Mathematical Problems in EngineeringVolume 2012 (2012), Article ID 718913, 17 pagesdoi:10.1155/2012/718913

Research Article

High-Order Stochastic Adaptive Controller Design with Application to Mechanical System

Jie Tian,1,2 Wei Feng,³ and Yuzhen Wang¹

¹School of Control Science and Engineering, Shandong University, Shandong 250100, China²School of Science, Shandong Jianzhu University, Shandong 250101, China³College of Information Science and Engineering, Shandong Agriculture University, Taian 271018, China

Received 30 October 2011; Accepted 29 November 2011

Academic Editor: Xue-Jun Xie

Copyright © 2012 Jie Tian 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.

Abstract.

The main purpose of this paper is to apply stochastic adaptive controller design to mechanical system. Firstly, by a series of coordinate transformations, the mechanical system can be transformed to a class of special high-order stochastic nonlinear system, based on which, a more general mathematical model is considered, and the smooth state-feedback controller is designed. At last, the simulation for the mechanical system is given to show the effectiveness of the design scheme.

1. Introduction

In recent years, the study for deterministic high-order nonlinear systems has achieved remarkable development, see, for example, [1–3] and references herein. Inspired by these interesting and important results, it is natural to generalize their results to the following stochastic high-order nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input: where , and are the measurable state and the input of system, respectively, , , is referred to as the state of the stochastic inverse dynamics, is an -dimensional standard Wiener process defined on a probability space with being a sample space, being a -algebra, and being a probability measure, , are odd integers, and the functions and , are assumed to be smooth, vanishing at the origin .

For (1.1) with , Xie and Tian in [4] considered the state-feedback stabilization problem for the first time. After considering the stabilization of high-order stochastic nonlinear systems, [5] further addressed the problem of state-feedback inverse optimal stabilization in probability, that is, the designed stabilizing backstepping controller is also optimal with respect to meaningful cost functionals. When , [6] designed an adaptive state-feedback controller for a class of stochastic nonlinear uncertain systems with , and [7] designed a smooth adaptive state-feedback controller for high-order stochastic systems with by using the parameter separation lemma and some flexible algebraic techniques. Recently, more excellent results [8–28] were achieved by Xie and his group.

However, all these theoretical results mentioned above are demonstrated only by some numerical simulation examples. Since many practical application systems in aerospace industry, industrial process control, and so forth, can be described by (or transformed to) stochastic high-order nonlinear systems, so it is very necessary to apply the control schemes to these systems. Based on this reason, we consider a practical example of mechanical movement in this paper. By a series of coordinate transformations, the mechanical system can be transformed to a high-order stochastic nonlinear system, based on which, we consider a more general mathematical model and design a smooth state-feedback control law. At last, the simulation for the mechanical system is given to show the effectiveness of the design scheme.

This paper is organized as follows. Section 2 gives a practical example. Section 3 provides preliminary knowledge and presents problem statement. Controller design and stability analysis are given in Section 4. The simulation for the practical example is provided to demonstrate the control scheme in Section 5. Section 6 gives some concluding remarks.

2. A Practical Example

Let us consider the following mechanical system which consists of two masses and on a horizontal smooth surface as shown in Figure 1. The mass is interconnected to the wall by a linear spring and to the mass by a nonlinear spring which has cubic force-deformation relation. Let be the displacement of mass and the displacement of mass such that at and , that is, the springs are unstretched. A control force acts on .

Figure 1: A mechanical system.

Where the units of , , and are “kg”, “m”, and “N”, respectively, and . The equations of motion for the system are described by where and are the spring coefficients, and their units are “N/m” and “N/m³”, respectively.

Introducing the smooth change of coordinates one gets

The linear spring constant has a specific nominal value which is considered uncertain, and . Let . For all , is the Gaussian white noise process with and . We can choose the value of parameter such that obeys the bound with a sufficiently high probability. This model of spring rate variations leads to an uncertain stochastic system. By (2.2), one chooses the smooth state-feedback control which together with the property of leads to where , and is standard Wiener process.

This stochastic high-order nonlinear systems can be generalized to a more general system which will be given in the following section.

3. Preliminary Knowledge and Problem Statement

3.1. Preliminary Knowledge

In this section, we will introduce the concept of input-to-state practical stability (ISpS) in probability.

Consider the following stochastic nonlinear system where , are the state and the input of system, respectively. The Borel measurable functions and are locally Lipschitz in , and is an -dimensional independent standard Wiener process defined on the complete probability space .

The following definitions and lemmas will be used throughout the paper.

