## Stochastic Systems: Modeling, Analysis, Synthesis, Control, and their Applications to Engineering

View this Special IssueResearch Article | Open Access

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

**Academic Editor:**Weihai Zhang

#### 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: where , , are the state and control input, respectively. , . is odd integer. is an -dimensional standard Wiener process defined in a complete probability space with being a sample space, being a -field, being a filtration, and being the probability measure. is an unknown bounded nonnegative definite Borel measurable matrix function and denotes the infinitesimal covariance function of the driving noise . and , , are assumed to be smooth with and .

When , 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 , 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 . 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.

stands for the set of all nonnegative real numbers, is the -dimensional Euclidean space, is the space of real -matrixes. denotes the family of all the functions with continuous second partial derivatives. is the usual Euclidean norm of a vector . , where is its trace when is a square matrix and denotes the transpose of . denotes the set of all functions: , which are continuous, strictly increasing and vanishing at zero; is the set of all functions which are of class and unbounded; denotes the set of all functions : , which are of class for each fixed and decrease to zero as for each fixed . To simplify the procedure, we sometimes denote by for any variable .

Consider the nonlinear stochastic system where is the state, is an -dimensional independent Wiener process with incremental covariance , that is, , where is a bounded function taking values in the set of nonnegative definite matrices, and are locally Lipschitz functions.

*Definition 2.1 (see [13]). *For any given associated with stochastic system (2.1), the differential operator is defined as

*Definition 2.2 (see [25]). *For the stochastic system (2.1) with , , the equilibrium is globally asymptotically stable (GAS) in probability if for any , there exists a class function such that

Lemma 2.3 (see [14]). *Consider the stochastic system (2.1). If there exist a function , class function and , constants and , and a nonnegative function such that for all , **
then,*(a)*there exists an almost surely unique solution on for each ,*(b)*when , , , and is continuous, then the equilibrium is globally stable in probability and the solution satisfies .*

Lemma 2.4 (see [12]). *Let be real variables, for any positive integers , positive real number and nonnegative continuous function , then
**
when , is a positive constant, then the above inequality is
*

Lemma 2.5 (see [12]). *Let and , , be real variables and let and be smooth mappings. Then for any positive integers , and real number , there exist two nonnegative smooth functions and such that the following inequalities hold:*(i)*,
*(ii)*. *

Lemma 2.6 (see [12]). *Let , , be positive real variables, then
*

#### 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 , , , such that

##### 3.1. Controller Design

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

First, we introduce the following coordinate change: where , , are smooth virtual controllers which will be designed later, is the estimation of , and Then, by Itô’s differentiation rule, one has where

*Step 1. *Define the first Lyapunov function
where is a positive constant, is the parameter estimation error. By (3.2)–(3.4) and Assumption 3.1, there exist nonnegative smooth functions and such that
Choosing the first smooth virtual controller
and the tuning function
one has
where is a design parameter.

*Step i (). *For notational coherence, denote . Assuming that at step , one has
where . In the sequel, we will prove that (3.11) still holds for the th Lyapunov function . By (3.4) and (3.11), one has
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
where , , , , , , , are positive constants and , , , , , , , , , , , , are nonnegative smooth functions. Substituting (3.13) into (3.12), one has
Choosing the th smooth virtual controller
and tuning function
and substituting (3.15) and (3.16) into (3.14), it follows that
where , .

Hence at step , the smooth adaptive state-feedback controller such that the th Lyapunov function satisfies where , , are nonnegative smooth functions, , , are constants, and

##### 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 for each ,*(ii)*the origin of the closed-loop system is globally stable in probability,*(iii)* and exists and is finite.*

*Proof. *It is easy to verify that is on and . For , choose the design parameter , , then by (3.21), , . Since is continuous, positive, and radially bounded, by (3.20), (3.21), and Lemma 4.3 in [25], there exist two class functions and such that . Hence, the condition of Lemma 2.3 holds.

By Lemma 2.3, it follows that conclusion (i), (ii) hold, and . In view of and , one has . By (3.20) and the definition of in (3.19), it holds that converges a.s. to a finite limit as , therefore converges a.s. to a finite limit as .

#### 4. A Simulation Example

Consider a two-order nonlinear stochastic system where , . By Lemma 2.4, one gets , , , . We choose , , , , Assumption 3.1 is satisfied.

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

*Step 1. *Define , . A smooth virtual controller
and the tuning function
yield , where

*Step 2. *Defining , , by (3.12), one has
where , . By Lemma 2.4, the definition of , and (4.2), one can obtain
by (4.3), Lemmas 2.4, 2.6, and the definitions of and , one has
where , , , , , , , , , , , , , , , , , , , , , , are positive constants.

