Mixed Feedforward/Feedback Based Adaptive Fuzzy ... - IEEE Xplore

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

Mixed Feedforward/Feedback Based Adaptive Fuzzy Control for a Class of MIMO Nonlinear Systems Chian-Song Chiu, Member, IEEE

Abstract—This paper proposes a mixed feedforward/feedback (FFB) based adaptive fuzzy controller design for a class of multiple-input–multiple-output (MIMO) uncertain nonlinear systems. By integrating both feedforward and feedback compensation, we introduce the FFB-based fuzzy controller composed of a feedforward fuzzy compensator and a robust error-feedback compensator. To achieve a forward compensation of uncertainties, the feedforward fuzzy compensator takes the desired commands as premise variables of fuzzy rules and adaptively adjusts the consequent part from an error measure. Meanwhile, the feedback control techniques controller part is constructed based on and nonlinear damping design. Then, the attenuation of both disturbances and estimated fuzzy parametric errors is guaranteed from a linear matrix inequality (LMI)-based gain design. The main advantages are: i) a simpler architecture for implementation is provided; and ii) the typical boundedness of assumption on fuzzy universal approximation errors is not required. Finally, an inverted pendulum system and a two-link robot are taken as application examples to show the expected performance. Index Terms— performance, adaptive fuzzy control, linear matrix inequality (LMI), multiple-input–multiple-output (MIMO).

I. INTRODUCTION ROM THE pioneering works of Wang and Mendel [1], [2], fuzzy logic control methods have been applied to many control problems of complex or poorly modeled systems. For example, adaptive fuzzy schemes [3],[4] were proposed for poorly modeled cases of single-input–single-output (SISO) nonlinear systems. As an extension of SISO cases, fuzzy controllers with complicated adaptive (training) algorithms (e.g., [5], [6]) were applied to multiple-input–multiple-output (MIMO) nonlinear systems. From current literatures, most fuzzy controller design is based on the fuzzy universal approximation theorem [1], [2]. Indeed, the fuzzy system is taken as an approximator to achieve feedback cancellation such that stability is assured by Lyapunov’s method. We call these methods the feedback (FB) based fuzzy control, which is often constructed with the configuration shown in Fig. 1. Unfortunately, the complexity of FB-based fuzzy controllers increases with regard to dimensions of plants, such as large number of fuzzy rules. In light of this, the Takagi–Sugeno fuzzy control schemes [7]–[9] and observer-based fuzzy control schemes [10], [11] are developed for the reduction of fuzzy rules and feedback states respectively. However, the control problem

F

Manuscript received March 5, 2004; revised May 18, 2005 and January 24, 2006. This work was supported by the National Science Council, R.O.C., under Grant NSC-93-2213-E-270-001. The author is with the Department of Electronic Engineering, Chien-Kuo Technology University, Changhua 50050, Taiwan, R.O.C. (e-mail: cschiu@ctu. edu.tw). Digital Object Identifier 10.1109/TFUZZ.2006.877357

Fig. 1. Configuration of FB-based adaptive fuzzy controller.

of MIMO nonlinear systems is still an uneasy task due to an uncertain input matrix. In detail, the assumption of an invertible fuzzy approximated input matrix needs to be imposed, such as [11]–[13]. If not, more information or limits are required for the input matrix (e.g., [14]–[16]). For example, when the upper and lower bounds of each component of the uncertain input matrix are known, a projection algorithm is applied into all entries of the fuzzy approximated input matrix to obtain a well-defined control law [14]. Thus, few works have solved the tracking control of uncertain MIMO nonlinear systems. Since typical adaptive schemes cannot efficiently eliminate or attenuate both external disturbances and fuzzy approximation errors, various strategies including robust adaptive fuzzy control [3], [4], [17], [18] or supervisor-based fuzzy control [19], [20] have been presented in recent years. References [17] and [18], combining both typical control techniques and adaptive fuzzy control, achieve an attenuation performance of disturbances and approximation errors from solving a Riccati-like equation. In addition, the supervisor-based fuzzy controllers are designed based on sliding mode control methods to guarantee some robustness. Therefore, the common disadvantages existing in the previously cited FB-based fuzzy control schemes are: i) the bounded fuzzy approximation errors are assumed even if they depend on states; ii) the practical implementation is difficult due to a large computational load and a vast amount of feedback data; and iii) the robustness to estimated fuzzy parametric errors has not been dealt with, i.e., the attenuation of both disturbances and fuzzy parameter errors has not been achieved in literatures. Considering MIMO nonlinear systems poorly modeled, this paper presents a mixed feedforward/feedback (FFB) based adaptive fuzzy control scheme to remove the disadvantages mentioned before. Instead of using feedback cancellation, we first introduce a FFB-based compensation concept by combining the feedforward and error-feedback compensation. Then, the concept is extended to the FFB-based adaptive fuzzy control method. Here a feedforward fuzzy compensator is developed to closely compensate the unknown forward terms required during steady state, while a robust error-feedback

