Somos uma comunidade de intercâmbio. Por favor, ajude-nos com a subida ** 1 ** um novo documento ou um que queremos baixar:

OU DOWNLOAD IMEDIATAMENTE

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

429

Design of a Nonlinear Disturbance Observer Xinkai Chen, Member, IEEE, Satoshi Komada, Member, IEEE, and Toshio Fukuda, Fellow, IEEE

Abstract—This paper presents a new disturbance observer based on the variable structure system theory for minimum-phase (with respect to the relationship between the disturbance and output) dynamical systems with arbitrary relative degrees. The model uncertainties and the nonlinear parts of the system are merged into the disturbance term and are regarded as a part of the disturbances. The upper and lower bounds of the disturbance are assumed to be known as a priori information. Simulation results are presented to show the robustness and effectiveness of the new disturbance observer. Experiment results show the practicality of the new observer. Index Terms—Minimum-phase systems, nonlinear disturbance observer, relative degree, variable structure system theory.

I. INTRODUCTION

I

N ALMOST all engineering control systems, the presence of disturbances, model uncertainties, and nonlinear model parts is inevitable. For example, when the robot manipulators grasp an unknown payload, they are affected by unknown inertia variation and gravity force, but these changes are rarely captured in the models. It is most desirable that the controller be insensitive to these uncertainties. Thus, in recent years, the problem of controlling uncertain dynamical systems subject to external disturbances has been a topic of considerable interest. Many robust control methods have been proposed. Among these control [1], [5], [12], methods, the most typical ones may be variable structure system (VSS) sliding-mode control [6], [16], [18], adaptive control [7], [8], [10], [11], -synthesis method [2], [14], dynamic versus parametric uncertainty control [4], [8], and robust process control [3], [13]. These methods can cope with different types of uncertain systems [4]. In this paper, the model uncertainties, the nonlinear parts of the system, and the external disturbances are merged into one term, and we refer it as the disturbances of the system. It might be argued that, if the disturbances can be estimated, the control problems of the systems with disturbances may become easier to solve. For example, the state observer and the controller with disturbance cancellation functions can be easily constructed by using the estimated disturbances. Among the many presented disturbance observer techniques, type the approximate differentiator type [15], [17] and [12] formulations have been popularly applied in the design of Manuscript received March 17, 1997; revised June 11, 1999. Abstract published on the Internet December 23, 1999. X. Chen is with the Department of Information Sciences, Tokyo Denki University, Saitama 350-0394, Japan (e-mail: [email protected]). S. Komada is with the Department of Electrical and Electronic Engineering, Mie University, Tsu 514, Japan. T. Fukuda is with the Department of Mechano-Informatics and Systems, Nagoya University, Nagoya 464-01, Japan (e-mail: [email protected]). Publisher Item Identifier S 0278-0046(00)02499-0.

tracking controllers for motion control systems. The procedure of the first approach closes an inner loop around the controlled plant to reject disturbances and force the input–output characteristics of this inner loop to approximate a “nominal” plant model at low frequencies. Tuning of the loop is accomplished through adjustment of a low-pass filter. Since the plant approximates a nominal model at low frequencies, overall closed-loop dynamics are usually well known and feedforward techniques are often applied, but there are some shortcomings with this approach. A fatal one is that a satisfactory control can hardly be obtained when the types of the disturbances are unknown and the model uncertainties exist [17]. Another is that the proposed formulation can only cope with some low-frequency disturbances. The second approach makes the best use of the control [12]. The shortcoming with this approach merits in is that it can only cope with step-type disturbances. From the point of view of robust control, the desirable properties of variable structure control systems are well documented in [16] and [18]. The advantage of this approach lies in that only the upper bounds of the uncertainties are required. It should be pointed out that the equivalent control method is very effective for estimating the unknown parts of a plant with known relative degrees. So, we are inspired to apply the VSS theory to the disturbance observer formulation. It is known that the disturbance can be estimated by using the VSS equivalent control theory for minimum-phase dynamical systems with relative degree one (with respect to the relationship between the disturbance and output) [16], but for the systems with higher relative degrees, no result has been reported. Usually, it is regarded as impossible. This is because the VSS-type estimation is constructed by discontinuous functions. In this paper, a disturbance estimation method based on the VSS equivalent control is proposed for the minimum-phase dynamical systems (with respect to the relationship between the disturbance and output) with arbitrary relative degrees. Only the upper and lower bounds of the disturbance are assumed to be known as the a priori information. By first estimating the disturbance through a higher order filter, the disturbance through a lower order filter is inductively estimated. Eventually, the disturbance is estimated. In the proposed formulation, the traditional discontinuous methods are approximated by differentiable approaches. The estimation error can be designed to be as small as it is needed. The organization of this paper is as follows. Section II gives the problem formulation. Section III presents the proposed robust disturbance observer. In Section IV, computer simulations and experiments are carried out to verify the practicality and effectiveness of the proposed approach. Also, the performances of the proposed observer and the traditional observer are compared.

0278–0046/00$10.00 © 2000 IEEE

430

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

Inspired by the formulation of the state observer for a possibly unstable system [9], we introduce a Hurwitz polynomial , . Thus, (6) can be rewritten as

II. PROBLEM STATEMENT Consider an uncertain system of the form (1) , and where denotes the differential operator are scalar input and output, respectively, is an unknown signal composed of the model uncertainties, the nonlinear parts , , and are of the system and the disturbances, and described by (2) (3) (4) In this paper, we make the following assumptions. , , and are known, and 1) The parameters in and are coprime. is a Hurwitz polynomial (i.e., the system is in 2) minimum phase with respect to the relation of disturand for . bance–output), where is bounded by an algebraic func3) The disturbance , , and , i.e., tion of

(7) Step 1: Corresponding to (7), consider the model (8) where

is the input chosen as

(the fact is used). Then, (as ). By using it is easy to prove that can be regarded as the esthe equivalent control method, . The above approach is the traditional timate of standard VSS-type disturbance estimation method. , we want to estimate the disturStep 2: By employing . Based on the following identical differential equabance tion: (9)

(5) is a known function. where is called the “disturbance” of For simplicity, the signal the system in the following sections of this paper. This paper deals with the disturbance estimation problem for minimum-phase dynamical systems with relative degree [by Assumption 2)].

