Sliding Mode Control With Time-varying

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 3, NO. 1, MARCH 1998 ... virtually maintenance free and attractive for various applica- tions [1]. In many AMB ... the transfer function of the LPV system becomes a linear time-invariant ..... Zurich, Switzerland: vdf Hochschuverlag AG an der ETH, 1994. [2] M. Fujita, K.
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 3, NO. 1, MARCH 1998

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Sliding Mode Control With Time-Varying Hyperplane for AMB Systems Selim Sivrioglu and Kenzo Nonami, Member, IEEE

Abstract— This paper deals with sliding mode hyperplane design for a class of linear parameter-varying (LPV) plants, the state-space matrices of which are an affine function of timevarying physical parameters. The proposed hyperplane, involving a linear matrix inequality (LMI) approach, has continuous dynamics due to scheduling parameters and provides stability and robustness against parametric uncertainties. We have designed a time-varying hyperplane for a rotor–magnetic bearing system with a gyroscopic effect, which can be considered an LPV plant due to parameter dependence on rotational speed. The obtained hyperplane is continuously scheduled with respect to rotational speed. We successfully carried out experiments using a commercially available turbomolecular pump system and results were reasonable and good. Index Terms—Active magnetic bearing system, gain-scheduling sliding mode control, linear parameter-varying system, turbomolecular pump.

A linear parameter-varying (LPV) plant definition can be described as a model of linear time-varying plants or nonlinear plants which are linearized using a vector of a time-varying . Using LPV plant definition, the gainparameter, such as control design is extensively presented in [10] scheduled and [11] in terms of linear matrix inequalities (LMI’s), the solution of which are within the scope of efficient convex optimization techniques. Here, our aims are to extend this approach to designing a sliding mode hyperplane and applying it to a practical system. All definitions and theorems given for LPV plants are also valid in our study and will not be repeated here. The state-space representation of LPV plants is described as (1)

I. INTRODUCTION

C

ONTROL system design for active magnetic bearings (AMB’s) is an advanced topic for control engineers, because of their highly complex structures, precise design requirements, and increasing applications in industry. The advantages of AMB’s, such as contactless and frictionless operation in normal running, without lubrication, make them virtually maintenance free and attractive for various applications [1]. In many AMB systems in use today, proportionalintegral-derivative (PID) controllers have been used because of certain practical advantages. However, it is not easy to satisfy the requirements for robust performance using PID control. Recently, robust control design approaches have been effectively studied for AMB control system design [2]–[4]. A variable structure system or sliding mode control uses a discontinuous control structure to control nonlinear systems [5], [6]. It is a powerful control method with increasing application in many areas of control engineering. Gain scheduling is a special type of nonlinear feedback used in a variety of engineering applications, such as flight control and process control [7]. Recent advances in robust control theory have offered a new theoretical framework and systematic gain scheduling [8]–[11].

where , and denote the state vector, the measured output vector, and the control input vector, respectively. is a vector of time-varying plant parameters, and all plant matrices are functions of the . For a frozen value of , such as , the transfer function of the LPV system becomes a linear time-invariant (LTI) system, such as (2) In practice, can be the time-varying physical parameters, such as velocity, damping, stiffness, etc., and be given between known extremal values such as (3) If measurements of the are available in real time during control operation, the designed controller has the same parameter dependence as an LPV plant. The controller form is (4) where is the measurement vector and is the control input. This controller has continuous adjustment with the parameter measurements against the variations in the plant dynamics and maintains stability and good performance. II. FREQUENCY-SHAPED HYPERPLANE DESIGN

Manuscript received March 3, 1997; revised June 20, 1997. Recommended by Technical Editor R. Isermann. The authors are with the Department of Mechanical Engineering, Chiba University, Chiba, 263 Japan (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1083-4435(98)02252-2.

