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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999
225
Letters The Adaptive Control of Nonlinear Systems Using the Sugeno-Type of Fuzzy Logic Dong-Ling Tsay, Hung-Yuan Chung, and Ching-Jung Lee Abstract— An adaptive fuzzy controller is synthesized from a collection of fuzzy IF–THEN rules. The parameters of the membership functions characterizing the linguistic terms in the fuzzy IF–THEN rules are changed according to some adaptive laws for the purpose of controlling a plant to track a reference trajectory. In this paper, a direct adaptive fuzzy control design method is developed for the general higher order nonlinear continuous systems. We use the Sugeno-type of the fuzzy logic system to approximate the controller. It is proved that the closedloop system using this adaptive fuzzy controller is globally stable in the sense that all signals involved are bounded. Finally, we apply the method of direct adaptive fuzzy controllers to control an unstable system. Index Terms— Adaptive fuzzy control, fuzzy control, linear state feedback control, nonlinear.
I. INTRODUCTION
F
UZZY logic controllers are generally considered applicable to plants that are mathematically poorly understood and where the experienced human operators are available. However, the fuzzy control has not been regarded as a rigorous science due to the lack of the guarantee of the global stability and acceptable performance, but some researchers propose the stability analysis of fuzzy control systems (e.g., [5]). The mathematical model of the plant is assumed to be known in [5]. Hence, this contradicts the very fundamental premise of fuzzy control systems. In fact, if the model of the plant is known, then we should give the conventional linear and nonlinear control methods high priority. Fuzzy controllers are assumed to work in situations where the plant parameters and structures have some uncertainties or unknown variations. The basic objective of adaptive control is to maintain the consistent performance of a system in the presence of uncertainties. So advanced fuzzy control might be adaptive. An adaptive fuzzy system is a fuzzy logic system equipped with an adaptation law. The major advantage of adaptive fuzzy controller over the conventional adaptive controller is that the adaptive fuzzy controller is capable of incorporating linguistic fuzzy information from human operators. The adaptive fuzzy controllers are divided into two classes. One is called the direct adaptive fuzzy control and the other Manuscript received December 20, 1996; revised March 27, 1998. This work was supported in part by the National Science Council of the Republic of China under Contract NSC-85-2213-E-008-020. The authors are with the Department of Electrical Engineering, National Central University, Chungli, Taiwan, 320 R.O.C. Publisher Item Identifier S 1063-6706(99)02804-0.
is called the indirect adaptive fuzzy control. In the direct adaptive fuzzy control, we view the fuzzy logic systems as controllers. However, in the indirect adaptive fuzzy control, the fuzzy logic systems are used to model the plant. Then the controller is constructed assuming that the fuzzy logic systems approximately represent the true plant. In this paper, an alternative direct adaptive fuzzy controller is developed. We use other fuzzy logic systems that are different from those used in [1] and [2]. This form of the fuzzy logic system is called the Sugeno type. And the consequent part of the fuzzy rules is based on the locally linear feedback control theory. In Section II, the considered problem formulation is shown. In Section III, the utilized fuzzy logic system is described. The main result is presented in Section IV. In Section V, the proposed design steps of the direct adaptive fuzzy controller are used to control an unstable system. Some conclusions of this paper are given in Section VI. II. PROBLEM FORMULATION Consider the th-order nonlinear systems of the form (1) is an unknown but bounded continuous function, where is a positive unknown constant, and and are the input and the output of the system, respectively. Let be the state vector of the system which is assumed to be available. The control objective is to force to follow a given bounded under the constraints that all signals reference signal involved must be bounded. Hence, a feedback control based on fuzzy logic systems and an adaptive law for adjusting the parameters of the fuzzy logic systems are both determined to satisfy the following conditions. 1) The closed-loop system must be globally stable in the sense that all variables must be uniformly bounded. should be as small as 2) The tracking error possible under the constraints in 1). Then our design objective is to impose an adaptive fuzzy control algorithm so that the following asymptotically stable tracking: (2) is achieved. The roots of polynomial in the characteristic equation of (2) are all in
1063–6706/99$10.00 1999 IEEE
226
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999
Hence, (6) will represent the mathematical models of the fuzzy logic system used in the following sections. IV. THE MAIN RESULT
Fig. 1. The basic configuration of a fuzzy logic system.
the open left-half plane via an adequate choice of coefficients .
