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Sliding mode controllers design for linear discrete-time systems with matching perturbations
Automatica 36 (2000) 1205}1211
Brief Paper
Sliding mode controllers design for linear discrete-time systems with matching perturbations夽 Chih-Chiang Cheng*, Ming-Hsiung Lin, Jia-Ming Hsiao Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC Received 28 April 1997; revised 26 October 1998; received in "nal form 22 November 1999
Abstract A simple methodology for designing sliding mode controllers is proposed in this paper for a class of linear multi-input discrete-time systems with matching perturbations. If the functions of perturbations are bounded, the proposed controllers cannot only satisfy the discrete sliding condition, but also drive the state trajectories toward a small bounded region, whose bound is dependent on the magnitude of perturbations. In order to stabilize the closed-loop system, the determination of designed parameter of controllers is discussed. Neither chattering phenomenon will occur nor the knowledge of upper bound of perturbations is required. Numerical example is also given for demonstrating the feasibility of the proposed control scheme. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Discrete systems; Linear systems; Sliding-mode control; Matching condition; Bounded perturbation
1. Introduction Sliding mode control technique has been studied since early 1960s. This technique is well known for its robustness against model uncertainties, parameter variations and external disturbances (Utkin, 1977). Over the past few years, it has been widely applied to many practical control systems, such as servo, power, and #ight control systems (Baily & Arapostathis, 1987; Chern, Chuang & Jiang, 1996; Utkin, 1978; Vadali, 1986). Due to the fast development of personal computers and DSP chips, using computer to implement the controllers becomes more and more important nowadays. Therefore, it is quite natural to extend the technique of continuous sliding mode control to discrete-time control systems. Sarpturk, Istefanopulos and Kaynak (1987) proposed a sliding and convergence condition for controlling discrete-time systems. Furuta (1990) used `sliding sectora concept to design sliding mode controller for
夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor T. Basar. * Corresponding author. Tel.: #886-7-5252000 ext. 4133; fax: #886-7-5254199. E-mail address: [email protected] (C.-C. Cheng).
linear single-input discrete-time systems, and successfully applied to inverted pendulum system (Pan & Furuta, 1994). Wang and Wu (1992), Wang, Wu and Yang (1994) and Lee and Wang (1995) used `switching regiona concept to design discrete variable structure controllers for a class of linear single-input systems with model uncertainties. However, Myszkprowski and Holmberg (1994) gave a counterexample showing the instability of the control scheme proposed by Wang et al. (1994). Pieper and Surgenor (1994), Chern et al. (1996) also proposed sliding mode controllers for a class of single-input discrete linear systems. For multi-input discrete-time systems, Spurgeon (1992) proposed a hyperplane design technique for discrete-time variable structure control systems in order to achieve global uniform asymptotic stability. Fujisaki, Togawa and Hirai (1994) designed a controller which consists of a linear state feedback and a switching feedback. Wang, Lee and Yang (1996) extended their design technique for single-input systems to multi-input systems. Chan (1991), Elmali and Olgac (1992) also proposed controllers with perturbation estimation ability so that a knowledge of upper-bound of perturbation is not required. The purpose of this paper is to provide a simple design technique of sliding mode controllers for a class of multiinput linear discrete-time systems with matching perturbations. The proposed technique uses sliding mode
0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 3 0 - 3
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control concept, but the designing idea is much simpler and with fewer limitations than those proposed by Wang et al. (1992, 1994, 1996), Lee and Wang (1995). The linear discrete-time systems considered in this paper can be described by the following dynamic equation: x(k#1)"Ax(k)#B[u(k)#v(k, x, u)],
(1)
where x3RL is the state vector, u3RK is the control input, m4n, and the constant matrices A, B are of appropriate dimensions. The vector v(k, x, u) is the lumped matching uncertainties, nonlinearities of the system and/or external disturbances, and its norm is bounded by an unknown positive constant d, i.e., ""v(k, x, u)"" 4d. Note that the N p-norm (p"1, 2, R) (Leon, 1994) is adopted for both matrix norm and vector norm. The objective of control is not only to stabilize the system, but also to drive the state trajectories into a small bounded region (as small as possible) when bounded perturbations v(k) exist.
