Global Adaptive Control of Stochastic Nonlinear Systems with Linearly Bounded Unmeasurable States by Output Feedback

Abstract

The problem of the output feedback stochastic stabilization is investigated for a class of stochastic nonlinear systems with linearly bounded unmeasurable states. Under the condition that the inverse dynamics is stochastic input-to-state stable and the nonlinear functions satisfy the linear growth conditions with unknown growth rate, an adaptive output feedback controller is proposed to make the closed-loop system globally stable in probability and the states of the closed-loop system converge to zero almost surely. A simulation example is provided to show the effectiveness of the theoretical results.

1. Introduction

The output feedback control for stochastic nonlinear systems is a challenging research subject in control theory. Since the design of output feedback control is more difficult and more practical in engineering than that of full-state feedback control, many output feedback controller design schemes for stochastic nonlinear systems have been proposed [1–8]. For the deterministic nonlinear systems, some counterexamples were given in [9] to indicate that the global output feedback stabilization of the nonlinear systems is usually impossible unless some growth conditions on the unmeasurable states of the systems are introduced. From then on, the problems of global output feedback stabilization for nonlinear systems with various growth conditions were studied [10–15]. Recently, some output feedback controllers were designed for the stochastic nonlinear systems in which nonlinear terms depend on the output and unmeasurable inverse dynamics or unmodeled dynamics [3, 16–19]; moreover, an output feedback controller was proposed for the stochastic nonlinear systems with the nonlinear terms dependent on the unmeasurable states and unmeasurable inverse dynamics [20]. However [20] considers only the case that the parameters in the bounding functions are known; when they are unknown constants, the global output feedback control of the stochastic system becomes much more difficult due to the lack of effective observer design techniques [21]. Recently, the output feedback stabilization for a class of stochastic nonlinear systems with stochastic iISS inverse dynamics was studied in [22–27].

In this paper, we focus on the output feedback stabilization for a class of stochastic nonlinear systems in which the drift and diffusion terms are dependent on unmeasurable states besides the output and unmeasurable inverse dynamics and without knowing the growth rate. Under the assumption that the inverse dynamics are SISS (stochastic input-to-state stable), an adaptive output feedback controller with a high-gain observer is constructed to make the closed-loop system globally stable in probability and the states of the closed-loop converge to zero almost surely. The contribution of this paper consists of two aspects. (i) By adopting the universal adaptive output feedback controller presented in [21], the output feedback control problem of stochastic nonlinear system with the nonlinear functions being bounded with unknown constants is investigated. (ii) The two assumptions in [20] are relaxed (see Remark 3.3).

The remained of this paper is organized as follows. Section 2 provides some preliminaries and Section 3 gives the problem to be studied. An adaptive output feedback controller is proposed in Section 4. The main results of this paper and the stability analysis of the closed-loop system are presented in Section 5. Section 6 gives a simulation example to demonstrate the correctness of the theoretical results and some concluding remarks are contained in Section 7.

2. Preliminaries

Consider the following stochastic nonlinear systems:𝑑𝑥=𝑓(𝑥,𝑡)𝑑𝑡+𝑔(𝑥,𝑡)𝑑𝑤, (2.1) where 𝑥∈𝑅^𝑛 is the state, 𝑤 is an 𝑚-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,𝑃) with Ω being a sample space, ℱ being a 𝜎-field, and 𝑃 being the probability measure, and 𝑓∶𝑅^𝑛×𝑅₊→𝑅^𝑛 and 𝑔∶𝑅^𝑛×𝑅₊→𝑅^𝑚 are locally Lipschitz continuous functions in 𝑥 uniformly in 𝑡 and satisfy the linear growth condition. Meanwhile, we assume that 𝑓(0,𝑡)=0 and 𝑔(0,𝑡)=0.

Throughout this paper, we define the differential operator 𝔏 as follows:𝔏𝑉=𝜕𝑉+𝜕𝑡𝜕𝑉𝑓()+1𝜕𝑥𝑥,𝑡2Tr𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔, (2.2) where 𝑉(𝑥,𝑡)∈𝐶^2,1(𝑅^𝑛×𝑅₊,𝑅₊). We also denote by 𝐷(𝑅₊;𝑅₊) the family of all functions 𝛾∶𝑅₊→𝑅₊ such that ∫₀^+∞𝛾(𝑡)𝑑𝑡<∞.

