∑^𝑛_𝑖=1𝑎_𝑖)²≤𝑛∑^𝑛_𝑖=1𝑎²_𝑖, Lemma 1, Assumption 5, and (4), we can get 𝐿𝑉₀=−(𝑛+1)𝐾|𝑥|²+(𝑛+1)TrΦ^𝑇(𝑥)(𝑥)()𝐾|𝑥|𝑃Φ≤−𝑛+1²+(𝑛+1)‖𝑃‖^𝑛_𝑖=1𝜑_𝑖(𝑥)𝐾^𝑖−12≤−(𝑛+1)𝐾|𝑥|²+(𝑛+1)‖𝑃‖^𝑛_𝑖=1𝜑_𝑖(𝑥)𝐾^𝑖−12≤−(𝑛+1)𝐾|𝑥|²+(𝑛+1)‖𝑃‖𝑑^2𝑛_𝑖=11𝐾^𝑖−12×𝑥₁^𝑝+𝑥₂^𝑝𝐾𝑥+⋯+^𝑛𝑝𝐾^𝑛−1≤−(𝑛+1)𝐾|𝑥|²+2^2𝑝−1𝑛𝑑^∗×^𝑛_𝑖=1𝑥^𝑝_𝑖𝐾^𝑖−12+^𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)}𝑥_𝑖^𝑝2≤−(𝑛+1)𝑛𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₁^2𝑝+𝑥₂^2𝑝𝐾²𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2, (12) where 𝑑^∗=(𝑛+1)‖𝑃‖𝑑^2𝑛_𝑖=11𝐾^𝑖−12=(𝑛+1)‖𝑃‖𝑑²∑^𝑛−1_𝑖=0(𝑖+1)𝐾^𝑖+∑^2𝑛−2_𝑖=𝑛(2𝑛−𝑖−1)𝐾^𝑖𝐾^2𝑛−2 (13) and ‖𝑥‖_∞=max_𝑖|𝑥_𝑖|.

We introduce a series of coordinate changes as follows: 𝑤₁=𝑥₁,𝑤_𝑖=𝑥_𝑖−𝛽_𝑖−1𝑥_[𝑖−1], (14) where 𝛽_𝑖−1(𝑥_[𝑖−1])  (𝑖=2,…,𝑛) is the virtual control law to be designed.

Step  1. Constructing the 1st Lyapunov function 𝑉₁𝐱,𝑤₁=𝑉₀(𝐱)+1𝑤𝑝+1^1𝑝+1, (15) using (3), (10), (12)–(15), and Young’s inequality [22], we can obtain 𝐿𝑉₁≤−(𝑛+1)𝑛𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2+2^2𝑝−1𝑛𝑑^∗𝑤₁^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₂^2𝑝𝐾²+𝑤^𝑝₁𝑤₂+𝑤^𝑝₁𝛽₁+𝐾𝑥²₁+𝐾4ℎ²₁𝑤₁^2𝑝. (16) Applying (14) and Lemma 1, choosing 𝐾≥2^2𝑝𝑛𝑑^∗, we can get 2^2𝑝−1𝑛𝑑^∗𝑤²₁≤𝐾2𝑤²₁,2^2𝑝−1𝑛𝑑^∗𝑥₂^2𝑝𝐾²≤4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝛽₁^2𝑝, (17) which one substitutes in (16) to obtain 𝐿𝑉₁≤−(𝑛+1)𝑛𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2+2^2𝑝−1𝑛𝑑^∗𝑤²₁+4^2𝑃−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝛽₁^2𝑝+𝑤^𝑝₁𝑤₂+𝑤^𝑝₁𝛽₁+𝐾𝑥²₁+𝐾4ℎ²₁𝑤₁^2𝑝≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₁^2𝑝+𝑤^𝑝₁𝑤₂+𝑤^𝑝₁𝛽₁1+𝐾2+ℎ²₁4𝑤₁^2𝑝≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝⋯𝑤₁^2𝑝_2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+𝑤^𝑝₁𝑤₂≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗𝑣₁^2𝑝2𝑝𝑤₁^2𝑝2+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+𝑤^𝑝₁𝑤₂=−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+𝑤^𝑝₁𝑤₂, (18) by choosing the 1st virtual control law 𝛽₁=−𝐾𝑣₁𝑤^𝑝₁,𝑣₁=12+ℎ²₁4+𝑛. (19)

