Sliding mode control design via reduced order model approach ...

International Journal of Automation and Computing

04(4), October 2007, 329-334 DOI: 10.1007/s11633-007-0329-4

Sliding Mode Control Design via Reduced Order Model Approach B. Bandyopadhyay1,∗ 1

S. Janardhanan2

Victor Sreeram3

Interdisciplinary Programme in Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India 2

3

Alemayehu G/Egziabher Abera1

Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India

School of Electrical, Electronic, and Computer Engineering, The University of Western Australia, Western Australia 6009, Australia

Abstract: This paper presents a design of continuous-time sliding mode control for the higher order systems via reduced order model. It is shown that a continuous-time sliding mode control designed for the reduced order model gives similar performance for the higher order system. The method is illustrated by numerical examples. The paper also introduces a technique for design of a sliding surface such that the system satisfies a cost-optimality condition when on the sliding surface. Keywords:

1

Sliding mode control, order reduction, reduced order model, higher order system, optimal control.

Introduction

Sliding mode control theory[1] has been the subject of vigorous studies during the past few years. The concept of sliding mode involves achieving desired system dynamics, by confining the system to a sub-manifold or subspace of the state space. Researchers have suggested a variety of approaches to obtain this goal[2−4] . The advantage of sliding mode control over other control strategies is that when the system is confined to the said sub-manifold, popularly termed as the sliding surface, the system is robust and insensitive to parameter uncertainties. Model order reduction and the reduced order modeling have received considerable attention in the literature in the last thirty years. However, very few research papers are available that deal with the controller design via reduced order model[5, 6] . The purpose of this paper is to show that continuous-time sliding mode control design can also be done via reduced order model. It will be also shown that if a sliding mode control is designed from the reduced order model and if applied to the higher order system by aggregation, it results in sliding mode motion for the high order system. The brief outline of this paper is as follows. In Section 2, a brief review on continuous-time sliding mode control is presented. In Section 3, the main result of this paper is included followed by the illustrative example and simulation results. Section 4 proposes the design of sliding mode control using linear quadratic regulator. Conclusions are drawn in Section 5.

2

A brief review on continuous-time sliding mode control

The design of sliding mode controller involves designing a switching surface s(t) = 0 to represent a desired system dynamics that has lower order than the given plant, and then designing a suitable control such that any state of the system outside the switching surface is driven to reach the Manuscript received January 5, 2007; revised May 16, 2007 *Corresponding Author. E-mail address: [email protected]

surface in finite time. Consider the n-th order, m-input continuous-time linear time invariability (LTI) system x(t)

=

Ax(k) + Bu(t)

y(t)

=

Cx(t).

(1)

Let the system be transformed into normal form through a transformation x ¯(t) = T x(t), A11 A12 0 ˙x ¯(t) = x ¯+ (2) ˜ u(t). A21 A22 B

2.1

Switching plane design

Consider a sliding surface with the form cˆT x ¯(t) = 0, where the sliding function parameter is cˆT = K I . The procedure for sliding surface design[7] can be described as follows. The normal form of the system in (2), when restricted on the sliding surface cˆT x ¯(t) = 0, would obey the relationship x ¯2 (t) = −K x ¯1 (t) ¯(t). where, x ¯2 (t) constitutes the last m states of x Thus, the dynamics of x ¯1 (t) can be represented as x ¯˙ 1 (t)

=

A11 x ¯1 (t) − A12 K x ¯1 (t) = (A11 − A12 K) x ¯1 (t).

(3)

From (3), it can be observed that if K is so chosen that the eigenvalues of (A11 − A12 K) are assigned in the desired location, then x ¯1 is stabilized when confined to the sliding surface. Consequently, due to the algebraic relationship x ¯2 (t) = K x ¯1 (t), x ¯2 (t) is also stable confined to the sliding surface. Thus, the stability requirement of the sliding surface is satisfied. The sliding surface can be expressed in terms of the original state co-ordinates as s = cˆT x ¯(t) = cˆT T x(t) = 0.