we consider the dynamical system described by (10) where input. If

is the estimate of is chosen as

, and

is the

III. NEW DISTURBANCE OBSERVER It should be pointed out that the discussions in this section, as well as the corresponding discussions in the other sections, are based on the transfer function method which inherently assumes zero initial conditions for all internal states of the system. It is known that this treatment does not lose any generality, since, for a stable closed-loop linear system, nonzero initial conditions only contribute to the solution of the state (or the system output) an additive term which decays to zero exponentially. Thus, the initial conditions of the filtered input, filtered output, and filtered disturbance can be assumed to be zero. In this paper, the starting time is supposed as . A. An Introductory Example The following example is given to introduce the proposed recursive procedure by using the basic VSS control theory. Consider the system described by (6) is unknown, , but is known. Our where by using the measurement of the purpose is to estimate and the a priori knowledge . output

then it can be easily proved that , is not an available where the problem is that is the estimate of , we signal, but, as by in the conare inspired to replace . Thus, the input should be chosen as trol input . Remark 1: For simplicity, the above analysis is carried out by using the discontinuous functions. It should be pointed out that, cannot be impleas the discontinuous function mented by a digital computer, it is approximately smoothened , where , in the following as sections. Thus, the estimation error which is determined by the parameter is still retained. Fig. 1 shows the design block dia, the funcgram for this disturbance observer. For a variable , , and in Fig. 1 are defined as tions

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

431

It should be noted that

is an available signal because it is composed of the output and the filters input and output. Hinted by the introduced example, based on (15), the general disturbance estimation method for system (1) is summarized in the next theorem. ( ), Theorem 1: For small positive constants construct the following dynamical systems described by

Fig. 1. Block diagram of the new observer for the two-order input unknown system with relative degree two.

(16) and where

, and

. for

B. New Disturbance Observer Motivated by the introduced example, the general recursive design method employing integrators is proposed in this section. th-order filter of the disturbance is In the th step, the estimated. In the final th step, the disturbance is estimated. As the VSS method will be employed, first of all, the upper bounds of the filters of the disturbance must be estimated. For positive constant , by applying (11) it is obvious that the upper bound of inductively estimated as

can be

(12) Remark 2: By the definitions in (12), it is obvious that . First, some operations are carried out on (1). Inspired by the techniques employed in [9], we introduce a Hurwitz polynomial as (13) determines the convergence where the design parameter speed and the precision of the estimating error, as we will see later in this section. yields Dividing both sides of (1) by (14) Multiplying (14) by

yields

(15)

where

and

(

(17)

) are given by (18)

and

for

(19)

and ( ) are signals generated respectively; by (16) and (17), respectively. It can be concluded that, when is very small, can be approximately regarded as for the corresponding estimates of as is large enough. Particularly, is the approximate , i.e., there exist and estimate of such that (20) as . where Proof: See Appendix A. Remark 3: Theorem 1 gives a disturbance estimation method for minimum phase dynamical systems with arbitrarily relative degrees by using the VSS theory. The smaller the parameters ( ) are chosen, the smaller the estimate error of the disturbance will be. For the high-frequency disturbances, the parameters should be chosen to be very small in order to get good estimates. Remark 4: The design parameter should be chosen to be large enough to raise the estimating speed [see (A2) and (A8)] if the controlled systems require, but if is chosen to be too large, the estimation precision may become bad. This is beand cause the estimation error is determined by . By referring to the proof of Theorem 1, it can and can be made very small, be see that, although and may not be very small when is chosen large. Simulation results will show how the performance of the new observer depends on the parameter .

432

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

Remark 5: Only the a priori information about the upper is required in our new formulation. The type of bound of the disturbance is not needed to be known as the a priori information. However, the traditional disturbance observers should be designed differently (i.e., different orders) for different types of disturbances in order to get good estimates [17]. As we can hardly know the types of the disturbances and the disturbances are usually composed of various types of signals, it is very difficult to determine the order of the traditional disturbance observer. Remark 6: We need not consider the variations of the model parameters in the new observer. The varying parts are regarded is not imporas a part of the disturbance. The polynomial is a tant to us; , only its order and the assumption that Hurwitz polynomial are important to us. As a matter of fact, is replaced by a designed Hurwitz polynomial in our formulation. Remark 7: From Remarks 5 and 6, we can conclude that the new disturbance observer is of high robustness with respect to the types of the disturbances and model uncertainties. By referring to [17], it can be concluded that the new observer works like the imaginary traditional observer with infinity order theoretically. Until now, the discussions are based on the theoretical analysis which is in the analog case. In Section IV, computer simulations and experimental results will be presented to illustrate the new formulation. IV. SIMULATION AND EXPERIMENTAL RESULTS In this section, the new disturbance observer is applied to control a linear motor. The model of the linear motor can be described by the following equation: (21) where position; control input (current); unknown term composed of the model uncertainties, the friction forces, the interaction forces, etc.; and unknown time-varying parameters. Now, we rewrite system (21) as (22) is the parameter of the nominal plant, and where is the unknown signal composed of the model uncertainties and the disturbances. Suppose and the starting time is . , the Hurwitz polynomial in (13) can be As . Then, we have chosen as (23) From Theorem 1, we construct the following two equations:

(24) (25)

where

and

are determined by

(26) So, mate of

and

can be regarded as the approximate estiand , respectively.

A. Simulation Test of the New Observer The behavior of the new disturbance observer can be clarified is canceled in by using computer simulations. As the signal the formulation of the disturbance observer, it can be assigned any signal (in the following simulation processes, it is assigned ). The units of the variables are not considered in the computer simulation process. The sampling period is chosen as s. : For the disturbance 1) Characteristics of , , and

(suppose its upper bound is known as )], Fig. 2 (with , is chosen as ) shows the the condition differences between the real disturbance and their estimates by . It can be seen that the using different parameters parameters determine the estimating precision. Remark 8: In order to get good estimates of the disturbances, the parameters in the new observer should be chosen to be very small. When the analog signals are implemented by a digital computer, the parameters should not be chosen to be much , where is the sampling smaller than is defined in (12). Othperiod, is the ending time, and defined in (18) erwise, as the variations of the functions and (19) will be too fast with respect to the sampling frequency, (16) and (17) cannot be precisely solved. Remark 9: Remark 8 also tells us that the new observer has some limitations for high-frequency disturbances and large amplitude disturbances if the digital computer is employed. (suppose its upper bound is For the disturbance ), Fig. 3 (with the condition , known as , and ) shows the differences between the real disturbance and its estimates by using different parameters . It can be seen that the parameter determines the estimating speed and precision. When the parameter is chosen to be too large, the estimating error will be large even though the parameters are chosen to be very small. Remark 10: For a controlled system, the parameters and should be appropriately chosen to meet the particular requirement. , Fig. 4 (with the initial condition For the disturbance and the choices , , ) shows the simulation results of the differences between . the disturbance and its estimates for different also influences the esRemark 11: Fig. 4 shows that timating speed. For definite parameters , appropriately large can result in a good estimation. The proofs of Relations

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

433

Fig. 2. Difference between the step disturbance and its estimates by using different parameters .