We will show the basic ideas of how to construct a direct adaptive fuzzy controller to achieve the control objectives and defined in Section II. First, let be such that all roots of the are in the open leftpolynomial hand plane. If the function and the constant are known, then the control law (7)
III. DESCRIPTION OF THE USED FUZZY LOGIC SYSTEM The basic configuration of the fuzzy logic systems considered in this paper is shown in Fig. 1. The fuzzy logic system to We assume that performs a mapping from where . The fuzzy rule base contains a collection of fuzzy IF–THEN rules IF is THEN
and
is (3)
and are the input and where is the the output of the fuzzy logic system, respectively. label of the fuzzy set in for and are the constant coefficients of the consequent part of the fuzzy rule. This fuzzy logic system is called the Sugeno type. Each fuzzy rule of (3) defines a fuzzy implication . In this paper, we would employ the product operation for the fuzzy implication and norm. The definition of the product operation is the same as that in [2]; besides, the singleton fuzzifier is used. Consequently, the final output value is
applied to the system (1) can result in
which implies that (the main objective of control). Since and are unknown, the optimal control cannot be implemented. Hence, the fuzzy logic controller will be designed to approximate this optimal control. is supposed to consist of a fuzzy control The control and a supervisory control , i.e., (8) is a fuzzy logic system in the form of (5) where is used to avoid the bigger control signal. In this and paper, however, the controller design approach is based on the concept of the locally linear feedback control theory. In [6], the design procedures of the fuzzy controllers based on the same concept are proposed. We should be able to find out that the consequent part of each rule is a linear state feedback control signal by the same procedures. is shown below. SubstitutThe method of designing ing (8) into the system (1), we will have (9)
(4)
After some straightforward manipulation, we can obtain the error equation of the closed-loop system (10)
are the membership functions for and . ’s and view the ’s as adjustable If we fix the parameters, then (4) can be rewritten as
where
where
(5) is a where parameter vector and is a regressive vector with the regressor defined as
(6)
(11)
, where is a symmetric Define a function positive definite matrix satisfying the Lyapunov equation (12)
TSAY et al.: ADAPTIVE CONTROL OF NONLINEAR SYSTEMS USING SUGENO-TYPE FUZZY LOGIC
where
. Differentiate the
with respect to , we have
227
Differentiated with respect to , we get
(13) such that . In order to Now, our work is to design satisfy this requirement, we need one assumption. and a Assumption 1: We can determine a function such that and . constant Under the above assumption, we could construct the superwith the following formula: visory control
(20) Let
be the last column of
. From (11) we can obtain (21)
We could choose the adaptive law (22)
(14) is a constant specified by the designer) where if if . Due to we can evaluate the and . Moreover, the other terms in (14) can value of of (14) be determined. Thus, the supervisory control . can be implemented. Now we will consider the case of Substituting (14) and (7) into (13), we have
(15) of (14), we always have So, using the supervisory control . Because , the boundedness of implies the boundedness of , which, in turn, implies the boundedness of . From (14), we can find that if the closed-loop system is well worked in the sense with the pure fuzzy controller ), then is zero. On that the error is not big (i.e., ), then the other hand, if the error is bigger (i.e., will operate to force . Hence, this is why we call a supervisory control. Next, we will develop an adaptive law to adjust the parameter vector . Define the optimal parameter vector (16) and are specified by the designer. The “minwhere imum approximation error” is defined as (17) into (10), we get the Substituting (17) and error equation of the closed-loop system
(18) where
. Define the Lyapunov function candidate (19)
We use the facts becomes
and
. Then (20) (23)
, we can use a projection In order to guarantee algorithm [7] to modify the basic adaptive law (22). The properties of this proposed direct adaptive fuzzy controller is shown in the following theorem. Theorem 1: Consider the nonlinear plant (1) with the conis given by (5) and is given by (14). Let trol (8), where the parameter vector be adjusted by the projection algorithm and let Assumption 1 be true. Then the overall control scheme guarantees the following properties: 1) 2)
(24)
3) (25)
, where is the minimum eigenvalue of for all and . The proof of Theorem 1 is shown in Appendix. Below, we will make a remark. Remark 1: For the practical control problems, the state and the control input are requested to be constrained within the certain regions. Based on (24) and (25), we can design the , , and to satisfy the given constraints parameters , and . Such design approach can guarantee the state and the control input are within the constraint sets. In order to achieve this objective, we should know some bounds and . Besides, from (11) and (12), is of determined by and . Hence, we can specify the and to satisfy the required . V. SIMULATION In this section, we apply the direct adaptive fuzzy controller to a system. The model is (26)
228
Fig. 2. Fuzzy membership functions defined over the state-space.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999
Fig. 4. The system state x(t) using the direct adaptive fuzzy control with = 1:5.