330

International Journal of Automation and Computing 04(4), October 2007

2.2

Sliding mode control design

Therefore, the equivalent sliding mode control is

Once a sliding surface is designed, the reaching-law-based sliding mode control proposed by Gao et al.[3] aims at designing a control so as to satisfy the reaching condition given as s(t) ˙ = −qs(t) − sgn(s(t)) (4) where the controller parameters q and are positive. A reaching-law-based on continuous-time control law has been derived in [3] for an LTI system in (1) and the stable sliding surface s(t) = cT x(t) = 0 with the form u(t) = F x(t) + γsgn(s(t)) where

3 3.1

F

=

γ

=

(5)

−1 − cT B cT (q + A) −1 − cT B .

Sliding mode control via reduced order model

Let the higher order continuous-time system of (1) be transformed into a Jordan form separating the dominant and non-dominant modes of the system using a transformation matrix x(t) = P x ˆ(t) x ˆ˙ (t)

=

P −1 AP x ˆ(t) + P −1 Bu(t)

y(t)

=

CP x ˆ(t)

x ˆ˙ (t)

Λ1 0

=

0 Λ2

x ˆ(t) +

B1 B2

u(t).

(6)

Let a reduced model be obtained by retaining only the dominant modes of the diagonal system[8] , i.e., Λ1 contains the dominant modes of the diagonal system, which is given below: z(t)

=

Λ1 z(t) + B1 u(t).

(7)

The relation between the state vector of the reduced model and higher order system is given as[9] z(t)

=

C¯a x(t)

(8)

where C¯a = [Ir : 0] P −1 . Let sz (t) = cT z(t) be a stable sliding surface with respect to the reduced model in (7). The design of stable sliding surface was already discussed in Section 2.1. Theorem 1. If sz (t) = cT z(t) be a stable sliding surface for the reduced model given in (7), s(t) = cT C¯a x(t) will be a stable sliding surface for the high order system (1). Proof. As sz (t) = cT z(t) is stable sliding surface for (7), then motion around sz (t) = 0 can be obtained by setting T

c z(t) ˙

=

0

c Λ1 z(t) + c B1 u(t)

=

0.

T

T

(9)

−(cT B1 )−1 cT Λ1 z(t).

=

(10)

Thus, the motion along sz (t) = 0 is given by z(t) ˙

=

Λ1 z(t) + B1 u(t) = Λ1 − B1 (cT B1 )−1 cT Λ1 z(t) = I − B1 (cT B1 )−1 cT Λ1 z(t).

(11)

As (11) is stable by design, eigenvalues of (I − B1 (cT B1 )−1 cT )Λ1 will be stable. Now let us find the motion around s(t) = cT C¯a x(t) = 0 for the system in (1). If the motion around sxˆ (t) = cT [Ir : 0]ˆ x(t) = c¯T x ˆ(t) for system (6) is stable, then it is clear that the motion around s(t) = cT C¯a x(t) = 0 for the system in (1) is also stable. Sliding motion around sxˆ (t) = 0 for the system in (6) can be obtained by setting s˙ xˆ (t) = 0, cT Ca

Controller design

or

u(t)

Λ1 0

0 Λ2 cT

cT Ca x ˆ˙ (t) B1 u(t) B2

=

0

=

0

x ˆ(t) + cT B1 u(t)

=

0

x ˆ(t) + cT Ca

Λ1

0

where Ca = [Ir : 0], and the equivalent control can be calculated as −1 u(t) = − cT B1 ˆ(t). (12) cT Λ1 0 x Thus, the motion around the switching line sxˆ (k) = 0 is Λ1 0 x ˆ(t) − x ˆ˙ (t) = 0 Λ2 B1 T −1 T c B1 ˆ(t) = c Λ1 0 x B2 −1 T c Λ1 0 Λ1 − B1 cT B1 −1 T x ˆ(t). (13) c Λ1 Λ2 −B2 cT B1 −1 T c Λ1 ) is stable by design and Λ2 is As (Λ1 − B1 cT B1 stable by assumption (modes ignored during model reduction), the sliding motion of (6) around sxˆ (t) is stable. So s(t) = cT C¯a x(t) is also a stable sliding surface for the system (1). Now the sliding mode control for the reduced model in (7) can be obtained from the following reaching law: s˙ z (t) = −qsz (t) − sgn(sz (t)).

