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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 3, JUNE 2003

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Robust Fuzzy Control of Mechanical Systems Yu Tang and Daniel Vélez-Díaz

Abstract—This paper considers the problem of controlling a mechanical system described by Euler–Lagrange equations to follow a desired trajectory in the presence of uncertainties. A fuzzy logic system (FLS) is used to approximate the unknown dynamics of the system. Based on the a priori information, the premise part of the FLS as well as a nominal weight matrix are designed first and are fixed. A compensation signal to the weight matrix error is designed based on Lyapunov analysis. To further reduce the tracking error due to the function reconstruction error, a second compensation signal is also synthesized. By running two estimators online for weight matrix error bound and function reconstruction error bound, the implementation of the proposed controller needs no a priori information on these bounds. Exponential tracking to a desired trajectory upto a uniformly ultimately bounded error is achieved with the proposed control. The effectiveness of this control is demonstrated through simulation and experiment results. These results also show that by incorporating a priori informations about the system, the fuzzy logic control can result good tracking behavior using a few fuzzy IF–THEN rules. Index Terms—Frictions, fuzzy control, Lyapunov stability, mechanical systems.

I. INTRODUCTION

D

URING recent years, the analytical study of nonlinear control using universal function approximators has received much attentions (see [17] and the references cited there). Mostly, a neural network (NN) or a fuzzy logic system (FLS) is used to approximate the nonlinearity of the system to be controlled and a controller is synthesized based on this approximation (indirect control), or a control law is directly designed using NNs, or FLSs based on stability theories (see, e.g., [6]–[8], [10], [13]–[16], [18][19], and [21]). In the class of approximators which are linear in the weights, FLS is much closer in spirit to human thinking and natural language [22]. It provides an effective means of capturing the approximate, inexact nature of the real world. In particular, this methodology appears very useful when the processes are too complex for analysis using conventional techniques or when the available sources of information are interpreted qualitatively, inexactly, or with large amount of uncertainty. In addition to the classical feedback control theory, adaptive control and robust control are effective techniques to treat system uncertainties [4], [9]. Adaptive control, by tuning on-line its parameters can deal with large uncertainties, but generally suffers from the disadvantage of being able to achieve asympManuscript received December 5, 2001; revised September 17, 2002 and October 21, 2002. This work was supported by Projects PAPIIT IN104700 and CONACyT 36154-A. Y. Tang is with the DEPFI-UNAM, 04510 Mexico City, Mexico, and also with PIMAyC-IMP, Mexico City, Mexico (e-mail: [email protected]). D. Vélez-Díaz is with the DEPFI-UNAM, 04510 Mexico City, Mexico (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2003.812700

totical convergence of the tracking error, also the on-line computation load is usual heavy. In robust control designs, on the other hand, a fixed control law based on a prior informations on the uncertainties (usually bounds on these uncertainties) is designed to compensate their effects, and exponential convergence of the tracking error to a (small) ball centered at the origin is obtained. However, if the uncertainties are larger than the assumed bounds, no stability or performance are guaranteed. By combining both techniques, large uncertainties can be dealt with, exponential convergence of the tracking error to a (small) ball centered at the origin is achievable and on-line computation load is kept to be minimum, since only uncertainty bounds are updated online. This paper considers the problem of controlling a mechanical system described by Euler–Lagrange equations to follow a desired trajectory in the presence of uncertainties. A fuzzy logic system is used to approximate the unknown dynamics of the system. Based on the a priori information, the premise part of the FLS as well as a nominal weight matrix are designed first and are fixed. A compensation signal to the weight matrix error is designed based on Lyapunov analysis [4]. To further reduce the tracking error due to the function reconstruction error, a second compensation signal is also synthesized. By running two estimators online for weight matrix error bound and function reconstruction error bound the implementation of the proposed controller needs no a priori information on these bounds. Exponential tracking to a desired trajectory upto a uniformly ultimately bounded error is achieved with the proposed control. The effectiveness of this control is demonstrated through simulation and experiment results. The results also show that by incorporating a priori informations about the system, the fuzzy logic control can result good tracking behavior using a few fuzzy IF–THEN rules. II. SYSTEM DESCRIPTION AND PROBLEM STATEMENT In this paper, we consider mechanical systems with gree-of-freedom described by Euler–Lagrange equations

de-

(2.1) are the generalized positions of the system, the velocities, the Lagragian function and the generalized forces applied to the system. the inertia matrix, by the Denoting by the gravity forces, Coriolis and centrifugal forces, and by with repreand considering the applied forces senting frictions, and the control inputs to the system, (2.1) takes the following form where