Fig. 4. Simulation results of the differences between the disturbance v(t) = t and its estimates for different (t).

Fig. 3. Differences between the real disturbance and its estimates by using different parameters .

Fig. 5. Difference between the disturbance (27) and its estimate by using the new observer.

(A2) and (A8) also imply this fact, but, in the practical applicannot be very large due to the energy limitation. cations, may reMoreover, by referring to Remark 8, a very large sult in a bad estimation when the signals are implemented by a digital computer (sinces the parameters should be chosen large for a definite sampling frequency). Thus, an appropriate a is necessary to get a good estimation. priori knowledge 2) Ability of Observing Complicated Disturand the imposed disturbance bances: Suppose is described by (27) where and

. is composed of varIt can be seen that the disturbance ious kinds of disturbances. Suppose the upper bound of is known as . Fig. 5 shows the difference between the real disturbance and its estimate by using the new observer , , where the parameters are chosen as . To get a similar good estimation, the tradiand , tional observer should be chosen as and . where 3) Performance in the Presence of Measurement Noises: As only the measurement of the output is employed in our formulation, corresponding to (1), we just consider the system with output measurement noise (Gaussian white noises). Suppose the measured output is (28)

where is the genuine system output and is a Gaussian . Under this assumption, white noise with the amplitude in the formulation (24)–(26) should be replaced the signal . by Suppose the disturbance is same as (27). In the presence of measurement noises, in order to get good estimates, the parame, ters should be chosen as in the new observer (when ), while the parameters in the traditional observer should be chosen as , (the response is same as the new observer). The simulation result is shown in Fig. 6. Remark 12: In the presence of measurement noises, the parameters in the new observer should be chosen to be much larger, while the parameters in the traditional observer should be chosen to be much smaller when comparing to the case in which the measurement noises are absent. Remark 13: For a strictly proper system, we do not consider the case when a very high-frequency noise is accompanied with the input or involved in the disturbance because the influence over the output of this kind noise is very small and can be omitted. B. Experimental Test of the New Observer is unknown, we As the real quantity of the disturbance cannot investigate the performance of our new disturbance observer directly by an experiment, but, by studying the response of a controlled system, the effectiveness of the new observer can be confirmed indirectly. The experimental system is shown in Fig. 7. The specifications of the linear motor are shown in

434

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

TABLE I SPECIFICATIONS OF LINEAR MOTOR

Fig. 6. Difference between the disturbance (27) and its estimate in the presence of measurement noises.

Fig. 7. Experimental system for the linear motor.

Table I. Thus, the parameter of the nominal model is calcu. For this linear motor, the lated as is known as . The control purupper bound of to track a desired position . pose is to force the output is determined as The control

Fig. 8. Experimental results of the position-controlled linear motor by using the new observers when the impulse disturbance is added.

(29) is the estimated disturbance used to cancel the diswhere , and is the estimated speed. The poles of the turbance rad/s. The sampling position controller are set to s. period is From Theorem 1, it can be seen that the new speed observer can be formulated as Fig. 9. Experimental results of the position-controlled linear motor by using the traditional observers when the impulse disturbance is added.

(30) By employing the controller (29), the best performance of the controlled motor using the new observer is very similar to that using the traditional observer, where the parameters in the new disturbance observer and the new speed observer are chosen as , , , , and , while the traditional disturbance observer and traditional speed observer are designed as and , respectively. Moreover, the impulse disturbance (width: 100 s; magnitude: 7.5) is added for every 210 ms during the position control. The tracking error of the position controlled motor employing the new nonlinear disturbance observer is better (about 35% than that employing the traditional disturbance observer).

Fig. 8 shows the experiment results where the new disturbance observer and new speed observer are used. Fig. 9 shows the experiment results where the traditional disturbance observer and traditional speed observer are used. The desired position (2.5 cm/div) in the top of the two figures are the same. The signals in the middle of the two figures are the tracking errors cm/div). The signals in the bottom show the estimated ( (6.7 cm/s/div). speeds V. CONCLUSIONS In this paper, based on the VSS equivalent control method, the disturbance observer has been constructed for minimumphase dynamical systems with arbitrarily relative degrees. Here, the term “disturbance” is referred to as the combination of the model uncertainties, the nonlinear parts of the system, and the external disturbances. Only the upper and lower bounds of the

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

435

disturbances are assumed as the a priori information. Simulation results show that the proposed observer is of high robustness to the types of the disturbances and the model uncertainties. Experimental results confirm the practicality of the new observer. ) and , By choosing small design parameters ( the performance of the new observer may become better, even for the high-frequency disturbances, but there is a limitation to the choice of these parameters when the measurement noise is present. When it is implemented by a digital computer, the design parameters are also limited by the sampling period. The proposed formulation is expected to be extended to multiinput multi-output systems. For systems with relatively large stochastic disturbances or measurement noises, the proposed observer is expected to be modified.

(A4) Consider the dynamical system described by (A5) where

is determined as (A6)

is a signal which can be obtained by solving (A5). . Then, from (A4) Let and (A5), we have (A7)

APPENDIX A PROOF OF THEOREM 1

It can be concluded that

Mathematical induction principle will be employed to prove this theorem. will be estimated. Step 1: Based on (15), For this purpose, let us consider the dynamical system described by

(16) where

Step 2: We will use to approximately estimate by appealing to the following identical differential equation:

is the input determined as

as

(A8)

and are positive constants. The proof of (A8) is where given in Appendix C. , we can see that as . Thus, as By the equivalent control method, from (A7), it is obvious that can be regarded as the estimate of as is sufficiently large. So, for very small positive constants ( be approximately regarded as the estimate of as is sufficiently large. Thus, there exist such that

),

and a quantity

(18) which can be computed by solving (16) is the estimate of . . Then, combining (15) and (16) yields Let

(A9) where Step a—(

as . ): Based on the identical equation

(A1) It can be proved that

is uniformly bounded and as

can

(A10) we consider the dynamical system described by

(A2)

is a small constant. The proof of relation (A2) is where given in Appendix B. , we can see that as . By the As equivalent control method, from (A1), it can be concluded that can be regarded as the as is sufficiently large. estimate of can be apSo, for a very small positive constant , as proximately regarded as the estimate of is sufficiently large. Thus, there exist and a quantity such that

(A11) where

is determined as (A12)

is a signal generated by (A11). In (A12), is the obtained in the th step. estimate of . Similarly Let can be controlled to to Appendix C, we can prove that be very small by choosing small . Thus, can be , approximately regarded as the estimate of and such that i.e., there exist (A13)

(A3) for all

, where

as

.