The following lemma provides a sufficient condition to ensure the existence and uniqueness of global solution for the system (2.1).

Lemma 2.1. For system (2.1), assume that both terms 𝑓(𝑥,𝑡) and 𝑔(𝑥,𝑡) are locally Lipschitz, and 𝑓(0,𝑡) and𝑔(0,𝑡) are bounded uniformly in 𝑡. If there exists a function 𝑉(𝑥,𝑡)∈𝐶^2,1(𝑅^𝑛×𝑅₊,𝑅₊) such that for a constant 𝐾>0 and any 𝑡≥0, 𝔏𝑉(𝑥,𝑡)≤𝐾(1+𝑉(𝑥,𝑡)),lim_|𝑥|→∞inf_𝑡≥0𝑉(𝑥,𝑡)=∞, (2.3) then, there exists a unique solution on [0,∞) for any initial value 𝑥(0)=𝑥₀∈𝑅^𝑛.

Proof. This is a special form of Theorem 1 in [28].

Lemma 2.2. For system (2.1), assume that there exist functions𝑉(𝑥,𝑡)∈𝐶^2,1(𝑅^𝑛×𝑅₊,𝑅₊), 𝜇₁∈𝒦,𝜇₂∈𝒦,𝜇₃∈𝒦,and 𝛾(𝑡)∈𝐷(𝑅₊;𝑅₊) such that 𝜇₁(𝑥)≤𝑉(𝑥,𝑡)≤𝜇₂(𝑥),𝔏𝑉(𝑥,𝑡)≤𝛾(𝑡)−𝜇₃(𝑥), (2.4) for all (𝑥,𝑡)∈(𝑅^𝑛,𝑅₊). Then, for any initial value 𝑥(0)=𝑥₀∈𝑅^𝑛, the solution of system (2.1) is bounded almost surely and has the properties that lim_𝑡→∞𝑥𝑡;𝑥₀=0a.s.,_0+∞𝛾(𝑡)−𝔏𝑉(𝑥,𝑡)𝑑𝑡<+∞a.s. (2.5)Proof. This Lemma can be proved by using Theorem  2.6 in [29] and Corollary  4.2 in [30].

3. Problem Statement

Consider the following stochastic system: 𝑑𝜂=𝑓₀𝜂,𝑦,𝑡𝑑𝑡+𝑔₀𝜂,𝑦,𝑡𝑑𝑤𝑑𝑥₁=𝑥₂+𝑓₁𝑥,𝜂,𝑡𝑑𝑡+𝑔₁⋮𝑥,𝜂,𝑡𝑑𝑤𝑑𝑥_𝑛−1=𝑥_𝑛+𝑓_𝑛−1𝑥,𝜂,𝑡𝑑𝑡+𝑔_𝑛−1𝑥,𝜂,𝑡𝑑𝑤𝑑𝑥_𝑛=𝑢+𝑓_𝑛𝑥,𝜂,𝑡𝑑𝑡+𝑔_𝑛𝑥,𝜂,𝑡𝑑𝑤𝑦=𝑥₁, (3.1) where 𝜂∈𝑅^𝑚 is the state of the unmeasurable inverse dynamics, 𝑥=(𝑥₁,…,𝑥_𝑛) is the state, 𝑢∈𝑅 is the control input, 𝑦∈𝑅 is the measured output, 𝑓₀∈𝑅^𝑚, 𝑔₀∈𝑅^𝑚×𝑠, 𝑓_𝑖∈𝑅, and 𝑔^𝑇_𝑖∈𝑅^𝑠,𝑖=1,…,𝑛 are uncertain locally Lipschitz in the first two arguments uniformly in 𝑡. Moreover𝑓_𝑖(0,0,𝑡)=0, 𝑔_𝑖(0,0,𝑡)=0, 𝑖=0,…,𝑛, and 𝑤 is 𝑚-dimensional standard Wiener process as defined in Section 2. For system (2.1), we give the following assumptions.