2. Design of controllers The sliding function of the proposed control scheme is designed as r(k)"Sx(k),
(2)
where the sliding coe$cient matrix S3RK"L is chosen such that SB is nonsingular (the determination of S will be discussed later). The design procedure of control law is divided into two parts. The "rst part is to determine the structure of the controllers. The second part is to determine the value of the designed parameter of controllers according to a set of preassigned closed-loop system's eigenvalues. If bounded perturbations exist, the resultant controllers will drive the state trajectories into a small, closed and bounded region, which is the intersection of two regions, named regions A and B. Region A is de"ned as
""P\B"" d N , R " z(k)3RL: ""z(k)"" 4 N 1!""K"" N
(3)
where z(k)OP\x(k), P is a diagonalizing transformation matrix, KOdiag[j j 2 j ] L
each eigenvalue of K satis"es "j "(1,j 3R, H H j"1, 2,2, n, and K Odiag[k k 2 k ], 0(k (2, K G b is a constant. Region B is de"ned as
LO[l ]"(SB)\SP, GH and ""v(k)#(b!1)Lz(k)"" 4o(k). N The following is the design procedure of the proposed controllers. 2.1. Determination of the structure of controllers Before designing the controllers, we "rst give a lemma proposed by Furuta (1990): Lemma. If the control satisxes p(k)*p(k#1)(![*p(k#1)] for p(k)O0, where *p(k#1)"p(k#1)!p(k), then p(k#1)( p(k). 䊐 The preceding lemma clearly gives a su$cient condition for the ful"llment of discrete sliding condition "p (k#1)"("p (k)". G G Now, the determination of the proposed control u(k) is stated in the following theorem: Theorem 1. Given the plant and the sliding function r as described in (1) and (2), respectively. Let r "(SB)\rO [p p 2 p ]2. If the control law is designed as K u(k)"u (k)#u (k), (5) C N where u (k)"(SB)\(bS!SA)x(k), C u (k)O[u u 2 u ]2"!K r (k) N N N NK and each eigenvalue of the diagonal matrix K given in (3) satisxes "j "(1, j 3R, j"1, 2,2, n. Then H H (a) the control law (5) will drive the state trajectories into R dexned in (3). (b) the discrete sliding condition "p (k#1)"("p (k)" will G G be satisxed outside R dexned in (4). Proof. (a) From (1) and using (5), we can obtain the dynamic equation of closed-loop system as x(k#1)"[A#B(SB)\(bS!SA)
"P\[A#B(SB)\(bS!SA)!BK (SB)\S]P,
L o(k) o(k) R " z(k)3RL: l z (k) 4max , GH H k 2!k G G H
where
, (4)
!BK (SB)\S]x(k)#Bv(k). (6) Since S is speci"ed by the designer, a nonsingular diagonalizing matrix P can be found such that it can transform (6) into z(k#1)"Kz(k)#P\Bv(k).
C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211
If the trajectories satisfy ""z"" '(""P\B"" d)/(1!""K"" ), N N N then ""z(k#1)"" """Kz(k)#P\Bv(k)"" N N ""P\B"" d N ""z"" 4 ""K"" # N N ""z"" N ((""K"" #1!""K"" )""z"" N N N 4""z(k)"" , N which indicates that the control law u(k) can drive the state trajectories into R . (b) From (1), (2) and (5) it is known that
*r(k#1)Or(k#1)!r(k) "SBu (k)#SBv(k)#(b!1)Sx(k). N The above equation indicates that *r "(SB)\*r"u #v#(b!1)Lz. (7) N If the closed-loop system (6) is stabilized, i.e., "j "(1, j"1, 2,2, n, then z(k) is bounded, and H v (k)Ov(k)#(b!1)Lz(k) is also bounded. Therefore, we can let ""v (k)"" 4o(k). Eq. (7) implies that N *p "u #v "!k p #v , i"1, 2,2, m. (8) G NG G G G G Now, we "rst suppose that
o(k) o(k) o(k) "p (k)"'max , " (9) G 2!k 2!k k G G G is satis"ed. This leads to the following two cases: Case 1: p '0. Eq. (9) implies that 2p !k p 'o, G G G G which also means that 1'(k p #o)/2p '(k p !v )/ G G G G G G 2p , i.e., !1((!k p #v )/2p . G G G G G Case 2: p (0. In this case (9) implies that !2p # G G k p 'o, which indicates that 1'(!k p #o)/ G G G G (!2p )'(!k p #v )/(!2p ), i.e., !1((!k p # G G G G G G G v )/2p . G G On the other hand, (9) also implies "p "'o/k , which G G indicates that
