Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 201432, 11 pages

http://dx.doi.org/10.1155/2013/201432

Research Article

Adaptive Fuzzy Robust Control for a Class of Nonlinear Systems via Small Gain Theorem

Xingjian Wang and Shaoping Wang

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Received 28 December 2012; Revised 23 February 2013; Accepted 23 February 2013

Academic Editor: Baocang Ding

Copyright © 2013 Xingjian Wang and Shaoping Wang. 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.

Practical nonlinear systems can usually be represented by partly linearizable models with unknown nonlinearities and external disturbances. Based on this consideration, we propose a novel adaptive fuzzy robust control (AFRC) algorithm for such systems. The AFRC effectively combines techniques of adaptive control and fuzzy control, and it improves the performance by retaining the advantages of both methods. The linearizable part will be linearly parameterized with unknown but constant parameters, and the discontinuous-projection-based adaptive control law is used to compensate these parts. The Takagi-Sugeno fuzzy logic systems are used to approximate unknown nonlinearities. Robust control law ensures the robustness of closed-loop control system. A systematic design procedure of the AFRC algorithm by combining the backstepping technique and small-gain approach is presented. Then the closed-loop stability is studied by using small gain theorem, and the result indicates that the closed-loop system is semiglobally uniformly ultimately bounded.

1. Introduction

High performance control algorithm is often required in modern industrial control systems, such as robot manipulators, precision manufacturing equipment, and automatic inspection machines. During the past two decades, mathematically rigorous design methods for adaptive control of nonlinear systems have developed rapidly, and many remarkable results have been obtained [1, 2]. As a breakthrough in the research field of nonlinear control systems, a recursive design technique named adaptive backstepping approach was proposed to obtain global stability and asymptotic tracking for nonlinear systems [3, 4]. Many researchers have made a lot of efforts in this field and achieved many significant results. For example, the adaptive robust control (ARC) approach for uncertain nonlinear systems proposed by Yao and Tomizuka belongs to this kind of work [5–7].

However, the common feature of the adaptive control algorithms discussed before is to deal with the nonlinearities of the control systems in the linearly parameterized forms; that is, these nonlinearities are assumed to be in the form of known nonlinear functions with unknown but constant parameters [8]. But in fact, in practical industrial control environment, uncertainties or nonlinearities of one system cannot always be totally repeatable, structured, or modeled. In order to deal with this kind of nonlinearities, many researchers focused on approximator-based approaches by using backstepping technique and Lyapunov stability theory [9–13]. Some advanced approximators such as fuzzy logic and neural network were used in control methods for nonlinear systems with unstructured uncertainties or nonlinearities [14–16]. Recently, Tong and his colleagues made many achievement on adaptive fuzzy controller design for large-scale nonlinear systems [17–20]. In these methods, all system dynamics are estimated by approximator, but the control performance is not necessarily better.

Actually, as the development of modern control theorem and system identification technique, part of the dynamics of control system can be modeled quite accurately, while some nonlinearities are still unknown or not replicable. In order to cope with the control problem of such class of nonlinear systems, we present a novel adaptive fuzzy robust control (AFRC) algorithm. Adaptive control law is used to compensate the linearizable part, and discontinuous-projection-based parameter adaptive law is employed to estimate unknown system parameters. The Takagi-Sugeno fuzzy logic systems are used to approximate unknown nonlinearities. Robust control law ensures the robustness of closed-loop control system. A mathematically design procedure of AFRC is developed by combining the adaptive backstepping technique and small-gain theorem [21, 22]. This proposed AFRC can combine technical characters of adaptive control and fuzzy control and improve the control performance for the class of nonlinear systems we considered.

The reminder of this paper is organized as follows. In Section 2, we will review some definitions about small gain theorem and Takagi-Sugeno fuzzy logic system, and the control problem is also formulated in this section. The systematic procedure for adaptive fuzzy robust control law is presented in Section 3. In Section 4, the closed-loop stability is studied by using small gain theorem. In Section 5, an illustrative example is presented to demonstrate the effectiveness of the proposed controller. Conclusions are given in Section 6.