Assumption 3.1. There exist unknown positive constants 𝜃₁, 𝜃₂, 𝜃₃, and 𝜃₄ such that 𝑓_𝑖𝑥,𝜂,𝑡≤𝜃₁𝑥₁+⋯+𝑥_𝑖+𝜃₂𝜂𝑔,𝑖=1,…,𝑛,_𝑖𝑥,𝜂,𝑡≤𝜃₃𝑥₁+⋯+𝑥_𝑖+𝜃₄𝜂,𝑖=1,…,𝑛. (3.2)

Assumption 3.2. There exist unknown positive constants 𝛼 and 𝑟 and a positive definite function 𝑉_𝜂(𝜂,𝑡) which is radially unbounded in 𝜂 uniformly in 𝑡, such that 𝔏𝑉_𝜂≤−𝛼𝜂²+𝑟𝑦². (3.3)

Remark 3.3. In [20] (B2 and B3), the positive constants 𝜃₁, 𝜃₂, 𝜃₃, 𝜃₄, 𝛼, and 𝑟 are known, but they are unknown in this paper. Thus, it is more difficult and meaningful to design an output feedback stabilizing control law for system (3.1).
The control problem in this paper is precisely formulated as follows.
For system (3.1), under Assumptions 3.1 and 3.2, the control objective is to design a smooth dynamic output feedback controller:,,𝜒̇=𝜛𝜒,𝑦𝑢=𝜇𝜒,𝑦 (3.4) such that the solution of closed-loop system (3.1) and (3.4) is bounded a.s., lim_𝑡→∞‖𝑥(𝑡)‖=0a.s., andlim_𝑡→∞‖𝜂(𝑡)‖=0a.s.

4. Adaptive Output Feedback Controller Design

First, we design a state observer for system (3.1): 𝑥̇₁=𝑥₂+𝐿𝑎₁𝑦−𝑥₁⋮𝑥̇_𝑛−1=𝑥_𝑛+𝐿^𝑛−1𝑎_𝑛−1𝑦−𝑥₁𝑥̇_𝑛=𝑢+𝐿^𝑛𝑎_𝑛𝑦−𝑥₁𝐿̇=𝑦−𝑥₁²,𝐿(0)=1, (4.1) where 𝑎_𝑖>0, 𝑖=1,…,𝑛 are the coefficients of the Hurwitz polynomial 𝑝(𝑠)=𝑠^𝑛+𝑎₁𝑠^𝑛−1+⋯+𝑎_𝑛.

Let 𝑒_𝑖=(𝑥_𝑖−𝑥_𝑖)/𝐿^𝑖−1, 𝑖=1,…,𝑛, and 𝑒=(𝑒₁,…,𝑒_𝑛)^𝑇. Then we can obtain the following error dynamic system:𝑑𝑒₁=𝐿𝑒₂−𝐿𝑎₁𝑒₁+𝑓₁𝑑𝑡+𝑔₁⋮𝑑𝑤𝑑𝑒_𝑛−1=𝐿𝑒_𝑛−𝐿𝑎_𝑛−1𝑒₁+𝑓_𝑛−1𝐿^𝑛−2−()𝐿̇𝑛−2𝐿𝑒_𝑛−1𝑔𝑑𝑡+^𝑛−1𝐿^𝑛−2𝑑𝑤𝑑𝑒_𝑛=−𝐿𝑎_𝑛𝑒₁+𝑓_𝑛𝐿^𝑛−1−()𝐿̇𝑛−1𝐿𝑒_𝑛𝑔𝑑𝑡+^𝑛𝐿^𝑛−1𝑑𝑤. (4.2) It can be written as the following compact form: 𝐿̇𝑑𝑒=𝐿𝐴𝑒+𝐹−𝐿𝐷𝑒𝑑𝑡+𝐺𝑑𝑤, (4.3) where 𝐴=−𝑎₁10⋮⋱−𝑎_𝑛−10⋯1−𝑎_𝑛{}𝑓0⋯0,𝐷=diag0,1,⋯,𝑛−1,𝐹=¹𝑓₂𝐿⋮𝑓_𝑛𝐿^𝑛−1𝑔,𝐺=¹𝑔₂𝐿⋮𝑔_𝑛𝐿^𝑛−1. (4.4) Since the polynomial 𝑝(𝑠)=𝑠^𝑛+𝑎₁𝑠^𝑛−1+⋯+𝑎_𝑛 is Hurwitz, there exists a positive definite matrix 𝑃 such that 𝐴^𝑇𝑃+𝑃𝐴=−𝐼,𝐷^𝑇𝑃+𝑃𝐷≥0. (4.5)