Choosing the smooth adaptive controller
and substituting (4.6)–(4.8) into (4.5), one has
where .

In simulation, we choose , the parameters , , , , , , , , , , , , , , the initial values , , , the sampling period . Figure 1 verifies the effectiveness of the control scheme.

**(a)**

**(b)**

**(c)**

**(d)**

#### 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.

#### References

- Z. G. Pan and T. Başar, “Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion,”
*SIAM Journal on Control and Optimization*, vol. 37, no. 3, pp. 957–995, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. G. Pan, K. Ezal, A. J. Krener, and P. V. Kokotović, “Backstepping design with local optimality matching,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 7, pp. 1014–1027, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. G. Liu, Z. G. Pan, and S. Shi, “Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 3, pp. 509–513, 2003. View at: Publisher Site | Google Scholar - Y.-G. Liu and J.-F. Zhang, “Reduced-order observer-based control design for nonlinear stochastic systems,”
*Systems & Control Letters*, vol. 52, no. 2, pp. 123–135, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y.-G. Liu and J.-F. Zhang, “Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics,”
*SIAM Journal on Control and Optimization*, vol. 45, no. 3, pp. 885–926, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Deng and M. Krstić, “Output-feedback stochastic nonlinear stabilization,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 2, pp. 328–333, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Deng, M. Krstić, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 8, pp. 1237–1253, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z.-J. Wu, X.-J. Xie, and S.-Y. Zhang, “Adaptive backstepping controller design using stochastic small-gain theorem,”
*Automatica*, vol. 43, no. 4, pp. 608–620, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S.-J. Liu, J.-F. Zhang, and Z.-P. Jiang, “Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems,”
*Automatica*, vol. 43, no. 2, pp. 238–251, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z.-J. Wu, X.-J. Xie, and S.-Y. Zhang, “Stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function,”
*International Journal of Control*, vol. 79, no. 12, pp. 1635–1646, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. Lin, 2010, http://nonlinear.cwru.edu/~linwei/.
- C. J. Qian,
*Global synthesis of nonlinear systems with uncontrollable linearization*, Ph.D. dissertation, 2001. - M. Krstić and H. Deng,
*Stabilization of Uncertain Nonlinear Systems*, Springer, New York, NY, USA, 1998. - H. Deng, M. Krstić, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 8, pp. 1237–1253, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. R. Ma,
*Stochastic Differential Equations and Their Applications*, Horwood, Chichester, UK, 2007. - X.-J. Xie and J. Tian, “State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics,”
*International Journal of Robust and Nonlinear Control*, vol. 17, no. 14, pp. 1343–1362, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Tian and X.-J. Xie, “Adaptive state-feedback stabilization for high-order stochastic non-linear systems with uncertain control coefficients,”
*International Journal of Control*, vol. 80, no. 9, pp. 1503–1516, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Tian and X. J. Xie, “Adaptive state-feedback stabilization for more general high-order stochastic nonlinear systems,”
*Acta Automatica Sinica*, vol. 34, no. 9, pp. 1188–1191, 2008. View at: Publisher Site | Google Scholar - X.-J. Xie and J. Tian, “Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization,”
*Automatica*, vol. 45, no. 1, pp. 126–133, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Liu and X. J. Xie, “State-feedback stabilization for stochastic high-order nonlinear systems with SISS inverse dynamics,”
*Asian Journal of Control*, vol. 14, no. 4, pp. 1–11, 2012. View at: Publisher Site | Google Scholar - X.-J. Xie, N. Duan, and X. Yu, “State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 8, pp. 1921–1926, 2011. View at: Publisher Site | Google Scholar - X.-J. Xie and W.-Q. Li, “Output-feedback control of a class of high-order stochastic nonlinear systems,”
*International Journal of Control*, vol. 82, no. 9, pp. 1692–1705, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Liu, N. Duan, and X.-J. Xie, “Output-feedback stabilization for stochastic high-order nonlinear systems with a ratio of odd integers power,”
*Acta Automatica Sinica*, vol. 36, no. 6, pp. 858–864, 2010. View at: Publisher Site | Google Scholar - W.-Q. Li and X.-J. Xie, “Inverse optimal stabilization for stochastic nonlinear systems whose linearizations are not stabilizable,”
*Automatica*, vol. 45, no. 2, pp. 498–503, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. K. Khalil,
*Nonlinear Systems*, Prentice-Hall, Upper Saddle River, NJ, USA, 3rd edition, 2002.

#### Copyright

Copyright © 2012 Cong-Ran Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.