(14)

Then the continuous-time sliding mode control can be obtained as u(t) where F

=

γ

=

=

F z(t) + γsgn(sz (t))

−1 − cT B1 cT Λ1 + qcT −1 − cT B1 .

(15)

331

B. Bandyopadhyay et al./ Sliding Mode Control Design via Reduced Order Model Approach

This control will bring sliding mode motion for the reduced model in (7). Theorem 2. If the control in (15) is expressed in terms of the states of the higher order system by aggregation matrix as −1 u(t) = F C¯a x(t) + cT B1 sgn(cT C¯a x(t)) (16) and applied to the higher order system, it results in a sliding mode motion for the same. Proof. If it can be shown that the control law u(t)

=

T

F Ca x ˆ(t) + γsgn(c Ca x ˆ(t))

(17)

is able to bring a sliding mode motion for the system given in (6), then u(t)

=

F C¯a x(t) + γsgn(cT C¯a x(t))

By a proper transformation, the reduced order model of the aforementioned higher order system is obtained as

z(t) ˙

=

0 0

z(t) +

3.8731 8.5212

u(t).

A continuous-time sliding mode control is designed for the reduced order system with cT = [0.3092 − 0.0232] and controller parameters q = 2 and = 0.05. The continuoustime control law is derived as

(18) u(t)

will bring a sliding mode motion for system in (1). Let us consider (14) ˙ = −qcT z(t) − sgn(cT z(t)). cT z(t)

5.0650 0

(19)

=

−2.1843 0.0463 z(t) − ⎛ ⎞ T 0.3092 0.500sgn ⎝ z(t)⎠ . −0.0232

(24)

ˆ(t), (19) can be written as Using the relation z(t) = Ca x cT Ca x ˆ(t) = −qτ cT Ca x ˆ(t) − τ sgn(cT Ca x ˆ(t)) ˆ˙ (t) − cT Ca x (20) or c¯T x ˆ(t) − sgn(¯ cT x ˆ(t)) ˆ˙ (t) = −q¯ cT x

(21)

s˙ xˆ (t) = −qsxˆ (t) − sgn(sxˆ (t)).

(22)

The continuous-time control designed for the reduced order system with the sliding mode control for the higher order system (1) can be constructed from (24) and aggregation matrix is as

or It can be easily found that the sliding mode control for the system (6) which satisfies the reaching law in (22) is

−1 B1 T u(t) = − c¯ c¯T Λ + q¯ ˆ(t) − cT x B2 −1

B1 T τ sgn(¯ cT x ˆ(t)) c¯ B2 which further can be written as −1 cT Λ1 + qcT z(t) − u(t) = − cT B1 −1 cT B1 τ sgn(cT z(t)).

u(t)

=

0.2397 − 9.8395 −1.8702 ⎛⎡ ⎞ ⎤T −0.0232 ⎜⎢ −0.0468 ⎥ ⎟ ⎜⎢ ⎟ ⎥ 0.500sgn ⎜⎢ ⎥ x(t)⎟ . ⎝⎣ 1.3927 ⎦ ⎠ 0.2387 0.0463

x(t) −

(25)

Figs. 1–3 give the evolution of the sliding function, the control input and the system states of the higher order system. (23)

Thus, it is shown that if the sliding mode control obtained via reduced model is expressed in terms of aggregation matrix to construct a sliding mode control for system (6) with its stable sliding surface c¯T , it results in a quasi sliding mode motion for system (6).

3.2

Illustrative example

Consider the continuous-time 4th order system (higher original system) as ⎡ ⎡ ⎤ ⎤ 1 1.0 0 0 0 ⎢ 0 −0.18 −42.67 ⎢ 1.8182 ⎥ 0 ⎥ ⎢ ⎢ ⎥ ⎥ x(t) ˙ =⎢ ⎥ x(t)+ ⎢ ⎥ u(t). ⎣0 ⎣ ⎦ 0 0 1.00 ⎦ 0 0 −0.55 21.18 0 4.5455

