Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 136863, 9 pages

http://dx.doi.org/10.1155/2013/136863

Research Article

Observer-Based Robust Tracking Control for a Class of Switched Nonlinear Cascade Systems

Ben Niu,¹ Deqing Zhao,² Xudong Zhao,³ Hongyi Li,³ Xiangyong Chen,⁴ and Xiaoxiao Dong⁵

¹College of Mathematics and Physics, Bohai University, Jinzhou 121001, China

²Jinzhou Heavy Water Pump Co., Ltd., Jinzhou 121001, China

³College of Information Science and Technology, Bohai University, Jinzhou 121001, China

⁴School of Scieces, Linyi University, Linyi 276005, China

⁵College of Science, Shenyang University of Industry, Shenyang 110001, China

Received 29 June 2013; Accepted 30 July 2013

Academic Editor: Rongni Yang

Copyright © 2013 Ben Niu 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.

This paper is devoted to robust output feedback tracking control design for a class of switched nonlinear cascade systems. The main goal is to ensure the global input-to-state stable (ISS) property of the tracking error nonlinear dynamics with respect to the unknown structural system uncertainties and external disturbances. First, a nonlinear observer is constructed through state transformation to reconstruct the unavailable states, where only one parameter should be determined. Then, by virtue of the nonlinear sliding mode control (SMC), a discontinuous nonlinear output feedback controller is designed using a backstepping like design procedure to ensure the ISS property. Finally, an example is provided to show the effectiveness of the proposed approach.

1. Introduction

Switched systems are a special class of hybrid systems in engineering applications and have attracted much attention from many researchers [1–8]. A switched system consists of a family of distinct active subsystems subject to a certain switching rule which chooses one of them being active during a certain time. The research on switched systems is motivated by two important practical considerations: (i) many real-world systems exhibit a fundamental characteristic of switching between different system structures; (ii) multicontroller switching provides an effective mechanism to handle highly complex systems and/or systems with large uncertainties. Therefore, switched systems have been a very active area of research in the past twenty years and have motivated a large and growing body of research work on a diverse array of issues, including modeling [9, 10], optimization [11, 12], stability analysis [13–17], and control [18–20].

Output feedback tracking control is a fundamentally important issue in control field and has been extensively studied over the last several decades. In the literature, several approaches have been developed to handle the output feedback control in the presence of structured or unstructured uncertainties: variable structure control approach [21], adaptive control approach [22], output dynamics controller with almost disturbance decoupling [23], and so forth. Inspired by these facts, for switched systems, output feedback tracking control is also a challenging issue for both theoretical investigation as well as practical applications [24–26]. Such a problem usually involves observer design [27], controller design [28], and switching law design [29]. However, to the best of the authors’ knowledge, the output feedback tracking control of switched nonlinear cascade systems by designing nonlinear state observer has not been investigated yet.

In sliding mode control (SMC), sliding mode surface design and discontinuous reaching control law are two of the basic control issues. A common practice in SMC is to design a sliding mode surface according to the null space dynamics, which must ensure a stable sliding manifold when the system is in the sliding mode [30]. However, if there exist uncertainties in the null space nonlinear dynamics, sliding mode surface design becomes extremely difficult. Traditionally, the reaching control law is to force the system to reach and stay on the sliding mode surface. Nevertheless, this feature alone is no longer sufficient in the presence of unmatched uncertainties. Due to the effect of the unmatched uncertainties, the nonlinear dynamics may become divergent in a period shorter than the reaching time, if the input-to-state stable (ISS) property does not hold during the reaching phase. Hence, ISS property should be guaranteed either in the sliding phase or in the reaching phase.

In this paper, a class of switched nonlinear cascade systems with null space dynamics and range space dynamics are addressed for the tracking control task. Assuming that the full states are not available for measurement, the main objective of the paper is to ensure the global ISS property of the tracking error nonlinear dynamics while achieving a small tracking error bound. The features of the proposed approach are the following: (i) a nonlinear observer is designed for the switched system in which only one parameter needs to be determined; (ii) the resulting sliding manifold in the sliding phase possesses the desired ISS property and to certain extent the optimality through solving a Hamilton-Jacoby inequality; (iii) associated with the sliding mode surface, SMC is applied to the second subsystem that achieves the desired tracking.