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If "p (k)"'max+o(k)/k , o(k)/(2!k ),"o(k)/k , we G G G G still can use the same analysis method to obtain (11). Note also that r "(SB)\Sx"Lz, i.e., p (k)" G L l z (k), i"1, 2,2, m. Therefore, if the control law H GH H u is used, then according to the preceding lemma, it is known that "p (k#1)"("p (k)", which indicates that G G the transformed sliding function p (k) is decreasing G outside R . 䊐 The purpose of using u (k) is to let the sliding function C r(k) satisfy r(k#1)"br(k), so that r(k) is a decreasing function if u (k)"v(k)"0 and "b"(1. In fact, this idea N of utilizing u (k) has been used in Pan and Furuta (1994) C for single input discrete-time systems. However, if we consider both u (k) and u (k) proposed in this paper, the C N sliding function r(k) will satisfy r(k#1)"br(k)!SBK r (k)"(b!1)r(k) (12) if K "I and v(k)"0. It indicates that the decreasing rate (if b is properly chosen) of the sliding function r(k) is b!1 (the choice of b will be discussed later). It is also noted from (5) that there is no switching action in the proposed controllers, which means that the chattering phenomenon caused by the sliding mode controllers as in the case of continuous systems will never happen. The other advantage of (5) is that the upper bound of perturbations v(k) need not to be known beforehand when implementing the controllers; which will also increase the applicability of the proposed control scheme. From Theorem 1 one also can know that the "nal region that the trajectories will enter is the intersection of R and R . Figs. 1 and 2 show two possible cases of two dimensional systems, respectively. Fig. 1 is the case when the width of R is smaller than the radius of bounded R . Fig. 2 shows the opposite situation. Note that since o(k) is a function of z(k), for simplicity of demonstration R of Figs. 1 and 2 is shown under the assumption
o "v " v 0'!k # 5!k # G 5!k # G G "p " G "p " G p G G G !k p #v G G G. " (10) p G From the preceding analysis of cases 1, 2 and (10), we obtain 0'(!k p #v )/2p '!1. Then according to G G G G (8), it is also equivalent to 2p !R( G (!1, *p G which also implies p *p (![*p ]. G G G
o(k) o(k) o o l z (k) 4max , 4max , , GH H k 2!k k 2!k G G G G H
(11) Fig. 1. The "nal bounded region of the trajectories (Case 1).
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exist. From Theorem 2 one can verify that k "1, G i"1, 2,2, m will render the minimum bound of R . Hence, we let K "I for the rest of this paper. 2.2. Determination of designed parameter b and S In the following the method for determination of designed parameter b and sliding coe$cient matrix S according to a set of preassigned eigenvalues j , j"1, 2,2, n of closed-loop system (6) is presented. In H fact, it has been proved by Pan and Furuta (1994) that for single-input systems, n!1 eigenvalues of the following closed-loop system: Fig. 2. The "nal bounded region of the trajectories (Case 2).
x(k#1)"[A#B(SB)\(bS!SA)]x(k) for bO0 is determined by the reduced order system
where o, the upper bound of o(k), is a constant, and k "1. Therefore, the stability of the proposed control G systems is guaranteed. One can see that if there are no perturbations, then all the state vectors will be driven to zero. Note also that the physical meaning of R plotted in Figs. 1 and 2 is totally di!erent from that of the switching region proposed by Furuta (1990). The former is the region in which all the trajectories of the closedloop system (6) will be constrained, the latter is the region where a proper control structure should be activated. It is also found that the least upper bound of "p (k)" is G o(k), this is proved in the following theorem. Theorem 2. If the control law u(k) is designed as stated in Theorem 1, then the least upper bound of "p (k)", G i"1, 2,2, m of discrete-time system (1) is equal to o(k). Proof. From Theorem 1 it is known that the discrete sliding condition "p (k#1)"("p (k)" will be satis"ed G G outside R if u(k) is used. Note that for each "xed k, o(k)/k is a decreasing function for 0(k (2, whereas G G o(k)/(2!k ) is an increasing function. The intersection G of these two functions is o(k). Hence,
o(k) o(k) max , 5o(k). k 2!k G IG G Therefore the least upper-bound of "p (k)" is G
o(k) o(k) min "p (k)"" min max , G k 2!k G G IG IG "o(k).