Step  2. Using (10), (14), and (18), we can get 𝑑𝑤₂=𝑥₃+𝐾²ℎ₂𝑥₁+𝐾𝑣₁𝑝𝑤₁^𝑝−1𝑥₂+𝐾ℎ₁𝑥₁=𝑥𝑑𝑡₃+𝐾²ℎ₂𝑥₁+𝐾²ℎ₁𝑣₁𝑝𝑤₁^𝑝−1𝑥₁+𝐾𝑣₁𝑝𝑤₁^𝑝−1𝑤₂+𝛽₁𝑑𝑡. (20) Constructing the 2nd Lyapunov function 𝑉₂𝑥,𝑤_[2]=𝑉₁𝑥,𝑤₁+1𝐾²⋅1𝑤𝑝+1^2𝑝+1, (21) applying (14), (18)–(21), 𝐾≥2^2𝑝𝑛𝑑^∗, Lemma 1, and Young’s inequality [20], we obtain 𝐿𝑉₂≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₄^2𝑝𝐾⁶𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+𝑤^𝑝₁𝑤₂+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴+1𝐾²𝑤^𝑝₂𝑥₃+𝐾²ℎ₂𝑥₁+𝐾²ℎ₁𝑣₁𝑝𝑤₁^𝑝−1𝑥₁+𝐾𝑣₁𝑝𝑤₁^𝑝−1𝑤₂−𝐾²𝑣²₁𝑝𝑤₁^2𝑝−1≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₄^2𝑝𝐾⁶𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+𝑤^𝑝₁𝑤₂+2^2𝑝−1𝑛𝑑^∗𝑥₃^2𝑝𝐾⁴+1𝐾²𝑤^𝑝₂𝑥₃+𝑤^𝑝₂ℎ₂𝑥₁+𝑤^𝑝₂ℎ₁𝑣₁𝑝𝑤₁^𝑝−1𝑥₁+1𝐾𝑤₂^𝑝+1𝑣₁𝑝𝑤₁^𝑝−1−𝑣²₁𝑝𝑤₁^2𝑝−1𝑤^𝑝₂≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₄^2𝑝𝐾⁶𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾⁴𝑤₃^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾⁴𝛽₂^2𝑝+1𝐾²𝑤^𝑝₂𝑤₃+1𝐾²𝑤^𝑝₂𝛽₂+𝑤^𝑝₂ℎ₂+ℎ₁𝑣₁𝑝𝑤₁^𝑝−1𝑥₁+4^2𝑝−1𝑛𝑑^∗1𝐾²𝑤₂^2𝑝+𝑤^𝑝₁𝑤₂+1𝐾𝑤₂^𝑝+1𝑣₁𝑝𝑤₁^𝑝−1−𝑣²₁𝑝𝑤₁^2𝑝−1𝑤^𝑝₂≤−((𝑛+1)𝑛−1)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₄^2𝑝𝐾⁶𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−𝑛𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾⁴𝑤₃^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾⁴𝛽₂^2𝑝+1𝐾²𝑤^𝑝₂𝑤₃+1𝐾²𝑤^𝑝₂𝛽₂+𝑤^𝑝₂ℎ₂+ℎ₁𝑣₁𝑝𝑤₁^𝑝−1𝑥₁+14𝑛𝑑^∗−1𝑤₂^2𝑝+𝑤₁^2𝑝+14𝑤²₂+1𝐾𝑤₂^2𝑝+2; (22) then we can get the 2nd virtual control law 𝛽₂𝑥_[2]=−𝐾²𝑣₂𝑤^𝑝₂,𝑣₂=ℎ²₂4+𝑣²₁ℎ²₁4+𝑣⁴₁4+𝑣₁+1+𝑛−1, (23) which satisfies 𝐿𝑉₂≤−((𝑛+1)𝑛−2)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥₄^2𝑝𝐾⁶𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−(𝑛−1)𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑤₁^2𝑝−1𝐾²𝑛−4^𝑝−1𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣₂^2𝑝𝑤₂^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾⁴𝑤₃^2𝑝+1𝐾²𝑤^𝑝₂𝑤₃. (24)

