Output Tracking Control for Fuzzy Systems Via Output Feedback Design
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Output Tracking Control for Fuzzy Systems Via Output Feedback Design Kuang-Yow Lian, Member, IEEE, and Jeih-Jang Liou
Abstract—Fuzzy observer-based control design is proposed to deal with the output tracking problem for nonlinear systems. For the purpose of tracking design, the new concept of virtual desired variables and, in turn the so-called generalized kinematics are introduced to simplify the design procedure. In light of this concept, the design procedure is split into two steps: i) Determine the virtual desired variables from the generalized kinematics; and ii) Determine the control gains just like solving linear matrix inequalities for stabilization problem. For immeasurable state variables, output feedback design is proposed. Here, we focus on a common feature held by many physical systems where their membership functions of fuzzy sets satisfy a Lipschitz-like property. Based on this setting, control gains and observer gains can be designed separately. Moreover, zero tracking error and estimation error are concluded. Three different types of systems, including nonlinear mass-spring systems, dc–dc converters, and induction motors are considered to demonstrate the design procedure. Their satisfactory simulation results verify the proposed approach. Index Terms—Control application, linear matrix inequality (LMI) approach, observer design, output tracking, Takagi–Sugeno (T–S) fuzzy systems.
I. INTRODUCTION An automatic controller is generally regarded as the device that can correct system behavior to conform to some preset desired values by using measurable information. According to the desired values, the tasks of control systems can be divided into three categories [1]: The stabilization problem, the regulation problem, and the tracking problem. In general, the tracking problem is the most difficult. Recently, the Takagi–Sugeno (T–S) fuzzy approach [2] has been extensively used to model nonlinear systems. The basic idea for the approach is to decompose the model of a nonlinear system into a set of linear subsystems with associated nonlinear weighting functions. Then by using well-formulated linear control theory, the control objective is achieved [3]. The stability analysis is carried out using Lyapunov direct method where the control problem is then formulated into solving linear matrix inequalities (LMIs) [4]. Then using powerful computational toolboxes, such as Matlab LMI Toolbox, we obtain the controller gains. Previous works have provided fruitful LMI-based approaches to control of dynamic systems [5]. In the pioneering works, the stabilization problem for T–S fuzzy models has been well practiced. Manuscript received April 29, 2004; revised December 23, 2004 and November 1, 2005. This work was supported by the National Science Council, R.O.C., under Grant NSC 93-2213-E-033-008. The authors are with the Department of Electrical Engineering, Chung-Yuan Christian University, Chung-Li 32023, Taiwan (e-mail: [email protected]. tw). Digital Object Identifier 10.1109/TFUZZ.2006.876725
The focus of tracking control, using the T–S fuzzy model based approach, has been limited to nonlinear model following, where all states are assumed known [6]; or output feedback tracking for a linear reference model, where the reference input is considered as disturbance and is attenuated using a robust criterion [7], [8]. For these situations, it often results in bounded tracking errors between the system states and reference states. We also note that the model following approach is not easily used to deal with some control tasks where only some outputs need to be controlled and other internal state behaviors are hard to be described. Hence, it is hard to define the reference model in this case. In light of this, we focus on the tasks of output tracking control in this work. The T–S fuzzy model-based control provides a systematic framework for the design of a state feedback controller [9]–[11]. An implicit assumption that the states are available for measurement is often made. However, we know that this is impossible in many practical physical systems. Measuring full states is difficult and costly. In traditional fuzzy observer design, the premise variables of fuzzy rules are assumed to be measurable. This is still a strict constraint and limits the application of such observer design [11], [12]. For some dynamic systems, it is inevitable to use immeasurable states as the premise variables. Some approaches are usually utilized to deal with the non-trivial problem of simultaneously solving the gains for both the controller and observer [13]. In this paper, we will discuss the output tracking control based on output feedback design. All the internal states including premise variables are immeasurable and need to be estimated by fuzzy observer. To this end, the new concepts, virtual desired variables and, in turn generalized kinematic constraint are introduced to make the design procedure become clear. Then the design procedure is split into two independent steps: i) Determine the virtual desired variables from the desired output equation and the generalized kinematic constraint; and ii) determine the control feedback gains and observer gains by solving a set of LMIs, the same type LMIs for stabilization problem. A common feature for many physical systems is specifically considered: The membership functions of fuzzy sets often satisfy a Lipschitz-like property. Based on this observation, the control gains and observer gains can be designed separately, i.e., the separation principle holds. Moreover, it is concluded that the output tracking error and estimation error converge to zero exponentially. The proposed approach has been found in widespread applications. To show this, three different types of systems and tasks are considered: i) Output tracking of a nonlinear mass-spring system: A mechanical system with state-dependent system matrix; ii) Output
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LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
regulation of a dc–dc converter: An electronic system with state-dependent input matrix and an immeasurable premise variable; and iii) Torque tracking of an induction motor: An electrical system with multiple inputs and multiple outputs, where the output equations are nonlinear functions of state variables. The control tasks for all the systems can be well done. The rest of this paper is organized as follows. In Section II, the basic concept for designing output tracking control is addressed. Then, how to determine the virtual desired variables is demonstrated in Section III. In Section IV, the fuzzy observer-based output tracking design is presented. In Section V, we show that the control gains and observer gains for many systems can be designed independently under a mild assumption. Simulation results for three examples are shown in Section VI. Finally, some conclusions are made in Section VII.