䊐
Theorem 2 clearly indicates that the least upper-bound of "p (k)", which is also the best result we can obtain, is G o(k) if the control law u(k) is used. In general, it is usually desired to have a minimum bound of R in order to increase the accuracy of control if the perturbations v(k)
x(k#1)"[A!B(SB)\SA)]x(k), p(k)"Sx(k)"0 and the rest one eigenvalue is b. For our multi-input control systems proposed in this paper, similar result can be obtained and is stated in the following theorem. Theorem 3. If the sliding coezcient matrix S has full rank, then n!m eigenvalues of the proposed closed-loop system described in (6) with K "I is determined by the reduced order system x(k#1)"[A!B(SB)\SA)]x(k)OAM x(k), r(k)"Sx(k)"0
(13)
and the rest m eigenvalues are b!1. Proof. If the sliding coe$cient matrix S has full rank, then it is possible to "nd a matrix H3RL\K"L such that the transform matrix T"[H2 S2]2 is invertible, and its inverse matrix is denoted as T\O[R Q ]. L"L\K L"K Hence, it is known that HR"I , HQ"0 , L\K L\K"K SR"0 , and SQ"I . Then we take the transK"L\K K formation w(k)"Tx(k), the closed-loop system (6) becomes
HA R HA Q w(k)#TBv(k) (14) SA R SA Q if K "I, where A "AM #B(SB)\(b!1)S. It is easy to verify that HA R"HAM R, SA R"0, SA Q"(b!1)I . K Therefore, (14) is equivalent to w(k#1)"
HAM R
HA Q w(k)#TBv(k), 0 (b!1)I K which indicates that the n!m eigenvalues of (6) is determined by the reduced order system (13), and the rest m eigenvalues are b!1. 䊐 w(k#1)"
C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211
From the preceding theorem, one can see that the range of b should be set to "b!1"(1 in order to stabilize the control systems, and the decaying rate of r(k) is equal to m eigenvalues of (6) if K "I and v(k)"0 according to (12). On the other hand, in Theorem 3 the assumption that the sliding coe$cient matrix S has full rank will ensure the existence of inverse of SB if B is also full rank, and the design of S for the presetting n!m eigenvalues of reduced order system (13) can refer to the method proposed by El-Ghezawi, Zinober and Billings (1983). Spurgeon (1992) has also presented a discrete-time variable structure control method using controller structure without switching action. In that paper the author proposed a hyperplane design technique which can achieve the property of global uniform asymptotic stability. The main di!erence of the hyperplane design philosophy between Spurgeon's method and ours is that the former's method cannot assign the desired closed-loop eigenvalues directly, whereas in our method one can utilize the methodology proposed by El-Ghezawi et al. (1983) to design the sliding coe$cient matrix S in order to assign the desired closed-loop eigenvalues directly. Then one can apply the controller given by (5) to achieve the property of global asymptotic stability. In other words, one has to use try and error method in order to obtain a desired system dynamic performance if Spurgeon's method is used. However, one of the advantages of Spurgeon's method is that it can deal with a more general class of matched perturbation with the form ""v(k)"" 4a #a ""x"" , where a and a are positive N N constants.
3. Example We consider the following multi-input discrete-time unstable plant
x(k#1)"
1
0.01
0
1
0
0.01
0.01
0
The eigenvalues of closed-loop system (6) are assigned to +0.6, 0.6, 0.8,. Then, according to Theorem 3, b"1.6. On the other hand, the sliding coe$cient matrix S is obtained by using the method proposed by El-Ghezawi et al. (1983) for the assigned eigenvalue 0.8. This sliding coe$cient matrix S is obtained as
S"
2065.2174 2065.2174 !1965.2174
2069.5652 2169.5652 !2069.5652
(u#v)(k)
with nonlinearities, parameter variations and/or disturbances v(k)
Fig. 4. State variable x .
.
1219.4782 1257.7826 !1219.4348 u " x. C 1221.0434 1322.4347 !1284.1304
0.01 !0.01 0
Fig. 3. State variable x .
In accordance with (5), the equivalent control law is designed as
0.01 x(k)
!0.01 0.02 0.99
#
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x x !0.2x #0.1x u !x cos(k)#0.2 sin(k) " . x !x !2x #0.1x #0.01x u #0.5 cos(k)
The simulation results with k "1, i"1, 2 and initial G condition x(0)"[1 !0.5 0.5] are shown in Figs. 3}9. Figs. 3}5 show that the state trajectories are all driven into a small bounded region rapidly. The control inputs u and u are shown in Figs. 6 and 7. The sliding functions p and p also decay rapidly into a small bounded region, as shown in Figs. 8 and 9. Note that the controllers proposed by Furuta (1990), Pan and Furuta (1994), Wang et al. (1992, 1994), Lee and Wang (1995), Chern et al. (1996), Pieper and Surgenor (1994) can be used directly only for single-input plant; the controller
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Fig. 5. State variable x .
Fig. 8. Sliding surface p .
Fig. 6. Control input u .
Fig. 9. Sliding surface p .
time systems in this paper. When the system contains matching uncertainties, disturbances and nonlinearities, it is shown that in general, unlike the results of controlling continuous system, the state trajectories can only be driven into a small bounded region. The bound of this region is dependent on the magnitude of perturbations. The main reason is that during the sampling time interval the powerful in"nite switching ability of traditional continuously sliding mode control cannot be applied. The idea of the proposed design technique is very easy, and a knowledge of bound of perturbations is not required beforehand. Fig. 7. Control input u .