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CHIU: MIXED FEEDFORWARD/FEEDBACK BASED ADAPTIVE FUZZY CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS

;

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de; , are, accordingly, unknown nonlinear function, input matrix, which satisfy for an appropriate desired tracking

where fined as

is a part of the state vector

command

; is the control input; and is an external disturbance assumed to be bounded. For (1), the following assumptions are made. Assumption 1: For an arbitrary variable , the nonlinear term satisfies the following inequality:

Fig. 2. Configuration of FFB-based adaptive fuzzy controller.

compensator is used to assure the stability during transient state. In detail, the feedforward fuzzy compensator is constructed with the desired commands as premise variables of fuzzy rules, while the consequent part is adaptively adjusted according to control errors of feedback. The effect of omitting the transient values of unknown dynamic terms is redeemed control by using error feedback techniques (which are the technique and nonlinear damping design). Thus, the FFB-based adaptive fuzzy controlled system is illustrated in Fig. 2. In addition, applying the LMI-based gain design (cf. [21]–[23]), both disturbance and fuzzy parametric error attenuation are achieved in the -gain sense. In comparison, both the FB and FFB-based adaptive fuzzy control methods belong to a class of the feedback-error-learning-based control methods since the feedback-error is used for tuning parameters of the compensator. But, there exist several differences on: i) the process of taming dynamic uncertainties, ii) the type of training signals, and iii) the use of error-feedback laws (i.e., a nonlinear damper is required in FFB). Although the merits of the FFB over the FB can not be quantitatively shown, the FFB-based fuzzy controller has a simpler architecture of implementation compared to traditional FB-based fuzzy controllers. This advantage arises from the fact that only desired commands are used in the premise variables of fuzzy rules. Moreover, since the bounded forward terms are considered in the FFB compensation, the boundedness assumption of the fuzzy universal approximation errors is naturally dropped from the design. In other words, the proposed method allows a flexible reduction of the number of premise variables, fuzzy rules, and adaptation laws. The remainder of this paper is organized as follows. First, the control problem is formulated in Section II. Then, the FFB-based adaptive fuzzy controller design is performed in Section III, where we present the basic FFB-based compensation concept, feedforward fuzzy compensator, error-feedback design, and robustness. Section IV shows the simulation results of controlling an inverted pendulum system and a two-link robot system. Finally, some conclusions are made in Section V. II. PROBLEM FORMULATION

(2) is the tracking error; and there exist symwhere parameters , metric positive semidefinite matrices , dependent on the nonlinearity of . (The proof is addressed in Appendix I). Assumption 2: The input matrix satisfies (the smallest eigenvalue of is larger than zero) and there exist such that for all in a certain controllability region , i.e., for . Note that most mechanical systems satisfy the above two assumptions, e.g., robots, inverted pendulum system, in Assumpmass-spring-damper systems, etc. Moreover, tion 1 is often held. For this, the robotic system is an obvious example, which has been proven with a similar property as (2) addressed in [25]. The proof of (2) for a class of nonlinear systems is given in Appendix I. The control objective is to track the desired command for a poorly modeled system (1). Thus, a robust tracking controller is derived in Section III. III. FFB-BASED ADAPTIVE FUZZY CONTROL A. FFB-Based Compensation Concept To carry out the controller design, some notations are defined as follows: , , and , where are the tracking error, auxiliary signal vector, auxiliary error signal, respectively; and , for , is a positive-definite diagonal matrix. From the previous definition and the system dynamics (1), the error dynamics can be described by .. .

..

.. .

Consider an th-order multivariable nonlinear system

(1)

.