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where denotes the desired trajectory or reference signal. In order to convert the output tracking problem into a stabilization problem, we introduce a set of virtual desired variables which will be tracked by the state variable . According , it is natural to require . Let to denote the tracking error for the state variyields ables. The time derivative of
(3) If the control input equation:
is assumed to satisfy the following
II. OUTPUT TRACKING CONTROL In this section, we give the basic architecture for output tracking control, whereas all states are temporarily assumed to be measurable. Consider a general nonlinear dynamic equation
(4) is a new control to be designed, then the tracking where error system (3) results in the following form: (5)
(1) where is the state vector; are the measured output and controlled output (variables), respectively; is the control input vector; and are nonlinear functions with appropriate dimensions. The measured output and controlled output may often be the same, but not always (for instance, see Example 3 described later). The nonlinear system (1) can be expressed by the fuzzy system Rule IF
is
and
and
is
The feasibility of (4) will be investigated in the next section. For the error system (5), we can find that the design for the new is similar to solve a stabilization problem. Our concontrol to zero, which means that the state trol purpose is to steer tracks . The new fuzzy controller is designed based on parallel distributed compensation (PDC) and represented as follows: Controller Rule IF is
Then
where are the premise variables which would are the consist of the states of the system; and are fuzzy sets; is the number of fuzzy rules; and system matrices with appropriate dimensions. For simplicity, we assume that the membership functions have been normalized, . i.e., Using singleton fuzzifier, product inferred, and weighted defuzzifier, the fuzzy system is inferred as follows:
and
and
is
THEN
where represent feedback gains. The inferred output of the PDC controller is with the following form:
(6) Substituting (6) into (5), we obtain the closed-loop system (7)
(2) where . Note that for all , where , for , are regarded as the grade functions. For output tracking control, the control objective is required to satisfy as
The exponential stability for system (7) is addressed later. and Theorem 1: Suppose that the virtual desired variable are bounded. The augmented error system (7) its derivative is exponentially stable if there exist a common positive–definite , and symmetric positive–definite matrices matrix such that D and (8) (9)
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.. .
.. .
..
.
.. .
(10)
, and . where Proof: Choose the Lyapunov function candidate as . Taking the derivative of with respect to , we have
.. .
.. .
..
.
(12)
.. .
where
Therefore, the LMIs, once solved, yields and . Then, the controller gains are obtained from . Remark 2: The stability conditions (8)–(10) for output tracking are the same as the stabilization problem. This means in (6) can be obtained by directly that the feedback gains solving stabilization problem. The main difference of the control law (6) for output tracking from stabilization problem , which appears in and in (4). In what comes from is to be discussed. follows, the virtual desired variables
According to (8) and (9), it follows that
III. CONSTRAINT OF GENERALIZED KINEMATICS The remaining design for the output tracking is to determine and then obtain the practical controller input . To this end, we use the fact and rewrite (4) as the following compact form: (13) where . Therefore, once the inequality , we (10) is satisfied and using the fact that have
. From (13), the exwhere depends on the form of . istence of the control input Here, the input matrix is assumed with full-column rank and after rearranging the coordinate frame in (1) if necessary, is with the following form:
where
where is a zero matrix and is nonsingular. Similarly, we partition and lowing form:
matrix
, and denote the minimal and maximal eigenvalue of , respectively. Hence, we obtain
Therefore, is concluded. Remark 1: The derivation of Theorem 1 is inspired by [15]. We, however, introduce the matrix to enhance the decay rate of the tracking error. In this form, the flexibility of assigning the decay rate is better than the method in [11]. The tracking response can be significantly improved by carefully choosing . Expanding the compact expression and taking the congruence transformation, we can convert the feasibility problem of , and from the (8)–(10) into finding following LMIs: (11)
into fol-
Consequently, the condition (13) is with the following form: (14) As a result, the virtual desired variables are determined according to the following constraints: (15) (16) where (15) rises from the output equation and (16) is due to (14). Also from (14), the practical control input
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
Fig. 1. Mass-spring mechanical system.
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Fig. 2. Equivalent circuit of a buck converter.
Formanyphysicalsystems, the virtualdesiredvariables tion and the generalized kinematics for yield can be exactly determined by the equations of (15) and (16). A good example is the class of strick feedback sys, there tems. It is also remarked here that even for a singular (18) still has some alternative methods to deal with the difficult situations. However, the derivation will become lengthy. Hence, we can easily obtain Condition (16) implies that must satisfy some type of con. straint if their associated states are not driven directly by control Example 2: Regulation of a PWM Buck Converter: Cominput. For mechanical systems, the constraint means that the vir- pared to the linearity in control input for Example 1, the dytual desired variables need to be governed by their kinematics. namics of the buck converter is with nonlinear input vector. The For this reason, we will call (16) as the constraint of generalized equivalent circuit of the conventional buck converter is illuskinematics. In the following, we consider three different type of trated in Fig. 2. Each period of the PWM buck converter consists physical systems and show the method to determine . of two stages. First, inductor is charged from the source while Example 1: Tracking of a Mass-Spring Mechanical System: the MOSFET is turned on. Second, when the power MOSFET We first consider the mass-spring mechanical system sketched is turned off, the inductor is discharged via the load. According in Fig. 1. Let denote the displacement from a reference point. to averaging method for one-time-scale discontinuous system According to Newton’s law, the equation of motion is methodology [16], we obtain the dynamic equations of the buck converter as follows:
where denotes the viscous damping force; is the restoring is the mass; and is external control force of the spring; input. The linear damping force is with and the with constants hardening spring force is and . As a result, we have the dynamic equation . Let us define . Then, we can represent the nonlinear system as the state–space form
where is the static drain to source resistance of the power is the forward voltage of the power diode; and MOSFET; is duty ratio of PWM buck converter. Let us define , where and denote the inductor current and and the voltage of the capacitor. After letting , we can represent the nonlinear system as follows:
(17) It is nature to choose as the premise variable, and the corresponding membership functions are and , where and are the upper bound and lower bound of , respectively, i.e., . The subsystem matrices are given by (19) as the premise variable, and the It is natural to choose corresponding membership functions are and , where we assume . The subsystem matrices for fuzzy rules are given by For (17), constraint (13) becomes
where is the desired velocity and is the desired position. According to (15) and (16), the constraints of the output equa-
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For (19), constraint (13) becomes
. The generated torque
is represented
by
where where is the desired output voltage. For the purpose of regis a conulating output voltage, the output (15) implies that stant and . According to (16), the constraint of the generalized kinematics leads to (20) Hence, we can easily obtain . Example 3: Torque Control for an Induction Motor: As a multiple-input–multiple-output (MIMO) nonlinear system, we consider the torque control problem for an induction motor. Based on the stationary frame, the model of three-phase induction motor is represented by a fourth-order model [17]
and For torque tracking, the control objective is required to satisfy as for a desired torque while the rotor flux is expected to be with constant magnitude. The constant magnitude of results in the optimal generated torque. In light of this, the controlled output equations are set to be (22) From (21), it is natural to choose as the premise variable. The corresponding membership functions are and , where we assume . Consequently, the subsystem matrices are given as shown in the equation at the bottom of the page. For (21), constraint (13) becomes