, where as . for all By combining the above analysis, Theorem 1 is proved.

436

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

Let

APPENDIX B PROOF OF RELATION (A2) From (A1), differentiating

be the upper bound of

yields Then, by using (A3) and (A17), (A16) can be estimated as

as

(A18)

Now, let (A19) (A14)

It follows that equation (A19) yields

as

. Solving the differential

Based on (A14), it can be easily obtained that

as

(A15)

Thus, relation (A2) is proved.

as Thus, relation (A8) is proved.

APPENDIX C PROOF OF RELATION (A8) From (A7), differentiating

(A20)

ACKNOWLEDGMENT

yields

The authors would like to thank the reviewers for their invaluable suggestions and criticisms, which were helpful in the improvement of the manuscript. REFERENCES

(A16) The second term of the right side of (A16) can be estimated as

(A17)

[1] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. Berlin, Germany: Springer-Verlag, 1995. [2] U. Christen, H. E. Musch, and M. Steiner, “Robust control of distillation columns: -vs H -synthesis,” J. Process Contr., vol. 7, no. 1, pp. 19–30, 1997. [3] W. R. Cluett, L. Wang, and A. Zivkovic, “Development of quality bounds for time and frequency domain models: Application to the shell distillation column,” J. Process Contr., vol. 7, no. 1, pp. 75–80, 1997. [4] S. J. Cusumano and K. Poolla, “Nonlinear feedback versus linear feedback for robust stabilization,” in Proc. 27th Conf. Decision and Control, Austin, TX, 1988, pp. 1776–1780. [5] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “Statespace solutions to standard H and H control problems,” IEEE Trans. Automat. Contr., vol. 34, pp. 831–847, Aug. 1989. [6] C. Edwards and S. K. Spurgeon, “Robust output tracking using a sliding-mode controller/observer scheme,” Int. J. Contr., vol. 64, no. 5, pp. 967–983, 1996. [7] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984. [8] P. A. Ioannou and K. S. Tsakalis, “A robust direct adaptive controller,” IEEE Trans. Automat. Contr., vol. 31, pp. 1033–1043, Nov. 1986. [9] G. Kreisselmeier, “Adaptive observer with exponential rate of convergence,” IEEE Trans. Automat. Contr., vol. 22, pp. 2–8, Jan. 1977. [10] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, “Nonlinear design of adaptive controllers for linear systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 738–752, Apr. 1994. [11] R. Lozano and X. H. Zhao, “Adaptive pole placement without excitation probing signals,” IEEE Trans. Automat. Contr., vol. 39, pp. 47–58, Jan. 1994. [12] T. Mita, M. Hirata, and K. Murata, “Theory of H control and disturbance observer” (in Japanese), Trans. Inst. Elect. Eng. Jpn., vol. 115-C, no. 8, pp. 1002–1011, 1995. [13] S. Stogestad, M. Morari, and J. C. Doyle, “Robust control of ill-conditioned plants: High-purity distillation,” IEEE Trans. Automat. Contr., vol. 33, pp. 1092–1105, July 1988.

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

437

[14] S. Tøffner-Clausen, System Identification and Robust Control: A Case Study Approach. Berlin, Germany: Springer-Verlag, 1996. [15] T. Umeno, T. Kaneko, and Y. Hori, “Robust servosystem design with two-degrees-of-freedom and its application to novel motion control of robust manipulators,” IEEE Trans. Ind. Electron., vol. 40, pp. 473–485, Oct. 1993. [16] V. I. Utkin, Sliding Modes in Control Optimization. New York: Springer-Verlag, 1992. [17] K. Yamada, S. Komada, M. Ishida, and T. Hori, “Characteristic of servo system using high order disturbance observer,” in Proc. 35th Conf. Decision and Control (CDC’96), Kobe, Japan, 1996, pp. 3252–3257. [18] A. S. I. Zinober, Ed., Variable Structure and Lyapunov Control. London, U.K.: Springer-Verlag, 1994.

Satoshi Komada (M’89) was born in Bangkok, Thailand, in 1964. He received the B.E., M.E., and Ph.D. degrees from Keio University, Yokohama, Japan, in 1987, 1989, and 1994, respectively, all in electrical engineering. Since 1989, he has been with Mie University, Tsu, Japan, where he is an Associate Professor of Electrical and Electronic Engineering. His research interests include robotics, motion control, remote control, and nonlinear control. Dr. Komada is a member of the IEEE Industrial Electronics, IEEE Robotics and Automation, IEEE Systems, Man, and Cybernetics, and IEEE Control Systems Societies.

Xinkai Chen (M’96) was born in Hebei, China, in 1966. He received the B.S. and M.S. degrees in mathematics from Hebei University, Hebei, China, in 1986 and 1989, respectively, the M.E. degree in electrical engineering from Mie University, Tsu, Japan, in 1995, and the Ph.D. degree in engineering from Nagoya University, Nagoya, Japan, in 1999. From 1989 to 1992, he was with North China University of Electric Power. From 1995 to 1999, he was with the Department of Electrical and Electronic Engineering, Mie University, as an Assistant Professor. In April 1999, he joined the Department of Information Sciences, Tokyo Denki University, Saitama, Japan. His current research interests include robust nonlinear control, adaptive control, motion control, and machine vision.

Toshio Fukuda (M’83–F’95) graduated from Waseda University, Tokyo, Japan, in 1971 and received the M.S. and Dr.Eng. degrees from the University of Tokyo, Tokyo, Japan, in 1973 and 1977, respectively. He also studied at the Graduate School, Yale University, New Haven, CT, from 1973 to 1975. In 1977, he joined the National Mechanical Engineering Laboratory, Japan. He was a Visiting Research Fellow at the University of Stuttgart, Stuttgart, Germany, from 1979 to 1980. He joined the Science University of Tokyo, Tokyo, Japan, in 1981 and then joined Nagoya University, Nagoya, Japan, in 1989. Currently, he is a Professor in the Center for Cooperative Research in Advanced Science and Technology, Nagoya University, mainly engaged in the research fields of intelligent robotic systems, mechatronics, and microrobotics. He was Vice President of the International Fuzzy Systems Association in 1997. Prof. Fukuda was the recipient of the IEEE Eugene Mittlemann Award in 1997. He was the Vice President of the IEEE Industrial Electronics Society from 1990 to 1999 and President of the IEEE Robotics and Automation Society during 1998–1999.