Acknowledgements proposed by Wang et al. (1996) can be used for the plant without the uncertainties in the input matrix, and the plant considered by Fujisaki et al. (1994) does not contain perturbations. Therefore, the controllers proposed in these papers cannot be used directly for this case.
4. Conclusions The designing of sliding mode controllers is successfully proposed for a class of multi-input linear discrete-
The authors would like to thank the Editor, Associate Editor, and the reviewers for their many helpful comments and suggestions that have helped to improve the quality of this paper. References Baily, E., & Arapostathis, A. (1987). Simple sliding mode control scheme applied to robot manipulators. International Journal of Control, 45(4), 1197}1209.
C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211 Chan, C. Y. (1991). Robust discrete-time sliding mode controller. Systems and Control Letters, 23, 371}374. Chern, T. -L., Chuang, C. W., & Jiang, R. -L. (1996). Design of discrete integral variable structure control systems and application to brushless dc motor control. Automatica, 32(5), 773}779. El-Ghezawi, O. M. E., Zinober, A. S. I., & Billings, S. A. (1983). Analysis and design of variable structure systems using a geometric approach. International Journal of Control, 38(3), 657}671. Elmali, H., & Olgac, N. (1992). Sliding mode control with perturbations estimation (SMCPE): A new approach. International Journal of Control, 56(4), 923}941. Fujisaki, Y., Togawa, K., & Hirai, K. (1994). Sliding mode control for multi-input discrete time systems. Proceedings of the 33rd conference on decision and control (pp. 1933}1935). Furuta, K. (1990). Sliding mode control of a discrete system. Systems and Control Letters, 14, 145}152. Lee, R.-C., & Wang, W.-J. (1995). Robust variable structure control synthesis in discrete-time uncertain system.. Control-Theory and Advanced Technology, Part 4, 10(4), 1785}1796. Leon, S.J. (1994). Linear algebra with applications (4th ed.). Macmillan College Publishing Company, Inc. Myszkprowski, P., & Holmberg, U. (1994). Comments on variable structure control design for uncertain discrete-time system. IEEE Transactions on Automatic Control, 39(11), 2366}2367. Pan, Y., & Furuta, K. (1994). Vss controller design for discrete-time system. Control-Theory and Advanced Technology, Part 1, 10(4), 669}687. Pieper, J. K., & Surgenor, B. W. (1994). Discrete time sliding mode control applied to a gantry crane. Proceedings of the 33rd conference on decision and control, 829}834. Sarpturk, S. Z., Istefanopulos, Y., & Kaynak, O. (1987). On the stability of discrete-time sliding mode control systems. IEEE Transactions on Automatic Control, AC-32(10), 930}932. Spurgeon, S. K. (1992). Hyperplane design techniques for discrete-time variable structure control systems. International Journal of Control, 55(2), 445}456. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE Transactions on Automatic Control, AC-22(2), 212}222. Utkin, V. I. (1978). Sliding Modes and Their Applications to Variable Structure System. Moscow: MIR Publishers. Vadali, S. R. (1986). Variable structure control of spacecraft large-angle maneuvers. Journal of Guidance and Control of Dynamics, 9(2), 235}239. Wang, W.-J., & Wu, G.-H. (1992). Variable structure control design on discrete-time systems from another viewpoint. Control-Theory and Advanced Technology, 8(1), 1}16.
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Wang, W.-J., Wu, G.-H., & Yang, D.-C. (1994). Variable structure control design for uncertain discrete-time systems. IEEE Transactions on Automatic Control, 39(1), 99}102. Wang, W.-J., Lee, R.-C., & Yang, D.-C. (1996). Sliding mode control design in multi-input perturbed discrete-time systems. Journal of Dynamic Systems, Measurement, and Control, 118, 322}327. Chih-Chiang Cheng was born in Taipei, Taiwan, Republic of China on February 27, 1957. He received the B.S. degree in Electrical Engineering from Chung Yuan Christian University, Chung-Li, Taiwan, in 1981, the M.S. and Ph.D. degree in Electrical Engineering from University of Texas at Arlington, Arlington, Texas, in 1983 and 1991, respectively, as well as Engineer degree in Electrical Engineering from University of Southern California, Los Angeles, in 1985. He is currently an Associate Professor in the department of Electrical Engineering at National Sun Yat-Sen University, Kaohsiung, Taiwan. His current research interests include system and control theory, with emphasis on design of nonlinear control systems such as variable structure control, adaptive control, fuzzy control and digital control. Ming-Hsiung Lin was born on December 12, 1972. He received the B.S. degree in Electronic Engineering from National Taiwan Institutes of Technology, in 1994, and the M.S. degree in Electrical Engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1996. He is currently a product engineer of Asus computer Co.. His main research interests include design of discrete-time variable structure systems and personal computer structure. Jia-Ming Hsiao was born on February 12, 1971. He received the B.S. degree in Electrical Engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1994. Currently he is working toward his Ph.D. at National Sun Yat-Sen University, Kaohsiung, Taiwan. His main research interests include design of variable-structure systems and fuzzy control theory.