Step 𝑖  (𝑖=3,…,𝑛−1). Suppose at (𝑖−1)th, that there are a set of virtual control laws 𝛽₁(𝑥₁),…,𝛽_𝑖−1(𝑥_[𝑖−1]): as follows 𝛽₁𝑥₁=−𝐾𝑣₁𝑤^𝑝₁,𝛽₂𝑥_[2]=−𝐾²𝑣₂𝑤^𝑝₂,⋮𝛽_𝑖−1𝑥_[𝑖−1]=−𝐾^𝑖−1𝑣_𝑖−1𝑤^𝑝_𝑖−1,𝑣₁=12+ℎ²₁4𝑣+𝑛,₂=ℎ²₂4+𝑣²₁ℎ²₁4+𝑣⁴₁4+𝑣₁⋮𝑣+1+𝑛−1,_𝑖−1=14ℎ_𝑖−1+𝑣_𝑖−2ℎ_𝑖−2+𝑣_𝑖−2𝑣_𝑖−3ℎ_𝑖−3+⋯+𝑣_𝑖−2𝑣_𝑖−3⋯𝑣₁ℎ₁²1+⋯+4𝑣_𝑖−3𝑣_𝑖−2−𝑣²_𝑖−2²+𝑣_𝑖−2+1+𝑛−(𝑖−2), (25) with 𝑣_𝑗>0  (𝑗=1,…,𝑖−1) being independent of 𝐾 such that the 𝑖th Lyapunov function 𝑉_𝑖−1𝑥,𝑤_[𝑖−1]=𝑉₀(𝑥)+1𝑝+1^𝑖−1_𝑗=11𝐾^𝑗𝑤_𝑗^𝑝+1 (26) satisfies 𝐿𝑉_𝑖−1≤−((𝑛+1)𝑛−(𝑖−1))𝐾−2^2𝑝−1𝑛𝑑^∗×^𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥^2𝑝_𝑖+1𝐾^2𝑖𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−^𝑖−1_𝑗=11𝐾^2𝑗−2𝑛−4^{(𝑝−1)(𝑗−1)}𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑗^2𝑝𝑤_𝑗^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾^2(𝑖−1)𝑤_𝑖^2𝑝+1𝐾^2(𝑖−2)𝑤^𝑝_𝑖−1𝑤_𝑖. (27) In the sequel, we will prove that (27) still holds for 𝑉_𝑖𝑥,𝑤_[𝑖]=𝑉_𝑖−1𝑥,𝑤_[𝑖−1]+1⋅1𝑝+1𝐾^𝑖𝑤_𝑖^𝑝+1. (28) Using (14) and (25), a direct calculation leads to 𝑑𝑧_𝑖=𝑥_𝑖+1+𝐾^𝑖ℎ_𝑖𝑥_𝑖+𝐾ℎ_𝑖−1𝑝𝑤^𝑝−1_𝑖−1𝑥_𝑖+𝐾^𝑖−1ℎ_𝑖−1𝑥₁+𝐾²𝑣_𝑖−1𝑣_𝑖−2𝑝𝑤^𝑝−1_𝑖−2𝑥_𝑖−1+𝐾^𝑖−2ℎ_𝑖−2𝑥₁+⋯+𝐾^𝑖−1𝑣_𝑖−1𝑣_𝑖−2⋯𝑣₁𝑝𝑤₁^𝑝−1𝑥₂+𝐾ℎ₁𝑥₁𝑑𝑡. (29) Using 𝐾≥2^2𝑝𝑛𝑑^∗, Lemma 1, Young’s inequality [22], (14), (27), (28), and (29), we obtain 