Let 𝑧_𝑖=𝑥_𝑖/𝐿^𝑖−1, 𝑖=1,…,𝑛, 𝑣=𝑢/𝐿^𝑛, and 𝑧=(𝑧₁,…,𝑧_𝑛)^𝑇. Then we can obtain𝑑𝑧₁=𝐿𝑧₂+𝐿𝑎₁𝑒₁⋮𝑑𝑡𝑑𝑧_𝑛−1=𝐿𝑧_𝑛+𝐿𝑎_𝑛−1𝑒₁−()𝐿̇𝑛−2𝐿𝑧_𝑛−1𝑑𝑡𝑑𝑧_𝑛=𝐿𝑣+𝐿𝑎_𝑛𝑒₁−()𝐿̇𝑛−1𝐿𝑧_𝑛𝑑𝑡. (4.6) Design the controller𝑢=𝐿^𝑛𝑣=−𝐿^𝑛𝑏₁𝑧₁+⋯+𝑏_𝑛𝑧_𝑛=−𝐿^𝑛𝑏₁𝑥₁+𝐿^𝑛−1𝑏₂𝑥₂+⋯+𝐿𝑏_𝑛𝑥_𝑛, (4.7) where 𝑏_𝑖>0,𝑖=1,…,𝑛, are the coefficients of the Hurwitz polynomial ℎ(𝑠)=𝑠^𝑛+𝑏_𝑛𝑠^𝑛−1+⋯+𝑏₁. Substituting (4.7) into (4.6), we have 𝑑𝑧=𝐿𝐵𝑧+𝐿𝑎𝑒₁−𝐿̇𝐿𝐷𝑧𝑑𝑡, (4.8) where 𝐵=01⋯0⋮⋮⋱⋮00⋯1−𝑏₁−𝑏₂⋯−𝑏_𝑛,𝑎=𝑎₁,…,𝑎_𝑛^𝑇. (4.9) For the polynomial ℎ(𝑠)=𝑠^𝑛+𝑏_𝑛𝑠^𝑛−1+⋯+𝑏₁ being Hurwitz, there exists a positive definite matrix 𝑄 such that 𝐵^𝑇𝑄+𝑄𝐵=−2𝐼,𝐷^𝑇𝑄+𝑄𝐷≥0. (4.10) Then the closed-loop system can be represented in the following compact form: 𝑑𝜂=𝑓₀𝜂,𝑦,𝑡𝑑𝑡+𝑔₀𝐿̇𝜂,𝑦,𝑡𝑑𝑤,𝑑𝑒=𝐿𝐴𝑒+𝐹−𝐿𝐷𝑒𝑑𝑡+𝐺𝑑𝑤,𝑑𝑧=𝐿𝐵𝑧+𝐿𝑎𝑒₁−𝐿̇𝐿̇𝐷𝑧𝑑𝑡,𝐿=𝑦−𝑥₁²=𝑒²₁. (4.11)

5. Stability Analysis of the Closed-Loop System

By Assumption 3.1 and 𝐿≥1, we have the following estimates: ‖𝐹‖²≤^𝑛_𝑖=1𝑓_𝑖𝐿^𝑖−12≤^𝑛_𝑖=11𝐿^𝑖−1𝜃₁𝑥₁+⋯+𝑥_𝑖+𝜃₂𝜂²≤2^𝑛_𝑖=11𝐿^𝑖−1𝜃₁𝑥₁+⋯+𝑥_𝑖²+2𝑛𝜃²₂𝜂²≤2𝑛𝜃²₁𝑥₁+𝑥₂𝐿𝑥+⋯+^𝑛𝐿^𝑛−12+2𝑛𝜃²₂𝜂²≤2𝑛𝜃²₁𝑒₁+𝑥₁+𝑒₂+𝑥₂𝐿+⋯+𝑒_𝑛+𝑥_𝑛𝐿^𝑛−12+2𝑛𝜃²₂𝜂²=2𝑛𝜃²₁𝑒₁+⋯+𝑒_𝑛+𝑧₁+⋯+𝑧_𝑛²+2𝑛𝜃²₂𝜂²≤4𝑛²𝜃²₁‖𝑒‖²+‖𝑧‖²+2𝑛𝜃²₂𝜂²,‖𝐺‖²≤^𝑛_𝑖=1𝑔_𝑖𝐿^𝑖−12≤^𝑛_𝑖=11𝐿^𝑖−1𝜃₃(𝑥₁+⋯+𝑥_𝑖)+𝜃₄𝜂²≤2^𝑛_𝑖=11𝐿^𝑖−1𝜃₃(𝑥₁+⋯+𝑥_𝑖)²+2𝑛𝜃²₄𝜂²≤2𝑛𝜃²₃(𝑥₁+𝑥₂𝐿𝑥+⋯+^𝑛𝐿^𝑛−1)²+2𝑛𝜃²₄𝜂²≤4𝑛²𝜃²₃‖𝑒‖²+‖𝑧‖²+2𝑛𝜃²₄𝜂². (5.1)