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(2.2)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 3, JUNE 2003

The friction depends predominantly on velocity, but generally also on position, time and external forces [11]. It constitutes one of the major source of the uncertainty in the mechanical system (2.2). For the purposes of designing the controller, we assume the dynamical (2.2) to have the following properties. : for some 1) Symmetry and boundedness of , , . : 2) Skew symmetry of , . be a given smooth ( , , bounded) deLet sired trajectory and define the tracking error as . The problem we consider in this paper is to design a control law for (2.2) to ensure the tracking error to be uniformly ultimately bounded, also the ultimate bound should be made arbitrarily small by choosing appropriately controller parameters, while maintaining all signals in the system uniformly bounded.

In the rest of the paper, will be referred to as the the fuzzy basis functions. weight matrix, and It is well known that the FLS (3.5) is a universal approximator in a in the sense that given any real continuous function and any there exists an FLS (3.5) compact set such that [22] (3.6) denotes Euclidean norm where (and throughout this paper) or its induced matrix norm. In light of this result, the function can be expressed as (3.7) where

is called function reconstruction error satisfying (3.8)

and

is the optimal weight matrix (3.9)

III. FUNCTIONAL APPROXIMATION USING FLS’S Consider a fuzzy logic system [19], [22] with the product-inference rule, singleton fuzzifier, center average defuzzifier, and Gaussian membership function given by fuzzy IF–THEN rules if

is

and then

and is is and and

is

(3.1)

th rule, , and are the input ( inputs) and the output ( outputs) of the fuzzy logic system, respectively, is the fuzzy singleton are for the th output in the th rule, and fuzzy sets with Gaussian membership functions where

denotes the

(3.2) with , given by

In the practice, the optimal weight matrix may be not unique and unknown. Several methods based the gradient of an error function are available to estimate it (see, e.g., [22]). Also, when some part of an FLS (number of rules , membership functions , or consequent parameters ) is fixed, the function reconstruction error bound is unknown.

design parameters. The th output of the FLS is

IV. CONTROL DESIGN In this section, we use an FLS to approximate the unknown dynamics of the system. We use the robust control technique [4] to design two compensation signals to compensate for the uncertainties mainly due to 1) structured (parameter) uncertainty, which arises from the unknown optimal weight matrix , and 2) unstructured uncertainty which arises from the function reconstruction error. To remove the assumption on the parameter error bound and the function reconstruction error bound two estimators are provided to estimate them online. A. Error Equations

(3.3) (3.4)

If the membership functions (i.e.,, , ) are fixed, the normalis a ized firing strength (activation degree) of the th rule function of only . Therefore, the output of an FLS allows a linear parametrization in its consequent parameters

For the subsequent analysis using Lyapunov stability theory, we first develop the error equations from the given plant dynamics (2.2). be the reference velocity and the filtered tracking Let error defined as (4.1) with (4.1)

being a positive–definite matrix. It is clear from (4.2)

is bounded and goes to a (small) ball centered at the that if , ([5, p. 59]). So, the control law will be origin, so do is uniformly designed such that the filtered tracking error ultimately bounded. The dynamics of are obtained from (2.2) (4.3) where (3.5)

(4.4)

TANG AND VÉLEZ-DÍAZ: ROBUST FUZZY CONTROL OF MECHANICAL SYSTEMS

In the previous equation, we assume that the frictions are a function of positions and velocities. From the results in Section III, can be approximated by an FLS in a compact set (recall that denotes the degree-of-freedom the mechanical system)

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Consider the second right-hand term of (4.14)

(4.5) and the function reconstruction error is bounded by a unknown bound (4.15)

(4.6)

Similarly, the third right-hand term of (4.14) is bounded by B. Control Law a nominal weight matrix, and Let , i.e., to the weight matrix error

a bound (4.7)

We assume first that both bounds on the weight matrix error and the function reconstruction error are known, and later we will remove this assumption by estimating them online. The proposed control for the mechanical system (2.2) is given in the following:

(4.16) Let (4.17)

(4.8) is a positive definite matrix gain, and (recall that denotes the number of IF–THEN rules in an FLS) and are designed to compensate for the weight matrix error and the function reconstruction error, respectively

where

where denotes the smallest eigenvalue of the matrix gain it follows from (4.14)–(4.16) that

,

(4.18) which implies

(4.9) (4.19) (4.10) with

,

By (4.13)

design parameters.