(21) , and denote the compowhere nents of the stator currents, rotor fluxes, stator voltages and , and rotor speed, respectively. The parameters are the stator resistance, rotor resistance, stator inductance, rotor inductance and mutual inductance, respectively,
where the virtual desired variables consist of the desired stator current and the desired rotor flux . The constraints of output
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
equations and generalized kinematics are expressed as follows: (23) (24)
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tunately, we find that for the membership functions satisfying Lipschitz-like condition, there is a totally different result: the separation principle holds and the tracking error tends to zero. In the fuzzy observer design, the estimation error is required to satisfy
(25) as
(26) for a constant . Hence, form (24), variable
for some
to be defined later. It follows that
(27)
where denotes the estimation of the state . To this end, we express the nonlinear system (1) with the measured output equation as the following fuzzy system: Rule IF
is
and
and
is
Then
Substituting (27) into (25) and (26), we obtain
(28) After premultiplying
on both sides of (28), it leads to
where . In light of this, the IF–THEN rules of a generic observer can be rewritten as Observer Rule IF is
and
and
is
THEN
where tively; and
denote the estimations of , respecare observer gains to be design later; is the premise variable vector depending . Therefore, the inferred output of observer system is as on follows:
which yields
Substituting (29) into (28), the desired stator current is simplified to (31)
(30) Finally, the practical control input is
where
and are fuzzy sets for . The observer-based control rule can be represented as follows: Controller Rule IF is
and
and
is
THEN
IV. OBSERVER-BASED OUTPUT TRACKING CONTROL In the previous section, we assume that all the states are available. This assumption restricts the applicability on practical situations. In this section, all the states including premise variables are immeasurable. Hence, we need to design the fuzzy observer to construct internal states for the purpose of state feedback. For a general situation, the immeasurable premise variables of fuzzy rules make the estimation error dynamics coupled by the tracking error dynamics. This fact leads to the fail of designing control gains and observer gains separately. Moreover, the residue tracking error and estimation error occur. For-
Then the inferred output of the PDC controller can be expressed as
(32) Compared to (13), the constraint for
becomes
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which results in the following generalized kinematics and practical control input, respectively
Combining (34) and (36), an augmented error system is obtained as (38)
where (33) is assumed with bounded norm. After substiwhere tuting the control input into (3), the tracking error system can be written as Further analysis on the error system (38) reveals that the . residue tracking error occurs due to the existence of can The common methods used to analyze the effect of refer the works of [5], [7], where the bounded tracking error -like performance are concluded. We also note that in or these methods, the control gains and observer gains cannot be designed separately and often suffer the problem of solving a set of nonlinear matrix inequalities. V. DESIGN BASED ON SEPARATION PRINCIPLE
(34)
For the error system (38), the conclusions we can arrive at of fuzzy is closely related to whether the perturbation term rules vanishes at . The observer-based control design yields a vanishing perturbation if the membership functions satisfy a Lipschitz-like condition. Assumption 1: The membership function satisfies the following condition:
where
(35) Here, we define the state estimation error as . Taking the derivative of , we have
for some bounded function vector . We remark here that this assumption is valid from investigation on many physical systems, including the three examples considered in this paper. It is also noted that the premise variables are often simple function of . Based on the assumption, we have the following property for grade functions. Property 1: The grade function error is proportional to the estimation error, i.e.,
for some bounded function vector . Proof: Since the grade function , it follows that (36) where
(37)
is with the form
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
By Assumption 1, it follows that
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feasible:
(41) (42) (43)
where . According to (37), the disturbance tion vector
can be rewritten as for some func-
. for all Proof: Choose the Lyapunov function candidate as , where is chosen in the following , where . Taking the time form: , we have derivative of
. By Property 1, it follows that
(39) Inspired by the form (39), we suppose that the following assumption holds. Assumption 2: The perturbation satisfies the following bounding fashion: (40)
(44) Here
. with a constant matrix On the other hand, from Property 1 and the facts
where Property 2 has been applied. From (40), it follows that and, hence
it follows that both and are proportional to the estimation error. In view of defined in (35), , we obtain the following since property. and are bounded Property 2: Suppose that and Assumption 1 succeeds, then the disturbance can be rewritten as follows:
for some bounded function matrix . and observer Based on these properties, the control gains gains can be separately designed according to the following theorem. and Theorem 2: Suppose that the virtual desired variable are bounded and Assumption 1 and 2 sucits derivative ceed, then the augmented error system (38) is exponentially stable if there exist common symmetric positive–definite maand such that the following LMIs are trices
where the last equality succeeds by (43). Therefore, the inequality for can be expressed as follows:
(45) where
and
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Using Schur’s complement, the first matrix is negative definite if and only if the following inequalities hold:
By letting small enough [13], the off-diagonal blocks of the first matrix in (45) become negligible. It follows that
if LMIs (41) and (42) are feasible. Here, notice that LMI (41) and letting is obtained by taking Schur’s complement on . LMI (42) is obtained by letting . Hence, the exponentially stability for the augmented error system is proven. Remark 3: In solving LMIs (41)–(43), the positive–definite and are given to prescribe the decay rates of matrixes tracking error and observation error, respectively. in the proof is Remark 4: The subtle treatment for to get a stronger result. Otherwise, the LMIs conditions will become restrictive. and observer gains can Accordingly, the control gains be obtained by solving (41) and (42) separately. The bounding can be regarded as how large the deviation of matrix the controller can tolerate. The LMI condition (41)–(42) can be relaxed if the premise variables are measurable. In this case, the and, hence, in (38). grade functions Theorem 3: Suppose that the virtual desired variable and its derivative are bounded and the premise variables are measurable. The augmented error system (38) is exponentially stable if there exist common symmetric positive–definite such that matrices
(46) (47)
VI. NUMERICAL SIMULATIONS To verify the theoretical derivation, the numerical simulations for Examples 1–3 are carried out. Example 1: Tracking of a Mass-Spring Mechanical System (Cont’d): In this problem, the membership functions and satisfy the Lipschitz-like condition. But, in fact the premise variable is the measured output and controlled output as well. Hence, Theorem 3 is applicable. The parameters are set as and the desired output is . The observer-based controller (33) is given as follows:
where according to (18). From (46) and observer and (47), we obtain the controller gains gains
In
solving
the
LMIs,
we
set
the matrices . In simulation, , the initial estimate state the initial states . The responses of , and and , and the control input are shown in Fig. 3. We remark here that the estimation error converges to zero rapidly and is not shown. Example 2: Regulation of a PWM Buck Converter (Cont’d): In this problem, the membership functions and , where is immeasurable, satisfy the Lipschitz-like condition. Hence, Theorem 2 is applied. mH, mF, The parameters are with V, and V. The controlled output (or the measured output) is to be regulated . The observer-based control law (33) is given by at
for all . and Remark 5: Comparing (7) and (38) (now, ), we can derive similar conditions such as (8)–(10) where according to (20). By (41)–(43), the control gains and observer gains obtained via LMI toolbox are given as follows:
.. .
