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Fuzzy Control

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 1, FEBRUARY 1998 137 Letters Stable and Optimal Fuzzy Control of Linear Systems Li-Xin Wang Abstract— In this letter, a number of stable and optimal fuzzy controllers are developed for linear systems. Based on some classical results in control theory, we design the structure and parameters of fuzzy controllers such that the closed-loop fuzzy control systems are stable, provided that the process under control is linear and satisfies certain conditions. It turns out that if stability is the only requirement, there is much freedom in choosing the fuzzy controller parameters. Therefore, a performance criterion is set to optimalize the parameters. Using the Pontryagin minimum principle, we design an optimal fuzzy controller for linear systems with quadratic cost function. Finally, the optimal fuzzy controller is applied to the ball-and-beam system. Index Terms—Closed loop, fuzzy control, linear, optimal fuzzy. I. INTRODUCTION F UZZY controllers are rule-based nonlinear controllers, therefore their main application should be the control of nonlinear systems. However, since linear systems are good approximations of nonlinear systems around the operating points, it is of interest to study fuzzy control of linear systems. Additionally, fuzzy controllers due to their nonlinear nature may be more robust than linear controllers even if the plant is linear. Furthermore, fuzzy controllers designed for linear systems may be used as initial controllers for nonlinear adaptive fuzzy control systems where on-line turning is employed to improve the controller performance. Therefore, a systematic study of fuzzy controllers for linear systems is of theoretical and practical interest. The goal of this letter is to explore the fundamentals of stable and optimal fuzzy control of linear systems and to show how fuzzy control can be enriched by transferring results from linear and nonlinear control theory. Stability and optimality are the most important requirements for any control system. Stable fuzzy control of linear systems has been studied by a number of researchers. Ray and Majumder [10] applied the circle criteria to analyze linear systems with fuzzy controllers. Their approach is limited to a special fuzzy controller, which is equivalent to a multilevel relay (it is well-known nowadays that fuzzy controllers are universal nonlinear controllers [15]). Langari and Tomizuka [7] designed a class of stable fuzzy controllers for linear systems, and Chiu and Chand [3] developed a fuzzy controller for a flexible wing aircraft model and analyzed the stability. All these studies are Manuscript received April 28, 1995; revised January 18, 1997. This work was supported in part by the Hong Kong RGC Grant HKUST684/95E. The author is with the Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Publisher Item Identifier S 1063-6706(98)00848-0. preliminary in nature and deeper studies can be done. For optimality, it seems that the field of optimal fuzzy control is totally open. In this letter, we will try to provide the skeleton of stable and optimal fuzzy control of linear systems by designing a number of stable or optimal fuzzy controllers for single-input-single-output (SISO) and multi-input-multi-output (MIMO) linear systems. In Section II, we develop two fuzzy controllers for SISO linear systems, which are exponentially stable and input-output stable, respectively. In Section III, we generalize the results in Section II to MIMO linear systems. In Section IV, an optimal fuzzy controller is designed for linear systems with quadratic cost function and it is tested for the ball-and-beam system. Section V concludes the letter. II. STABLE FUZZY CONTROL OF SISO LINEAR SYSTEMS For any control systems (including fuzzy control systems), stability is the most important requirement because an unstable control system is typically useless and potentially dangerous. Conceptually, there are two types of stability: Lyapunov stability (a special case of which is exponential stability) and input–output stability. We now develop fuzzy controllers, which are either exponentially stable or input–output stable. A. Exponential Stability of Fuzzy Control Systems Assume that the process under control is a single-inputsingle-output (SISO) time-invariant linear system represented by the following state variable model: (1) (2) is the control, is the output and where is the state vector. Suppose that the control is a fuzzy system whose input is ; that is (3) where is a fuzzy system [15]. Substituting (3) into (1), we obtain