C. Stability and Performance Analysis

(4.20)

in (4.5) into the Substituting the control law (4.8) and error equations (4.3) gives the closed-loop system in terms of the filtered tracking error (4.11) and denote the structured uncertainty and where unstructured uncertainties, respectively (Equations (4.7) and (4.5)). Consider the following positive–definite function: (4.12) It is clear from Property 1 that (4.13) Taking the time derivative of erty 2 results

along (4.11) and using Prop(4.14)

Let condition

be a compact set which contains the initial , and the corresponding ball centered at the origin with ra-

dius

Therefore, for all , the trajectory of (4.11) will re. This means that for all , the corremain in sponding trajectory of the closed-loop system will remain in . Consequently, all signals in the closed-loop system are unitends to a formly bounded, and the filtered tracking error . This ball centered at the origin with radius implies that [see (4.2)] the tracking error will tend to a ball cen(where tered at the origin with radius denotes the smallest eigenvalue of the matrix gain ), which can be made arbitrarily small by adjusting control gains , and the design parameter .

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In what follows, we design two estimators for the uncertainty and . bounds

where

D. Estimation of the Uncertainty Bounds and denote the estimate of and Let instant , respectively, which are updated according to

(4.30)

at the which implies (4.21) (4.22)

where , , sation signals

, and

are design parameters. The compenare now given by

(4.31) By (4.13)

(4.23)

and Let the following positive–definite function:

(4.24)

(4.32)

. Consider

be a compact set which contains the initial conLet , and dition the corresponding ball centered at the origin with radius

(4.25) where

is defined in (4.12). As preceeded in Section IV-C

(4.26) Following the steps in (4.15), the second right-hand term of (4.26) is bounded by

Therefore, for all , the trajectory of (4.11) will , which implies that all signals in the closed-loop remain in system are uniformly bounded, and the filtered tracking tends to a ball centered at the origin with radius error . Consequently, the tracking error will tend to , a ball centered at the origin with radius which can be made arbitrarily small by adjusting control gains , and the design parameter and , . E. Remarks on the Proposed Control

(4.27) The last inequality is because that . Similarly, the third right-hand term of (4.14) is bounded by (4.28) It follows from (4.26)–(4.28) that (4.29)

Remark 4.1: By combining advantages of FLS, adaptive control and robust control techniques, the proposed controller may incorporate easily a priori information about the system through IF–THEN rules, and achieves exponential convergence of the tracking error to a small neighborhood of the origin instead of asymptotical convergence as in most existing control schemes based on universal approximators [6]–[8], [13]–[16], [18], and [21]. It is important to obtain the exponential convergence since it is more robust with respect to unmodeled dynamics and/or external disturbances. Remark 4.2: In [6] a robust control was developed for robot manipulators based on its state space representation, and global exponential convergence to a ball of the origin was obtained. The main differences between the results in [6] and the present work are as follows. 1) the proposed control is designed based on the Lagrangian equations (2.2) without transforming it into the first-order state space form. Therefore, physical properties are conserved, which were used to facilitate the control design. 2) The uncertainty bounds in the proposed control are constants,

TANG AND VÉLEZ-DÍAZ: ROBUST FUZZY CONTROL OF MECHANICAL SYSTEMS

which are easy to be estimated, whereas in [6] the uncertainty in [6, eq. (15)]). bound is a state-dependent quantity ( 3) The control implementation in [6] needs parameters as the , and bounds on the uncerupper bound on the initial matrix in each of the fuzzy sets ( in [6, eq. (25)]), tainty bound which appear to be difficult to get and take very high values in the simulation. Remark 4.3: In [8] and [18], two adaptive tracking controls using universal approximators (NN and FLS, respectively) were proposed and asymptotical convergence of the tracking error to a small neighborhood of the origin was obtained. In order to reduced the approximation error, a robustifying term like (4.9) and (4.10) was also used. The implementation of this robustifying term, however, requires informations on the approximation error bound, and bound on the unknown optimal weights of the universal approximator. By introducing an estimator of the uncertainty bounds, the control in this work needs no a priori information on these bounds. Also, since there are only two parameters estimated online independent of the degree-offreedom of the mechanical system, the computational burden is much less than that in an adaptive control, which updates online the weight matrix typically of high dimension. Remark 4.4: One of the advantages of using FLS as an approximator of the uncertainties is its ease to incorporate a priori informations through IF–THEN rules, and no needs on the structure information of the uncertainties as needed in compensation schemes based on a model of the uncertainty (see, e. g., [12] and [20]).

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Fig. 1. Without friction compensation. (Top) Desired q (dashed line) and rotor angle position q (solid line). (Middle) Desired q_ (dashed line) and motor angle speed q_ (solid). (Bottom) Control input u.

