Research Article | Open Access
Adaptive Output Feedback Stabilization of Random Nonlinear Systems with Unmodeled Dynamics Driven by Colored Noise
This paper focuses on the problem of adaptive output feedback stabilization for random nonlinear systems with unmodeled dynamics and uncertain nonlinear functions driven by colored noise. Under the assumption of unmodeled dynamics having enough stability margin, an adaptive output feedback stabilization controller is designed based on a reduced-order observer such that the state of the closed-loop system has an asymptotic gain in the 2-th moment (AG-2-M) and the mean square of the output can be made arbitrarily small by tuning parameters. A simulation example is used to illustrate the effectiveness of the control scheme.
During the past decades, the control problem for systems with unmodeled dynamics mainly caused by simplifications in modeling process has received much attention. To deal with the unmodeled dynamics, an available dynamic signal is constructed to bound the unmodeled dynamics by exploiting prior information on the system in . By the aid of small-gain technique in [2, 3], the unmodeled dynamics were treated by a worst-case design on the basis of small-gain arguments in . By introducing input-to-state stability for time-varying control systems, sufficient conditions for global stabilization of triangular systems are given in . For time-varying control systems, various equivalent characterizations of the nonuniform in time input-to-state stability (ISS) property are established in . By using a suitable extension of the small-gain theorem, a uniform input-to-state stabilization controller is given in . For MIMO nonlinear systems, a robust adaptive observer was given in . By introducing K-filters and dynamic signal,  designed an adaptive output feedback controller such that the closed-loop control system is semiglobally uniformly ultimately bounded. By combining small-gain theorem and changing supply function techniques,  proposed a robust adaptive output feedback controller. However, most of the existing references mainly focus on the deterministic case.
With the development of stochastic theory [11–15], many researchers pay great attention to the control problem of stochastic differential equations (SDEs) in the presence of unmodeled dynamics. Under the assumption of unmodeled dynamics having enough stability margin,  presented an output feedback controller based on minimal-order observer. In , a unifying solution to stochastic adaptive output feedback stabilization was presented by introducing dynamic signal and changing supply function technique. Also, some other control strategies such as adaptive neural networks control [18–20], adaptive fuzzy control [21, 22], and adaptive tracking control  were proposed.
Since the stochastic disturbance is more reasonably described as colored noise than white noise in practical engineering, the models described by SDEs may not suit very well. In , a theoretical framework on stability of random nonlinear systems where stochastic disturbance is colored noise and applications to controller design are given. For a class of random nonlinear systems with unmodeled dynamics, under the assumption of unmodeled dynamics having enough stability margin,  designed a feedback stabilization controller by backstepping method. Until now, to our knowledge, there are no many results on adaptive output feedback control for random nonlinear systems with unmodeled dynamics and uncertain nonlinear functions in the literature. In this paper, motivated by [16, 17, 24, 25], the purpose of this paper is to solve this problem under some milder assumptions. The main work consists of the following aspects.
(i) Because it is difficult to deal with Hessian terms caused by Wiener process, the stochastic disturbance is regarded as colored noise whose second-order moment is bounded in this paper. Distinguished from the existing stochastic analysis method, the method of ordinary differential equations is used to analyze the stability of the closed-loop system.
(ii) For easier-to-implement and more reliable practical purposes, a reduced-order observer and -dimension adaptive law are introduced, which lead to a simple controller. The state of the closed-loop system has an asymptotic gain in the -th moment (AG--M) and the mean square of the output can be regulated to an arbitrarily small neighborhood of the origin.
The paper is organized as follows: in Section 2, the mathematical preparation is given and the problem is formulated. Section 3 gives the observer-based backstepping adaptive controller design procedure. The main result is presented in Section 4. A simulation example is included in Section 5 to illustrate effectiveness of the proposed design method. Section 6 concludes the paper.
Notations: the following notations are used throughout the paper. For a vector , denotes its transpose; for a matrix , and denote its smallest and largest eigenvalue, respectively; denotes the usual Euclidean norm of “”; denotes the mathematical expectation; denotes the set of all nonnegative real numbers; denotes the real -dimensional space; denotes the real matrix space; denotes the set of all functions with continuous -th partial derivative. For simplicity, sometimes the arguments of functions are dropped.
2. Mathematical Preliminaries and Problem Formulation
2.1. Mathematical Preliminaries
Consider the following random nonlinear affine system:where is the state of system, is a stochastic process, and the underlying complete probability space is taken to be the quartet with a filtration satisfying the usual condition (i.e., it is increasing and right continuous while contains all -null sets). Both functions and are locally Lipschitz in piecewise continuous in ; i.e., for each , there exists a constant such thatfor any and . Moreover, and are bounded a.s. Process is a -adapted process and piecewise continuous, and there exists a positive constant such that
The following definition, criterion, and inequality are represented now for the stability analysis.
Proof. Following the same lines as the proof of [24, Theorem 3], the result can be obtained.