the closed-loop fuzzy control system, which is shown in Fig. 1. We now cite a famous result in control theory [14] and use it to design an exponentially stable fuzzy control system. Proposition 1 [14] Consider the closed-loop control system (1)–(3) and suppose that: 1) all eigenvalues of lie in the open left half of the complex plane; 2) the system (1)–(2) is controllable and observable; and 3) the transfer function of 1063–6706/98$10.00  1998 IEEE 138 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 1, FEBRUARY 1998 Fig. 1. Closed-loop fuzzy control system. the system (1)–(2) is strictly positive real. If the nonlinear function satisfies and Fig. 2. Membership functions for the fuzzy controller. • Step 3—Design the fuzzy controller from the 2 fuzzy IF-THEN rules (5) using singleton fuzzifier, centeraverage defuzzifier and product inference engine [15]; that is, the designed fuzzy controller is (4) then the equilibrium point of the closed-loop system (1)–(3) is globally exponentially stable. The nice feature of this proposition is that the conditions and are simple and clear and it is easy to design a controller that satisfies these conditions. Conditions 1)–3) in Proposition 1 are imposed on the process under control, not on the controller . They are strong conditions, essentially requiring that the open-loop system is stable and well-behaved. Conceptually, these systems are not difficult to control, therefore the requirements on the fuzzy controller, and (4), are not very strong. Proposition 1 guarantees that if we design a fuzzy controller , which satisfies and (4), then the closed-loop system is globally exponentially stable, provided that the process under control is linear and satisfies conditions 1)–3) in Proposition 1. We now design such a fuzzy controller. Design of a Stable Fuzzy Controller: • Step 1—Suppose that the output takes values in the . Define fuzzy sets interval in whose membership functions are shown in Fig. 2; that is, we use the fuzzy sets to cover the negative interval and the other fuzzy sets to cover the positive interval . The shape and relationship among these membership functions are as shown in Fig. 2. The center of fuzzy set , is at zero and the centers of other fuzzy sets can be freely chosen provided that the relationship shown in Fig. 2 is preserved. • Step 2—Consider the following fuzzy IF–THEN rules IF is where the fuzzy sets centers of fuzzy sets for for for THEN is (5) are defined in Fig. 2 and the are chosen such that (7) where are shown in Fig. 2 and satisfy (6). We now prove that the fuzzy controller designed from the three steps above produces a stable closed-loop system. Theorem 1: Consider the closed-loop fuzzy control system in Fig. 1. If the fuzzy controller is designed through the above three steps (that is, is given by (7)) and the process under control satisfies conditions 1)–3) in Proposition 1, then the equilibrium point of the closed-loop fuzzy control system is guaranteed to be globally exponentially stable. Proof: According to Proposition 1, we only need to prove and for all . From (7), (6) and Fig. 2 we have that . If , then from (7) and Fig. 2 we have (8) or for some . Since and the membership functions are nonnegative, we have ; therefore, . Similarly, we can prove that if . From the three steps of the design procedure we see that in designing the fuzzy controller , we do not need to know the specific values of the process parameters and . Also, there is much freedom in choosing the parameters of the fuzzy controller. Specifically, we only require that the ’s satisfy (6) and the membership functions are in the form shown in Fig. 2. B. Input–Output Stability of Fuzzy Control Systems (6) The closed-loop fuzzy control system in Fig. 1 does not have an explicit input. In order to study input–output stability, WANG: STABLE AND OPTIMAL FUZZY CONTROL OF LINEAR SYSTEMS 139 which is a linear function of . Since is continuous, it is a piecewise linear function. Combining Lemma 1 and Proposition 2, we obtain the following theorem. Theorem 2: Consider the closed-loop fuzzy control system in Fig. 3. Suppose that the fuzzy controller is designed as in (7) and that all the eigenvalues of lie in the left-half complex plane. Then, the closed-loop fuzzy control system in Fig. 3 is -stable. Proof: Since a continuous, bounded, and piecewise linear function satisfies the Lipschitz condition (9), this theorem follows from Proposition 2 and Lemma 1. Fig. 3. Closed-loop fuzzy control system with external input. III. STABLE FUZZY CONTROL OF MIMO LINEAR SYSTEMS we introduce an extra input and the system is shown in Fig. 3. We now cite a well-known