.. .
..
.
.. .
where . However, we notice that these inequalities cannot be transformed to LMIs.
In solving the LMIs, we set the bounding matrices to be . and the In simulation, the initial states . The responses of initial estimate state
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
Fig. 3. Responses of x
;x
;x ;x
Fig. 4. Responses of x
;x
;x ;x ;x
^
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and control input u for mass-spring system.
and control input u for mass-spring system.
and , and , and the controller input are given in Fig. 4. Here the estimation of immeasurable state is also shown, which converges to rapidly. Example 3: Torque Control for an Induction Motor (Cont’d): For convenience to demonstrate the approach, a technique of
simplification is taken here. The rotor speed available time-varying signal driven by
is regarded as an
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Fig. 5. Responses of i
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 5, OCTOBER 2006
; i ; ; ; T , and T
for induction motor.
where is the moment of inertial and is a damping coefficient. Therefore, Theorem 3 is applicable. Notice that the associated membership functions also satisfy Lipschiz-like condition if is regarded as a state variable. The parameters are with . In this problem, the controlled outputs have been . given in (22), whereas the measured outputs are The desired torque and the derived magnitude of rotor flux are and , respectively. The obgiven by server-based control law (33) is given in (48), as shown at the bottom of the page, where the virtual desired variables derived in Section III are summarized as follows:
According to (46) and (47), the control gains and the observer gains obtained via LMI toolbox are given as follows:
In simulation, the initial states and the initial estimated state . The responses and , and , torque and desired of state are shown in Fig. 5. Both the torque tracking and the torque regulation of rotor flux are achieved. VII. CONCLUSION An observer-based fuzzy controller has been proposed to achieve output tracking. To this end, the new concepts, namely virtual desired variables and generalized kinematic constraint are introduced to benefit the control design. Under Lipschitz-like property, a common feature for many physical systems, exponentially output tracking for nonlinear systems has been concluded. Especially, the control gains and observer gains are designed separately and are obtained by solving LMIs. The proposed scheme is applicable for many physical systems. In this works, three different types of systems, including a nonlinear mass-spring system, a dc–dc converter, and an induction motor have been investigated in detail. The
(48)
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
satisfactory result of numerical simulations further reveals the validity of the proposed scheme. REFERENCES [1] H.K. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [2] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan. 1985. [3] H. O. Wang, K. Tanaka, and M. F. Griffin, “Parallel distributed compensation of nonlinear systems by Takagi-Sugeno fuzzy model,” in Proc. FUZZY-IEEE/IFES, 1995, vol. 2, pp. 531–538. [4] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [5] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. New York: Wiley, 2001. [6] C. S. Tseng and B. S. Chen, “Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 381–392, Jun. 2001. / fuzzy output [7] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed feedback control design for nonlinear dynamic systems: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 249–265, Jun. 2000. [8] B. S. Chen, C. S. Tseng, and H. J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, no. 5, pp. 571–585, Oct. 1999. [9] J. Joh, Y.-H Chen, and R. Langari, “On the stability issues of linear Takagi–Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 6, no. 3, pp. 402–410, Aug. 1998. [10] Z. X. Han, G. Feng, B. L. Walcott, and J. Ma, “Dynamic output feedback controller design for fuzzy system,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 30, no. 1, pp. 204–210, Feb. 2000. [11] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observer: Relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 250–265, May 2000. [12] K.-Y. Lian, C.-S. Chiu, and P. Liu, “LMI-based fuzzy chaotic synchronization and communications,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 539–553, Aug. 2001. [13] S. S. Farinwata, D. Filev, and R. Langari, Fuzzy Control: Synthesis and Analysis. New York: Wiley, 2000, pp. 246–247.
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[14] A. Jadbabaie, A. Titli, and M. Jamshidi, “Fuzzy observer-based control of nonlinear systems,” in Proc. 36th IEEE Conf. Decision and Control, 1997, vol. 4, pp. 3347–3349. [15] E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 523–533, Oct. 2000. [16] J. Sun and H. Grotstollen, “Averaged modelling of switching power converters: Reformulation and theoretical basis,” in Proc. IEEE PESC’92, 1992, pp. 1165–1172. [17] P. C. Krause, Analysis of Electric Machinery. New York: McGrawHill, 1986. Kuang-Yow Lian (S’91–M’94) was born in Taiwan in 1961. He received the B.S. degree in engineering science from National Cheng-Kung University, Taiwan, in 1984, and the Ph.D. degree in electrical engineering from National Taiwan University, Taiwan, in 1993. From 1986 to 1988, he served as a Control Engineer at ITRI, Taiwan. He joined Chung-Yuan Christian University, Taiwan, in 1994, where he is currently a Professor and Chair of the Department of Electrical Engineering. His research interests include fuzzy control, nonlinear control systems, stability analysis, chaotic systems, and control applications. Dr. Lian was the recipient of the Outstanding Research Award of Chung-Yuan Christian University.