Brief Paper
Sliding mode controllers design for linear discrete-time systems with matching perturbations夽 Chih-Chiang Cheng*, Ming-Hsiung Lin, Jia-Ming Hsiao Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC Received 28 April 1997; revised 26 October 1998; received in "nal form 22 November 1999
Abstract A simple methodology for designing sliding mode controllers is proposed in this paper for a class of linear multi-input discrete-time systems with matching perturbations. If the functions of perturbations are bounded, the proposed controllers cannot only satisfy the discrete sliding condition, but also drive the state trajectories toward a small bounded region, whose bound is dependent on the magnitude of perturbations. In order to stabilize the closed-loop system, the determination of designed parameter of controllers is discussed. Neither chattering phenomenon will occur nor the knowledge of upper bound of perturbations is required. Numerical example is also given for demonstrating the feasibility of the proposed control scheme. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Discrete systems; Linear systems; Sliding-mode control; Matching condition; Bounded perturbation
1. Introduction Sliding mode control technique has been studied since early 1960s. This technique is well known for its robustness against model uncertainties, parameter variations and external disturbances (Utkin, 1977). Over the past few years, it has been widely applied to many practical control systems, such as servo, power, and #ight control systems (Baily & Arapostathis, 1987; Chern, Chuang & Jiang, 1996; Utkin, 1978; Vadali, 1986). Due to the fast development of personal computers and DSP chips, using computer to implement the controllers becomes more and more important nowadays. Therefore, it is quite natural to extend the technique of continuous sliding mode control to discrete-time control systems. Sarpturk, Istefanopulos and Kaynak (1987) proposed a sliding and convergence condition for controlling discrete-time systems. Furuta (1990) used `sliding sectora concept to design sliding mode controller for
夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor T. Basar. * Corresponding author. Tel.: #886-7-5252000 ext. 4133; fax: #886-7-5254199. E-mail address: [email protected] (C.-C. Cheng).
linear single-input discrete-time systems, and successfully applied to inverted pendulum system (Pan & Furuta, 1994). Wang and Wu (1992), Wang, Wu and Yang (1994) and Lee and Wang (1995) used `switching regiona concept to design discrete variable structure controllers for a class of linear single-input systems with model uncertainties. However, Myszkprowski and Holmberg (1994) gave a counterexample showing the instability of the control scheme proposed by Wang et al. (1994). Pieper and Surgenor (1994), Chern et al. (1996) also proposed sliding mode controllers for a class of single-input discrete linear systems. For multi-input discrete-time systems, Spurgeon (1992) proposed a hyperplane design technique for discrete-time variable structure control systems in order to achieve global uniform asymptotic stability. Fujisaki, Togawa and Hirai (1994) designed a controller which consists of a linear state feedback and a switching feedback. Wang, Lee and Yang (1996) extended their design technique for single-input systems to multi-input systems. Chan (1991), Elmali and Olgac (1992) also proposed controllers with perturbation estimation ability so that a knowledge of upper-bound of perturbation is not required. The purpose of this paper is to provide a simple design technique of sliding mode controllers for a class of multiinput linear discrete-time systems with matching perturbations. The proposed technique uses sliding mode
0005-1098/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 3 0 - 3
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control concept, but the designing idea is much simpler and with fewer limitations than those proposed by Wang et al. (1992, 1994, 1996), Lee and Wang (1995). The linear discrete-time systems considered in this paper can be described by the following dynamic equation: x(k#1)"Ax(k)#B[u(k)#v(k, x, u)],
(1)
where x3RL is the state vector, u3RK is the control input, m4n, and the constant matrices A, B are of appropriate dimensions. The vector v(k, x, u) is the lumped matching uncertainties, nonlinearities of the system and/or external disturbances, and its norm is bounded by an unknown positive constant d, i.e., ""v(k, x, u)"" 4d. Note that the N p-norm (p"1, 2, R) (Leon, 1994) is adopted for both matrix norm and vector norm. The objective of control is not only to stabilize the system, but also to drive the state trajectories into a small bounded region (as small as possible) when bounded perturbations v(k) exist.