Lihat lebih banyak...
429

Design of a Nonlinear Disturbance Observer Xinkai Chen, Member, IEEE, Satoshi Komada, Member, IEEE, and Toshio Fukuda, Fellow, IEEE

Abstract—This paper presents a new disturbance observer based on the variable structure system theory for minimum-phase (with respect to the relationship between the disturbance and output) dynamical systems with arbitrary relative degrees. The model uncertainties and the nonlinear parts of the system are merged into the disturbance term and are regarded as a part of the disturbances. The upper and lower bounds of the disturbance are assumed to be known as a priori information. Simulation results are presented to show the robustness and effectiveness of the new disturbance observer. Experiment results show the practicality of the new observer. Index Terms—Minimum-phase systems, nonlinear disturbance observer, relative degree, variable structure system theory.

I. INTRODUCTION

I

N ALMOST all engineering control systems, the presence of disturbances, model uncertainties, and nonlinear model parts is inevitable. For example, when the robot manipulators grasp an unknown payload, they are affected by unknown inertia variation and gravity force, but these changes are rarely captured in the models. It is most desirable that the controller be insensitive to these uncertainties. Thus, in recent years, the problem of controlling uncertain dynamical systems subject to external disturbances has been a topic of considerable interest. Many robust control methods have been proposed. Among these control [1], [5], [12], methods, the most typical ones may be variable structure system (VSS) sliding-mode control [6], [16], [18], adaptive control [7], [8], [10], [11], -synthesis method [2], [14], dynamic versus parametric uncertainty control [4], [8], and robust process control [3], [13]. These methods can cope with different types of uncertain systems [4]. In this paper, the model uncertainties, the nonlinear parts of the system, and the external disturbances are merged into one term, and we refer it as the disturbances of the system. It might be argued that, if the disturbances can be estimated, the control problems of the systems with disturbances may become easier to solve. For example, the state observer and the controller with disturbance cancellation functions can be easily constructed by using the estimated disturbances. Among the many presented disturbance observer techniques, type the approximate differentiator type [15], [17] and [12] formulations have been popularly applied in the design of Manuscript received March 17, 1997; revised June 11, 1999. Abstract published on the Internet December 23, 1999. X. Chen is with the Department of Information Sciences, Tokyo Denki University, Saitama 350-0394, Japan (e-mail: [email protected]). S. Komada is with the Department of Electrical and Electronic Engineering, Mie University, Tsu 514, Japan. T. Fukuda is with the Department of Mechano-Informatics and Systems, Nagoya University, Nagoya 464-01, Japan (e-mail: [email protected]). Publisher Item Identifier S 0278-0046(00)02499-0.

tracking controllers for motion control systems. The procedure of the first approach closes an inner loop around the controlled plant to reject disturbances and force the input–output characteristics of this inner loop to approximate a “nominal” plant model at low frequencies. Tuning of the loop is accomplished through adjustment of a low-pass filter. Since the plant approximates a nominal model at low frequencies, overall closed-loop dynamics are usually well known and feedforward techniques are often applied, but there are some shortcomings with this approach. A fatal one is that a satisfactory control can hardly be obtained when the types of the disturbances are unknown and the model uncertainties exist [17]. Another is that the proposed formulation can only cope with some low-frequency disturbances. The second approach makes the best use of the control [12]. The shortcoming with this approach merits in is that it can only cope with step-type disturbances. From the point of view of robust control, the desirable properties of variable structure control systems are well documented in [16] and [18]. The advantage of this approach lies in that only the upper bounds of the uncertainties are required. It should be pointed out that the equivalent control method is very effective for estimating the unknown parts of a plant with known relative degrees. So, we are inspired to apply the VSS theory to the disturbance observer formulation. It is known that the disturbance can be estimated by using the VSS equivalent control theory for minimum-phase dynamical systems with relative degree one (with respect to the relationship between the disturbance and output) [16], but for the systems with higher relative degrees, no result has been reported. Usually, it is regarded as impossible. This is because the VSS-type estimation is constructed by discontinuous functions. In this paper, a disturbance estimation method based on the VSS equivalent control is proposed for the minimum-phase dynamical systems (with respect to the relationship between the disturbance and output) with arbitrary relative degrees. Only the upper and lower bounds of the disturbance are assumed to be known as the a priori information. By first estimating the disturbance through a higher order filter, the disturbance through a lower order filter is inductively estimated. Eventually, the disturbance is estimated. In the proposed formulation, the traditional discontinuous methods are approximated by differentiable approaches. The estimation error can be designed to be as small as it is needed. The organization of this paper is as follows. Section II gives the problem formulation. Section III presents the proposed robust disturbance observer. In Section IV, computer simulations and experiments are carried out to verify the practicality and effectiveness of the proposed approach. Also, the performances of the proposed observer and the traditional observer are compared.

0278–0046/00$10.00 © 2000 IEEE

430

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

Inspired by the formulation of the state observer for a possibly unstable system [9], we introduce a Hurwitz polynomial , . Thus, (6) can be rewritten as

II. PROBLEM STATEMENT Consider an uncertain system of the form (1) , and where denotes the differential operator are scalar input and output, respectively, is an unknown signal composed of the model uncertainties, the nonlinear parts , , and are of the system and the disturbances, and described by (2) (3) (4) In this paper, we make the following assumptions. , , and are known, and 1) The parameters in and are coprime. is a Hurwitz polynomial (i.e., the system is in 2) minimum phase with respect to the relation of disturand for . bance–output), where is bounded by an algebraic func3) The disturbance , , and , i.e., tion of

(7) Step 1: Corresponding to (7), consider the model (8) where

is the input chosen as

(the fact is used). Then, (as ). By using it is easy to prove that can be regarded as the esthe equivalent control method, . The above approach is the traditional timate of standard VSS-type disturbance estimation method. , we want to estimate the disturStep 2: By employing . Based on the following identical differential equabance tion: (9)

(5) is a known function. where is called the “disturbance” of For simplicity, the signal the system in the following sections of this paper. This paper deals with the disturbance estimation problem for minimum-phase dynamical systems with relative degree [by Assumption 2)].