result in control theory [14]. Proposition 2 [14]: Consider the control system in Fig. 3 and suppose that the nonlinear controller is globally Lipschitz continuous; that is (9) for some constant . If the open-loop unforced system is globally exponentially stable (or equivalently, the eigenvalues of lie in the left-half complex plane), then the forced closed-loop system in Fig. 3 is -stable for all . The nice feature of Proposition 2 is that condition (9) is simple, and it is easy to design a controller that satisfies this condition. It is interesting to see whether the fuzzy controller (7) designed through the three steps in Section IIA satisfies the Lipschitz condition (9). We first show that the fuzzy controller of (7) is continuous, bounded, and piecewise linear, from which we conclude that it satisfies the Lipschitz condition. Lemma 1: The fuzzy controller of (7) is continuous, bounded, and piecewise linear. Proof: Let be the center of fuzzy set whose membership function is shown in Fig. 2, where equals the largest value among all such that and equals the smallest value among all such that . Since the membership functions in Fig. 2 are continuous, the of (7) is continuous. Since for and for where and are finite numbers, the is bounded. To show that the is a piecewise linear function, we partition the real line into . For and we have and , respectively, which are linear functions. For with , we have from Fig. 2 that (10) A. Exponential Stability Consider the multi-input multi-output (MIMO) timeinvariant linear system (11) (12) , the output , and the where the input state . We assume that the number of input variables equals the number of output variables; this is called “squared” systems. In this case, the control consists of fuzzy systems; that is, (13) and are -input one-output where fuzzy systems. The closed-loop fuzzy control system is still of the structure in Fig. 1, except that the vector is replaced by the matrix , the vector is replaced by the matrix , and the scalar function is replaced by the vector function . As before, we first cite an important result in control theory and then design a fuzzy controller that satisfies the conditions. The following proposition can be found in [14]. Proposition 3: Consider the closed-loop system (11)–(13) and suppose that: 1) all eigenvalues of lie in the open lefthalf complex plane; 2) the system (11)–(12) is controllable and observable; and 3) the transfer matrix is strictly positive real. If the control vector satisfies and (14) of the closed-loop system then the equilibrium point (11)–(13) is globally exponentially stable. Condition (14) is a generalization of condition (4), and its nice feature is that a controller can be easily designed to satisfy this condition. Note that in order to satisfy (14), the fuzzy systems cannot be designed fuzzy systems such independently. We now design these that the resulting fuzzy controller satisfies (14). 140 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 1, FEBRUARY 1998 Design of a Stable Fuzzy Controller: • Step 1—suppose that the output takes values in the interval where . Define fuzzy sets in whose membership functions 2 are shown in Fig. 2 with the subscribe added to all the variables. • Step 2—consider groups of fuzzy IF–THEN rules where the th group consists of the following rules: IF is where centers that and and is of the fuzzy sets if if if THEN is (15) and the are chosen such (16) for with can take any where values from . • Step 3—design the fuzzy controllers each from the rules in (15) using singleton fuzzifier, center-average defuzzifier and product inference engine [15]; that is, the designed fuzzy controllers are (17) . where We see from these three steps that as in the SISO case, we do not need to know the process parameters and in order to design the fuzzy controller; we only require that the membership functions are in the form of Fig. 2 and the parameters satisfy (16). There is much freedom in choosing these parameters. We now show that the fuzzy with designed as in (17) controller produces a stable closed-loop system. Theorem 3: Consider the closed-loop fuzzy control system (11)–(13). If the fuzzy controller is designed through the three steps above (that is, is given by (17)) and the process under control satisfies the conditions 1)–3) in Proposition 3, then the equilibrium point of the closed-loop system is globally exponentially stable. Proof: If we can show that and for all where , then this theorem follows from Proposition 3. From Fig. 2, (17) and (16) we have that for . Since can show then we have Fig. 2 we have for all for all for and all . If , if we , , then