2. Design of controllers The sliding function of the proposed control scheme is designed as r(k)"Sx(k),
(2)
where the sliding coe$cient matrix S3RK"L is chosen such that SB is nonsingular (the determination of S will be discussed later). The design procedure of control law is divided into two parts. The "rst part is to determine the structure of the controllers. The second part is to determine the value of the designed parameter of controllers according to a set of preassigned closed-loop system's eigenvalues. If bounded perturbations exist, the resultant controllers will drive the state trajectories into a small, closed and bounded region, which is the intersection of two regions, named regions A and B. Region A is de"ned as
""P\B"" d N , R " z(k)3RL: ""z(k)"" 4 N 1!""K"" N
(3)
where z(k)OP\x(k), P is a diagonalizing transformation matrix, KOdiag[j j 2 j ] L
each eigenvalue of K satis"es "j "(1,j 3R, H H j"1, 2,2, n, and K Odiag[k k 2 k ], 0(k (2, K G b is a constant. Region B is de"ned as
LO[l ]"(SB)\SP, GH and ""v(k)#(b!1)Lz(k)"" 4o(k). N The following is the design procedure of the proposed controllers. 2.1. Determination of the structure of controllers Before designing the controllers, we "rst give a lemma proposed by Furuta (1990): Lemma. If the control satisxes p(k)*p(k#1)(![*p(k#1)] for p(k)O0, where *p(k#1)"p(k#1)!p(k), then p(k#1)( p(k). 䊐 The preceding lemma clearly gives a su$cient condition for the ful"llment of discrete sliding condition "p (k#1)"("p (k)". G G Now, the determination of the proposed control u(k) is stated in the following theorem: Theorem 1. Given the plant and the sliding function r as described in (1) and (2), respectively. Let r "(SB)\rO [p p 2 p ]2. If the control law is designed as K u(k)"u (k)#u (k), (5) C N where u (k)"(SB)\(bS!SA)x(k), C u (k)O[u u 2 u ]2"!K r (k) N N N NK and each eigenvalue of the diagonal matrix K given in (3) satisxes "j "(1, j 3R, j"1, 2,2, n. Then H H (a) the control law (5) will drive the state trajectories into R dexned in (3). (b) the discrete sliding condition "p (k#1)"("p (k)" will G G be satisxed outside R dexned in (4). Proof. (a) From (1) and using (5), we can obtain the dynamic equation of closed-loop system as x(k#1)"[A#B(SB)\(bS!SA)
"P\[A#B(SB)\(bS!SA)!BK (SB)\S]P,
L o(k) o(k) R " z(k)3RL: l z (k) 4max , GH H k 2!k G G H
where
, (4)
!BK (SB)\S]x(k)#Bv(k). (6) Since S is speci"ed by the designer, a nonsingular diagonalizing matrix P can be found such that it can transform (6) into z(k#1)"Kz(k)#P\Bv(k).
C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211
If the trajectories satisfy ""z"" '(""P\B"" d)/(1!""K"" ), N N N then ""z(k#1)"" """Kz(k)#P\Bv(k)"" N N ""P\B"" d N ""z"" 4 ""K"" # N N ""z"" N ((""K"" #1!""K"" )""z"" N N N 4""z(k)"" , N which indicates that the control law u(k) can drive the state trajectories into R . (b) From (1), (2) and (5) it is known that
*r(k#1)Or(k#1)!r(k) "SBu (k)#SBv(k)#(b!1)Sx(k). N The above equation indicates that *r "(SB)\*r"u #v#(b!1)Lz. (7) N If the closed-loop system (6) is stabilized, i.e., "j "(1, j"1, 2,2, n, then z(k) is bounded, and H v (k)Ov(k)#(b!1)Lz(k) is also bounded. Therefore, we can let ""v (k)"" 4o(k). Eq. (7) implies that N *p "u #v "!k p #v , i"1, 2,2, m. (8) G NG G G G G Now, we "rst suppose that
o(k) o(k) o(k) "p (k)"'max , " (9) G 2!k 2!k k G G G is satis"ed. This leads to the following two cases: Case 1: p '0. Eq. (9) implies that 2p !k p 'o, G G G G which also means that 1'(k p #o)/2p '(k p !v )/ G G G G G G 2p , i.e., !1((!k p #v )/2p . G G G G G Case 2: p (0. In this case (9) implies that !2p # G G k p 'o, which indicates that 1'(!k p #o)/ G G G G (!2p )'(!k p #v )/(!2p ), i.e., !1((!k p # G G G G G G G v )/2p . G G On the other hand, (9) also implies "p "'o/k , which G G indicates that