V. SIMULATION AND EXPERIMENT RESULTS To verify the theoretical results, simulations and experiments were carried out in a single degree-of-freedom mechanical system consisting of a dc motor connected to a gearbox with significant friction: (5.1) where is the motor angular position, the control input, and represents the frictions, which is core of the aforementioned model. There is many recent research about these phenomena (see, e.g., [1] and [2] for a survey). For the simulation purpose, the frictions are represented by the LuGre model [3], [11] (5.2) (5.3) (5.4) where is an internal state of the model. The parameters used in the simulation are taken from [11] which were identified from an experiment setup1

This example was also used in [12], where an adaptive friction compensator was proposed. 1SI

units are used in this paper.

Fig. 2. With friction compensation. (Top) Desired q (dashed line) and rotor angle position q (solid line). (Middle) Desired q_ (dashed line) and motor angle speed q^ (solid). (Bottom) Control input u.

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Fig. 4.

Estimate of the error bounds. (Top) bound on the weight matrix error

^ . (Bottom) Bound on the function reconstruction error ^ .

This FLS is motivated by the simplest friction model (5.8)

Fig. 3. (Top) Position tracking error q~. (Middle) Friction torque F . (Bottom) Compensation signal (B + B )W (x) + u .

Since the major source of uncertainty in the (5.1) is frictions, to exhibit their effects we first closed the loop with a fixed controller which neglects the friction (5.5) The desired trajectory and its derivatives are generated by a (where overdamped filter with transfer function ) fed by , and the controller parameters . The position and speed, starting from the initial values , the desired position and velocity as well as the corresponding control input are shown in Fig. 1. Large tracking errors observed in the figures can of cause be reduced with larger gains , but at a cost of larger input effort. The robust fuzzy control (4.8) with compensation signals (4.21)–(4.24) was then implemented with two IF–THEN rules if is where , functions

then

is

(5.6)

are Gaussian membership functions (3.2) with and , giving the fuzzy basis

(5.7) with the corresponding weight matrix

the unknown friction level, which captures only being the stiction phenomena [11]. The nominal weight matrix in the control law (4.8) was , and the design parameters in the compensation sig, , nals (4.21)–(4.24) , and the initial conditions . Several simulations were taken over and the performance is quite robust to these design parameters and initial conditions. The evolution of relevant signals are depicted in Figs. 2–4. Notice the following. • Even thought the robust fuzzy controller has two IF–THEN rules, which models only the stiction effects, the tracking performance is quite good. in the middle section of Fig. 3 • For the friction level (about 0.3 (Nm)), it gives a parameter uncertainty (with ) of 200%. respect • Exponential convergence of position and velocity tracking errors to a small neighborhood of the origin is obtained. Compared with the simulation results in [12] (where a slightly different set of parameter was used) and those of [1] and repeated in [12], the proposed control gives a similar performance, although in the theoretical aspects, the controls in [1] and [12] give the global asymptotical tracking while in this paper the tracking is semiglobal exponential upto a uniformly ultimately bounded error. The tracking control in the aforementioned simulation was repeated in an experimental setup consisting of a dc motor from Feedback Inc., which was purchased in 1990. Due to lack of maintenance, this motor contains large amount of friction. The control algorithm was implemented in a dSPACE 1103 card, inserted into a Pentium III computer. The sampling time was 0.001 s. As in the simulation, the experiments stared with the control , . in (5.5) with controller parameters Due to the large amount of friction, the motor did not even

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Fig. 7. Estimate of the error bounds. (a) Bound on the weight matrix error ^ . (b) Bound on the function reconstruction error ^ .

Fig. 5. Without friction compensation. (Top) Desired q (dashed line) and rotor angle position q (solid line). (Bottom) Control input u.

with the same parameters as in the simulation and the uncer, tainty bound estimator (4.21) and (4.22) with , , and the initial conditions , . The results are depicted in Figs. 6 and 7. Notice that the control signal responses quickly to the friction, and gives a significantly improved tracking performance, specially when the velocity changes the direction. VI. CONCLUSION This paper has presented a robust fuzzy control for mechanical systems to track a smooth desired trajectory. By combining advantages of FLS, adaptive controls and robust controls, the proposed control may incorporate easily a priori information about the system, and achieves exponential convergence of the tracking error to a ball centered at the origin, whose radius can be made arbitrary small. Simulations on an experimentally validated model and experimental results have shown the effectiveness of the proposed control. ACKNOWLEDGMENT The authors would like to thank Anonymous Reviewer B and the Associate Editor whose suggestions have contributed to the improvements of this paper.