Mx
Fig. 3. The system state x(t) using the direct adaptive fuzzy control with = 3.
Mx
Fig. 5. The control inputs = 1 :5 .
u
=
uc
+
us
for two cases
Mx
= 3 and
Mx
We want to regulate the plant to the origin, i.e., . We can find that the plant (26) is unstable if without control , then for and for input because if . We choose , , , and . Define six fuzzy sets over with labels . The the interval membership functions are
which are shown in Fig. 2. and . Hence, we can obtain We choose . Substituting these values into the algorithm (14), then we can get the supervisory control . Finally, we use the SIMULINK of the MATLAB TOOLBOX to simulate the overall system. is chosen. Fig. 3 shows the state The initial state which does not hit the boundary . Therefore,
the supervisory control never fired. Then we consider the and keep all other parameters the second case of same as in the simulation in Fig. 3. The simulation result for this case is shown in Fig. 4. We can see that the supervisory control does force the state to be inside the constraint set . is proportional to the upper From (14), we can see that , which is usually very large. Large control is bound undesirable because the implementation cost may increase. Fig. 5 shows the variation of the control inputs for the two of the case cases. We can find that the variation of is stronger. Therefore, we choose the to operate in the supervisory fashion. VI. CONCLUSION In this paper, the Sugeno-type fuzzy logic system is used in the direct adaptive fuzzy controller. The major advantage is that the accurate mathematical model of the system is not required to be known. The proposed method can guarantee the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. Besides, the
TSAY et al.: ADAPTIVE CONTROL OF NONLINEAR SYSTEMS USING SUGENO-TYPE FUZZY LOGIC
specific formula of the bounds is also given. Finally, we use the direct adaptive fuzzy controller to regulate an unstable system to the origin. We also showed explicitly how the supervisory control forced the state to be within the constraint set.
From
229
and (14) we can have
APPENDIX PROOF OF THEOREM 1 The properties 1) and 2) could be proved by the same steps that are utilized in [2]. Hence, we only show the proof of property 3) in this Appendix. Proof of Property 3): The output of the considered fuzzy logic system can be rewritten as (A.1) where Schwarz inequality, we get
. According to the
(A.2) Since
is a weighted average of the element , we may obtain (A.3)
(A.4) Q.E.D. REFERENCES
H1
[1] B. S. Chen, C. H. Lee, and Y. C. Chang, “ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 32–43, Feb. 1996. [2] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 146–155, May 1993. [3] B. Kosko, Neural Network and Fuzzy Systems. Englewood Cliffs, NJ: Prentice-Hall, 1992. [4] M. Jamshidi, N. Vadiee, and T. J. Ress, Fuzzy Logic and Control. Englewood Cliffs, NJ: Prentice-Hall, 1993. , “Stability analysis and design of fuzzy control system,” Fuzzy [5] Sets Syst., vol. 45, no. 2, pp. 135–156, 1992. [6] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: Stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14–23, Feb. 1996. [7] G. C. Goodwin and D. Q. Mayne, “A parameter estimation prespective of continuous time model reference adaptive control,” Automatica, vol. 23, pp. 57–70, 1987.