we consider the dynamical system described by (10) where input. If

is the estimate of is chosen as

, and

is the

III. NEW DISTURBANCE OBSERVER It should be pointed out that the discussions in this section, as well as the corresponding discussions in the other sections, are based on the transfer function method which inherently assumes zero initial conditions for all internal states of the system. It is known that this treatment does not lose any generality, since, for a stable closed-loop linear system, nonzero initial conditions only contribute to the solution of the state (or the system output) an additive term which decays to zero exponentially. Thus, the initial conditions of the filtered input, filtered output, and filtered disturbance can be assumed to be zero. In this paper, the starting time is supposed as . A. An Introductory Example The following example is given to introduce the proposed recursive procedure by using the basic VSS control theory. Consider the system described by (6) is unknown, , but is known. Our where by using the measurement of the purpose is to estimate and the a priori knowledge . output

then it can be easily proved that , is not an available where the problem is that is the estimate of , we signal, but, as by in the conare inspired to replace . Thus, the input should be chosen as trol input . Remark 1: For simplicity, the above analysis is carried out by using the discontinuous functions. It should be pointed out that, cannot be impleas the discontinuous function mented by a digital computer, it is approximately smoothened , where , in the following as sections. Thus, the estimation error which is determined by the parameter is still retained. Fig. 1 shows the design block dia, the funcgram for this disturbance observer. For a variable , , and in Fig. 1 are defined as tions

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

431

It should be noted that

is an available signal because it is composed of the output and the filters input and output. Hinted by the introduced example, based on (15), the general disturbance estimation method for system (1) is summarized in the next theorem. ( ), Theorem 1: For small positive constants construct the following dynamical systems described by

Fig. 1. Block diagram of the new observer for the two-order input unknown system with relative degree two.

(16) and where

, and

. for

B. New Disturbance Observer Motivated by the introduced example, the general recursive design method employing integrators is proposed in this section. th-order filter of the disturbance is In the th step, the estimated. In the final th step, the disturbance is estimated. As the VSS method will be employed, first of all, the upper bounds of the filters of the disturbance must be estimated. For positive constant , by applying (11) it is obvious that the upper bound of inductively estimated as

can be

(12) Remark 2: By the definitions in (12), it is obvious that . First, some operations are carried out on (1). Inspired by the techniques employed in [9], we introduce a Hurwitz polynomial as (13) determines the convergence where the design parameter speed and the precision of the estimating error, as we will see later in this section. yields Dividing both sides of (1) by (14) Multiplying (14) by

yields

(15)

where

and

(

(17)

) are given by (18)

and

for

(19)

and ( ) are signals generated respectively; by (16) and (17), respectively. It can be concluded that, when is very small, can be approximately regarded as for the corresponding estimates of as is large enough. Particularly, is the approximate , i.e., there exist and estimate of such that (20) as . where Proof: See Appendix A. Remark 3: Theorem 1 gives a disturbance estimation method for minimum phase dynamical systems with arbitrarily relative degrees by using the VSS theory. The smaller the parameters ( ) are chosen, the smaller the estimate error of the disturbance will be. For the high-frequency disturbances, the parameters should be chosen to be very small in order to get good estimates. Remark 4: The design parameter should be chosen to be large enough to raise the estimating speed [see (A2) and (A8)] if the controlled systems require, but if is chosen to be too large, the estimation precision may become bad. This is beand cause the estimation error is determined by . By referring to the proof of Theorem 1, it can and can be made very small, be see that, although and may not be very small when is chosen large. Simulation results will show how the performance of the new observer depends on the parameter .

432

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

Remark 5: Only the a priori information about the upper is required in our new formulation. The type of bound of the disturbance is not needed to be known as the a priori information. However, the traditional disturbance observers should be designed differently (i.e., different orders) for different types of disturbances in order to get good estimates [17]. As we can hardly know the types of the disturbances and the disturbances are usually composed of various types of signals, it is very difficult to determine the order of the traditional disturbance observer. Remark 6: We need not consider the variations of the model parameters in the new observer. The varying parts are regarded is not imporas a part of the disturbance. The polynomial is a tant to us; , only its order and the assumption that Hurwitz polynomial are important to us. As a matter of fact, is replaced by a designed Hurwitz polynomial in our formulation. Remark 7: From Remarks 5 and 6, we can conclude that the new disturbance observer is of high robustness with respect to the types of the disturbances and model uncertainties. By referring to [17], it can be concluded that the new observer works like the imaginary traditional observer with infinity order theoretically. Until now, the discussions are based on the theoretical analysis which is in the analog case. In Section IV, computer simulations and experimental results will be presented to illustrate the new formulation. IV. SIMULATION AND EXPERIMENTAL RESULTS In this section, the new disturbance observer is applied to control a linear motor. The model of the linear motor can be described by the following equation: (21) where position; control input (current); unknown term composed of the model uncertainties, the friction forces, the interaction forces, etc.; and unknown time-varying parameters. Now, we rewrite system (21) as (22) is the parameter of the nominal plant, and where is the unknown signal composed of the model uncertainties and the disturbances. Suppose and the starting time is . , the Hurwitz polynomial in (13) can be As . Then, we have chosen as (23) From Theorem 1, we construct the following two equations:

(24) (25)

where

and

are determined by

(26) So, mate of

and

can be regarded as the approximate estiand , respectively.

A. Simulation Test of the New Observer The behavior of the new disturbance observer can be clarified is canceled in by using computer simulations. As the signal the formulation of the disturbance observer, it can be assigned any signal (in the following simulation processes, it is assigned ). The units of the variables are not considered in the computer simulation process. The sampling period is chosen as s. : For the disturbance 1) Characteristics of , , and

(suppose its upper bound is known as )], Fig. 2 (with , is chosen as ) shows the the condition differences between the real disturbance and their estimates by . It can be seen that the using different parameters parameters determine the estimating precision. Remark 8: In order to get good estimates of the disturbances, the parameters in the new observer should be chosen to be very small. When the analog signals are implemented by a digital computer, the parameters should not be chosen to be much , where is the sampling smaller than is defined in (12). Othperiod, is the ending time, and defined in (18) erwise, as the variations of the functions and (19) will be too fast with respect to the sampling frequency, (16) and (17) cannot be precisely solved. Remark 9: Remark 8 also tells us that the new observer has some limitations for high-frequency disturbances and large amplitude disturbances if the digital computer is employed. (suppose its upper bound is For the disturbance ), Fig. 3 (with the condition , known as , and ) shows the differences between the real disturbance and its estimates by using different parameters . It can be seen that the parameter determines the estimating speed and precision. When the parameter is chosen to be too large, the estimating error will be large even though the parameters are chosen to be very small. Remark 10: For a controlled system, the parameters and should be appropriately chosen to meet the particular requirement. , Fig. 4 (with the initial condition For the disturbance and the choices , , ) shows the simulation results of the differences between . the disturbance and its estimates for different also influences the esRemark 11: Fig. 4 shows that timating speed. For definite parameters , appropriately large can result in a good estimation. The proofs of Relations

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

433

Fig. 2. Difference between the step disturbance and its estimates by using different parameters .

Fig. 4. Simulation results of the differences between the disturbance v(t) = t and its estimates for different (t).