Fig. 6. Without friction compensation. (a) Desired q (dashed line) and rotor angle position q (solid line). (b) Control input u.

move with these controller gains. So, the controller gains were , . The desired and actual position increased to as well as control input are depicted in Fig. 5. Observe that the closed-loop system fails to track the desired signal when the velocity changes the direction. Further increasing controller gains did not improve the tracking performance, and even deteriorated it due to control signal saturation. Next, we added to (5.5) the compensation component, implemented by (4.23) and (4.24)

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[5] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input–Output Properties. New York: Academic, 1975. [6] C. Ham, Z. Qu, and R. Johnson, “Robust fuzzy control for robot manipulators,” IEE Proc. Control Theory Appl., vol. 147, no. 2, pp. 212–216, 2000. [7] H. Lee and M. Tomizuka, “Robust adaptive control using a universal approximator for SISO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 95–106, Feb. 2000. [8] F. L. Lewis, A. Yesildirek, and K. Liu, “Multilayer neural-net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Networks, vol. 7, pp. 388–399, Apr. 1996. [9] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Upper Saddle River, NJ: Prentice-Hall, 1989. [10] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Networks, vol. 1, pp. 4–27, Feb. 1990. [11] H. Olsson, K. J. Astrom, C. Canudas, M. Gafvert, and P. Lischinsky, “Friction models and friction compensation,” Euro. J. Control, vol. 4, no. 3, 1998. [12] E. Panteley, R. Ortega, and M. Gafvert, “An adaptive friction compensator for global tracking in robot manipulators,” Syst. Control Lett., vol. 33, pp. 307–313, 1998. [13] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 447–451, Mar. 1996. [14] A. Poznyak, W. Yu, E. N. Sanchez, and J. P. Perez, “Nonlinear adaptive trajectory tracking using dynamical neural networks,” IEEE Trans. Neural Networks, vol. 10, pp. 1402–1411, Dec. 1999. [15] R. M. Sanner and J. J. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Networks, vol. 3, pp. 837–863, Dec. 1992. [16] S. Seshagiri and H. K. Khalil, “Output feedback control of nonlinear systems using RBF neural networks,” IEEE Trans. Neural Networks, vol. 11, pp. 69–79, Feb. 2000. [17] J. T. Spooner and K. M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 339–359, June 1996. [18] C. Su and Y. Stepanenko, “Adaptive control of a class of nonlinear systems with fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 2, pp. 285–294, Aug. 1994. [19] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, Jan. 1985. [20] Y. Tang and M. Tomizuka, “Decentralized robust control for a class of nonlinear systems,” presented at the 1997 Amer. Control Conf., Albuquerque, NM, 1997.

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[21] L. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 146–155, May 1993. [22] , Adaptive Fuzzy Systems and Control, Design and Stability Analysis. Upper Saddle River, NJ: Prentice-Hall, 1994.

Yu Tang was born in Beijing, China, in 1960. He received the B.S. degree (with honors) in computer engineering and the M.E. and Ph.D. degrees in electrical engineering, all from the National University of Mexico, Mexico City, in 1984, 1985, and 1988, respectively. He joined the same university in 1989, and is currently a Professor at the Faculty of Engineering. He was a Guest Professor at the Beijing Polytechnic University from July 1997 to July 2001. He has held a research position with the University of California, Berkeley, from January to December 1996, and teaching and research positions in CINVESTAV-IPN, Mexico, from January 1991 to July 1994. He was a Consultant for the Mexican Institute of Research in Electricity from January 1994 to December 1995. His research interests involve adaptive control, intelligent control, and robotics, and their applications to industrial problems. Dr. Tang is a Regular Member of the Mexican Academy of Sciences and System of National Investigators (SNI-2). He received the Weizmann Prize from the Mexican Academy of Sciences for the Best Ph.D. Dissertation in 1991, the “National University Distinction for Young Academics” in 1995, and the “National University Recognized Professor” in 1997, both from the National University of Mexico.

Daniel Vélez-Díaz was born in Mexico City, Mexico. He received the B.S. degree in electronic engineering from the Metropolitan Autonomous University, Mexico City, Mexico, and the M.S. degree in computational sciences from the Technologic Institute of Toluca, Toluca, Mexico, in 1989 and 1999, respectively. He is currently working toward the D.Eng. degree in electrical engineering at the National Autonomous University of Mexico, Mexico City. His research interests include fuzzy control and intelligent control of nonlinear systems with uncertainties.