from . Hence, (17) becomes (18), shown at the bottom of the page. From (16) we have for and with and . Hence, and . Similarly, we can prove if . that B. Input–Output Stability Consider again the closed-loop fuzzy control system (11)–(13). Similar to Proposition 2, we have the following result concerning the stability of the control system. Proposition 4 [14]: Consider the closed-loop control system (11)–(13) and suppose that the open-loop unforced system is globally exponentially stable. If the nonlinear controller is globally Lipschitz continuous; that is, (19) where is a constant, then the closed-loop system (11)–(13) is stable for all . Again, the nice feature of this proposition is that it is relatively easy to use for controller design. Using the same arguments as for Lemma 1, we can prove that the fuzzy systems (17) are continuous, bounded, and piecewise linear functions. Since a vector of continuous, bounded, and piecewise linear functions satisfies the Lipschitz condition (19), we have the following result according to Proposition 4. Theorem 4: The fuzzy control system (11)–(13), with the fuzzy systems given by (17), is stable for all provided that the eigenvalues of lie in the open left-half complex plane. IV. OPTIMAL FUZZY CONTROL OF LINEAR SYSTEMS In Sections II and III, we gave conditions on which the closed-loop fuzzy control systems are stable. Usually, we determine ranges for the fuzzy controller parameters such that stability is guaranteed if the parameters lie inside these ranges. We did not show how to choose the parameters within these ranges. In this section, we study how to determine the specific values of the fuzzy controller parameters such that certain performance criterion is optimalized; that is, we are now considering the design of optimal fuzzy controllers for linear systems. This is a more difficult problem than designing stableonly fuzzy controllers. The approach in this section is very preliminary and much work remains to be done. Additionally, in designing the optimal controller, we must know the values of and , which are not required in designing the stable-only fuzzy controllers in Sections II and III. (18) WANG: STABLE AND OPTIMAL FUZZY CONTROL OF LINEAR SYSTEMS We will first state the Pontryagin minimum principle for solving the optimal control problem. Then we will constrain the controller to be a fuzzy system and use the Pontryagin minimum principle to design the fuzzy controller parameters such that a quadratic cost function is minimized. 141 Now assume that the controller is constructed from the fuzzy systems in the form of (17) except that we change the system output in (17) by the state ; that is, with A. The Pontryagin Minimum Principle Consider the system (20) (29) where is the state, with initial condition is the control input, and is a linear or nonlinear function. The optimal control problem for the system (20) is as follows: determine the control such that the performance criterion We assume that the membership functions are fixed; they can be in the form of Fig. 2 or in other forms. Our task is to determine the parameters such that of (28) is minimized. Define the fuzzy basis function as (21) (30) is minimized, where and are given functions and the final time may be given. The Pontryagin minimum principle for solving this optimal control problem proceeds as follows. First, define the so-called Hamilton function where and . Define the parameter matrix as (31) (22) and find this . Substituting such that is minimized with into (22) and define (23) Then, solve the 2 differential equations (with the two-point boundary condition) (24) (25) and denote the solution of (24) and (25) and let (which is called the optimal trajectory). Finally, the optimal control is obtained as where consists of the parameters for in the same ordering as for . Using these notations, we can rewrite the fuzzy controller of (29) as (32) To achieve maximum optimality, we assume that the parameter matrix is time-vary; that is, . Substituting (32) into (27) and (28), we obtain the closedloop system (33) and the performance criterion (26) (34) B. Design of an Optimal Fuzzy Controller Suppose that the system under control is the time-invariant linear system (27) and and that the performance criterion where is the following quadratic function: Hence, the problem of designing the optimal fuzzy controller becomes the problem of determining the optimal such that of (34) is minimized. Viewing the as the control in the Pontryagin minimum principle, we can determine the optimal from (22)–(26). Specifically, define the Hamilton function (35) (28) where the matrices are symmetric