A. LTI Control Systems The frequency-shaped-based hyperplane design is proposed control in [12], and this approach is extended using

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theory in [13]. If a plant has some unknown or neglected dynamics, such as truncation of high-frequency modes in a flexible system, the frequency-shaped-based hyperplane for sliding mode maintains stability and good performance. Suppose that a transformed system is given by

(5) , , , and is assumed full where rank. and represent the measured output vector and the disturbance input vector, respectively. Specifically, the control system is considered as a servo type and the error signal is defined by (6) where is the reference input signal. Now, we can offer a switching function for this control system as follows: (7) is a linear operator of . The offered switchwhere ing function will have some dynamics comparing with a conventional switching function in terms of the defined new variable :

(8) After combining (5) and (8), the extended system can be obtained by

Fig. 1. Sliding mode hyperplane as an

If the system is controllable, (9) and (10) are . The sliding mode hyuniquely decided with control for an LTI system is shown in Fig. perplane as an and are used for frequency 1. In this figure, filters shaping. B. LPV Control Systems Frequency-shaped hyperplane design can also be extended to LPV plants. Although the LPV plant is described in a very general form, the control design for an LPV plant requires some limitations, such as parameter independence of control inputs and measured outputs due to tractability reasons. Therefore, the canonical form will be obtained directly from the state-space equation of the considered plant by using a prefilter. Let us consider the state-space representation of the LPV plant:

(9) (10) Using the known equation for sliding mode equivalent control input is

, the

H1 control for an LTI system.

(14) Choose a filter, such as

(15) If we combine the state-space model of the plant with the above filter, we can obtain an augmented structure:

(11) From (7), we have

Supposing that the sliding mode occurs on of sliding mode is given by

(12)

(16)

, the equation

0, (16) reduces the canonical form, as given in (5). For If measurements of the time-varying parameter are available in real time, then the switching function will have parameter dependence as follows:

(13) (17)

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Fig. 2. Reduced-order system for LPV plants.

Finally, in the general form, the extended system can be written as

The derivative of the hyperplane can be obtained by

(18) (25) (19)

From (21),

The reduced-order system with scheduling parameter is shown in Fig. 2 for LPV plants. III. VSS CONTROLLER DESIGN FOR LPV PLANTS We seek a Lyapunov function such that (20) The following condition should be satisfied for existence of sliding mode and stabilization of the closed-loop system: (26)

(21) For affine parameter-dependent plants, the fixed Lyapunov function is replaced by a parameter-dependent Lyapunov function [14]. Affine parameter-dependent systems can be converted to polytopic parameter ones. In this paper, the compensator matrices are obtained as polytopic ones. For simplicity, we have supposed that the compensator is strictly proper, . The same design procedure is also valid for the proper case. For a single parameter case, such as , the affine parameter-dependent case is

where the equivalent control input the condition

is obtained by using

(27) The condition of (21) becomes

(22) and are known fixed matrices. This can be conwhere verted to a polytopic parameter-dependent matrix as follows: (23) where

,

,

, and

(28) The control input , which satisfies the condition of (28), can be chosen by

are obtained by

(29) Discontinuous control input

is given by

(24) (30)

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Fig. 3. Turbomolecular pump system.

Fig. 5. Bode plot of prefilter.

Fig. 4.

Model of rotor–active bearing system.

where and are selected as 0.1 and 0.01, respectively. is used for a smooth discontinuous control input. is specified with experimental experiences. For slow parameter variation, the derivative of parameter becomes very small and term can be neglected.

a lower side permanent magnetic bearing which has constant stiffness and damping effect and an upper side AMB which produces control inputs. Gyroscopic couples have considerable effects on the rotor and natural frequencies of the system vary with rotational frequency. A. Modeling of AMB System With Gyroscopic Effect The equations of motion of the rotor–magnetic bearing system given in Fig. 4 can be written by

IV. CONTROL OBJECT The control object presented here is a commercially available turbomolecular pump system (Fig. 3). Gain-scheduled control design has been previously studied for this system in [15]. We designed the controller only for radial directions. Axial direction is controlled by PID controllers. The plant has

Cos Sin (31)

SIVRIOGLU AND NONAMI: SLIDING MODE CONTROL WITH TIME-VARYING HYPERPLANE

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(a)

(b) Fig. 6. Bode plot of gain-scheduled compensator. (a) 2D. (b) 3D.

Control forces produced by the upper side AMB can be given by

displacement variables of the rotor with respect to the center of gravity measured in the fixed coordinate as follows: (35)

(32) and lower side forces are

(33)

The gyroscopic effects are characterized by a skew, which contains the rotor speed symmetric matrix as a linear factor. The unbalance effects also vary with the square function of rotational speed . The gyroscopic matrix and linearized unbalance matrix using some approximation are

Table I shows the parameters of the AMB system and the above equations can be rewritten in a more compact form by (34) where , and are the mass matrix, the damping matrix, the gyroscopic matrix, and the stiffness matrix, respectively. and indicate the control input and the disturbance represents the input locations, respectively. The vector

(36)

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TABLE I PARAMETERS OF AMB SYSTEM

Fig. 9. Control input for initial step input.