Select 𝑉₁=𝛿𝑒^𝑇𝑃𝑒, where 𝛿>0 is a parameter to be specified later. Then, by (2.2), (4.5), (4.11), and (5.1) we have 𝔏𝑉₁=𝛿𝑒^𝑇𝐿𝐴^𝑇𝑃+𝑃𝐴𝑒+2𝛿𝑒^𝑇𝑃𝐹−𝛿𝑒^𝑇()𝑒𝐿̇𝐷𝑃+𝑃𝐷𝐿+𝛿𝑡𝑟𝐺^𝑇‖𝑒‖𝑃𝐺≤−𝛿𝐿²+𝛿𝑒^𝑇𝑃²+𝛿‖𝐹‖²−𝛿𝑒^𝑇()𝑒𝐿̇𝐷𝑃+𝑃𝐷𝐿+𝛿𝜆_max(𝑃)‖𝐺‖²≤−𝛿𝐿−‖𝑃‖²‖𝑒‖²+𝛿4𝑛²𝜃²₁‖𝑒‖²+‖𝑧‖²+2𝑛𝜃²₂𝜂²−𝛿𝑒^𝑇()𝑒𝐿̇𝐷𝑃+𝑃𝐷𝐿+𝛿𝜆_max(𝑃)4𝑛²𝜃²₃‖𝑒‖²+‖𝑧‖²+2𝑛𝜃²₄𝜂²≤−𝛿𝐿−‖𝑃‖²−𝑐₁‖𝑒‖²+𝑐₁𝛿‖𝑧‖²+𝑐₂𝛿𝜂², (5.2) where 𝑐₁=4𝑛²(𝜃²₁+𝜆_max(𝑃)𝜃²₃), and 𝑐₂=2𝑛(𝜃²₂+𝜆_max(𝑃)𝜃²₄).

Select 𝑉₂=𝑧^𝑇𝑄𝑧; then 𝑉̇₂=𝑧^𝑇𝐿𝐵^𝑇𝑄+𝑄𝐵𝑧+2𝐿𝑧^𝑇𝑄𝑎𝑒₁−𝑧^𝑇()𝑧𝐿̇𝐷𝑄+𝑄𝐷𝐿. (5.3) Notice that 2𝐿𝑧^𝑇𝑄𝑎𝑒₁≤𝐿‖𝑧‖²+𝐿‖𝑄𝑎‖²𝑒²₁. (5.4) By (4.10), we get 𝑉̇₂≤−𝐿‖𝑧‖²+𝐿‖𝑄𝑎‖²𝑒²₁. (5.5) Select 𝑉=((𝛿𝑐₂+𝜏)/𝛼)𝑉_𝜂+𝑉₁+𝑉₂, where 𝜏>0; then from (3.3), (5.2), and (5.5), we can obtain that 𝔏𝑉≤−𝜏𝜂²+𝛿𝑐₂+𝜏𝛼𝑟𝑦²−𝛿𝐿−‖𝑃‖²−𝑐₁−𝐿‖𝑄𝑎‖²‖𝑒‖²−𝐿−𝛿𝑐₁‖𝑧‖²≤−𝜏𝜂²+2𝛿𝑐₂+𝜏𝛼𝑟𝑒²₁+2𝛿𝑐₂+𝜏𝛼𝑟𝑧²₁−𝛿𝐿−‖𝑃‖²−𝑐₁−𝐿‖𝑄𝑎‖²‖𝑒‖²−𝐿−𝛿𝑐₁‖𝑧‖²=−𝜏𝜂²−𝛿𝐿−‖𝑃‖²−𝑐₁−𝐿‖𝑄𝑎‖²−2𝛿𝑐₂+𝜏𝛼𝑟‖𝑒‖²−𝐿−𝛿𝑐₁−2𝛿𝑐₂+𝜏𝛼𝑟‖𝑧‖². (5.6) Let 𝛿=‖𝑄𝑎‖²+1, 𝑐₃=𝛿‖𝑃‖²+𝛿𝑐₁+2((𝛿𝑐₂+𝜏)/𝛼)𝑟; then 𝔏𝑉≤−𝜏𝜂²−𝐿−𝑐₃‖𝑒‖²+‖𝑧‖². (5.7)