..

.

.. .

.. .

(3) (4)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

where

; and are defined from the above associated components. This means that the auxiliary provides a stable manifold , which error signal exponentially conrenders the tracking errors . Hence, we call an error measure verging to zero as deviating from the stable manifold. As a result, the control objective will be accomplished by forcing the error subsystem . (4) to the manifold and the terms , are exClearly, if actly known, we can apply a traditional FB-based compensation concept and set the control law as (5)

with a symmetric positive–definite matrix . This will lead to an exponentially tracking result for an exactly modeled system without disturbances. However, the terms and are often poorly understood such that the fuzzy approximator is considered to realize the ideal control law (5) in FB-based fuzzy control methods. Note that when the tracking and accordingly converge to goal is achieved, terms and . functions The FB-based control law (5) converges to

the stability of the closed-loop error system can be easily proven by using the Lyapunov method along the candidate function with . Therefore, the FFB-based control method is also able to achieve the tracking objective. Remark 1: In comparison, the FB and FFB based control methods have the same form in the steady state but they are difand ferent in the transient state. Due to omitting the errors , the FFB-based controller needs a nonlinear damping term and conservative gains to assure the stability. Although the merits of the FFB over the FB can not be quantitatively shown, the FFB-based controller has a simpler architecture of implementation compared to traditional FB-based controllers in (6) can be precalculated using (because the forward term computer-based control). Unfortunately, practical systems are often poorly modeled such that the ideal feedforward compensation law (6) can not be realized straightforwardly. Inspired by the FB-based fuzzy control, the controller (7) is further improved to a FFB-based adaptive fuzzy controller. Based on this mentioned concept, the configuration of the overall controller is thus illustrated in Fig. 2. The proposed controller mainly consists of a feedforward fuzzy compensator and an error-feedback compensator to be designed in the following. B. Feedforward Fuzzy Compensator

(6) which is only dependent on the pre-planned desired command . In other words, the FB-based control law becomes a feedforward compensation law during steady state. This encourages us to introduce a FFB-based compensation concept. Different to the FB-based compensation, a FFB-based control will mix both feedforward and error-feedback compensations and in the design. Instead of using the feedback terms , we directly generate the feedforward compensation law (6), while a error-feedback law is designed to counteract the and . effect of errors If Assumptions 1 and 2 are satisfied and the model is exactly known, the FFB-based control law is set to (7) (8) (9) where defined in (6) is a feedforward compensation term to provide the required force in the steady state; and are, compensator, a nonlinear damper to cope accordingly, an , ; and with the effect from omitting the errors . Then, the FFB-based controller results in the following error subsystem: (10) with . Based on Assumptions 1 and 2, the perturbed term has an upper bound functional of tracking errors and . This implies that

To realize the feedforward compensation law (6), a novel fuzzy system is proposed to closely approximate the unknown terms and . Since we only need the desired commands as premise variables of fuzzy rules, the fuzzy system is called the feedforward fuzzy approximator or feedforward fuzzy compensator. In detail, the fuzzy system is set with the following rules: Rule

If

is

and

is is

and and

is

is

is

Then and

is

is and (11)

where

are the premise variables composed of since and are functional of the desired commands; with denoting the total number of rules; ; is the th element of is the th entry of the approxthe approximation of ; ; are properly chosen fuzzy sets deimation of pendent only on the desired commands; and , are fuzzy sets with tunable center points , , respectively. Using the singleton fuzzifier, product fuzzy inference and center-average defuzzifier, the inferred outputs of the fuzzy system (11) are

where compact sets ;

for some ;

; and is a regression vector composed of the fuzzy basis functions dewith fined as

CHIU: MIXED FEEDFORWARD/FEEDBACK BASED ADAPTIVE FUZZY CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS

for all . Combining the associated entries from the inferred outputs (i.e., and ), the approximations of and are obtained accordingly as (12) (13)

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C. Error-Feedback and Adaptive Algorithm Although the fuzzy system (11) is used to closely realize the feedforward compensation law (6), the effect of fuzzy approximation errors and external disturbances still need to be considered. Therefore, the error-feedback part of the FFB-based controller (7) and the adaptive algorithm of the feedforward fuzzy system are rederived. Based on the fuzzy inferred outputs (12) and (13), the FFB-based fuzzy controller is set to

and ; and blockis the regresand sion matrix. Here, the tunable fuzzy parameters are conformed to specified regions where