Fig. 7.

Fig. 10.

Closed-loop step response with sliding mode control.

Fig. 11.

Configuration of the experimental setup.

Displacement in the x direction for initial step input.

is the disturbance input vector, and vector. The system matrices are

Fig. 8. Switching function for initial step input.

is the measured output

where is a constant. Finally, the state-space model of this system can be obtained by (37) (38) where

is the state vector,

is the control input vector,

SIVRIOGLU AND NONAMI: SLIDING MODE CONTROL WITH TIME-VARYING HYPERPLANE

Fig. 12.

Simulink diagram of sliding mode controller.

Fig. 14. Fig. 13.

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Simulink diagram of

K1 .

B. Augmentation of State Space by Using a Prefilter We used a prefilter to augment the state space of the plant to canonical form. The filter given in Fig. 5 is selected, so as to give integral action to the sliding mode controller. Combining the filter and the plant, the augmented system is obtained as

(39)

C. Gain-Scheduled Compensator Design The control system accelerates from 0 to 10 000 r/min in 30 s and decelerates in breaking mode in the same amount of time. We define parameter dependence of the plant due to rotational speed as follows: or

r/min

Simulink diagram of

K2 .

We specified two frequency shaping filters for robust stability and sensitivity reduction. Our plant is open-loop unstable and has right-half-plane poles and zeros. A good way to select the corner frequency of the filters is to consider the unstable poles and zeros of the plant [16]. Unfortunately, this rule does not work completely for multi-input–multi-output (MIMO) systems. Therefore, the following way is chosen from experimental experience:

where is the unstable pole of the plant and , , . For every direction, the same frequency-shaping and filter is used. Finally, using the parameter-dependent plant with the above filters, the gain-scheduled compensator is computed using the LMI Control Toolbox in MATLAB1: (42) The interpolated structure of this compensator has the form

(40)

In this parameter variation range, the plant can be defined by (43) (41)

1 P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox For Use with MATLAB, MathWorks, Natick, MA, 1995.

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Fig. 15. Orbits. (a)

!z

(a)

(b)

(c)

(d)

= 4000 r/min. (b) !z = 6000 r/min. (c) !z = 8000 r/min. (d) !z = 10 000 r/min. V. SIMULATIONS

The order of the plant is eight. Using two weighting functions, the compensator is computed as twelfth order. The Bode plot of the compensator is given in Fig. 6 as twodimensional (2-D) and three-dimensional (3-D) plots. As can be seen in this figure, the compensator dynamics vary with rotational speed. The displacement of the rotor in the direction, the switching function, and the control input are shown in Figs. 7–9, respectively. The closed-loop system step response is given in Fig. 10. VI. EXPERIMENTAL RESULTS The configuration of the experimental setup is schematically shown in Fig. 11. The actual turbomolecular pump is used for the experiment.The sliding mode controller (Fig. 12) is installed on the digital signal processor (DSP) (TMS320C40), and and experiments are carried out. Figs. 13 and 14 show

compensators structure, respectively. Two displacements measured by two position sensors in the and directions go to the DSP through A/D and rotational speed signal converters. Two control inputs are supplied to electromagnets through D/A converters and power amplifiers. The sampling time of the controller is 0.275 ms. Generally speaking, it is impossible for rotating machinery to remove unbalance effects completely. Here, our aim is to maintain stability and robustness, even if the rotor has some unbalance effects. For this reason, we tested the controller by mounting a 0.6-g unbalance mass on the rotor. For four different rotational speeds, the orbits of the rotor central axis 6000 r/min, the obtained are shown in Fig. 15. Except orbits are synchronized and reasonable. The critical speed of the rigid rotor occurred at 6000 r/min and, for this reason, the diameter of the orbit at this rotational speed was increased, but the closed-loop system was still stable.