225
Letters The Adaptive Control of Nonlinear Systems Using the Sugeno-Type of Fuzzy Logic Dong-Ling Tsay, Hung-Yuan Chung, and Ching-Jung Lee Abstract— An adaptive fuzzy controller is synthesized from a collection of fuzzy IF–THEN rules. The parameters of the membership functions characterizing the linguistic terms in the fuzzy IF–THEN rules are changed according to some adaptive laws for the purpose of controlling a plant to track a reference trajectory. In this paper, a direct adaptive fuzzy control design method is developed for the general higher order nonlinear continuous systems. We use the Sugeno-type of the fuzzy logic system to approximate the controller. It is proved that the closedloop system using this adaptive fuzzy controller is globally stable in the sense that all signals involved are bounded. Finally, we apply the method of direct adaptive fuzzy controllers to control an unstable system. Index Terms— Adaptive fuzzy control, fuzzy control, linear state feedback control, nonlinear.
I. INTRODUCTION
F
UZZY logic controllers are generally considered applicable to plants that are mathematically poorly understood and where the experienced human operators are available. However, the fuzzy control has not been regarded as a rigorous science due to the lack of the guarantee of the global stability and acceptable performance, but some researchers propose the stability analysis of fuzzy control systems (e.g., [5]). The mathematical model of the plant is assumed to be known in [5]. Hence, this contradicts the very fundamental premise of fuzzy control systems. In fact, if the model of the plant is known, then we should give the conventional linear and nonlinear control methods high priority. Fuzzy controllers are assumed to work in situations where the plant parameters and structures have some uncertainties or unknown variations. The basic objective of adaptive control is to maintain the consistent performance of a system in the presence of uncertainties. So advanced fuzzy control might be adaptive. An adaptive fuzzy system is a fuzzy logic system equipped with an adaptation law. The major advantage of adaptive fuzzy controller over the conventional adaptive controller is that the adaptive fuzzy controller is capable of incorporating linguistic fuzzy information from human operators. The adaptive fuzzy controllers are divided into two classes. One is called the direct adaptive fuzzy control and the other Manuscript received December 20, 1996; revised March 27, 1998. This work was supported in part by the National Science Council of the Republic of China under Contract NSC-85-2213-E-008-020. The authors are with the Department of Electrical Engineering, National Central University, Chungli, Taiwan, 320 R.O.C. Publisher Item Identifier S 1063-6706(99)02804-0.
is called the indirect adaptive fuzzy control. In the direct adaptive fuzzy control, we view the fuzzy logic systems as controllers. However, in the indirect adaptive fuzzy control, the fuzzy logic systems are used to model the plant. Then the controller is constructed assuming that the fuzzy logic systems approximately represent the true plant. In this paper, an alternative direct adaptive fuzzy controller is developed. We use other fuzzy logic systems that are different from those used in [1] and [2]. This form of the fuzzy logic system is called the Sugeno type. And the consequent part of the fuzzy rules is based on the locally linear feedback control theory. In Section II, the considered problem formulation is shown. In Section III, the utilized fuzzy logic system is described. The main result is presented in Section IV. In Section V, the proposed design steps of the direct adaptive fuzzy controller are used to control an unstable system. Some conclusions of this paper are given in Section VI. II. PROBLEM FORMULATION Consider the th-order nonlinear systems of the form (1) is an unknown but bounded continuous function, where is a positive unknown constant, and and are the input and the output of the system, respectively. Let be the state vector of the system which is assumed to be available. The control objective is to force to follow a given bounded under the constraints that all signals reference signal involved must be bounded. Hence, a feedback control based on fuzzy logic systems and an adaptive law for adjusting the parameters of the fuzzy logic systems are both determined to satisfy the following conditions. 1) The closed-loop system must be globally stable in the sense that all variables must be uniformly bounded. should be as small as 2) The tracking error possible under the constraints in 1). Then our design objective is to impose an adaptive fuzzy control algorithm so that the following asymptotically stable tracking: (2) is achieved. The roots of polynomial in the characteristic equation of (2) are all in
1063–6706/99$10.00 1999 IEEE
226
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999
Hence, (6) will represent the mathematical models of the fuzzy logic system used in the following sections. IV. THE MAIN RESULT
Fig. 1. The basic configuration of a fuzzy logic system.
the open left-half plane via an adequate choice of coefficients .
