Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 734934, 9 pages
http://dx.doi.org/10.1155/2015/734934

Research Article

Piecewise Static Output Feedback Controller Design for Nonlinear Systems Based on T-S Affine Fuzzy Models

Xingang Zhao

State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China

Received 23 October 2014; Accepted 27 January 2015

Academic Editor: Hak-Keung Lam

Copyright © 2015 Xingang Zhao. 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 concerned with the problem of designing controllers via static output feedback controller for a class of complex nonlinear systems, which is approximated by continuous-time affine fuzzy models. A decomposition method is presented to divide the output-space into different operating regions and interpolation regions. Based on this partition, a novel piecewise controller with affine terms via static output feedback is designed. By using a dilated linear matrix inequality (LMI) characterization, some nonconvex conditions are converted into convex ones to make the asymptotic stability and performance of the closed-looped system. The effectiveness of the proposed method is illustrated by a numerical example.

1. Introduction

As a well universal approximator, Takagi-Sugeno (T-S) fuzzy models provide an effective way to deal with nonlinearities and uncertainties in practical engineering areas since 1985 [1–3]. Some locally linear time-invariant systems in the form of IF-THEN rules are used to describe the prominent feature of T-S fuzzy models. During the past few years, the T-S fuzzy control systems have been studied extensively, and many significant advances have been achieved [4–9].

Both linear fuzzy models and affine fuzzy models are usually used to describe T-S Fuzzy systems [10–12]. Compared with linear fuzzy models, T-S affine fuzzy dynamic system with offset terms has much improved function approximation capabilities [3]. This has ignited enormous research activities in search of new methodologies to investigated affine T-S fuzzy systems (see [13, 14] and the references therein).

Most of the above results are mainly focused on state feedback control, which requires all state variables to be accessible. This may simplify the considered problem but cause the obtained results to be restrictive in engineering. Since only a partial information through a measured output can be available in many practical nonlinear control systems, many researchers focus their attention on the output stabilization [15, 16]. The main advantage of the static output feedback is the simplicity of its implementation and reduction of real time computational cost when implementing practical applications. Recently, static output feedback (SOF) problem for T-S fuzzy systems has attracted considerable attention of many researchers. Until now, most of the existing fuzzy SOF control results are obtained for linear fuzzy systems. Many efforts have been done to convert the nonconvex problem to convex problem, such as iterative linear matrix inequality (ILMI) approaches (see [17–19]) and linear matrix inequality (LMI) approaches [20–22].

Compared with linear fuzzy systems, the constant bias terms in affine fuzzy systems bring more challenge to the controller design. For continuous-time affine fuzzy systems, most existing results about controller design via output feedback are derived in the framework of bilinear matrix inequalities (BMIs) [12, 23, 24]. In [25], some sufficient conditions for the robust output feedback controller design problem for discrete-time affine fuzzy systems are presented in the frameworks of LMIs. However, they supposed the transitions of both the plant and the controller could happen among all regions, having the controller with conservative designing conditions and heavy computations. To the best of our knowledge, few results have been developed about controller design via static output feedback for continuous-time affine fuzzy systems based on output partition.

In this paper, the considered nonlinear system is described by affine fuzzy models, and the system outputs are chosen as premise variables. There is no doubt that this modeling approach may be coarse to represent a complex nonlinear system because less information can be used for modeling. However, it is more convenient and less conservative in the case where state variables are not all available, for example, static output feedback problem. Moreover, static output feedback is simply to be implemented. Based on such fuzzy models, the static output feedback control design problem is discussed in this paper. A partition is presented to divide the output-space into different operating regions and interpolation regions. Then based on this partition, a new piecewise static output feedback controller with affine terms is proposed, whose design conditions are derived in the framework of LMIs by introducing additional slack matrix variables. Finally, a numerical example is given to illustrate the effectiveness and merits of the obtained method.

The rest of this paper is organized as follows. Section 2 introduces the preliminaries and design objective. The static output feedback controller design problem is addressed in Section 3, and a solution of this problem is given in terms of LMIs. A numerical example is given in Section 4 to show the effectiveness of proposed method. Finally, conclusions are drawn in Section 5.

The following notations are used throughout the paper. The superscript “” stands for matrix transposition. Then notation where is symmetric matrix indicates that is positive definite. In block symmetric matrices or long matrix expressions, the symbol is used to represent a term that is induced by symmetry.