Definition 3.1 (see [29]). For any given , associated with stochastic system (3.1), the differential operator is defined as follows:

Definition 3.2 (see [30]). The stochastic system (3.1) is input-to-state practically stable (ISpS) in probability if for any , there exist a class -function , a class -function , and a constant such that

Lemma 3.3 (see [30]). For system (3.1), if there exist a function , class functions , , , a class function , and a constant such that then (1)There exists an almost surely unique solution on ;(2)The system (3.1) is ISpS in probability.

Lemma 3.4 (see [6]). Let and be real variables. Then, for any positive integers , and any nonnegative smooth function , the following inequality holds:

Lemma 3.5 (see [2]). For real variables , , and real number , the following inequality holds:

3.2. Problem Statement

From (2.5), we introduce a more general class of stochastic nonlinear systems as follows: where , are the state, the input, and the measurable output of system, respectively, , , , are positive odd integers, and are smooth functions with and is unknown control coefficient with known sign, and is an -dimensional standard Wiener process defined on the complete probability space .

The following assumptions are made on system (3.8).

A1: for each , there exist unknown constant and known nonnegative smooth functions and such that

A2: for functions , , , there exist known nonnegative smooth functions and such that

A3: the reference signal and its derivative are bounded.

The objective of this paper is to design an adaptive controller such that the closed-loop system is ISpS in probability and the tracking error can be regulated to a neighborhood of the origin with radius as small as possible.

4. Controller Design and Stability Analysis

With the aid of Lemmas 3.3–3.5, we are ready to present the main results of this paper. In this section, we show that under A1–A3, it is possible to construct a globally stabilizing, state-feedback smooth controller for system (3.8). Introduce the odd positive integer , and the following coordinate change where , , are virtual smooth controllers to be designed later, , and denotes the estimate of . Then, according to It differentiation rule, one has Let , . Next, we design the controller step by step by backstepping.

Step 1. Consider the 1st Lyapunov candidate function where is the parameter estimation error. In view of (3.2), (4.1), and (4.2), one has By Lemma 3.4 and A2, there exist nonnegative smooth functions and such that which together with the boundedness of imply that where is a nonnegative smooth function, . Then, for any real number , choosing , , , by Lemma 3.5, there is a smooth function such that where . Substituting (4.5) and (4.7) into (4.4), and adding and subtracting on the right-hand side of (4.4), we have where . Suppose the virtual smooth controller with , which together with A1 lead to Substituting (4.9) into (4.8), one can obtain where is a nonnegative smooth function. Choose as follows: where is a smooth function. Then,

Step i. : Assume that at Step , there exists a smooth state-feedback virtual control such that where is a smooth function, and . We will prove that (4.14) still holds for Step .Define the th Lyapunov candidate function From (4.2) and (4.14), it follows that By A2 and Lemma 3.4, there is a smooth nonnegative function such that then, where is a smooth function. By A2, (4.1) and (4.13), there exists a nonnegative smooth function such that By (4.13), we have where and are two smooth functions. From A1, (4.1), and (4.13), it follows that where and are two smooth functions. From (4.19), one can obtain where is a smooth nonnegative function. Substituting (4.20)–(4.23) into (4.16), one gets where Suppose the virtual smooth controller with , which together with A2 render Substituting (4.26) into (4.24) leads to where . For (4.27), we have where is a design parameter, and are the smooth functions. Let , . (4.27) becomes by choosing where is a smooth function.Finally, when is the actual control. By choosing the actual control law and the adaptive law: where and are smooth functions, one gets

Theorem 4.1. If A1–A3 hold for the high-order stochastic nonlinear system (3.8), under the smooth adaptive state-feedback controller (4.32), the closed-loop system is ISpS in probability, and the tracking error can be regulated to a neighborhood of the origin in probability with radius as small as possible (Figure 2).

(a)

(b)

Figure 2: Gives the response of the closed-loop system, from which, the effectiveness of the controller is demonstrated.

Proof. For , it is obvious that satisfies (3.4). Choosing all the design parameters and to satisfy such that (3.5) holds, and then using Lemma 3.3, one can prove Theorem 4.1.

5. Simulation

Now, we apply the control scheme to the mechanical system (2.5). Let be the tracking error, where is a bounded smooth reference signal. For (2.5), , and .

Choose . Then, The smooth virtual controller can be chosen as , which renders

Next, defining , a direct calculation gives where . By Lemma 3.5, choosing , one can obtain that for any constant , Then, by (5.4) and (5.5), it is easy to see that by choosing .

Defining and the Lyapunov function , one gets by choosing . At last, choosing , a direct calculation gives by choosing

Choose the design parameters , and . Moreover, to satisfy (5.3), we choose and . Choose the initial values .