Fig. 3. Differences between the real disturbance and its estimates by using different parameters .

Fig. 5. Difference between the disturbance (27) and its estimate by using the new observer.

(A2) and (A8) also imply this fact, but, in the practical applicannot be very large due to the energy limitation. cations, may reMoreover, by referring to Remark 8, a very large sult in a bad estimation when the signals are implemented by a digital computer (sinces the parameters should be chosen large for a definite sampling frequency). Thus, an appropriate a is necessary to get a good estimation. priori knowledge 2) Ability of Observing Complicated Disturand the imposed disturbance bances: Suppose is described by (27) where and

. is composed of varIt can be seen that the disturbance ious kinds of disturbances. Suppose the upper bound of is known as . Fig. 5 shows the difference between the real disturbance and its estimate by using the new observer , , where the parameters are chosen as . To get a similar good estimation, the tradiand , tional observer should be chosen as and . where 3) Performance in the Presence of Measurement Noises: As only the measurement of the output is employed in our formulation, corresponding to (1), we just consider the system with output measurement noise (Gaussian white noises). Suppose the measured output is (28)

where is the genuine system output and is a Gaussian . Under this assumption, white noise with the amplitude in the formulation (24)–(26) should be replaced the signal . by Suppose the disturbance is same as (27). In the presence of measurement noises, in order to get good estimates, the parame, ters should be chosen as in the new observer (when ), while the parameters in the traditional observer should be chosen as , (the response is same as the new observer). The simulation result is shown in Fig. 6. Remark 12: In the presence of measurement noises, the parameters in the new observer should be chosen to be much larger, while the parameters in the traditional observer should be chosen to be much smaller when comparing to the case in which the measurement noises are absent. Remark 13: For a strictly proper system, we do not consider the case when a very high-frequency noise is accompanied with the input or involved in the disturbance because the influence over the output of this kind noise is very small and can be omitted. B. Experimental Test of the New Observer is unknown, we As the real quantity of the disturbance cannot investigate the performance of our new disturbance observer directly by an experiment, but, by studying the response of a controlled system, the effectiveness of the new observer can be confirmed indirectly. The experimental system is shown in Fig. 7. The specifications of the linear motor are shown in

434

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

TABLE I SPECIFICATIONS OF LINEAR MOTOR

Fig. 6. Difference between the disturbance (27) and its estimate in the presence of measurement noises.

Fig. 7. Experimental system for the linear motor.

Table I. Thus, the parameter of the nominal model is calcu. For this linear motor, the lated as is known as . The control purupper bound of to track a desired position . pose is to force the output is determined as The control

Fig. 8. Experimental results of the position-controlled linear motor by using the new observers when the impulse disturbance is added.

(29) is the estimated disturbance used to cancel the diswhere , and is the estimated speed. The poles of the turbance rad/s. The sampling position controller are set to s. period is From Theorem 1, it can be seen that the new speed observer can be formulated as Fig. 9. Experimental results of the position-controlled linear motor by using the traditional observers when the impulse disturbance is added.

(30) By employing the controller (29), the best performance of the controlled motor using the new observer is very similar to that using the traditional observer, where the parameters in the new disturbance observer and the new speed observer are chosen as , , , , and , while the traditional disturbance observer and traditional speed observer are designed as and , respectively. Moreover, the impulse disturbance (width: 100 s; magnitude: 7.5) is added for every 210 ms during the position control. The tracking error of the position controlled motor employing the new nonlinear disturbance observer is better (about 35% than that employing the traditional disturbance observer).

Fig. 8 shows the experiment results where the new disturbance observer and new speed observer are used. Fig. 9 shows the experiment results where the traditional disturbance observer and traditional speed observer are used. The desired position (2.5 cm/div) in the top of the two figures are the same. The signals in the middle of the two figures are the tracking errors cm/div). The signals in the bottom show the estimated ( (6.7 cm/s/div). speeds V. CONCLUSIONS In this paper, based on the VSS equivalent control method, the disturbance observer has been constructed for minimumphase dynamical systems with arbitrarily relative degrees. Here, the term “disturbance” is referred to as the combination of the model uncertainties, the nonlinear parts of the system, and the external disturbances. Only the upper and lower bounds of the

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

435

disturbances are assumed as the a priori information. Simulation results show that the proposed observer is of high robustness to the types of the disturbances and the model uncertainties. Experimental results confirm the practicality of the new observer. ) and , By choosing small design parameters ( the performance of the new observer may become better, even for the high-frequency disturbances, but there is a limitation to the choice of these parameters when the measurement noise is present. When it is implemented by a digital computer, the design parameters are also limited by the sampling period. The proposed formulation is expected to be extended to multiinput multi-output systems. For systems with relatively large stochastic disturbances or measurement noises, the proposed observer is expected to be modified.

(A4) Consider the dynamical system described by (A5) where

is determined as (A6)

is a signal which can be obtained by solving (A5). . Then, from (A4) Let and (A5), we have (A7)

APPENDIX A PROOF OF THEOREM 1

It can be concluded that

Mathematical induction principle will be employed to prove this theorem. will be estimated. Step 1: Based on (15), For this purpose, let us consider the dynamical system described by

(16) where

Step 2: We will use to approximately estimate by appealing to the following identical differential equation:

is the input determined as

as

(A8)

and are positive constants. The proof of (A8) is where given in Appendix C. , we can see that as . Thus, as By the equivalent control method, from (A7), it is obvious that can be regarded as the estimate of as is sufficiently large. So, for very small positive constants ( be approximately regarded as the estimate of as is sufficiently large. Thus, there exist such that

),

and a quantity

(18) which can be computed by solving (16) is the estimate of . . Then, combining (15) and (16) yields Let

(A9) where Step a—(

as . ): Based on the identical equation

(A1) It can be proved that

is uniformly bounded and as

can

(A10) we consider the dynamical system described by

(A2)

is a small constant. The proof of relation (A2) is where given in Appendix B. , we can see that as . By the As equivalent control method, from (A1), it can be concluded that can be regarded as the as is sufficiently large. estimate of can be apSo, for a very small positive constant , as proximately regarded as the estimate of is sufficiently large. Thus, there exist and a quantity such that

(A11) where

is determined as (A12)

is a signal generated by (A11). In (A12), is the obtained in the th step. estimate of . Similarly Let can be controlled to to Appendix C, we can prove that be very small by choosing small . Thus, can be , approximately regarded as the estimate of and such that i.e., there exist (A13)

(A3) for all

, where

as

.

, where as . for all By combining the above analysis, Theorem 1 is proved.