2. Preliminaries and Problem Statement

2.1. System Model

The following continuous-time affine T-S model is described by fuzzy IF-THEN rules, whose collection represents the approximation of a complex nonlinear system.

 Plant Rule : if is ,  and is , then

where is the state space vector, is the control input, denotes the external disturbance, is the measurable premise variable, and is the regulated output. , , , , ,    are constant matrices with appropriate dimension. are the fuzzy sets that are characterized by membership function; is the number of premise variables; and is the number of IF-THEN rules.

The overall affine fuzzy model achieved by fuzzy-blending of each individual plant rule (local model) is given by

where

Denote

and then

is the grade of membership of in , and is the normalized membership function.

Remark 1. Similar to [12], a constant affine term is included in (1) to approximate the original nonlinear system, which makes the function approximation capabilities of fuzzy systems improve substantially.

Assumption 2. For each from 1 to , assume that are of full column rank.
A decomposition method will be used to divide the output-space into different operating regions and interpolation regions. In a certain operating region, it follows and   , and then the system can be described by . Between different operating regions, that is, interpolation regions, ; then a convex combination of different affine systems can describe the system.
Denote as a polyhedral partition of the output, and as the set of region indexes. and are the set of indexes for region that contain or do not contain the origin, respectively. For each local region , the set contains the indexes for the system matrices used in the interpolation within that region. For operating regions, only contains a single element. By a blending of subsystems, the global system in (2) is represented as follows in each cell:

where

The range of is represented by the following system of at most linear inequalities while :

 for , or or , for , or or ,  for , or or .

2.2. Problem Statement

Based on output partition, the following affine piecewise controller via static output feedback is chosen:

where , are the controller gains to be designed later. Here we assume that the equilibrium point of closed-loop system is located at the origin and therefore we have for all . Applying the piecewise static output feedback controller (8) to system (6), the resulting closed-loop system is given by the following:

Remark 3. Since in many practical control systems, the system sates of system (6) are not completely available. Thus, the static output feedback controller is more applicable than that via state feedback controller.

Problem 4. For the closed-loop system (9), design a piecewise affine via static output feedback in the form of (8), such that the following requirements are satisfied.

(1)The closed-loop system (9) is asymptotically stable when .(2)Under zero initial conditions, the -norm of the operator from disturbance to the controlled output is less than ,

 for any external disturbance .

3. Main Results

In this section, we will give some sufficient conditions to solve static output feedback controller design problem for the closed-loop system (9).

The following lemma will be used to derive the main results of this paper.

Lemma 5. The following two statements are equivalent for the closed-loop affine fuzzy system (9) being asymptotically stable and satisfying an performance index .

(1)If there exist a symmetric matrix and scalars such that for , ,

 where , and for , ,

 where and , , are similar to those in [12] such that while , .(2)For large enough positive constants , if there exist , matrices , , and scalars    such that for ,

 and for ,

 where

Proof. Firstly, we prove that if the statement (1) holds, then the closed-loop affine fuzzy system (9) is asymptotically stable and satisfying an performance index .
Chose a Lyapunov function as

Then from the derivative of along system (9), it follows

with .
For , it can be easily seen that the following inequality implies . Consider

where .
Then condition (12) implies (18), by expanding the fuzzy-basis functions. For the case , condition (11) can be easily obtained without using the S-procedure, where .
Secondly, : for the case if (11) is satisfied, then rewrite (11) as . It implies that there exists a sufficiently large positive scalar such that :

The above-mentioned inequality can be rewritten as

Applying Schur complement, we get

By taking , then condition (13) is obtained. In the same way, if (12) is satisfied for the case , we can still prove (13) holds.
For the case , multiplying (13) by and its transpose from the left side and the right one implies (11) holds. Similarly for the case of , the conclusion that (14) implies (12) holds.
Then the proof is completed.

Notice that conditions (13) and (14) in Lemma 5 are nonconvex. Next theorem will give some convex conditions for the affine controller design problem via static output feedback for (6).