𝐿𝑉_𝑖≤−((𝑛+1)𝑛−(𝑖−1))𝐾−2^2𝑝−1𝑛𝑑^∗×^𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥^2𝑝_𝑖+2𝐾^2(𝑖+1)𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−^𝑖−1_𝑗=11𝐾^2𝑗−2𝑛−4^{(𝑝−1)(𝑗−1)}𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑗^2𝑝𝑤_𝑗^2𝑝+2^2𝑝−1𝑛𝑑^∗1𝐾^2𝑖𝑥^2𝑝_𝑖+1+1𝐾^2(𝑖−1)𝑤_𝑖^2𝑝𝑥_𝑖+1+1𝐾^𝑖𝑤^𝑝_𝑖×𝑥_𝑖+1+𝐾^𝑖ℎ_𝑖𝑥_𝑖+𝐾𝑣_𝑖−1𝑝𝑤^𝑝−1_𝑖−1𝑥_𝑖+𝐾^𝑖−1ℎ_𝑖−1𝑥₁+⋯+𝐾^𝑖−1𝑣_𝑖−1𝑣_𝑖−2⋯𝑣₁𝑝𝑤₁^𝑝−1𝑥₂+𝐾ℎ₁𝑥₁≤−((𝑛+1)𝑛−𝑖)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥^2𝑝_𝑖+2𝐾^2(𝑖+1)𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−^𝑖_𝑗=11𝐾^2𝑗−2𝑛−4^{(𝑝−1)𝑗}𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑗^2𝑝𝑤_𝑗^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾^2𝑖𝑤^2𝑝_𝑖+1+4^2𝑝−1𝑛𝑑^∗1𝐾^2𝑖𝛽_𝑖^2𝑝+1𝐾^2(𝑖−1)𝑤^𝑝_𝑖𝑤_𝑖+1+1𝐾^2(𝑖−1)𝑤^𝑝_𝑖𝛽_𝑖+1𝐾^2𝑖−3𝑤_𝑖^2𝑝14ℎ_𝑖+𝑣_𝑖−1ℎ_𝑖−1+𝑣_𝑖−1𝑣_𝑖−2ℎ_𝑖−2+⋯+𝑣_𝑖−1𝑣_𝑖−2⋯𝑣₁ℎ₁²1+⋯+4𝑣_𝑖−3𝑣_𝑖−2−𝑣²_𝑖−2²+𝑣_𝑖−1+1, (30) then we can choose the 𝑖th smooth virtual control law 𝛽_𝑖𝑥_[𝑖]=−𝐾^𝑖𝑣_𝑖𝑤^𝑝_𝑖,𝑣_𝑖=14ℎ_𝑖+𝑣_𝑖−1ℎ_𝑖−1+𝑣_𝑖−1𝑣_𝑖−2ℎ_𝑖−2+⋯+𝑣_𝑖−1𝑣_𝑖−2⋯𝑣₁ℎ₁²1+⋯+4𝑣_𝑖−3𝑣_𝑖−2−𝑣²_𝑖−2²+𝑣_𝑖−1+1+𝑛−(𝑖−1) (31) and get 𝐿𝑉_𝑖≤−((𝑛+1)𝑛−𝑖)𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}×‖𝑥‖_∞^2𝑝+2^2𝑝−1𝑛𝑑^∗𝑥^2𝑝_𝑖+2𝐾^2(𝑖+1)𝑥+⋯+^𝑛2𝑝𝐾^2𝑛−2−^𝑖_𝑗=11𝐾^2𝑗−2𝑛−4^{(𝑝−1)𝑗}𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑗^2𝑝𝑤_𝑗^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾^2𝑖𝑤^2𝑝_𝑖+1+1𝐾^2(𝑖−1)𝑤^𝑝_𝑖𝑤_𝑖+1. (32)