436

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 2, APRIL 2000

Let

APPENDIX B PROOF OF RELATION (A2) From (A1), differentiating

be the upper bound of

yields Then, by using (A3) and (A17), (A16) can be estimated as

as

(A18)

Now, let (A19) (A14)

It follows that equation (A19) yields

as

. Solving the differential

Based on (A14), it can be easily obtained that

as

(A15)

Thus, relation (A2) is proved.

as Thus, relation (A8) is proved.

APPENDIX C PROOF OF RELATION (A8) From (A7), differentiating

(A20)

ACKNOWLEDGMENT

yields

The authors would like to thank the reviewers for their invaluable suggestions and criticisms, which were helpful in the improvement of the manuscript. REFERENCES

(A16) The second term of the right side of (A16) can be estimated as

(A17)

[1] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. Berlin, Germany: Springer-Verlag, 1995. [2] U. Christen, H. E. Musch, and M. Steiner, “Robust control of distillation columns: -vs H -synthesis,” J. Process Contr., vol. 7, no. 1, pp. 19–30, 1997. [3] W. R. Cluett, L. Wang, and A. Zivkovic, “Development of quality bounds for time and frequency domain models: Application to the shell distillation column,” J. Process Contr., vol. 7, no. 1, pp. 75–80, 1997. [4] S. J. Cusumano and K. Poolla, “Nonlinear feedback versus linear feedback for robust stabilization,” in Proc. 27th Conf. Decision and Control, Austin, TX, 1988, pp. 1776–1780. [5] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “Statespace solutions to standard H and H control problems,” IEEE Trans. Automat. Contr., vol. 34, pp. 831–847, Aug. 1989. [6] C. Edwards and S. K. Spurgeon, “Robust output tracking using a sliding-mode controller/observer scheme,” Int. J. Contr., vol. 64, no. 5, pp. 967–983, 1996. [7] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984. [8] P. A. Ioannou and K. S. Tsakalis, “A robust direct adaptive controller,” IEEE Trans. Automat. Contr., vol. 31, pp. 1033–1043, Nov. 1986. [9] G. Kreisselmeier, “Adaptive observer with exponential rate of convergence,” IEEE Trans. Automat. Contr., vol. 22, pp. 2–8, Jan. 1977. [10] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, “Nonlinear design of adaptive controllers for linear systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 738–752, Apr. 1994. [11] R. Lozano and X. H. Zhao, “Adaptive pole placement without excitation probing signals,” IEEE Trans. Automat. Contr., vol. 39, pp. 47–58, Jan. 1994. [12] T. Mita, M. Hirata, and K. Murata, “Theory of H control and disturbance observer” (in Japanese), Trans. Inst. Elect. Eng. Jpn., vol. 115-C, no. 8, pp. 1002–1011, 1995. [13] S. Stogestad, M. Morari, and J. C. Doyle, “Robust control of ill-conditioned plants: High-purity distillation,” IEEE Trans. Automat. Contr., vol. 33, pp. 1092–1105, July 1988.

CHEN et al.: DESIGN OF A NONLINEAR DISTURBANCE OBSERVER

437

[14] S. Tøffner-Clausen, System Identification and Robust Control: A Case Study Approach. Berlin, Germany: Springer-Verlag, 1996. [15] T. Umeno, T. Kaneko, and Y. Hori, “Robust servosystem design with two-degrees-of-freedom and its application to novel motion control of robust manipulators,” IEEE Trans. Ind. Electron., vol. 40, pp. 473–485, Oct. 1993. [16] V. I. Utkin, Sliding Modes in Control Optimization. New York: Springer-Verlag, 1992. [17] K. Yamada, S. Komada, M. Ishida, and T. Hori, “Characteristic of servo system using high order disturbance observer,” in Proc. 35th Conf. Decision and Control (CDC’96), Kobe, Japan, 1996, pp. 3252–3257. [18] A. S. I. Zinober, Ed., Variable Structure and Lyapunov Control. London, U.K.: Springer-Verlag, 1994.

Satoshi Komada (M’89) was born in Bangkok, Thailand, in 1964. He received the B.E., M.E., and Ph.D. degrees from Keio University, Yokohama, Japan, in 1987, 1989, and 1994, respectively, all in electrical engineering. Since 1989, he has been with Mie University, Tsu, Japan, where he is an Associate Professor of Electrical and Electronic Engineering. His research interests include robotics, motion control, remote control, and nonlinear control. Dr. Komada is a member of the IEEE Industrial Electronics, IEEE Robotics and Automation, IEEE Systems, Man, and Cybernetics, and IEEE Control Systems Societies.

Xinkai Chen (M’96) was born in Hebei, China, in 1966. He received the B.S. and M.S. degrees in mathematics from Hebei University, Hebei, China, in 1986 and 1989, respectively, the M.E. degree in electrical engineering from Mie University, Tsu, Japan, in 1995, and the Ph.D. degree in engineering from Nagoya University, Nagoya, Japan, in 1999. From 1989 to 1992, he was with North China University of Electric Power. From 1995 to 1999, he was with the Department of Electrical and Electronic Engineering, Mie University, as an Assistant Professor. In April 1999, he joined the Department of Information Sciences, Tokyo Denki University, Saitama, Japan. His current research interests include robust nonlinear control, adaptive control, motion control, and machine vision.

Toshio Fukuda (M’83–F’95) graduated from Waseda University, Tokyo, Japan, in 1971 and received the M.S. and Dr.Eng. degrees from the University of Tokyo, Tokyo, Japan, in 1973 and 1977, respectively. He also studied at the Graduate School, Yale University, New Haven, CT, from 1973 to 1975. In 1977, he joined the National Mechanical Engineering Laboratory, Japan. He was a Visiting Research Fellow at the University of Stuttgart, Stuttgart, Germany, from 1979 to 1980. He joined the Science University of Tokyo, Tokyo, Japan, in 1981 and then joined Nagoya University, Nagoya, Japan, in 1989. Currently, he is a Professor in the Center for Cooperative Research in Advanced Science and Technology, Nagoya University, mainly engaged in the research fields of intelligent robotic systems, mechatronics, and microrobotics. He was Vice President of the International Fuzzy Systems Association in 1997. Prof. Fukuda was the recipient of the IEEE Eugene Mittlemann Award in 1997. He was the Vice President of the IEEE Industrial Electronics Society from 1990 to 1999 and President of the IEEE Robotics and Automation Society during 1998–1999.

Somos uma comunidade de intercâmbio. Por favor, ajude-nos com a subida ** 1 ** um novo documento ou um que queremos baixar:

OU DOWNLOAD IMEDIATAMENTE