Fig. 1

Sliding function

332


quadratic cost function[10] ∞ xT Qx + uT Ru dt J =

0

∞

= 0

Fig. 2

x ˆT T −T QT −1 x ˆ + uT Ru dt

(27)

where Q is positive semi-definite matrix and R is positive definite matrix. The aim is to design the sliding mode control in (15) which is equivalent to the optimal control in (26), when the system is in sliding mode. For the control to be equivalent to the optimal control (26), when the system is in sliding mode., i.e., when sz (t) = 0, −1 cT Λ1 + qcT . Fopt = Fs = − cT B1

Control input for the original model

Use the definitions of cT and Λ1 in normal form

Fopt

=

0 ¯ B

T

− c

0 ¯ B

T

c

−1

−1 q

Fig. 3

4

F2

System states

Sliding surface design using linear quadratic regulator

In this paper, we propose a method by which a sliding mode control is designed so as to emulate an optimal control when in sliding mode and then derive the appropriate sliding manifold that would guarantee the optimality.

4.1

Proposed sliding mode control algorithm

Consider the n-th order, m-input reduced order system given in (7). Let u(t) = Fopt x(t)

(26)

be an optimal control designed so as to minimize the

K

K

I

I

A11 A21

A12 A22

−

=

T ¯ −1 (KA11 + A21 + qK) T −(B) . ¯ −1 (KA12 + A22 + q) T −(B)

This implies that if Fopt = F1

= =

F1

F2

, then

¯ −1 (KA11 + A21 + qK) −(B) ¯ −(B)

−1

(KA12 + A22 + qI) .

(28) (29)

From (29), the value of q can be derived in terms of the sliding function parameter K as ¯ 2 − KA12 − A22 . q = −BF

(30)

Substituting the value of q from (30) into (28), we get the equation for K as ¯ 2 + A22 K −KA11 − A21 + BF ¯ 1 = 0. (31) KA12 K + BF Solving the matrix quadratic equation in (31) using matrix optimization techniques, one can obtain the sliding function parameter cT . However, since the sliding function should be real valued, only optimal control Fopt such that the equation in (31) yields real valued K as a solution, is permissible. Once K is obtained, the matrix q can be obtained from (30). Since, the controls are equivalent only on the sliding surface, it should be ensured that the state trajectory reaches the sliding surface. The necessary condition for this is q > 0. Hence, this puts up an additional restriction on the choice of K. K has to be such that q is positive definite.

333

B. Bandyopadhyay et al./ Sliding Mode Control Design via Reduced Order Model Approach

4.2

Illustrative example

Consider the continuous-time system x˙

0 1

=

1 −1

x+

0 1

u.

An optimal control u = Fopt x is designed for this system with Fopt

=

F1

F2

=

−2.2247

−1.4392

.

Fig. 5

Phase trajectory

Determining K using (31), we obtain two real valued solutions. K=

1.7321,

0.7071

.

Using (30), the values of q are found to be as follows. For K = 1.7321, q = 0.7071 and for K = 0.7071, q = 1.7321. K has to be such that q is positive definite; both the values of q are positive. Thus, we consider the smaller value of q. Therefore, the state feedback sliding mode control gain Fs =

−2.2247

−1.4392

.

Obtain the sliding function parameter cT = as c=

Fig. 4

K

I

Fig. 6

5

1.7327 1.0000

.

Sliding function

Control input

Conclusions

A method for designing continuous-time sliding mode control for higher order system via reduced order model is presented. The sliding mode control design from the reduced order model is expressed in terms of aggregation matrix to construct a sliding mode control for the higher order system. It has been proven that if sliding mode control designed from the reduced order model is applied to the higher order system by aggregation, it will result in a sliding mode motion for the higher order system. The illustrative example has been considered to validate the proposed method. The results are found to be highly satisfactory. Thus, this designing technique can be considered as an acceptable control design methodology. An optimal control-based method for determining the sliding surface and reaching law parameters for sliding mode control has been also proposed. The advantage of this method is that when the system is in sliding mode, the behavior of the system is equivalent to that of a system controlled by a linear quadratic regulator. Furthermore, the method assures that the sliding surface has this additional property of optimality in addition to

334


stability. A property that has not been taken into consideration in the traditionally used methods of sliding surface design, which guarantees only the stability of the surface.