Step 𝑛. Using repeatedly the previous arguments, at the 𝑛−1th step, we can get 𝐿𝑉_𝑛−1≤−((𝑛+1)𝑛−(𝑛−1))𝐾−2^2𝑝−1𝑛𝑑^∗×^𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}‖𝑥‖_∞^2𝑝−^𝑛−1_𝑗=11𝐾^2𝑗−2×𝑛−4^{(𝑝−1)(𝑗−1)}𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑗^2𝑝𝑤_𝑗^2𝑝+4^2𝑝−1𝑛𝑑^∗1𝐾^2(𝑖−1)𝑤_𝑛^2𝑝+1𝐾^2(𝑖−2)𝑤^𝑝_𝑛−1𝑤_𝑛, (33) where 𝑉_𝑛−1𝑥,𝑤_[𝑛−1]=𝑉_𝑛−2𝑥,𝑤_[𝑛−2]+1𝐾^𝑛−1⋅1𝑤𝑝+1^𝑛2𝑝. (34) At the end of the recursive procedure, choosing the controller 𝑢𝑥_[𝑛]=−𝐾^𝑛𝑣_𝑛𝑤^𝑝_𝑛, (35) where 𝑣_𝑛>0 satisfies (25) and is independent of 𝐾, we can get 𝐿𝑉_𝑛≤−𝑛²𝐾−2^2𝑝−1𝑛𝑑^∗𝑛_𝑖=1𝐾^{(𝑖−1)𝑝−(𝑖−1)2}‖𝑥‖_∞^2𝑝−^𝑛−1_𝑗=11𝐾^2𝑗−2×𝑛−4^{(𝑝−1)(𝑗−1)}𝐾−2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑗^2𝑝𝑤_𝑗^2𝑝−1𝐾^2𝑛−5𝑤_𝑛^2𝑝, (36) where 𝑉_𝑛𝑥,𝑤_[𝑛]=(𝑛+1)𝑥^𝑇𝑃1𝑥+𝑝+1^𝑛_𝑗=11𝐾^𝑗𝑤_𝑗^𝑝+1. (37)