2. Problem Formulation

2.1. Preliminaries

In this section, some definitions about input-to-state stable (ISS) and small gain theorem [21–23] will be introduced, which have been widely used in nonlinear control problems for stability analysis [24]. First, the class , , and functions [25] will be reviewed. Then the structure of fuzzy logic system will be briefly introduced.

2.1.1. ISS and Small Gain Theorem

Definition 1. A function is said to belong to class if it is continuous, strictly increasing, and . It is said to belong to class if additionally is unbounded.

Definition 2. A function is of class if is said to belong to class for each fixed , and is strictly decreasing for each fixed and .

Definition 3. For a nonlinear system , it is said to be input-to-state practically stable (ISpS) if there exist a class function and a class function , such that for any initial condition , each bounded control input defined for all and a constant , the associated solution satisfies

where and is a truncated function defined as

When in (1), the ISpS property becomes ISS property [21, 26].

Definition 4. A function is said to be an ISpS-Lyapunov function for the system if

(1)there exist class functions , , such that

(2)there exist class function , and a constant , such that

When in (4), is said to be an ISS-Lyapunov function [26]. Then nonlinear gain in (1) can be chosen as

Proposition 5. The nonlinear system is ISpS if and only if there exists an ISpS-Lyapunov function [24].

Theorem 6 (small gain theorem). Consider a system in composite feedback form of two ISpS systems [23, 24]:

In particular, if there exist class functions , , class functions , , and two constants , , for all , the solutions and are satisfied as follows:

If

then the solution of the composite systems (6) is ISpS.

2.1.2. Fuzzy Logic System

In the past several years, fuzzy logic system has been extensively studied [24]. In this paper, it will be used as a approximator to estimate these system dynamics which cannot be linearly parameterized in adaptive fuzzy torque controller design.

Takagi-Sugeno (T-S) fuzzy system [27] is applied in this study, which can be described as the following form which consists of rules:

where are fuzzy subsets, denote designed constant parameters in fuzzy rule conclusion step, is fuzzy output, and is the total number of fuzzy rules.

If we choose product inference engine, singleton fuzzifier, and center average defuzzifier (which is very commonplace choice), the output of T-S fuzzy system can be given as

where , ,

and are fuzzy basis vector and fuzzy basis function, respectively, and , . Usually, the membership function of fuzzy set is given by [26]

where , and are adjustable parameters of , , .

Lemma 7. For any continuous function defined in a compact set and for all , there exists an ideal fuzzy logic, such that [28]

2.2. Problem Formulation

We consider a class of SISO nonlinear systems given as

where is system state vector, is system input, and is system output. Denote , . is the linearly parameterized system model where is the vector of unknown but constant parameters and is the vector of known system function. is unknown but continuous system function which cannot be linearly parameterized. is external unknown disturbance.

is the desired trajectory and is assumed to be bounded with bounded derivatives up to th order (i.e., ). Control objective is to design a bounded control law for the input , such that output tracks as closely as possible.

Assumption 8. The extents of parametric uncertainties are known, and unknown disturbances are bounded by known function; that is, parametric uncertainties and unknown disturbances are assumed to satisfy

where and are known lower and upper bound constant vectors of . In this paper, the operation for two vectors is performed in terms of corresponding elements (e.g., means that , for all ).

Remark 9. When , if we treat as the control input of the dynamics, the dynamics would only depend on the states of its previous dynamics, that is, . In other words, only the feedback signals determine the dynamics. Such a form is called strict feedback form. Equation (14) is called a semistrict feedback form in which the bounding function is required to be function of , and only, but may contain some bounded functions of , , thus violating the strict feedback property [6, 7].

3. Adaptive Fuzzy Robust Controller Design

3.1. Discontinuous Projection-Based Adaptive Law

The discontinuous projection-type parameter adaptation law is used to update parameters in AFRC controller. Let denote the estimate of , and let denote estimation error (i.e., ). Specifically, viewing (15), parameter estimate is updated through a parameter adaptation law having the form of

where is a diagonal matrix of adaptation rate and is an adaptation function to be designed according to system model. The discontinuous projection mapping can be defined as

It can be shown that for any adaptation function , the projection mapping used in (18) has the following properties [5]:

3.2. Backstepping Design of Adaptive Fuzzy Robust Control

In this section, we give out adaptive fuzzy robust controller design procedure by using backstepping techniques, and it contains steps. At each step, an intermediate control signal will be developed by using an appropriate Lyapunov function .