, for

with adjustable parameters and . The parameters and will be properly chosen to assure that have a specified upper bound, for example, . Inside the specified set, there exist the so-called optimal approximation parameters defined as

(16) (17) (18) where , , are, accordingly, a feedforward fuzzy concompensator, and a nonlinear damper detertroller part, an mined later; and the last two terms , denote the error feedback compensation part (i.e., the block diagram of the overall controller is illustrated in Fig. 2). Then, the FFB-based fuzzy controller (16) renders the error dynamics (4) to (19) where (12)–(18) have been used; ; and

such that the approximation of and is most accurate. This yields the minimum approximation errors

is a perturbed term of the closed-loop system. To choose a suitable tuning algorithm for the fuzzy system ad the nonlinear damping law , we consider the Lyapunov function candidate

(14) (15) is only dependent on the Since the regression matrix , we have , bounded desired command i.e., , in the specified sets. As a result, it is reasonably concluded that , are , upper bounded for all (from the facts that and ). Moreover, based , on the universal approximation theorem [1], can be arbitrarily small. In addition, a special characteristic of the feedforward fuzzy approximator is that all terms, except the consequent parameters, are able to be calculated offline after given the signal . This implies that the proposed fuzzy system (11) allows offline calculation of premise inference using less computational power. Overall, the following remark is made. Remark 2: The proposed feedforward fuzzy system (11) has three important characteristics: a) the premise variables only consist of desired commands such that some fuzzy inference ) can be performed offline; steps (e.g., calculation of b) an assumption on the bounded approximation error is not needed; and c) due to the naturally bounded approximation er, , the total number of fuzzy rules can be flexibly rors reduced if a large approximation error is acceptable. Therefore, the feedforward fuzzy system allows less computation and the synthesized controller has simpler implementation.

with a symmetric positive-definite matrix and . The along the error dynamics (3) and (19) is time derivative of

(20) By applying the fact tr , the update laws of the adaptive fuzzy parameters can be chosen as (21) (22) Meanwhile, the nonlinear damping law is set as

(23)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 6, DECEMBER 2006

where

is chosen for

. Then,

,

is further simplified to

, for all

; and b)

, for

(24)

diag ; , are projection criterion functions defined as

. As a result,

holds the inequality (24)

and

is always well-defined (from the facts

and for all ). In addition, if is in a strictly diagonally dominant form, the projection algorithm in (26) can be further simplified. Compared to [11], [14]–[16], and we do not require the assumption on the existence of . extra projections on all entries for Remark 3: For simplification, we give the following guidelines in choosing the parameters of the above adap, , and tive algorithm: i) large values of are used to obtain better approximation (the reason is that leads to a large training rate whereas large a large , and yield large space of training data); ii) the , , determine the smooth projection parameters regions, i.e., if they are dropped, then the update laws betunes the come a traditional projection; iii) the parameter

Furthermore, to assure the fuzzy parameters lies in its own constraint region and avoid a singular con, the update laws (21) and (22) are troller due to modified by using the projection techniques to get (25) and (26), as shown at the bottom of the page, where the initial conditions are properly chosen for satisfying , , and ;

and

and

; ,

upper bound of

, i.e., tr

;

and iv) the initial fuzzy parameters are set to and block-diag proper for the nonsingular (i.e.,

with and

is diagonal and

positive-definite). Complete stability analysis is given after stating the following main theorem. Theorem: Consider the highly unknown system (1) using the FFB-based adaptive fuzzy controller (16) composed of [(17), (18), and (23)] with update laws (25)–(26). If there exist and symmetric positive–definite matrices , satisfying the LMI problem shown in (27) at the bottom of the page, where ; ; ;

with adjustable parameters , , , accordingly con, , , formed to and . The aforementioned update laws are extensions of a smooth projection algorithm addressed in [24]. The , , would be properly chosen such that the parameters situation of both and does not occur during the tuning procedure. Then, the update laws as, sure the following properties: a)

if

and

;

(25)

otherwise if if otherwise

and and

; ;

(26)

Given subject to (27)