Remark 7. The item (𝑥₁^2𝑝+𝑥₂^2𝑝/𝐾²+⋯+𝑥_𝑛^2𝑝/𝐾^2𝑛−2) is canceled at step 𝑛−1. By the following analysis, we obtain the maximum value interval of 𝐾 to ensure the system to be globally asymptotically stable in probability at the origin.

5. Performance Analysis

Next, we give the main result in this paper.

Theorem 8. If Assumption 5 holds for stochastic nonlinear system (9) under the controllers (10) and (35), then there always exists a constant 𝐾^∗≥0, such that for any 𝐾>𝐾^∗,(1)the closed-loop system has an almost surely unique solution on [0,∞) for any 𝑥₀;(2)the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability.

Proof. Using 𝐾≥0, (18), (23), and (31), obviously, if 2𝐾>max^4𝑝−1𝑝𝑛𝑑^∗𝑣₁^2𝑝𝑛,2^4𝑝−1𝑝𝑛𝑑^∗𝑣₂^2𝑝𝑛−4^𝑝−12,…,_4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑛^2𝑝𝑛−4^{(𝑝−1)(𝑛−1)}=2^4𝑝−1𝑝𝑛𝑑^∗𝑣_𝑛^2𝑝𝑛−4^{(𝑝−1)(𝑛−1)} (38) holds, then conclusions (1) and (2) follow from (36), (37), and Lemma 3. In the following, we analyze (38). From (13), (25), and (31), it is easy to find that 𝑑^∗ depends on 𝐾. Substituting (13) into (38) leads to 2𝐾>^4𝑝−1𝑝𝑛𝑣_𝑛^2𝑝𝑛−4^{(𝑝−1)(𝑛−1)}⋅(𝑛+1)‖𝑃‖𝑑²𝐾^2𝑛−2⋅^𝑛−1_𝑖=0(𝑖+1)𝐾^𝑖+^2𝑛−2_𝑖=𝑛(2𝑛−𝑖−1)𝐾^𝑖; (39) equivalently, 𝐾^2𝑛−1>𝑛𝑛−4^{(𝑝−1)(𝑛−1)}⋅2^4𝑝−1⋅𝑝⋅𝑣_𝑛^2𝑝⋅(𝑛+1)‖𝑃‖𝑑^2𝑛−1_𝑖=0(𝑖+1)𝐾^𝑖+^2𝑛−2_𝑖=𝑛(2𝑛−𝑖−1)𝐾^𝑖, (40) which is equivalent to 𝐾^2𝑛−1+^2𝑛−2_𝑖=0𝑎_𝑖𝐾^𝑖>0 (41) with the real numbers 𝑎₀𝑎=−Δ,₁⋮𝑎=−2Δ,_𝑛−1𝑎=−𝑛Δ,_𝑛()𝑎=−𝑛−1Δ,_𝑛+1()⋮𝑎=−𝑛−2Δ,_2𝑛−2=−Δ−(𝑛+1)‖𝑃‖²,Δ=2^4𝑝−1𝑝𝑛(𝑛+1)𝑑²𝑣^2𝑝_𝑛−1. (42) According to the factorization theorem of real coefficient polynomial, (41) can be further expressed as 𝐾−𝐾₁^𝑚1⋯𝐾−𝐾_𝑟^𝑚𝑟𝐾²+𝑝₁𝐾+𝑞₁^𝑛1⋯𝐾²+𝑝_𝑠𝐾+𝑞_𝑠^𝑛𝑠>0, (43) where 𝑚_𝑖,  𝑛_𝑗 are positive integers with ∑^𝑟_𝑖=1𝑚_𝑖+2∑^𝑠_𝑗=1𝑛_𝑗=2𝑛−1, 𝐾_𝑖, 𝑖≤𝑟, are different real numbers, and (𝑝_𝑗,𝑞_𝑗), 𝑗≥𝑠, satisfy 𝑝²_𝑗−4𝑞_𝑗<0 for all 𝑗=1,…,𝑠. Obviously, 𝐾²+𝑝_𝑗𝐾+𝐾_𝑗>0 for all 𝑗=1,…,𝑠. Now, we divide into two cases to discuss the choice of 𝐾_𝑖. (1) If there is at least one positive number for 𝐾₁,…,𝐾_𝑟 under the condition of appropriate value of 𝑝, one chooses 𝐾^∗=max_{1≤𝑖≤𝑟}{𝐾_𝑖}. (2) Otherwise, 𝐾^∗=0. Thus there always exists 𝐾^∗≥0, such that for any 𝐾>𝐾^∗, (38) holds.

6. Inverse Optimal Controller Design

In this section we will design the inverse optimal controller on the basis of Theorem 8 to meet specific performance indicators besides achieving control objectives.

Theorem 9. The control law 𝑢^∗=−𝜃𝐾^𝑛𝑣_𝑛𝑤^𝑝_𝑛,𝜃≥2, (44) solves the problem of inverse optimal stabilization in probability for (9) by minimizing the cost function 𝐽(𝑢)=𝐸^∞₀𝑙(𝑥)+1𝑥,𝐾^2(𝑛−1)𝑣_𝑛⁻¹(𝑥)𝐾^𝑛𝑢²𝑑𝑟, (45) where 𝜑₁(𝑥,𝑥)=(Φ^𝑇(𝑥),0,…,0)^𝑇,  𝜑₂(𝑥,𝑥)=(0,…,0,1)^𝑇, 𝑉=𝑉_𝑛.

Proof. Equations (10) and (11) can be represented as 𝑑𝑥𝑑𝑥=𝜑₁(𝑥,𝑥)𝑑𝜔+𝜑₂(𝑥,𝑥)𝑢𝑑𝑡, (46) where 𝜑₁,  𝜑₂ are identified in Theorem 9. Choosing 𝛾(𝑟)=(1/2𝐾^2(𝑛−1))𝑟², we can get (𝛾^′)⁻¹(𝑟)=𝐾^2(𝑛−1)𝑟 and ℓ𝛾(𝑟)=(1/2)𝐾^2(𝑛−1)𝑟². Applying Lemma 4, we get 𝑢=𝛽(𝑥)=−𝑅₂⁻¹(𝑥)1𝐾^2(𝑛−1)𝑤^𝑝_𝑛12𝐾^2(𝑛−1)1=−2𝑅₂⁻¹(𝑥)𝑤^𝑝_𝑛. (47) According to Theorem 8 and Lemma 4, the inverse optimal controller can be designed as follows: 𝑢^∗=𝛽^∗(𝑥)𝜃=−2𝑅₂⁻¹(𝑥)1𝐾^2(𝑛−1)𝑤^𝑝_𝑛𝐾^2(𝑛−1)𝜃=−2𝑅₂⁻¹(𝑥)𝑤^𝑝_𝑛=𝜃𝛽(𝑥)=𝜃𝑢,𝜃≥2, (48) where 𝑅₂(𝑥)=1/2𝐾^𝑛𝑣_𝑛.