Step 1. Consider the first subsystem of (14), which is given as

In (20), by viewing as the control input of this subsystem, we can design an intermediate control law for , such that tracks its desired trajectory . Define the tracking error variables as

The time derivative of can be given as

Since is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, can be expressed as

where

, represents fuzzy approximating error, and ; here is unknown but bounded positive constant.

Let , . It is clear that ; then let , and (23) can be rewritten as

Define the 2th tracking error variable as

Then substitute (25) into (22),

where

where is a positive constant and .

The intermediate control law consists of three parts as

At first, we design the adaptive control item as

Substituting (29) and (30) into (22) leads to

Then, we design the fuzzy control law as

where , , and is an online estimate of parameter with estimate error . , , and are positive design parameters.

Define a positive semidefinite (p.s.d.) function as

Considering , time derivative of can be given as

Applying Young’s inequality, the following inequality can be obtained:

Considering that and , from (35), the following inequality is holding:

Substituting (32) and (36) into (34) leads to

Now we can get the following robust performance conditions for :

Remark 10. One smooth example of robust control function satisfying (38) can be found in the following way. Let be any smooth function satisfying where . is positive design parameter.

Then we can design a robust control law as

Thus, (37) can be given as

Step 2. Consider the second subsystem of (14), which is given as

In (41), by viewing as the control input of this subsystem, we can design an intermediate control law for , such that tracks the intermediate control law .

The time derivative of can be given as

And the time derivative of is

Then substituting (43) into (42) leads to

where

Since is an unknown continuous function, according to fuzzy logic theorem as mentioned in Lemma 7, can be expressed as

where

, represents fuzzy approximating error, and ; here is unknown but bounded positive constant.

Let , . It is clear that ; then let , and (23) can be rewritten as

Define the 3th tracking error variable as

Substituting (48) and (49) into (44), we have

where

where is a positive constant and .

The control law consists of three parts as

At first, we design the adaptive control item as

Substituting (52) and (53) into (50) leads to

Then, we design the fuzzy control law as

where , is an online estimate of parameter with estimate error . , , and are positive design parameters.

Define a positive semidefinite (p.s.d.) function as

Considering , time derivative of can be given as

Applying Young’s inequality, the following inequality can be obtained:

Considering that and , from (58), the following inequality is holding:

Substituting (55) and (59) into (57) leads to

Now we can get the following robust performance conditions for :

According to Remark 10, we can design a robust control law as

where can be any smooth function satisfying and is positive design parameter.

Thus, (60) can be given as

Step i. We can use the similar procedure for each Step i . Consider the th subsystem of (14), which is given as

We can design a control law for subsystem (64) as

Define a p.s.d. function as

By using the same technique, the time derivative of can be given as

Step n. Consider the th subsystem of (14), which is given as

Define the tracking error variable as

The time derivative of can be given as

The time derivative of is

Substituting (71) into (70) leads to

where

According to fuzzy logic theorem as mentioned in Remark 10, (73) can be expressed as

where

, represents fuzzy approximating error, and , and here is unknown but bounded positive constant.

Let , . It is clear that ; then let , and (74) can be rewritten as

Substitute (76) into (72), we have

where

where is a positive constant and .

The control law consists of three parts giving as

At first, we design the adaptive control item as

Substituting (79) and (80) into (77) leads to

Design the fuzzy control law as

where , , and is an online estimate of parameter with estimate error . , and are positive design parameters.

Define a p.s.d. function as

Considering , the time derivative of can be given as

Applying Young’s inequality, the following inequality can be obtained:

Considering that and , from (85), the following inequality is holding:

Substituting (82) and (86) into (84) leads to

Now we can get the following robust performance conditions for :

Then we can design a robust control law as

where can be any smooth function satisfying and is positive parameter.