and positive definite. and From ; that is (36) 142 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 6, NO. 1, FEBRUARY 1998 we obtain, approximately, that (37) where is a full-rank matrix with very small norm; we introduce to make invertible ( may be generated by a small random number generator). Substituting (37) into (35), we have Fig. 4. The ball-and-beam system. (38) where is defined as (39) Using this in (24) and (25), we obtain (40) • Step 4—the optimal fuzzy controller is obtained as (43). Note that Steps 1–3 are off-line operations; that is, we first compute following Steps 1–3 and store the for in the computer. Then, in on-line operation we simply substitute the stored into (43) to obtain the optimal fuzzy controller. The most difficult part in designing this optimal fuzzy controller is to solve the two-point boundary differential equations (40) and (41). Since these differential equations are nonlinear, numerical integration is usually used to solve them. C. Application to the Ball-and-Beam System The ball-and-beam system, as shown in Fig. 4, can be modeled by the following state equation: (41) (44) with boundary condition and . Let and be the solution of (40) and (41), then the optimal fuzzy controller parameters are (45) (42) and the optimal fuzzy controller is (43) Note that the optimal fuzzy controller (43) is a state feedback controller with time-varying coefficients. The design procedure of this optimal fuzzy controller is summarized as follows. Design of the Optimal Fuzzy Controller: • Step 1—specify the membership functions to cover the state space where and . We may not choose the membership functions as in Fig. 2 because the function with these membership functions is not differentiable [we need in (41)]. We may choose to be Gaussian functions. • Step 2—compute the fuzzy basis functions from (30) and the function from (39). Compute the . derivative • Step 3—solve the two-point boundary differential equations (40) and (41) and let the solution be and . Compute from (42). is the state vector, where the control equals the acceleration of the angle , the output is the angle , and and are parameters reflecting the mass of the ball and the beam [15]. In our simulations below, we and . Clearly, the ball-and-beam choose system is nonlinear. To apply our optimal fuzzy controller, we linearize it around the equilibrium point . The linearized system is in the form of (27) with (46) We now design the optimal fuzzy controller for the linearized ball-and-beam system and apply the designed controller to the original nonlinear system (44)–(45). We choose and for . The matrix is created by a small random number generator. The membership functions are chosen as (47) where with and and . WANG: STABLE AND OPTIMAL FUZZY CONTROL OF LINEAR SYSTEMS Fig. 5. Closed-loop output y (t) with the optimal fuzzy controller for the three initial conditions of case 1). 143 Although fuzzy control of linear systems could be a good starting point for a better understanding of some issues in fuzzy control synthesis, it does not have much practical implications. Using the fuzzy controller designed for linear systems directly as the controller may not be a good choice; a better way is to use it as a rough or initial controller based on which further turning and improvement could be conducted. When applying adaptive fuzzy controllers to nonlinear systems, a good initial controller becomes crucial for good performance [15]. Usually, an initial fuzzy controller is constructed from heuristic rules. In case there are no good heuristic rules, using the fuzzy controller designed for a rough linear model as the initial controller may be a good choice. Of course, further research is needed to validate these proposals. Overall, fuzzy control of linear systems should not be emphasized, but some simple and systematic design procedures might be helpful for those who want to get a rough fuzzy controller quickly. ACKNOWLEDGMENT The author would like to thank the reviewers for their constructive comments. REFERENCES Fig. 6. Closed-loop output y (t) with the optimal fuzzy controller for the three initial conditions of case 2). We simulated the control system for two sets of initial conditions: 1) and 2) . Figs. 5 and 6 show the closed-loop output for the two cases, respectively. We see from Figs. 5 and 6 that our optimal fuzzy controller, although designed based on the linearized model, could smoothly regulate the ball to the origin from a number of initial positions. V. CONCLUSION This letter provided a systematic treatment for fuzzy control of linear systems. 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