Jeih-Jang Liou was born in I-Lan, Taiwan. He received the B.S. degree from the National Taiwan Ocean University, in 1995, and M.S. degree from the National Taiwan Institute of Technology, in 1997. He is currently working toward the Ph.D. degree in electrical engineering at Chung-Yuan Christian University. He is a Lecturer in Department of Automatic Engineering in the Lan-Yang Institute of Technology, Taiwan. His areas of research interests include nonlinear control, fuzzy theory, and control applications.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 5, OCTOBER 2006
Output Tracking Control for Fuzzy Systems Via Output Feedback Design Kuang-Yow Lian, Member, IEEE, and Jeih-Jang Liou
Abstract—Fuzzy observer-based control design is proposed to deal with the output tracking problem for nonlinear systems. For the purpose of tracking design, the new concept of virtual desired variables and, in turn the so-called generalized kinematics are introduced to simplify the design procedure. In light of this concept, the design procedure is split into two steps: i) Determine the virtual desired variables from the generalized kinematics; and ii) Determine the control gains just like solving linear matrix inequalities for stabilization problem. For immeasurable state variables, output feedback design is proposed. Here, we focus on a common feature held by many physical systems where their membership functions of fuzzy sets satisfy a Lipschitz-like property. Based on this setting, control gains and observer gains can be designed separately. Moreover, zero tracking error and estimation error are concluded. Three different types of systems, including nonlinear mass-spring systems, dc–dc converters, and induction motors are considered to demonstrate the design procedure. Their satisfactory simulation results verify the proposed approach. Index Terms—Control application, linear matrix inequality (LMI) approach, observer design, output tracking, Takagi–Sugeno (T–S) fuzzy systems.
I. INTRODUCTION An automatic controller is generally regarded as the device that can correct system behavior to conform to some preset desired values by using measurable information. According to the desired values, the tasks of control systems can be divided into three categories [1]: The stabilization problem, the regulation problem, and the tracking problem. In general, the tracking problem is the most difficult. Recently, the Takagi–Sugeno (T–S) fuzzy approach [2] has been extensively used to model nonlinear systems. The basic idea for the approach is to decompose the model of a nonlinear system into a set of linear subsystems with associated nonlinear weighting functions. Then by using well-formulated linear control theory, the control objective is achieved [3]. The stability analysis is carried out using Lyapunov direct method where the control problem is then formulated into solving linear matrix inequalities (LMIs) [4]. Then using powerful computational toolboxes, such as Matlab LMI Toolbox, we obtain the controller gains. Previous works have provided fruitful LMI-based approaches to control of dynamic systems [5]. In the pioneering works, the stabilization problem for T–S fuzzy models has been well practiced. Manuscript received April 29, 2004; revised December 23, 2004 and November 1, 2005. This work was supported by the National Science Council, R.O.C., under Grant NSC 93-2213-E-033-008. The authors are with the Department of Electrical Engineering, Chung-Yuan Christian University, Chung-Li 32023, Taiwan (e-mail: [email protected]. tw). Digital Object Identifier 10.1109/TFUZZ.2006.876725
The focus of tracking control, using the T–S fuzzy model based approach, has been limited to nonlinear model following, where all states are assumed known [6]; or output feedback tracking for a linear reference model, where the reference input is considered as disturbance and is attenuated using a robust criterion [7], [8]. For these situations, it often results in bounded tracking errors between the system states and reference states. We also note that the model following approach is not easily used to deal with some control tasks where only some outputs need to be controlled and other internal state behaviors are hard to be described. Hence, it is hard to define the reference model in this case. In light of this, we focus on the tasks of output tracking control in this work. The T–S fuzzy model-based control provides a systematic framework for the design of a state feedback controller [9]–[11]. An implicit assumption that the states are available for measurement is often made. However, we know that this is impossible in many practical physical systems. Measuring full states is difficult and costly. In traditional fuzzy observer design, the premise variables of fuzzy rules are assumed to be measurable. This is still a strict constraint and limits the application of such observer design [11], [12]. For some dynamic systems, it is inevitable to use immeasurable states as the premise variables. Some approaches are usually utilized to deal with the non-trivial problem of simultaneously solving the gains for both the controller and observer [13]. In this paper, we will discuss the output tracking control based on output feedback design. All the internal states including premise variables are immeasurable and need to be estimated by fuzzy observer. To this end, the new concepts, virtual desired variables and, in turn generalized kinematic constraint are introduced to make the design procedure become clear. Then the design procedure is split into two independent steps: i) Determine the virtual desired variables from the desired output equation and the generalized kinematic constraint; and ii) determine the control feedback gains and observer gains by solving a set of LMIs, the same type LMIs for stabilization problem. A common feature for many physical systems is specifically considered: The membership functions of fuzzy sets often satisfy a Lipschitz-like property. Based on this observation, the control gains and observer gains can be designed separately, i.e., the separation principle holds. Moreover, it is concluded that the output tracking error and estimation error converge to zero exponentially. The proposed approach has been found in widespread applications. To show this, three different types of systems and tasks are considered: i) Output tracking of a nonlinear mass-spring system: A mechanical system with state-dependent system matrix; ii) Output
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LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
regulation of a dc–dc converter: An electronic system with state-dependent input matrix and an immeasurable premise variable; and iii) Torque tracking of an induction motor: An electrical system with multiple inputs and multiple outputs, where the output equations are nonlinear functions of state variables. The control tasks for all the systems can be well done. The rest of this paper is organized as follows. In Section II, the basic concept for designing output tracking control is addressed. Then, how to determine the virtual desired variables is demonstrated in Section III. In Section IV, the fuzzy observer-based output tracking design is presented. In Section V, we show that the control gains and observer gains for many systems can be designed independently under a mild assumption. Simulation results for three examples are shown in Section VI. Finally, some conclusions are made in Section VII.