We will show the basic ideas of how to construct a direct adaptive fuzzy controller to achieve the control objectives and defined in Section II. First, let be such that all roots of the are in the open leftpolynomial hand plane. If the function and the constant are known, then the control law (7)
III. DESCRIPTION OF THE USED FUZZY LOGIC SYSTEM The basic configuration of the fuzzy logic systems considered in this paper is shown in Fig. 1. The fuzzy logic system to We assume that performs a mapping from where . The fuzzy rule base contains a collection of fuzzy IF–THEN rules IF is THEN
and
is (3)
and are the input and where is the the output of the fuzzy logic system, respectively. label of the fuzzy set in for and are the constant coefficients of the consequent part of the fuzzy rule. This fuzzy logic system is called the Sugeno type. Each fuzzy rule of (3) defines a fuzzy implication . In this paper, we would employ the product operation for the fuzzy implication and norm. The definition of the product operation is the same as that in [2]; besides, the singleton fuzzifier is used. Consequently, the final output value is
applied to the system (1) can result in
which implies that (the main objective of control). Since and are unknown, the optimal control cannot be implemented. Hence, the fuzzy logic controller will be designed to approximate this optimal control. is supposed to consist of a fuzzy control The control and a supervisory control , i.e., (8) is a fuzzy logic system in the form of (5) where is used to avoid the bigger control signal. In this and paper, however, the controller design approach is based on the concept of the locally linear feedback control theory. In [6], the design procedures of the fuzzy controllers based on the same concept are proposed. We should be able to find out that the consequent part of each rule is a linear state feedback control signal by the same procedures. is shown below. SubstitutThe method of designing ing (8) into the system (1), we will have (9)
(4)
After some straightforward manipulation, we can obtain the error equation of the closed-loop system (10)
are the membership functions for and . ’s and view the ’s as adjustable If we fix the parameters, then (4) can be rewritten as
where
where
(5) is a where parameter vector and is a regressive vector with the regressor defined as
(6)
(11)
, where is a symmetric Define a function positive definite matrix satisfying the Lyapunov equation (12)
TSAY et al.: ADAPTIVE CONTROL OF NONLINEAR SYSTEMS USING SUGENO-TYPE FUZZY LOGIC
where
. Differentiate the
with respect to , we have
227
Differentiated with respect to , we get
(13) such that . In order to Now, our work is to design satisfy this requirement, we need one assumption. and a Assumption 1: We can determine a function such that and . constant Under the above assumption, we could construct the superwith the following formula: visory control
(20) Let
be the last column of
. From (11) we can obtain (21)
We could choose the adaptive law (22)
(14) is a constant specified by the designer) where if if . Due to we can evaluate the and . Moreover, the other terms in (14) can value of of (14) be determined. Thus, the supervisory control . can be implemented. Now we will consider the case of Substituting (14) and (7) into (13), we have
(15) of (14), we always have So, using the supervisory control . Because , the boundedness of implies the boundedness of , which, in turn, implies the boundedness of . From (14), we can find that if the closed-loop system is well worked in the sense with the pure fuzzy controller ), then is zero. On that the error is not big (i.e., ), then the other hand, if the error is bigger (i.e., will operate to force . Hence, this is why we call a supervisory control. Next, we will develop an adaptive law to adjust the parameter vector . Define the optimal parameter vector (16) and are specified by the designer. The “minwhere imum approximation error” is defined as (17) into (10), we get the Substituting (17) and error equation of the closed-loop system
(18) where
. Define the Lyapunov function candidate (19)
We use the facts becomes
and
. Then (20) (23)
, we can use a projection In order to guarantee algorithm [7] to modify the basic adaptive law (22). The properties of this proposed direct adaptive fuzzy controller is shown in the following theorem. Theorem 1: Consider the nonlinear plant (1) with the conis given by (5) and is given by (14). Let trol (8), where the parameter vector be adjusted by the projection algorithm and let Assumption 1 be true. Then the overall control scheme guarantees the following properties: 1) 2)
(24)
3) (25)
, where is the minimum eigenvalue of for all and . The proof of Theorem 1 is shown in Appendix. Below, we will make a remark. Remark 1: For the practical control problems, the state and the control input are requested to be constrained within the certain regions. Based on (24) and (25), we can design the , , and to satisfy the given constraints parameters , and . Such design approach can guarantee the state and the control input are within the constraint sets. In order to achieve this objective, we should know some bounds and . Besides, from (11) and (12), is of determined by and . Hence, we can specify the and to satisfy the required . V. SIMULATION In this section, we apply the direct adaptive fuzzy controller to a system. The model is (26)
228
Fig. 2. Fuzzy membership functions defined over the state-space.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999
Fig. 4. The system state x(t) using the direct adaptive fuzzy control with = 1:5.