CHIU: MIXED FEEDFORWARD/FEEDBACK BASED ADAPTIVE FUZZY CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS

; and , then the closed-loop error system has the following properties: i) all error signals and fuzzy parameters are bounded; ii) the tracking performance criterion

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with . The trajectory of

is solved and shaped

by

(28)

is assured for and ; and iii) asymptotically converges to zero for . Proof: Consider the time derivative of in (24). If update laws (25)–(26) are applied and Assumptions 1 and 2 are satisfied, then

with . Since the fuzzy parameters have and , been conformed to the specified regions has an adjustable uniform ultimate bound the error . Finally, due to , and (if ), the result the fact that , is concluded by Barbalat’s lemma. From the previos description, the highly unknown dynamics of the plant has been coped with by the feedforward fuzzy compensator. Meanwhile, the effect from omitting the errors and have been eliminated by an compensator and nonlinear damper such that the feedforward fuzzy compensation and is achieved. Therefore, the FFB-based of uncertain adaptive fuzzy controller has been synthesized as Fig. 2. D. Attenuation of Fuzzy Parametric Errors

(29) where

; the expressions have been applied

and

and the detailed derivation is stated in Appendix II. Then, a feasible solution of the LMI (27) yields

Although the disturbance attenuation problem has been solved, poor fuzzy approximations may lead to an unexpected transient response. This worst case will occur especially when using a smaller number of fuzzy rules. To avoid this, the attenuation of fuzzy parametric errors is taken into consideration later. Corollary: Consider the highly unknown system (1) using the FFB-based adaptive fuzzy controller (16) composed of [(17), (18)] with update laws [(25), (26)], and the nonlinear damping

(30) and is negative semidefinite outside the comSince pact set for , and . As a result, , we have is assured from the boundedness of all terms on right-hand side of (3) and (19). In turn, . Moreover, by integrating the inequality (30), the tracking performance criterion (28) is assured. In other is attenuated to a prescribed words, the disturbance level . Also, if is integrable. In addition, according to the fact with and , the inequality (30) can be further rewritten as

(31) If there exist and symmetric positive–definite matrices , such that the LMI problem (27) has a feasible solution, then tracking performance the closed-loop system achieves

(32) is a quadratic term dependent on the iniwhere is a prescribed tial values of tracking errors; attenuation level for the fuzzy parametric error . Proof: Consider the Lyapunov function candidate with . Similar to the proof of

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the main theorem, the feasibility of LMI (27) and the design of (31) renders

length of the pole; is the applied force; and ternal disturbance assumed to be , the working space have along with

is an ex. Consider , we

The inequalities

where

with

, yields

(33) Integrating both sides of (33), the closed-loop system guarantees the robust performance criterion (32). In addition, using the similar arguments as the proof of the main theorem, the boundedness and the convergence of the controlled system are assured. Remark 4: Compared to FB-based fuzzy control methods, which are constructed only based on state feedback, the FFBbased fuzzy controller emphasizes both the feedforward term and the error-feedback compensation. Although the FFB-based adaptive fuzzy controller needs a nonlinear damping term and conservative gains, the feedforward fuzzy compensator (only dependent on the desired commands except the tuning parameters) renders to an easier implementation form. Moreover, the compensation and nonlinear damper provide more robustness to fuzzy approximation errors and uncertainties. In addition, from the fact that fuzzy approximation errors are naturally bounded, the results are assured in a global manner.

and . In this simulation, the reference signal . When parameters , , and are is , , and assumed accordingly as , the parameters , , and in (2) are found. Moreover, the feedforward fuzzy system (11) is constructed with linguistic and , which accordingly are classified into five variables and three fuzzy sets. Since the behavior of the linguistic variables is exactly known, the fuzzy sets are easily characterized by the following membership functions:

IV. SIMULATION RESULTS To verify the validity of the proposed controller, we take an inverted pendulum system and a two-link robot as application examples that follow. 1) Example 1: The dynamics of the inverted pendulum system is

where is the angle of the pole; is the acceleration due to gravity; , , are parameters functional accordingly to the mass of the cart, the mass of the pole, and the

The total number of fuzzy rules is 15, i.e., and . The update laws (25) and (26) are set with , , , , , and the initial values along with for and . According to the main corollary, the LMI problem (27) is solved and for given with a feasible solution , , , and . When the iniand , the position and tial states are velocity tracking results are illustrated in Fig. 3. In addition, we construct a FB-based adaptive fuzzy controller using the similar , , and . Then, a design except for letting

CHIU: MIXED FEEDFORWARD/FEEDBACK BASED ADAPTIVE FUZZY CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS

Fig. 3. (a) Position tracking response. (b) Velocity tracking response. (actual trajectory

, desired trajectory

).