Lemma 3 (). Consider the continuous functions , , and they are integrable over every finite interval. If a continuous function satisfies the inequalitythen
2.2. Problem Formulation
Consider the following systemwhere , , and are the state, the unmodeled dynamics, the input and the output of system, respectively. and are -adapted stochastic processes. In this paper, it is assumed that only the output can be measured and uncertain functions , , , and are smooth.
The objective in this paper is to design an adaptive output feedback controller such that the state of the closed-loop system has an AG--M, and the mean square of the output can be regulated to an arbitrarily small neighborhood of the origin.
Throughout the paper, the following assumptions are made on system (9).
Assumption 4. and are -adapted processes and piecewise continuous, and there exists a positive constant such that
Assumption 5. For the -system, there exists a function , a -function , a smooth function , and constants , , , , such that
Assumption 6. For each , there exits an unknown positive constant such thatwhere , are known nonnegative smooth functions and are assumed.
Assumption 7. There exists a constant such that
Remark 8. Assumption 5 describes the dynamical behavior of the unobservable state . It has some stability margin with respect to the unmodeled dynamics. The term imposes restrictions on the influence of state on the stability of the unobservable state .
Remark 9. In Assumption 6, using the identity to , there exist smooth functions such thatwhich will be frequently used in the subsequent sections.
Remark 10. Assumption 7 depicts the connection between the stability margin of the unobservable state and the unmodeled dynamics.
3. Controller Design
3.1. Reduced-Order Observer Design
To counteract the unavailable state , a reduced-order partial-state observer is introduced as follows:where is chosen such thatis asymptotically stable. For each , denote the observer error aswhere . From (9), (15), and (17), one haswith , , and , whose compact from iswhere and .
From Assumption 4, we havewhere and .where . Since is stable, there exists a symmetric positive definite matrix such that . Along the solution of (19), differentiating the quadratic function yieldswhere is a design parameter.
3.2. Adaptive Backstepping Controller Design
From (9) and (17), the derivative of output is represented aswhich, together with (9), (15), and (19), consists of the following systemAn adaptive backstepping controller based on the system of (24) will be developed.
Step 1. Introduce the first two error variableswhere will be given in later. Consider the Lyapunov function candidatewhere , is the estimate of . In view of (24) and (25), the derivative of satisfiesApplying Young’s inequality, one haswhere is a design parameter. By Assumption 4, one haswhere and are design parameters. Substituting (28)-(30) into (27), one haswhere . The stabilizing function and the tuning function are designed aswhere is a design parameter and will be chosen in later. Then
Step . Introduce the coordinate changewith and . The differential of is given as follows:where . Assume that one has designed smooth function , such that the following inequality holds for ,where , , () are design parameter. In the following, we will prove that (36) holds for the -th Lyapunov function candidateThe derivative of satisfiesApplying Young’s inequality, one haswhere is a design parameter. By Assumption 4, one haswhere and are design parameters. With (39)-(41), it is obtained thatBy choosing the -th tuning function as with , one hasChoosewhere is a design parameter. ThenAt the end of the recursive procedure, the control law and adaptive law are chosen asBy (45) and (46), one gets
Remark 11. Because the system of this paper contains the colored noise and uncertain nonlinear functions (see (9)), the small-gain technique in [1–7] which is applied to deterministic nonlinear system is not applicable any more. Because the system of this paper is not an Itô type stochastic differential equation and the changing supply function technique in  cannot deal with the colored noise, the unmodeled dynamics is assumed to have enough stability margin motivated by  in this paper
Remark 12. Different from the deterministic nonlinear systems in [1–10], a class of random nonlinear systems driven by colored noise is considered in this paper. The stochastic disturbance is regarded as colored noise other than the white noise in [16, 17] which is more reasonable. Different from  only considering the problem of the state feedback stabilization, an adaptive output feedback stabilization controller is designed in this paper.
4. Stability Analysis
Theorem 13. For the random system (9), under Assumptions 4–7, the control law and adaptive law (46), the closed-loop system has a unique solution on , and the state of the closed-loop system has an AG--M. Furthermore, the output satisfieswhere and the right-hand can be made small enough by tuning parameters.
Proof. Define . From the definition of , satisfieswith and . Since the functions of the closed-loop system satisfy the local Lipschitz condition, from Lemma 2, (53), and (51), then the closed-loop system has a unique solution on , and the state of the closed-loop system has an AG--M.
Furthermore, by defining , from (51), one hasBy Lemma 3 and (3), it is obtained thatwhich together with (48) implieswhich leads to (52).
Noting and , it is clear that the right-hand sides of (52) can be made small enough by choosing large enough and small enough.
Remark 14. By Chebyshev’s inequality, for any and , there exists a moment such that when , , where can be regulated to small enough, which implies the asymptotically stabilization in probability in some sense.
5. A Simulation Example
Consider the following nonlinear systemwhere , , , , , with , .
For -subsystem of (57), by choosing the Lyapunov function , one can verifywhich implies that Assumption 5 holds for , , , , . By verifying Assumptions 6 and 7, it is easy to obtain that , , , , , , and .
The following observer is neededwhere is a design parameter. By defining two error variables (25), the adaptive output feedback control law is given by the recursive design procedure in Section 3, i.e.,where , , and .
In the simulation, the disturbance is produced by