Notations. We use standard notations throughout this paper. and stand for the maximum and minimum eigenvalues of a symmetric matrix , respectively. denotes the first rows and columns in , and denotes the last rows and columns in . denotes the set of nonnegative real numbers, denotes an -dimension real vector space, is the Euclidean norm and induced matrix norm, and is the space of uniformly bounded functions on . and are row vectors, and denotes the largest singular value of a matrix.

2. System Description and Problem Statement

This paper is concerned with the following switched nonlinear cascade system described by

where is a physically measurable state vector, is the null space dynamics, is the range space dynamics, and and are the external disturbance. is the right continuous piecewise constant switching signal to be designed; stands for the control input of the th subsystem, the mappings ,  ,  ,  ,  , and are known and smooth with respect to and continuous with respect to time , and ,  , denote the uncertainties in the control input. The relation holds for the system (1).

Corresponding to the switching signal , we have the switching sequence

which means that the th subsystems are active when . In addition, we assume that the state of the system (1) does not jump at the switching instants; that is, the trajectory is everywhere continuous.

In this paper, the following assumptions are adopted to develop the main results.

Assumption 1. There exist two positive constants and such that for all ,  ,

where is the identity matrix. Moreover, ,   are assumed to be invertible.

Assumption 2. The uncertainties ,   and in (1) are bounded as

where ,  , and are known positive constants.

In this paper, the output of the system (1) is required to track a given reference model: ; that is, the subpart is required to track the desired reference model

where is a smooth reference input. Define the tracking error as . Then, the error dynamics of the -subpart can be transformed into

Assumption 3. There exists a smooth function such that the following matching condition holds:

where is asymptotically stable.

According to Assumption 3, the error dynamics (6) and system (1) with the tracking objective (5) can be rewritten as

Definition 4 (input-to-state stable (ISS) [31, 32]). Consider a nonlinear dynamical system of the form

where and are the states and the inputs of (9), respectively. The system (9) is said to be locally input-to-state stable if there exist a class function , a class function , and constants such that

for all and satisfying and ,  . It is said to be input-to-state stable or globally ISS if ,  , and (10) is satisfied for any initial state and any bounded input .

Control Objective. Under Assumptions 1–3, design a nonlinear observer for the system (1). Based on the observer, design a controller and a switching law such that

(i) the tracking error norm in (8) tends to a ball in finite time, where the ball is defined as

  where is a positive constant;(ii) the closed-loop system (8) possesses ISS property with respect to the disturbances ,  .

3. Nonlinear Observer Design

This section is devoted to the design of a nonlinear observer for the system (1). Motivated by the work in [32, 33], a nonlinear observer is constructed through a state transformation which converts the system (1) into a new form such that the observer gain can be designed in a straightforward manner.

First, the system in (1) can be rewritten as the following:

where and

Define the transformation matrices ,   and the matrices ,  , and as

Therefore, we obtain ,  , and

Denote as the left inverse of the matrix . Then, the system can be written as

Thus, the observer for the transformed system in (16) can be constructed as

where is the symmetric positive definite solution of the following algebraic Lyapunov equation:

Theorem 5. Assume that the system in (12) satisfies Assumptions 1-2. Then, under arbitrary switchings, the estimation error of the states has the following property:

where ,  ,  , and ,  ,  , and are positive constants.

Proof. Define . From (16) and (17), the estimation error dynamics of becomes

Consider a transformation on the error as . Then, we have

Choosing , where is the solution of (18), we obtain

For any , we can infer that ,  ,  , and , where ,  , and do not depend on . Then, (16) is transformed into

where and . If is selected, then (23) becomes

Using , (24) becomes

where and . Thus, . Furthermore, from Assumption 1, we have and , where and are constants.

Based on and , we get

with ,  . This completes the proof.

From (16) and , the observer to the original coordinate is

Hence, the estimation error dynamics in the -coordinate with becomes

4. Controller Design and Stability Analysis

Before the controller design, we would like to rewrite the observer dynamics in (27) as

where

Define . In terms of the observer dynamics (22) and the desired trajectory (4), we have the following error dynamics:

In what follows, we first choose a sliding mode surface for the error dynamics of the null space dynamics . Second, we design a controller for the augmented system in (21) and (32) such that ISS property is achieved.