Fig. 4. Comparison of position tracking results for the FFB ( (

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, while the forces, the viscous friction is added into Columbus and Stribeck friction is taken into the consid. The term eration in the first term of disturbance exhibits the effect of high frequency measurement noise. Dy, , namic parameters , , and depend , and the length , of two robot links, on mass while are friction parameters. Each parameter is assumed of having a 15% deviation, where we set the actual , , value of system parameters as , , , , , and . Then, the bounds of the matrix are found and . Using the same arguments for as the proof of (2) (cf. Appendix I), Assumption 1 is held with , block, , (for ), where , , , , , , , , , and denotes . Note an upper bound of unknown , for , , and are conservative arising that the values of from strict analysis. Alternatively, the values can be properly chosen from trial and error methods on tuning tracking perand formances. In this simulation, we let . The desired motion tracking trajectories are set as . Since the proposed approach allows a flexible reduction of fuzzy rules, the feedforward fuzzy system (11) only takes and as the premise variables. From the exactly known behavior of the premise variables, three fuzzy sets are used and characterized by the following membership functions (for ):

) and FB

) based adaptive fuzzy controllers.

comparison of position tracking results is given in Fig. 4, where a larger residual position error exists in the traditional FB-based control. 2) Example 2: Consider a two-link robot expressed in the form (1) with

, and . and denote the first and second link Here, the states angles, respectively. In the presence of the joint friction

For simplification, the fuzzy rule-base is constructed with the three rules shown in the equation at the bottom of the and . Meannext page. This renders while, the fuzzy consequent parts are adjusted by (25) and , , (26), where , , , , and blockfor and . On the other hand, after given the control parameters , , , and , the control gain is obtained from solving the LMI (27). Based on the main corollary, the position and velocity tracking results for the initial states, and , are shown in Figs. 5 and 6, respectively. The control inputs and the min(which is positive for all time, i.e., imum eigenvalue of ) are illustrated in Fig. 7. The control effort of the

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Fig. 5. (a) Position response of Link 1. (b) Position response of Link 2. (actual trajectory

, desired trajectory

Fig. 7. (a) Control input of Link 1 ( imum eigenvalue of

).

G (t) (

) and Link 2 (

) and

G (t) (

). (b) The min-

).

Fig. 6. (a) Velocity response of Link 1. (b) Velocity response of Link 2. (actual trajectory

, desired trajectory

). Fig. 8. Control efforts induced by (a) feedforward fuzzy compensator, (b) control, and (c) nonlinear damper. (Link 1

feedforward fuzzy compensator, the controller part, and the nonlinear damper is shown in Fig. 8. To show more benefits of the proposed approach, a FB-based fuzzy controller is also applied to the robot. Since the bounded and are taken approximation errors are assumed, both

Rule

If

Rule

and

and is

is and

and

is

and is

is and

and

Rule

is

and

is

and

If If

).

as premise variables for an admissible approximation. If all premise variables are classified by three fuzzy sets, the FB-based fuzzy controller needs 81 fuzzy rules, i.e.,

is is

, Link 2

H

is

Then is

is

and

is

and

is

and

is

Then is

is

Then is

CHIU: MIXED FEEDFORWARD/FEEDBACK BASED ADAPTIVE FUZZY CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS

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V. CONCLUSIONS This paper has solved the tracking control problem of a class of MIMO uncertain nonlinear systems by the FFB-based adaptive fuzzy controller. In the proposed controller, the feedforward fuzzy compensation part provides advantages of easier implecompensator and mentation. Meanwhile, by combining the nonlinear damper, the feedback compensation part assures some robust tracking properties and the validity of using the feedforward fuzzy compensation. Thus, the mixed feedforward/feedback based controller design is more convincing than traditional FB-based control. Moreover, by applying an LMI technique, tracking performance is guaranteed with the attenuation of disturbances, approximation errors, and estimated fuzzy parameter errors. In addition, the developed projection algorithm has removed the nonsingularity assumption on the fuzzy approximated input matrix. Finally, simulation results and some comparisons have shown the expected performance of the proposed approach. Fig. 9. Using the FFB (

) and FB (

) based adaptive fuzzy con-

trollers, the position tracking errors for (a) Link 1 and (b) Link 2.