The main result of the paper is summarized in the following theorem.

Theorem 5.1. Suppose that the system (3.1) satisfies Assumptions 3.1 and 3.2; then under the control law (4.7) and the observer (4.1), the closed-loop system has a unique solution on [0,∞) and the solution process is bounded almost surely; moreover, lim_𝑡→∞‖𝑥(𝑡)‖=0a.s.,andlim_𝑡→∞‖𝜂(𝑡)‖=0a.s.

Proof. The proof process can be divided three parts.
First, we prove that 𝐿(𝑡) is bounded on [0,∞). This can be done by a contradiction argument. Suppose lim_𝑡→∞𝐿(𝑡)=∞. Since𝐿̇(𝑡)=𝑒²₁≥0, 𝐿(𝑡) is a monotone nondecreasing function. Thus, there exists a finite time 𝑡^⋅<∞, such that𝐿(𝑡)≥𝑐₃+1,∀𝑡∈[𝑡^⋅,∞). (5.8) From (5.7), it follows that 𝔏𝑉≤−𝜏𝜂²−‖𝑒‖²−‖𝑧‖²,∀𝑡∈[𝑡^⋅,∞). (5.9) With Lemma 2.2, we can obtain that ^∞_𝑡⋅𝑒²₁𝑑𝑡≤^∞_𝑡⋅‖𝑒‖²𝑑𝑡<+∞a.s. (5.10) By this and the definition of 𝐿̇(𝑡), we have +∞=𝐿(∞)−𝐿(𝑡^⋅)=^∞_𝑡⋅𝐿̇(𝑡)𝑑𝑡=^∞_𝑡⋅𝑒²₁𝑑𝑡<+∞a.s., (5.11) which leads to a contradiction. Thus, the dynamic gain 𝐿(𝑡) is well defined and bounded on [0,∞). Moreover, from 𝐿̇(𝑡)=𝑒²₁, it is concluded that ∫^∞₀𝑒²₁𝑑𝑡<∞.
From (5.7) and the boundedness of 𝐿(𝑡), it can be proved that 𝔏𝑉≤𝛽𝑉 for a positive constant 𝛽 and for ∀𝑡∈[0,∞). From Lemma 2.1, we can show the existence and uniqueness of global solution for the system (4.11).
Next, we claim that 𝑧 is well defined and bounded on the interval [0,+∞).
From (5.5) and 𝐿̇(𝑡)=𝑒²₁, one can obtain that 𝑉̇₂≤−𝐿‖𝑧‖²+𝐿‖𝑄𝑎‖²𝑒²₁≤−‖𝑧‖²+‖𝑄𝑎‖²𝐿𝐿̇. (5.12) Then, for ∀𝑡∈[0,∞), integrate two sides of the above inequality: 𝜆_min(𝑄)‖𝑧‖²−𝑧^𝑇(0)𝑄𝑧(0)≤𝑉₂(𝑡)−𝑉₂(0)≤−^𝑡₀‖𝑧‖²𝑑𝑡+^𝑡₀‖𝑄𝑎‖²𝐿𝐿̇𝑑𝑡=−^𝑡₀‖𝑧‖²1𝑑𝑡+2‖𝑄𝑎‖²𝐿²(𝑡)−1. (5.13) From (5.13) we can obtain that, for ∀𝑡∈[0,∞), ‖𝑧‖²≤1𝜆_min(𝑄)𝑧^𝑇(0)(0)+1𝑄𝑧2‖𝑄𝑎‖²𝐿²(𝑡)−1_𝑡0,‖𝑧‖²𝑑𝑡≤𝑧^𝑇(0)(0)+1𝑄𝑧2‖𝑄𝑎‖²𝐿²(𝑡)−1. (5.14) Since 𝐿(𝑡) is bounded on [0,+∞), the previous inequalities imply the boundedness of 𝑧 and ∫^∞₀‖𝑧‖²𝑑𝑡.