Mx
Fig. 3. The system state x(t) using the direct adaptive fuzzy control with = 3.
Mx
Fig. 5. The control inputs = 1 :5 .
u
=
uc
+
us
for two cases
Mx
= 3 and
Mx
We want to regulate the plant to the origin, i.e., . We can find that the plant (26) is unstable if without control , then for and for input because if . We choose , , , and . Define six fuzzy sets over with labels . The the interval membership functions are
which are shown in Fig. 2. and . Hence, we can obtain We choose . Substituting these values into the algorithm (14), then we can get the supervisory control . Finally, we use the SIMULINK of the MATLAB TOOLBOX to simulate the overall system. is chosen. Fig. 3 shows the state The initial state which does not hit the boundary . Therefore,
the supervisory control never fired. Then we consider the and keep all other parameters the second case of same as in the simulation in Fig. 3. The simulation result for this case is shown in Fig. 4. We can see that the supervisory control does force the state to be inside the constraint set . is proportional to the upper From (14), we can see that , which is usually very large. Large control is bound undesirable because the implementation cost may increase. Fig. 5 shows the variation of the control inputs for the two of the case cases. We can find that the variation of is stronger. Therefore, we choose the to operate in the supervisory fashion. VI. CONCLUSION In this paper, the Sugeno-type fuzzy logic system is used in the direct adaptive fuzzy controller. The major advantage is that the accurate mathematical model of the system is not required to be known. The proposed method can guarantee the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. Besides, the
TSAY et al.: ADAPTIVE CONTROL OF NONLINEAR SYSTEMS USING SUGENO-TYPE FUZZY LOGIC
specific formula of the bounds is also given. Finally, we use the direct adaptive fuzzy controller to regulate an unstable system to the origin. We also showed explicitly how the supervisory control forced the state to be within the constraint set.
From
229
and (14) we can have
APPENDIX PROOF OF THEOREM 1 The properties 1) and 2) could be proved by the same steps that are utilized in [2]. Hence, we only show the proof of property 3) in this Appendix. Proof of Property 3): The output of the considered fuzzy logic system can be rewritten as (A.1) where Schwarz inequality, we get
. According to the
(A.2) Since
is a weighted average of the element , we may obtain (A.3)
(A.4) Q.E.D. REFERENCES
H1
[1] B. S. Chen, C. H. Lee, and Y. C. Chang, “ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 32–43, Feb. 1996. [2] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 146–155, May 1993. [3] B. Kosko, Neural Network and Fuzzy Systems. Englewood Cliffs, NJ: Prentice-Hall, 1992. [4] M. Jamshidi, N. Vadiee, and T. J. Ress, Fuzzy Logic and Control. Englewood Cliffs, NJ: Prentice-Hall, 1993. , “Stability analysis and design of fuzzy control system,” Fuzzy [5] Sets Syst., vol. 45, no. 2, pp. 135–156, 1992. [6] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: Stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14–23, Feb. 1996. [7] G. C. Goodwin and D. Q. Mayne, “A parameter estimation prespective of continuous time model reference adaptive control,” Automatica, vol. 23, pp. 57–70, 1987.