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where denotes the desired trajectory or reference signal. In order to convert the output tracking problem into a stabilization problem, we introduce a set of virtual desired variables which will be tracked by the state variable . According , it is natural to require . Let to denote the tracking error for the state variyields ables. The time derivative of
(3) If the control input equation:
is assumed to satisfy the following
II. OUTPUT TRACKING CONTROL In this section, we give the basic architecture for output tracking control, whereas all states are temporarily assumed to be measurable. Consider a general nonlinear dynamic equation
(4) is a new control to be designed, then the tracking where error system (3) results in the following form: (5)
(1) where is the state vector; are the measured output and controlled output (variables), respectively; is the control input vector; and are nonlinear functions with appropriate dimensions. The measured output and controlled output may often be the same, but not always (for instance, see Example 3 described later). The nonlinear system (1) can be expressed by the fuzzy system Rule IF
is
and
and
is
The feasibility of (4) will be investigated in the next section. For the error system (5), we can find that the design for the new is similar to solve a stabilization problem. Our concontrol to zero, which means that the state trol purpose is to steer tracks . The new fuzzy controller is designed based on parallel distributed compensation (PDC) and represented as follows: Controller Rule IF is
Then
where are the premise variables which would are the consist of the states of the system; and are fuzzy sets; is the number of fuzzy rules; and system matrices with appropriate dimensions. For simplicity, we assume that the membership functions have been normalized, . i.e., Using singleton fuzzifier, product inferred, and weighted defuzzifier, the fuzzy system is inferred as follows:
and
and
is
THEN
where represent feedback gains. The inferred output of the PDC controller is with the following form:
(6) Substituting (6) into (5), we obtain the closed-loop system (7)
(2) where . Note that for all , where , for , are regarded as the grade functions. For output tracking control, the control objective is required to satisfy as
The exponential stability for system (7) is addressed later. and Theorem 1: Suppose that the virtual desired variable are bounded. The augmented error system (7) its derivative is exponentially stable if there exist a common positive–definite , and symmetric positive–definite matrices matrix such that D and (8) (9)
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.. .
.. .
..
.
.. .
(10)
, and . where Proof: Choose the Lyapunov function candidate as . Taking the derivative of with respect to , we have
.. .
.. .
..
.
(12)
.. .
where
Therefore, the LMIs, once solved, yields and . Then, the controller gains are obtained from . Remark 2: The stability conditions (8)–(10) for output tracking are the same as the stabilization problem. This means in (6) can be obtained by directly that the feedback gains solving stabilization problem. The main difference of the control law (6) for output tracking from stabilization problem , which appears in and in (4). In what comes from is to be discussed. follows, the virtual desired variables
According to (8) and (9), it follows that
III. CONSTRAINT OF GENERALIZED KINEMATICS The remaining design for the output tracking is to determine and then obtain the practical controller input . To this end, we use the fact and rewrite (4) as the following compact form: (13) where . Therefore, once the inequality , we (10) is satisfied and using the fact that have
. From (13), the exwhere depends on the form of . istence of the control input Here, the input matrix is assumed with full-column rank and after rearranging the coordinate frame in (1) if necessary, is with the following form:
where
where is a zero matrix and is nonsingular. Similarly, we partition and lowing form:
matrix
, and denote the minimal and maximal eigenvalue of , respectively. Hence, we obtain
Therefore, is concluded. Remark 1: The derivation of Theorem 1 is inspired by [15]. We, however, introduce the matrix to enhance the decay rate of the tracking error. In this form, the flexibility of assigning the decay rate is better than the method in [11]. The tracking response can be significantly improved by carefully choosing . Expanding the compact expression and taking the congruence transformation, we can convert the feasibility problem of , and from the (8)–(10) into finding following LMIs: (11)
into fol-
Consequently, the condition (13) is with the following form: (14) As a result, the virtual desired variables are determined according to the following constraints: (15) (16) where (15) rises from the output equation and (16) is due to (14). Also from (14), the practical control input
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
Fig. 1. Mass-spring mechanical system.
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Fig. 2. Equivalent circuit of a buck converter.
Formanyphysicalsystems, the virtualdesiredvariables tion and the generalized kinematics for yield can be exactly determined by the equations of (15) and (16). A good example is the class of strick feedback sys, there tems. It is also remarked here that even for a singular (18) still has some alternative methods to deal with the difficult situations. However, the derivation will become lengthy. Hence, we can easily obtain Condition (16) implies that must satisfy some type of con. straint if their associated states are not driven directly by control Example 2: Regulation of a PWM Buck Converter: Cominput. For mechanical systems, the constraint means that the vir- pared to the linearity in control input for Example 1, the dytual desired variables need to be governed by their kinematics. namics of the buck converter is with nonlinear input vector. The For this reason, we will call (16) as the constraint of generalized equivalent circuit of the conventional buck converter is illuskinematics. In the following, we consider three different type of trated in Fig. 2. Each period of the PWM buck converter consists physical systems and show the method to determine . of two stages. First, inductor is charged from the source while Example 1: Tracking of a Mass-Spring Mechanical System: the MOSFET is turned on. Second, when the power MOSFET We first consider the mass-spring mechanical system sketched is turned off, the inductor is discharged via the load. According in Fig. 1. Let denote the displacement from a reference point. to averaging method for one-time-scale discontinuous system According to Newton’s law, the equation of motion is methodology [16], we obtain the dynamic equations of the buck converter as follows:
where denotes the viscous damping force; is the restoring is the mass; and is external control force of the spring; input. The linear damping force is with and the with constants hardening spring force is and . As a result, we have the dynamic equation . Let us define . Then, we can represent the nonlinear system as the state–space form
where is the static drain to source resistance of the power is the forward voltage of the power diode; and MOSFET; is duty ratio of PWM buck converter. Let us define , where and denote the inductor current and and the voltage of the capacitor. After letting , we can represent the nonlinear system as follows:
(17) It is nature to choose as the premise variable, and the corresponding membership functions are and , where and are the upper bound and lower bound of , respectively, i.e., . The subsystem matrices are given by (19) as the premise variable, and the It is natural to choose corresponding membership functions are and , where we assume . The subsystem matrices for fuzzy rules are given by For (17), constraint (13) becomes
where is the desired velocity and is the desired position. According to (15) and (16), the constraints of the output equa-
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For (19), constraint (13) becomes
. The generated torque
is represented
by
where where is the desired output voltage. For the purpose of regis a conulating output voltage, the output (15) implies that stant and . According to (16), the constraint of the generalized kinematics leads to (20) Hence, we can easily obtain . Example 3: Torque Control for an Induction Motor: As a multiple-input–multiple-output (MIMO) nonlinear system, we consider the torque control problem for an induction motor. Based on the stationary frame, the model of three-phase induction motor is represented by a fourth-order model [17]
and For torque tracking, the control objective is required to satisfy as for a desired torque while the rotor flux is expected to be with constant magnitude. The constant magnitude of results in the optimal generated torque. In light of this, the controlled output equations are set to be (22) From (21), it is natural to choose as the premise variable. The corresponding membership functions are and , where we assume . Consequently, the subsystem matrices are given as shown in the equation at the bottom of the page. For (21), constraint (13) becomes