Finally, we prove the boundedness of 𝑒 and 𝜂 on [0,+∞).
Let 𝜉_𝑖=𝑥_𝑖−𝑥_𝑖𝐿^𝑖−1,𝑖=1,…,𝑛,𝜉=𝜉₁,…,𝜉_𝑛^𝑇, (5.15) where 𝐿 is a constant satisfying 𝐿=max3+‖𝑃‖²+𝑐₁,𝐿(+∞). (5.16) Then 𝑑𝜉=𝐿𝐴𝜉+𝐿𝑎𝜉₁−𝐿Γ𝑎𝜉₁+𝐹𝑑𝑡+𝐺𝑑𝑤, (5.17) where 𝐹=𝑓₁𝑓₂𝐿⋯𝑓_𝑛𝐿^𝑛−1𝑇,𝐺=𝑔₁𝑔₂𝐿⋯𝑔_𝑛𝐿^𝑛−1𝑇1𝐿,Γ=diag𝐿⋯𝐿^𝑛−1𝐿^𝑛−1. (5.18) Similar to (5.1), we have 𝐹²≤^𝑛_𝑖=1𝑓_𝑖𝐿^𝑖−12≤4𝑛²𝜃²₁𝜉²+‖𝑧‖²+2𝑛𝜃²₂𝜂²,𝐺²≤^𝑛_𝑖=1𝑔_𝑖𝐿^𝑖−12≤4𝑛²𝜃²₃𝜉²+‖𝑧‖²+2𝑛𝜃²₄𝜂². (5.19) Now we consider the Lyapunov function 𝑉₃=𝜉^𝑇𝑃𝜉+((𝑐₂+𝜏)/𝛼)𝑉_𝜂. A straightforward calculation shows that along the trajectories of (4.11), we can obtain 𝔏𝑉₃≤−𝐿𝜉²+2𝜉₁𝐿𝑎^𝑇𝑃𝜉−2𝜉₁𝐿𝑎^𝑇Γ𝑃𝜉+2𝐹^𝑇𝑃𝜉+𝑡𝑟𝐺^𝑇𝑃𝐺−𝑐₂+𝜏𝜂²+𝑐₂+𝜏𝛼𝑟𝑦². (5.20) Noticing that 2𝜉₁𝐿𝑎^𝑇𝑃𝜉≤𝐿²𝑎^𝑇𝑃²𝜉²₁+𝜉²,−2𝜉₁𝐿𝑎^𝑇Γ𝑃𝜉≤𝐿²𝑎^𝑇Γ𝑃²𝜉²₁+𝜉²,2𝐹^𝑇𝑃𝜉≤‖𝑃‖²𝜉²+𝐹²≤‖𝑃‖²+4𝑛²𝜃²₁𝜉²+4𝑛²𝜃²₁‖𝑧‖²+2𝑛𝜃²₂𝜂²,𝑡𝑟𝐺^𝑇𝑃𝐺≤𝜆_max(𝑃)𝐺²≤4𝑛²𝜆_max(𝑃)𝜃²₃𝜉²+‖𝑧‖²+2𝑛𝜆_max(𝑃)𝜃²₄𝜂², (5.21) we have 𝔏𝑉₃≤−𝐿−2−‖𝑃‖²−𝑐₁𝜉²−𝑐₂+𝜏𝜂²+𝐿²𝑎^𝑇𝑃²+𝐿²𝑎^𝑇Γ𝑃²𝜉²₁+𝑐₁‖𝑧‖²+𝑐₂+𝜏𝛼𝑟𝑦²+𝑐₂𝜂²≤−𝜉²−𝜏𝜂²+𝑐₁‖𝑧‖²𝑐+2²+𝜏𝛼𝑟𝑒²₁+𝑧²₁+𝐿²𝑎^𝑇𝑃²+𝑎^𝑇Γ𝑃²𝑒²₁. (5.22) Since ∫^∞₀‖𝑧‖²𝑑𝑡 and ∫^∞₀𝑒²₁𝑑𝑡 are bounded, from Lemma 2.2, we can obtain that ‖𝜂(𝑡)‖ and ‖𝜉(𝑡)‖ are bounded almost surely, lim_𝑡→∞‖𝜉(𝑡)‖=0a.s., and lim_𝑡→∞‖𝜂(𝑡)‖=0a.s. From the definition of 𝑒 and 𝜉, we have that ‖𝑒(𝑡)‖ is bounded almost surely and lim_𝑡→∞‖𝑒(𝑡)‖=0a.s. Moreover, lim_𝑡→∞‖𝑧(𝑡)‖=0a.s. is followed from (5.5); then it leads to lim_𝑡→∞‖𝑥(𝑡)‖=0a.s.. Thus, we obtain that lim_𝑡→∞‖𝜂(𝑡)‖=0a.s., and lim_𝑡→∞‖𝑥(𝑡)‖=0a.s. Up to now, the proof is completed.