Theorem 6. If there exist positive definite, radially unbounded, and smooth functions and functions ,   such that

then, under the nonlinear sliding mode

and the switching law

the tracking error norm tends to a ball in finite time, where the ball is defined as

where and are positive constants.

Proof. First, we now define the following piecewise Lyapunov function candidate:

where is switched among the solution ’s of (34) in accordance with the piecewise constant switching signal .

Using the sliding mode surface constructed in (35) and under the switching law (36), the derivative of is

If there exist solutions of such that the inequality in (34) is satisfied, (39) becomes

From (22) and (40), we have

When is selected to be

then the inequality (41) becomes

Using , , and and according to , (43) becomes

From (17), is bounded as

where according to (42). Hence, (44) becomes

where and . Note that . Equation (47) shows that the tracking error norm in (8) tends to a ball in finite time, which is defined by

where , .

Remark 7. In the nonlinear uncertain system (32), if can be expressed as , when is a matrix-valued smooth function, then the HJI inequality (34) can be simplified into the following differential Riccati inequality:

where ,  , are symmetric positive definite smooth matrices.

Remark 8. In the observer design, the parameter is the only key parameter to be determined. It should be designed such that the two conditions are satisfied in (42) and simultaneously.

Remark 9. Since the estimation error of the states in Theorem 5 has the property (19), the tracking error simply converges to a ball showed in (37).

We are now in a position to design the controller to ensure the ISS stability.

Theorem 10. With the sliding mode surface (35), the switching law (36), and the following sliding mode controller

where and is a positive constant, the system (8) is globally ISS stable with respect to the external disturbance inputs, and the tracking error norm is bounded in as in Theorem 6.

Proof. Define . Choose the following piecewise Lyapunov function candidate:

where is switched among the solution ’s of (34) in accordance with the piecewise constant switching signal .

Then, we have

From (44) and (54), we have

which implies that the system (8) is globally ISS with respect to the external disturbance input, and the tracking error norm is bounded in in finite time.

5. Illustrative Example

In this section, we present a simulation example to illustrate the applicability and effectiveness of the proposed approach.

Example 1. Consider a switched nonlinear cascade system as in (1), where

The nonlinear observer is designed as in (27). Based on (18), we have the symmetric positive definite solution

Then, is selected based on Remark 8 with ,  ,  , and .

The target trajectory is and . From (8), the error dynamics of the -subpart can be expressed as

where .

Let . In -subpart, according to Remark 7, we first choose ,  , where are determined by the differential Riccati inequality (48). When and , from the linear algebraic matrix inequality

and using the singular values of the matrices and , we can get two symmetric positive definite smooth matrices

Therefore, the switching surface is . Moreover, the switching law is chosen as

according to (36) in Theorem 6, and the controller is constructed according to (49) in Theorem 10.

Let the initial states be . Figures 1 and 2 show the responses of the states and , respectively. The tracking errors and are shown in Figures 3 and 4, respectively, which demonstrate the tracking errors of the states and that are bounded with fast convergence. All the figures indicate the feasibility of our results.

Figure 1: The response of the state .

Figure 2: The response of the state .

Figure 3: The evolution of the tracking error .

Figure 4: The evolution of the tracking error .

6. Conclusions

In this paper, we have investigated the tracking control problem for a class of switched nonlinear cascade systems with unknown system uncertainties and external disturbances. A new robust output feedback control approach based on a nonlinear observer is proposed for the switched system. Through solving a Hamilton-Jacoby inequality, the nonlinear control law for the first subsystem specifies a nonlinear sliding mode surface. By virtue of nonlinear control for the first subsystem, the resulting sliding manifold in the sliding phase possesses the desired ISS property. Furthermore, sufficient conditions for the solvability of the tracking control problem of the switched systems and design of both switching law and output feedback controller are presented.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants (nos. 61304054, 61203002, 61304102, and 61203123), the Program for New Century Excellent Talents in University, the Program for Liaoning Excellent Talents in University, and the Project of Shandong Province Higher Educational Science and Technology Program under Grant (no. J13LI11).

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