(21) , and denote the compowhere nents of the stator currents, rotor fluxes, stator voltages and , and rotor speed, respectively. The parameters are the stator resistance, rotor resistance, stator inductance, rotor inductance and mutual inductance, respectively,
where the virtual desired variables consist of the desired stator current and the desired rotor flux . The constraints of output
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
equations and generalized kinematics are expressed as follows: (23) (24)
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tunately, we find that for the membership functions satisfying Lipschitz-like condition, there is a totally different result: the separation principle holds and the tracking error tends to zero. In the fuzzy observer design, the estimation error is required to satisfy
(25) as
(26) for a constant . Hence, form (24), variable
for some
to be defined later. It follows that
(27)
where denotes the estimation of the state . To this end, we express the nonlinear system (1) with the measured output equation as the following fuzzy system: Rule IF
is
and
and
is
Then
Substituting (27) into (25) and (26), we obtain
(28) After premultiplying
on both sides of (28), it leads to
where . In light of this, the IF–THEN rules of a generic observer can be rewritten as Observer Rule IF is
and
and
is
THEN
where tively; and
denote the estimations of , respecare observer gains to be design later; is the premise variable vector depending . Therefore, the inferred output of observer system is as on follows:
which yields
Substituting (29) into (28), the desired stator current is simplified to (31)
(30) Finally, the practical control input is
where
and are fuzzy sets for . The observer-based control rule can be represented as follows: Controller Rule IF is
and
and
is
THEN
IV. OBSERVER-BASED OUTPUT TRACKING CONTROL In the previous section, we assume that all the states are available. This assumption restricts the applicability on practical situations. In this section, all the states including premise variables are immeasurable. Hence, we need to design the fuzzy observer to construct internal states for the purpose of state feedback. For a general situation, the immeasurable premise variables of fuzzy rules make the estimation error dynamics coupled by the tracking error dynamics. This fact leads to the fail of designing control gains and observer gains separately. Moreover, the residue tracking error and estimation error occur. For-
Then the inferred output of the PDC controller can be expressed as
(32) Compared to (13), the constraint for
becomes
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which results in the following generalized kinematics and practical control input, respectively
Combining (34) and (36), an augmented error system is obtained as (38)
where (33) is assumed with bounded norm. After substiwhere tuting the control input into (3), the tracking error system can be written as Further analysis on the error system (38) reveals that the . residue tracking error occurs due to the existence of can The common methods used to analyze the effect of refer the works of [5], [7], where the bounded tracking error -like performance are concluded. We also note that in or these methods, the control gains and observer gains cannot be designed separately and often suffer the problem of solving a set of nonlinear matrix inequalities. V. DESIGN BASED ON SEPARATION PRINCIPLE
(34)
For the error system (38), the conclusions we can arrive at of fuzzy is closely related to whether the perturbation term rules vanishes at . The observer-based control design yields a vanishing perturbation if the membership functions satisfy a Lipschitz-like condition. Assumption 1: The membership function satisfies the following condition:
where
(35) Here, we define the state estimation error as . Taking the derivative of , we have
for some bounded function vector . We remark here that this assumption is valid from investigation on many physical systems, including the three examples considered in this paper. It is also noted that the premise variables are often simple function of . Based on the assumption, we have the following property for grade functions. Property 1: The grade function error is proportional to the estimation error, i.e.,
for some bounded function vector . Proof: Since the grade function , it follows that (36) where
(37)
is with the form
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
By Assumption 1, it follows that
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feasible:
(41) (42) (43)
where . According to (37), the disturbance tion vector
can be rewritten as for some func-
. for all Proof: Choose the Lyapunov function candidate as , where is chosen in the following , where . Taking the time form: , we have derivative of
. By Property 1, it follows that
(39) Inspired by the form (39), we suppose that the following assumption holds. Assumption 2: The perturbation satisfies the following bounding fashion: (40)
(44) Here
. with a constant matrix On the other hand, from Property 1 and the facts
where Property 2 has been applied. From (40), it follows that and, hence
it follows that both and are proportional to the estimation error. In view of defined in (35), , we obtain the following since property. and are bounded Property 2: Suppose that and Assumption 1 succeeds, then the disturbance can be rewritten as follows:
for some bounded function matrix . and observer Based on these properties, the control gains gains can be separately designed according to the following theorem. and Theorem 2: Suppose that the virtual desired variable are bounded and Assumption 1 and 2 sucits derivative ceed, then the augmented error system (38) is exponentially stable if there exist common symmetric positive–definite maand such that the following LMIs are trices
where the last equality succeeds by (43). Therefore, the inequality for can be expressed as follows:
(45) where
and
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Using Schur’s complement, the first matrix is negative definite if and only if the following inequalities hold:
By letting small enough [13], the off-diagonal blocks of the first matrix in (45) become negligible. It follows that
if LMIs (41) and (42) are feasible. Here, notice that LMI (41) and letting is obtained by taking Schur’s complement on . LMI (42) is obtained by letting . Hence, the exponentially stability for the augmented error system is proven. Remark 3: In solving LMIs (41)–(43), the positive–definite and are given to prescribe the decay rates of matrixes tracking error and observation error, respectively. in the proof is Remark 4: The subtle treatment for to get a stronger result. Otherwise, the LMIs conditions will become restrictive. and observer gains can Accordingly, the control gains be obtained by solving (41) and (42) separately. The bounding can be regarded as how large the deviation of matrix the controller can tolerate. The LMI condition (41)–(42) can be relaxed if the premise variables are measurable. In this case, the and, hence, in (38). grade functions Theorem 3: Suppose that the virtual desired variable and its derivative are bounded and the premise variables are measurable. The augmented error system (38) is exponentially stable if there exist common symmetric positive–definite such that matrices
(46) (47)
VI. NUMERICAL SIMULATIONS To verify the theoretical derivation, the numerical simulations for Examples 1–3 are carried out. Example 1: Tracking of a Mass-Spring Mechanical System (Cont’d): In this problem, the membership functions and satisfy the Lipschitz-like condition. But, in fact the premise variable is the measured output and controlled output as well. Hence, Theorem 3 is applicable. The parameters are set as and the desired output is . The observer-based controller (33) is given as follows:
where according to (18). From (46) and observer and (47), we obtain the controller gains gains
In
solving
the
LMIs,
we
set
the matrices . In simulation, , the initial estimate state the initial states . The responses of , and and , and the control input are shown in Fig. 3. We remark here that the estimation error converges to zero rapidly and is not shown. Example 2: Regulation of a PWM Buck Converter (Cont’d): In this problem, the membership functions and , where is immeasurable, satisfy the Lipschitz-like condition. Hence, Theorem 2 is applied. mH, mF, The parameters are with V, and V. The controlled output (or the measured output) is to be regulated . The observer-based control law (33) is given by at
for all . and Remark 5: Comparing (7) and (38) (now, ), we can derive similar conditions such as (8)–(10) where according to (20). By (41)–(43), the control gains and observer gains obtained via LMI toolbox are given as follows:
.. .