SIVRIOGLU AND NONAMI: SLIDING MODE CONTROL WITH TIME-VARYING HYPERPLANE

VII. CONCLUSIONS In this study, we have proposed a new type of hyperplane control. The LPV plant design based on gain-scheduled eigenvalues move by means of linear parameters, and control of this class of system using the conventional method is generally difficult. The gain-scheduled compensator has the same parameter dependence, and the eigenvalues of the compensator also vary with the parameters. As a result of the synchronized behavior of the plant and compensator, stability and robustness can be maintained. Due to the parameter dependence of the LPV plant, the existence of the sliding mode by a static hyperplane may not be guaranteed. The offered hyperplane has continuous dynamics with respect to time-varying parameters and the sliding mode occurs on this hyperplane in every case. REFERENCES [1] G. Schweitzer, H. Bleuler, and A. Traxler, Active Magnetic Bearings. Zurich, Switzerland: vdf Hochschuverlag AG an der ETH, 1994. [2] M. Fujita, K. Hatake, F. Matsumura, and K. Uchida, “An experimental evaluation and comparison of H = control for a magnetic bearing,” in Proc. 12th IFAC World Congr., 1993, vol. 4, p. 393. [3] F. Matsumura, T. Namerikawa, K. Hagiwara, and M. Fujita, “Application of gain scheduled H robust controllers to a magnetic bearing,” IEEE Trans. Contr. Syst. Technol., vol. 4, pp. 484–493, Sept. 1996. [4] K. Nonami and T. Ito, “ synthesis of flexible rotor bearing systems,” IEEE. Trans. Contr. Syst. Technol., vol. 4, pp. 503–512, Sept. 1996. [5] V. I. Utkin, Sliding Modes in Control Optimization. Berlin, Germany: Springer-Verlag, 1992. [6] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” in Proc. IEEE Int. Workshop on VSS, 1996, p. 1. [7] K. J. Astrom and B. Wittenmark, Adaptive Control. Reading, MA: Addison-Wesley, 1989. [8] J. Shamma and M. Athans, “Guaranteed properties of gain scheduled control of linear parameter-varying plants,” Automatica, vol. 27, no. 3, pp. 559–564, 1991. [9] A. Packard, “Gain scheduling via linear fractional transformations,” Syst. Contr. Lett., vol. 22, pp. 79–92, 1994. [10] P. Apkarian and P. Gahinet, “A convex characterization of gain scheduled H controllers,” IEEE. Trans. Automat. Contr., vol. 40, pp. 853–863, May 1995. [11] P. Apkarian, P. Gahinet, and G. Becker, “Self-scheduled H control of linear parameter-varying systems: A design example,” Automatica, vol. 31, no. 9, pp. 1251–1261, 1995.

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[12] D. Young and U. Ozguner, “Frequency shaping compensator design for sliding mode control,” Int. J. Contr., vol. 57, no. 5, p. 1005, 1993. [13] K. Nonami, H. Nishimura, and H. Tian, “H = control-based frequency-shaped sliding mode control for flexible structures,” JSME Int. J., ser. C, vol. 39, no. 3, pp. 493–501, 1996. [14] P. Gahinet, P. Apkarian, and M. Chilali, “Affine parameter-dependent Lyapunov functions and real parametric uncertainty,” IEEE. Trans. Automat. Contr., vol. 41, pp. 436–442, Mar. 1996. [15] S. Sivrioglu and K. Nonami, “LMI based gain scheduled H controller design for AMB systems under gyroscopic and unbalance disturbance effect,” in Proc. ISMB, 1996, p. 191. [16] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1992.

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Selim Sivrioglu received the B.S. and M.S. degrees in mechanical engineering from Yildiz Technical University, Istanbul, Turkey, in 1985 and 1988, respectively. He is currently working towards the Ph.D. degree at Chiba University, Chiba, Japan. From 1987 to 1994, he was a Research Assistant in the Mechanical Engineering Department, Yildiz Technical University. Since 1994, he has been conducting research as a Monbusho (Ministry of Education of Japan) Scholar in the Control and Robotics Laboratory, Mechanical Engineering Department, Chiba University. His research interests include linear matrix inequalities (LMI’s), gain-scheduled H control, and sliding mode control applications in mechatronics systems.

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Kenzo Nonami (M’97) received the Ph.D. degree from Tokyo Metropolitan University, Tokyo, Japan, in 1979. He is currently a Professor in the Department of Mechanical Engineering, Chiba University, Chiba, Japan, where he was a Research Associate from 1980 to 1988 and an Associate Professor from 1988 to 1994. He was a NASA NRC Research Associate from 1985 to 1986 and, also, in 1988. His current research interests are active vibration control for flexible structures, active control for magnetic bearings and magnetic levitation, control for locomotion robots and space robots, active noise control, robust control theory, and their applications.