Theorem 6. Consider the affine fuzzy system (6) with the designed static output feedback controller (8), for large enough positive constants , if there exist , matrices , , , , , , and with appropriate dimensions, where , , , , and scalars    such that

 for , ,

 and for , ,

 where

The closed-loop system (9) is asymptotically stable with a performance index . Moreover, the controller gains are given by

Proof. Let and have the following form:

with

Based on Assumption 2, we can take some invertible matrices for such that

Here the rows of are the ones which are mutually independent and perpendicular to the columns of . It can be easily seen that is not unique. Denote that function is defined as

After some matrix computation in (13) and (14), (22) and (23) are easily obtained.

Remark 7. When and in (22) and (23) are set to be fixed parameters, the problems become convex and can be solved by employing the LMI Toolbox. To find the optimal values of corresponding parameters, in this paper, we will first solve the feasibility problem of LMIs in (22) and (23) by using LMI Toolbox under a set of initial scaling parameters. Then, for the optimization problem, apply a numerical optimization algorithm, such as the program minsearch, and then a locally convergent solution to the problem is obtained.

When in (8), a linear static output feedback controller can be given as follows:

Based on Theorem 6, the following corollary can be easily obtained.

Corollary 8. Consider the affine fuzzy system (6) with the designed static output feedback controller (30), for large enough positive constants , if there exist , matrices , , , , , , and , where , , , and scalars    such that

 for , ,

 and for , ,

 where .

The corresponding closed-loop system is asymptotically stable with a performance index . Moreover, The controller gains are given by

Remark 9. Notice that the results given in this paper are obtained based on the assumption that the input matrix is of full column rank, which leads to much wider scope of the applicability than the common input matrix. Nevertheless, if there exist invertible matrices , , such that

where , which means that the input matrices are not full column rank, then a novel piecewise affine state feedback controller is desirable with the form

where , . The corresponding static output feedback control synthesis method is proposed as follows.

Corollary 10. The continuous affine fuzzy system (6) is quadratically stabilizable at large by the piecewise controller (35) if there exist , matrices , , , , , , with appropriate dimensions, where , , and , , and scalars    such that for given positive constants , the linear matrix inequalities in (31) and (32) are satisfied, and the local gains of the fuzzy controller are given by

4. Numerical Example

In order to demonstrate the effectiveness and merits of the proposed piecewise affine static output feedback controller over the existing results, the following numerical example is considered.

 : if is , then

 where

Here, the system state is assumed to be not completely available for feedback and thus cannot be chosen as the premise of the fuzzy model. The membership functions , , and are depicted in Figure 1.

Figure 1: Membership functions.

Due to the fact that system state cannot be used for controlling the system, the existing controller design results for T-S affine fuzzy systems in [14, 24] are not suitable for this class of fuzzy systems. Besides, the existing output feedback controller design method in [25] is proposed for discrete-time affine fuzzy systems and not suitable for continuous-time affine fuzzy systems.

In this simulation, both the piecewise affine static output feedback controller (8) and the piecewise linear static output feedback controller (30) are used to guarantee the resulting closed-loop system asymptotically stable with performance under and zero initial conditions. By applying Theorem 6 and the minsearch, the controller gains of piecewise static output feedback controller are obtained as follows:

Table 1 shows the minimum performance based on the two abovementioned approaches (Theorem 6 and Corollary 8), respectively, with scaling parameters and . It can be seen that the proposed piecewise affine static output feedback controller (8) can guarantee a smaller optimal performance index over the piecewise linear static output feedback controller (30). The state responses curves of the closed-loop system via the proposed static output feedback controller design method are given in Figures 2–4. The square root of ratio of the regulated output energy to the disturbance input noise energy is shown in Figure 5.

Table 1: performance .

Approach
Performance

Theorem 6
7.6819

Corollary 8
9.4937

Figure 2: Response curve of .

Figure 3: Response curve of .

Figure 4: Response curve of .

Figure 5: The ratio of .

5. Conclusion

This paper addressed the static output feedback controller design problem for a class of continuous-time affine fuzzy systems. Based on a partition of output-space technique, a new piecewise controller with affine terms via static output feedback is designed. The corresponding design conditions to guarantee the closed-loop system asymptotic stable with performance are derived in a dilated linear matrix inequality (LMI) characterization. Finally, an example has been given to illustrate the effectiveness of the proposed approach.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61273355, 61273356, and 61333019), the Foundation of State Key Laboratory of Robotics (Grant no. Z2013-06), and Natural Science Foundation of Liaoning Province, China (Grant no. 2014020082).

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