.. .
..
.
.. .
where . However, we notice that these inequalities cannot be transformed to LMIs.
In solving the LMIs, we set the bounding matrices to be . and the In simulation, the initial states . The responses of initial estimate state
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
Fig. 3. Responses of x
;x
;x ;x
Fig. 4. Responses of x
;x
;x ;x ;x
^
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and control input u for mass-spring system.
and control input u for mass-spring system.
and , and , and the controller input are given in Fig. 4. Here the estimation of immeasurable state is also shown, which converges to rapidly. Example 3: Torque Control for an Induction Motor (Cont’d): For convenience to demonstrate the approach, a technique of
simplification is taken here. The rotor speed available time-varying signal driven by
is regarded as an
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Fig. 5. Responses of i
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; i ; ; ; T , and T
for induction motor.
where is the moment of inertial and is a damping coefficient. Therefore, Theorem 3 is applicable. Notice that the associated membership functions also satisfy Lipschiz-like condition if is regarded as a state variable. The parameters are with . In this problem, the controlled outputs have been . given in (22), whereas the measured outputs are The desired torque and the derived magnitude of rotor flux are and , respectively. The obgiven by server-based control law (33) is given in (48), as shown at the bottom of the page, where the virtual desired variables derived in Section III are summarized as follows:
According to (46) and (47), the control gains and the observer gains obtained via LMI toolbox are given as follows:
In simulation, the initial states and the initial estimated state . The responses and , and , torque and desired of state are shown in Fig. 5. Both the torque tracking and the torque regulation of rotor flux are achieved. VII. CONCLUSION An observer-based fuzzy controller has been proposed to achieve output tracking. To this end, the new concepts, namely virtual desired variables and generalized kinematic constraint are introduced to benefit the control design. Under Lipschitz-like property, a common feature for many physical systems, exponentially output tracking for nonlinear systems has been concluded. Especially, the control gains and observer gains are designed separately and are obtained by solving LMIs. The proposed scheme is applicable for many physical systems. In this works, three different types of systems, including a nonlinear mass-spring system, a dc–dc converter, and an induction motor have been investigated in detail. The
(48)
LIAN AND LIOU: OUTPUT TRACKING CONTROL FOR FUZZY SYSTEMS VIA OUTPUT FEEDBACK DESIGN
satisfactory result of numerical simulations further reveals the validity of the proposed scheme. REFERENCES [1] H.K. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [2] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116–132, Jan. 1985. [3] H. O. Wang, K. Tanaka, and M. F. Griffin, “Parallel distributed compensation of nonlinear systems by Takagi-Sugeno fuzzy model,” in Proc. FUZZY-IEEE/IFES, 1995, vol. 2, pp. 531–538. [4] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [5] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. New York: Wiley, 2001. [6] C. S. Tseng and B. S. Chen, “Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model,” IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 381–392, Jun. 2001. / fuzzy output [7] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed feedback control design for nonlinear dynamic systems: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 249–265, Jun. 2000. [8] B. S. Chen, C. S. Tseng, and H. J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, no. 5, pp. 571–585, Oct. 1999. [9] J. Joh, Y.-H Chen, and R. Langari, “On the stability issues of linear Takagi–Sugeno fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 6, no. 3, pp. 402–410, Aug. 1998. [10] Z. X. Han, G. Feng, B. L. Walcott, and J. Ma, “Dynamic output feedback controller design for fuzzy system,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 30, no. 1, pp. 204–210, Feb. 2000. [11] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observer: Relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 250–265, May 2000. [12] K.-Y. Lian, C.-S. Chiu, and P. Liu, “LMI-based fuzzy chaotic synchronization and communications,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 539–553, Aug. 2001. [13] S. S. Farinwata, D. Filev, and R. Langari, Fuzzy Control: Synthesis and Analysis. New York: Wiley, 2000, pp. 246–247.
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[14] A. Jadbabaie, A. Titli, and M. Jamshidi, “Fuzzy observer-based control of nonlinear systems,” in Proc. 36th IEEE Conf. Decision and Control, 1997, vol. 4, pp. 3347–3349. [15] E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 523–533, Oct. 2000. [16] J. Sun and H. Grotstollen, “Averaged modelling of switching power converters: Reformulation and theoretical basis,” in Proc. IEEE PESC’92, 1992, pp. 1165–1172. [17] P. C. Krause, Analysis of Electric Machinery. New York: McGrawHill, 1986. Kuang-Yow Lian (S’91–M’94) was born in Taiwan in 1961. He received the B.S. degree in engineering science from National Cheng-Kung University, Taiwan, in 1984, and the Ph.D. degree in electrical engineering from National Taiwan University, Taiwan, in 1993. From 1986 to 1988, he served as a Control Engineer at ITRI, Taiwan. He joined Chung-Yuan Christian University, Taiwan, in 1994, where he is currently a Professor and Chair of the Department of Electrical Engineering. His research interests include fuzzy control, nonlinear control systems, stability analysis, chaotic systems, and control applications. Dr. Lian was the recipient of the Outstanding Research Award of Chung-Yuan Christian University.
Jeih-Jang Liou was born in I-Lan, Taiwan. He received the B.S. degree from the National Taiwan Ocean University, in 1995, and M.S. degree from the National Taiwan Institute of Technology, in 1997. He is currently working toward the Ph.D. degree in electrical engineering at Chung-Yuan Christian University. He is a Lecturer in Department of Automatic Engineering in the Lan-Yang Institute of Technology, Taiwan. His areas of research interests include nonlinear control, fuzzy theory, and control applications.