Thus, (87) can be given as

4. The Closed-Loop System Stability Analysis

Consider the system (14) with partly linearizable system dynamics, unknown nonlinearities, and external disturbances, and suppose that the unknown system functions can be approximated by T-S fuzzy systems. If we design the adaptive fuzzy robust control law with the intermediate control law , , and adaptive laws for and , then the overall closed-loop control system is semiglobally uniformly ultimately bounded in the sense that all signals in this system is bounded. If control parameters are appropriately selected, the tracking error can be smaller than a prescribed error bound, and it means the tracking error asymptotically converges zero.

Proof. Rewrite the closed-loop system into two composited subsystems, namely, and as follows:

where is given as input of subsystem and is output, , , and .

Design a Lyapunov function as

Choosing , noting the time derivative of can be obtained as

where

According to small gain theorem [8, 13, 21], the subsystem satisfies ISpS, and there exist class functions , , such that , and , , then the gain of subsystem can be obtained as

Consider subsystem as

where is the input of subsystem, output is . , , , and . Considering and , , we have

Let ,

Therefore , the gain of subsystem .

According to small gain theorem, if , then the closed-loop system (91) is ISpS. Then,

Because , if we choose , the closed-loop system satisfies ISpS conditions. Therefore, there exist a class function and a positive condition , such that

Therefore, , . There exist and ; such that, , for all , that is the closed-loop system is uniformly ultimately bounded.

Furthermore, according to (93), if , the subsystem is ISS, and the closed-loop system is also ISS. Therefore, there exists a class function, such that

That means ; that is, the asymptotic output tracking is achieved.

5. Illustrative Example

In order to illustrate the control performance of the proposed AFRC, the following simulation example is considered in this section:

where is system state vector, is system input, and is system output. is the vector of parameters. and represent unknown functions. and are nonlinear dynamic disturbances. In the simulation, we set , , , , , and .

Let the desired trajectory as which is shown in Figure 1. The control objective is to guarantee that (a) all signals in the closed-loop system are bounded and (b) the output follows the trajectory as closely as possible.

Figure 1: The desired trajectory.

Define fuzzy membership functions for each error dynamics , as

The first intermediate control function can be given as

Then we design the controller law as

where and , . The discontinuous projection-type parameter adaptation laws are given as follows:

with lower bound and upper bound . The initial values of parameter estimates are chosen as .

The online estimation laws of parameter and are given as

In the simulation, three controllers are compared.

 (C1): the proposed adaptive fuzzy robust controller as described previously. (C2): by setting , the controller is equivalent to the adaptive robust control law in [6]. (C3): by setting , the controller becomes a fuzzy controller.

Simulation results are shown in Figures 2–5. Figure 2 shows the output tracking error , and Figure 3 shows the intermediate error . According to Figures 2 and 3, it is obvious that all three controllers have very good tracking ability; however, AFRC (C1) has the best performance among the three controllers. Figure 4 shows the online estimations of system parameters, Figure 5 shows the online updated fuzzy parameters , and the figures indicate that both of them are all bounded.

Figure 2: The output tracking error of three compared controllers.

Figure 3: The error of three compared controllers.

(a)

(b)

Figure 4: The online estimations of system parameters.

(a)

(b)

Figure 5: The online updated fuzzy parameters .

6. Conclusion

The AFRC algorithm is proposed for a class of nonlinear systems which can be partly linearly parameterized and also have unknown system nonlinearities and external unknown disturbances. Essentially, AFRC is a combination of adaptive control and fuzzy control, and the algorithm is developed by using backstepping technique. By using small-gain approach, the closed-loop system is proved to be semiglobally uniformly ultimately bounded. The comparative simulation results demonstrate the effectiveness of the proposed AFRC algorithm. This proposed that AFRC effectively combines the technical characters of adaptive control and fuzzy control and improves the control performance by retaining the advantages of both control methods.

Acknowledgment

The authors would like to appreciate the support of National Natural Science Foundation of China (51175014) and Program 111 of China.

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