References

revisited the Control Engineering Laboratory of Ruhr University of Bochum during May-July 2000. He has authored and coauthored more than 200 journal articles and conference papers. His research interests include the areas of large-scale systems, model reduction, reactor control, smart structures, periodic output feedback control, fast output feedback control and sliding mode control.

[1] V. I. Utkin. Sliding Modes and their Applications in Variable Structure Systems, Mir, Moscow, 1974. Alemayehu G/Egziabher Abera received his M.Sc. degree in aircraft systems from Keiv Higher Aviation Education Institute, Keiv, USSR in 1989. Presently he is a Ph.D. candidate in systems and control engineering at Indian Institute of Technology Bombay, India. His research interests include aircraft systems control, induction motor control, sliding mode control, and multirate sam-

[2] K. Furuta, Y. Pan. Variable Structure Control with Sliding Sector. Automatica, vol. 36, no. 2, pp. 211–228, 2000. [3] J. Y. Hung, W. B. Gao, J. C. Hung. Variable Structure Control: A Survey. IEEE Transactions on Industrial Electronic, vol. 40, no. 1, pp. 2–21, 1993. [4] V. I. Utkin. Variable Structure Systems with Sliding Modes. IEEE Transactions on Automatic Control, vol. 22, no. 2, pp. 212–222, 1977. [5] S. V. Rao, S. S. Lamba. Suboptimal Control of Linear System via Simplified Model of Chidambara. Proceedings of the Institution of Electrical Engineers, vol. 121, no. 8, pp. 879–883, 1974. [6] B. Bandyopadhyay, H. Unbehauen, B. M. Patre. A New Algorithm for Compensator Design for Higher Order System via Reduced Order Model. Automatica, vol. 34, no. 7, pp. 917–920, 1998. [7] C. Edwards, S. Spurgeon. Sliding Mode Control: Theory and Applications. Taylor and Francis, London, 1998. [8] E. J. Davison. A Method for Simplifing Linear Dynamic Systems. IEEE Transactions on Automatic Control, vol. 11, no. 1, pp. 93–101, 1966. [9] M. Aoki. Control of Large-scale Dynamic Systems by Aggregation. IEEE Transactions on Automatic Control, vol. 13, no. 3, pp. 246–253, 1968. [10] D. E. Kirk. Optimal Control Theory, Printice-Hall, New Jersey, 1970.

B. Bandyopadhyay received his B.Eng degree in electronics and communication engineering from the University of Calcutta, Calcutta, India, and Ph.D. degree in electrical engineering from the Indian Institute of Technology, Delhi, India in 1978 and 1986, respectively. In 1987, he joined the interdisciplinary programme in systems and control engineering, Indian Institute of Technology Bombay, India, as a faculty member, where he is currently a professor. He visited the Center for System Engineering and Applied Mechanics, Universite Catholique de Louvain, Louvain-laNeuve, Belgium, in 1993. In 1996, he was with the Lehrstuhl fur Elecktrische Steuerung und Regelung, Ruhr Universitat Bochum, Bochum, Germany, as an Alexander von Humboldt Fellow. He

pling based control.

S. Janardhanan received his B.Eng. degree in instrumentation and control engineering from the University of Madras in 2000, Master and Ph.D. degrees in systems and control engineering from Indian Institute of Technology Bombay in 2002 and 2006, respectively. He joined the Department of Electrical Engineering, Indian Institute of Technology Delhi in 2007, where he is now an assistant professor. His research interests incldue discrete-time control systems, sliding mode control, multirate sampling, and output feedback control.

Victor Sreeram received his B.Eng. degree in 1981 from Bangalore University, India, M.Eng. degree in 1983 from Madras University, India, and Ph.D. degree from University of Victoria, B. C., Canada in 1989, all in electrical engineering. He worked as a project engineer in the Indian Space Research Organization from 1983 to 1985. He joined the Department of Electrical & Electronic Engineering, University of Western Australia in 1990 and he is now an associate professor. He held visiting appointments at Department of Systems Engineering, Australian National University during 1994, 1995 and 1996 and at the Australian Telecommunication Research Institute in Curtin University of Technology during 1997 and 1998. His research interests include control and signal processing.