7. Simulation Examples

In this section, for a numerical example, we design the output-feedback controller by using two methods, where one method is introduced in this paper and the other is introducted in [19, 20].

Consider the following stochastic system: 𝑑𝑥₁=𝑥₂1𝑑𝑡+𝑥10³¹sin𝑥₂𝑑𝜔,𝑑𝑥₂1=𝑢𝑑𝑡+𝑥10³¹+𝑥³₂𝑑𝜔,𝑦=𝑥₁, (49) where 𝜑₁(𝐱=1)𝑥10³¹sin𝑥₂≤1𝑥10¹³,𝜑₂(𝐱=1)𝑥10³¹+𝑥³₂≤1𝑥10¹³+𝑥₂³. (50) With the notation of Assumption 5, one can take 𝑝=3, 𝑐=1/10.

In line with design method discussed in Section 4, we can design the observer states as follows: 𝑥̇₁=𝑥₂+𝐾𝑦−𝑥₁,𝑥̇₂=𝑢+𝐾²𝑦−𝑥₁. (51) According to the design procedure in Section 4, we construct the controller as follows: 𝛽₁𝑥₁=−𝐾𝑣₁𝑤³₁,𝑣₁=2.75,𝑢=−𝐾²𝑣₂𝑤³₂,𝑣₂=8.27, (52) where 𝐾 will be chosen later, ℎ₁=ℎ₂=1, and 𝑑^∗=(1/12)×10⁻⁴‖𝑃‖(1+1/𝐾)², ‖𝑃‖=((15+5√5)/8)^1/2. With Theorem 8, one gets 𝐾>42.35. According to the design procedure in Section 6, we choose 𝜃=2 and construct the inverse optimal controller as follows: 𝑢^∗=−2𝐾²𝑣₂𝑤³₂. (53)

In simulation, we choose the initial values 𝑥₁(0)=0.02, 𝑥₂(0)=0.01, 𝑥₁(0)=0.01, 𝑥₂(0)=−0.01, and 𝐾=50. Figure 1 shows the responses of the closed-loop system (49)~(53), which demonstrate the effectiveness of the control scheme.

(a)

(b)

(c)

Figure 1 

The responses of the closed-loop systems (49)~(53).

If the method in [19, 20] is adopted for the same systems, Figure 2 gives the corresponding responses of the systems (here the controller design theory of [19, 20] is not tackled details, and the interested readers can consult the relevant literature).

(a)

(b)

(c)

Figure 2 

The responses of the closed-loop systems (49)~(53) when adopting the method in [19, 20].

Remark 10. By comparing the two figures, we can observe that the value of the control of Figure 1 is far less than Figure 2. In other words, our method requires less control effort to ensure the closed-loop system to be globally asymptotically stable in probability, and it demonstrates the advantage of this method clearly.

8. Concluding and Outlook

In this paper, we have studied the output-feedback stabilization for a class of stochastic nonlinear systems. We have given a design of the output-feedback controller so as to make the equilibrium at the origin of the closed-loop system globally asymptotically stable in probability by using the backstepping design technique and choosing a high-gain parameter, and the inverse optimal stabilization in probability is achieved. Our main contribution is extending the linear growth conditions to the more general power growth conditions so as to enable the result to be more general and to have a broader field of use.

There are two problems to be investigated.(1)By extending the value of 𝑝 in Assumption 5 from positive integers to the rationales, it can further weaken the conditions of the system (5). For this system, output-feedback problem deserves further research.(2)Another is to extend stochastic nonlinear systems in this paper to delay systems and study the design of controller.