6. Simulation

Considering the following stochastic nonlinear system: 𝑑𝜂=−𝜂+𝜆₁𝑦𝑑𝑡+𝜆₂𝑦sin𝑦𝑑𝑤,𝑑𝑥₁=𝑥₂+𝜂+𝜆₃𝑦𝜆𝑑𝑡+⁴𝑥²₁1+𝑥²₁𝜂𝑑𝑤,𝑑𝑥₂=𝑢+𝜂+𝜆₅𝑥₂𝑑𝑡,𝑦=𝑥₁, (6.1) where 𝑤 is an 𝑚-dimensional standard Wiener process, and 𝜆_𝑖, 𝑖=1,…,5 are unknown parameters.

By Theorem 5.1, we can design the output feedback controller as follows: 𝑑𝑥₁=𝑥₂+𝐿𝑎₁𝑦−𝑥₁𝑑𝑡,𝑑𝑥₂=𝑢+𝐿²𝑎₂𝑦−𝑥₁̇𝑑𝑡,𝐿=𝑦−𝑥₁²,𝑢=−𝐿²𝑏₁𝑥₁+𝐿𝑏₂𝑥₂. (6.2) The initial values of the states are 𝜂(0)=1, 𝑥(0)=(0.5,0.5)^𝑇, 𝑥(0)=(0,0)^𝑇, and 𝜆₁=1, 𝜆₂=0.5, 𝜆₃=0.6, 𝜆₄=0.6, and 𝜆₅=0.5. Choosing the design parameters 𝑎₁=2, 𝑎₂=1, 𝑏₁=1, and 𝑏₂=2, we can obtain Figures 1, 2, 3, 4, and 5. From the figures, we can see that 𝐿 is bounded and 𝜂,𝑥, and 𝑥 converge to zero almost surely.

Figure 1 

System states 𝑥1 and 𝑥2.

Figure 2 

Observer states 𝑥1 and 𝑥2.

Figure 3 

System states 𝜂.

Figure 4 

Dynamic gain 𝐿.

Figure 5 

Control 𝑢.

7. Conclusion

In this paper, the adaptive output feedback control has been considered for a class of stochastic nonlinear systems with stochastic inverse dynamic and linearly bounded unmeasurable states. Like [20], a quadratic Lyapunov function is adopted instead of a quartic function and this makes the design process simple. Under some weak conditions, an adaptive high-gain observer has been designed and the closed-loop system is globally stable in probability. Moreover, the states of the closed-loop system converge to zero almost surely.

Acknowledgments

This work is partially supported by the National Natural Science Foundation under Grant no. 61074040, Natural Science Foundation of Shandong Province under Grant no. ZR2009GL009, China Postdoctoral Science Foundation under Grant no. 20110491559, Shandong, Province Postdoctoral Innovation Foundation under Grant no. 201003041, Research Startup Foundation of Qufu Normal University and Research Foundation of Qufu Normal University under Grant no. XJ200846.