6. Concluding Remarks

In this paper, a mechanical system is firstly introduced. Then, by a series of coordinate transformations, the mechanical system can be transformed to a class of high-order stochastic nonlinear system, based on which, a more general mathematical model is considered and the smooth state-feedback controller is designed which guarantees that the tracking error can be regulated to a neighborhood of the origin in probability with radius as small as possible. At last, the simulation is given to show the effectiveness of the design scheme.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no.61004003) and Research Supporting Foundation of Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2009DX015).

References

W. Lin and C. Qian, “Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems,” Systems & Control Letters, vol. 39, no. 2, pp. 339–351, 2000.

C. Qian and W. Lin, “Practical output tracking of nonlinear systems with uncontrollable unstable linearization,” IEEE Transactions on Automatic Control, vol. 47, no. 1, pp. 21–36, 2002.

C. Qian and W. Lin, “Nonsmooth output feedback stabilization of a class of genuinely nonlinear systems in the plane,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1824–1829, 2003.

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.

W.-Q. Li and X.-J. Xie, “Inverse optimal stabilization for high-order stochastic nonlinear systems,” Automatica, vol. 45, no. 2, pp. 498–503, 2009.

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.

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.

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.

X. Yu and X.-J. Xie, “Output feedback regulation of stochastic nonlinear systems with stochastic iISS inverse dynamics,” IEEE Transactions on Automatic Control, vol. 55, no. 2, pp. 304–320, 2010.

N. Duan and X.-J. Xie, “Further results on output-feedback stabilization for a class of stochastic nonlinear systems,” IEEE Transactions on Automatic Control, vol. 56, no. 5, pp. 1208–1213, 2011.

X. Yu, X.-J. Xie, and N. Duan, “Small-gain control method for stochastic nonlinear systems with stochastic iISS inverse dynamics,” Automatica, vol. 46, no. 11, pp. 1790–1798, 2010.

N. Duan, X. Yu, and X.-J. Xie, “Output feedback control using small-gain conditions for stochastic nonlinear systems with SiISS inverse dynamics,” International Journal of Control, vol. 84, no. 1, pp. 47–56, 2011.

X. Yu, X.-J. Xie, and Y.-Q. Wu, “Further results on output-feedback regulation of stochastic nonlinear systems with SiISS inverse dynamics,” International Journal of Control, vol. 83, no. 10, pp. 2140–2152, 2010.

X.-J. Xie and N. Duan, “Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1197–1202, 2010.

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, Article ID 5740952, pp. 1921–1926, 2011.

L. Liu and X.-J. Xie, “Output-feedback stabilization for stochastic high-order nonlinear systems with time-varying delay,” Automatica, vol. 47, no. 12, pp. 2772–2779, 2011.

W. Li, X.-J. Xie, and S. Zhang, “Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions,” SIAM Journal on Control and Optimization, vol. 49, no. 3, pp. 1262–1282, 2011.

Z. Wu, M. Cui, X. Xie, and P. Shi, “Theory of stochastic dissipative systems,” IEEE Transactions on Automatic Control, vol. 56, no. 7, pp. 1650–1655, 2011.

Z.-J. Wu, X.-J. Xie, P. Shi, and Y.-Q. Xia, “Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching,” Automatica, vol. 45, no. 4, pp. 997–1004, 2009.

X.-J. Xie, “A robust model reference adaptive control without strictly positive real condition,” International Journal of Control, vol. 75, no. 14, pp. 1136–1144, 2002.

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.

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.

W. Feng, J. Tian, and P. Zhao, “Stability analysis of switched stochastic systems,” Automatica, vol. 45, no. 1, pp. 148–157, 2011.

W. Zhang, H. Zhang, and B.-S. Chen, “Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion,” IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 1630–1642, 2008.

W. Zhang and G. Feng, “Nonlinear stochastic control with -dependent noise: infinite horizon case,” IEEE Transactions on Automatic Control, vol. 53, no. 5, pp. 1323–1328, 2008.

W. Zhang, Y. Huang, and L. Xie, “Infinite horizon stochastic control for discrete-time systems with state and disturbance dependent noise,” Automatica, vol. 44, no. 9, pp. 2306–2316, 2008.

W. Zhang and B.-S. Chen, “Stochastic affine quadratic regulator with applications to tracking control of quantum systems,” Automatica, vol. 44, no. 11, pp. 2869–2875, 2008.

X. Yu, X.-J. Xie, and Y. Q. Wu, “Decentralized adaptive output-feedback control for stochastic interconnected systems with stochastic unmodeled dynamic interactions,” International Journal of Adaptive Control and Signal Processing, vol. 25, no. 8, pp. 740–757, 2011.

M. Krstić and H. Deng, Stabilization of Uncertain Nonlinear Systems, Springer, New York, NY, USA, 1998.

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.