TABLE I COMPARISONS BETWEEN FFB AND FB BASED SCHEMES ON CONTROLLING A TWO-LINK ROBOT

and . In other words, the proposed approach in this paper requires less numbers of fuzzy rules and tuning parameters. In addition, the FB-based controller is constructed with the same error-feedback design and parametric , , and . Fig. 9 illussetting except for trates the comparison of position tracking responses, where the proposed controller leads to better results (because the feedback gain in FB-based control is obtained from a rough analysis). Other differences for controlling the robot are summarized in Table I. Obviously, the contrast would increase along the raise of dimensions of plants. Therefore, the expected performance are verified from the previous statements.

APPENDIX I PROOF OF INEQUALITY (2) Consider a class of nonlinear systems with the continuous grouped as dynamic term , were is a finite term; is a quasi-linear term; is a quadratic term; and is a high-order term. Here, we assume: i) all the terms and their derivative with respect to are bounded for all ; and its derivative with respect to are bounded for ii) ; and iii) a term with the same property as can . In other words, be extracted from all components of may consist of , , , etc. In turn, the quasi-linear term may consist of , , etc.; the quadratic term may consist , , Coriolis/centripetal force term in of may mechanical systems, etc.; and the high-order term , etc. Accordingly, has the correconsist of , . Then, the boundedness of the four sponding partition as terms is discussed as follows. as and rewrite First, denote the error term it by using the mean value theorem [25] as shown in the equation , deat the bottom of the page, where , for notes the th component of . Due to the fact that all are upper bounded, there exist and such that

where quasi-linear term

. Then, consider the error of the . Analogously, the error

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is re-expressed as shown in the first equation at the bottom and its derivative, there of the page. Due to the bounded and such that exist

where . Next, consider the error of the . Since the th compoquadratic term can be rewritten as shown in the second equation nent of at the bottom of the page. We further have .. .

in (2) is held. In [25], the authors have proven the property for robotic manipulators. In addition, since the above proof is very , conservative and the plant is highly uncertain, parameters , are usually found from using some experiments or trial and error methods. APPENDIX II PROOF OF INEQUALITY (29) First, let us consider the boundedness of the perturbed term . From the error subsystem (3), the term rewritten as

can be

.. .

Taking the norm on the previous equation, it yields where

; the definitions and have been used. Based on the update law (26), and

we have

. This

implies that there exists a parameter where and satisfy and , respectively; and . Finally, using the similar arguments as before, we are able to obas tain the bounded fashion of the high-order term

where rameter

, , and accompanies the nonlinearity

satisfying (here Hence, under Assumption 1 on the error perturbed term satisfies

). , the

; and the pa. For example,

with and scalar , , , . By summarizing the above results, the inequality (2) is thus obtained for a class of nonlinear systems. To the best of our knowledge, most mechanical systems (e.g., robots, inverted pendulum system, etc.) only consist of the first , and , i.e., the parameter three terms

where , By applying

are intermediate parameters. and letting

CHIU: MIXED FEEDFORWARD/FEEDBACK BASED ADAPTIVE FUZZY CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS

,

we

obtain

the

following inequality:

where the fact has been used. holds Then, from the previous inequality,

727

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H

Thus, after expressing the errors inequality (29) is obtained.

and

in terms of

, the

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H

Chian-Song Chiu (M’04) was born in Taiwan, R.O.C., in 1975. He received the B.S. degree in electrical engineering and the Ph.D. degree in electronic engineering from the Chung-Yuan Christian University, Chung-Li, Taiwan, R.O.C., in 1997 and 2001, respectively. Since 2003, he has been with the Department of Engineering, Chien-Kuo Technology University, Changhua, Taiwan, R.O.C., where he is currently an Assistant Professor. He is the chapter coauthor of Fuzzy Chaotic Synchronization and Communication—Signal Masking Encryption (Soft Computing for Communication, New York: Springer-Verlag, 2004). His current research interests include fuzzy control, robotics, and nonlinear control.