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If "p (k)"'max+o(k)/k , o(k)/(2!k ),"o(k)/k , we G G G G still can use the same analysis method to obtain (11). Note also that r "(SB)\Sx"Lz, i.e., p (k)" G L l z (k), i"1, 2,2, m. Therefore, if the control law H GH H u is used, then according to the preceding lemma, it is known that "p (k#1)"("p (k)", which indicates that G G the transformed sliding function p (k) is decreasing G outside R . 䊐 The purpose of using u (k) is to let the sliding function C r(k) satisfy r(k#1)"br(k), so that r(k) is a decreasing function if u (k)"v(k)"0 and "b"(1. In fact, this idea N of utilizing u (k) has been used in Pan and Furuta (1994) C for single input discrete-time systems. However, if we consider both u (k) and u (k) proposed in this paper, the C N sliding function r(k) will satisfy r(k#1)"br(k)!SBK r (k)"(b!1)r(k) (12) if K "I and v(k)"0. It indicates that the decreasing rate (if b is properly chosen) of the sliding function r(k) is b!1 (the choice of b will be discussed later). It is also noted from (5) that there is no switching action in the proposed controllers, which means that the chattering phenomenon caused by the sliding mode controllers as in the case of continuous systems will never happen. The other advantage of (5) is that the upper bound of perturbations v(k) need not to be known beforehand when implementing the controllers; which will also increase the applicability of the proposed control scheme. From Theorem 1 one also can know that the "nal region that the trajectories will enter is the intersection of R and R . Figs. 1 and 2 show two possible cases of two dimensional systems, respectively. Fig. 1 is the case when the width of R is smaller than the radius of bounded R . Fig. 2 shows the opposite situation. Note that since o(k) is a function of z(k), for simplicity of demonstration R of Figs. 1 and 2 is shown under the assumption
o "v " v 0'!k # 5!k # G 5!k # G G "p " G "p " G p G G G !k p #v G G G. " (10) p G From the preceding analysis of cases 1, 2 and (10), we obtain 0'(!k p #v )/2p '!1. Then according to G G G G (8), it is also equivalent to 2p !R( G (!1, *p G which also implies p *p (![*p ]. G G G
o(k) o(k) o o l z (k) 4max , 4max , , GH H k 2!k k 2!k G G G G H
(11) Fig. 1. The "nal bounded region of the trajectories (Case 1).
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C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211
exist. From Theorem 2 one can verify that k "1, G i"1, 2,2, m will render the minimum bound of R . Hence, we let K "I for the rest of this paper. 2.2. Determination of designed parameter b and S In the following the method for determination of designed parameter b and sliding coe$cient matrix S according to a set of preassigned eigenvalues j , j"1, 2,2, n of closed-loop system (6) is presented. In H fact, it has been proved by Pan and Furuta (1994) that for single-input systems, n!1 eigenvalues of the following closed-loop system: Fig. 2. The "nal bounded region of the trajectories (Case 2).
x(k#1)"[A#B(SB)\(bS!SA)]x(k) for bO0 is determined by the reduced order system
where o, the upper bound of o(k), is a constant, and k "1. Therefore, the stability of the proposed control G systems is guaranteed. One can see that if there are no perturbations, then all the state vectors will be driven to zero. Note also that the physical meaning of R plotted in Figs. 1 and 2 is totally di!erent from that of the switching region proposed by Furuta (1990). The former is the region in which all the trajectories of the closedloop system (6) will be constrained, the latter is the region where a proper control structure should be activated. It is also found that the least upper bound of "p (k)" is G o(k), this is proved in the following theorem. Theorem 2. If the control law u(k) is designed as stated in Theorem 1, then the least upper bound of "p (k)", G i"1, 2,2, m of discrete-time system (1) is equal to o(k). Proof. From Theorem 1 it is known that the discrete sliding condition "p (k#1)"("p (k)" will be satis"ed G G outside R if u(k) is used. Note that for each "xed k, o(k)/k is a decreasing function for 0(k (2, whereas G G o(k)/(2!k ) is an increasing function. The intersection G of these two functions is o(k). Hence,
o(k) o(k) max , 5o(k). k 2!k G IG G Therefore the least upper-bound of "p (k)" is G
o(k) o(k) min "p (k)"" min max , G k 2!k G G IG IG "o(k).
䊐
Theorem 2 clearly indicates that the least upper-bound of "p (k)", which is also the best result we can obtain, is G o(k) if the control law u(k) is used. In general, it is usually desired to have a minimum bound of R in order to increase the accuracy of control if the perturbations v(k)
x(k#1)"[A!B(SB)\SA)]x(k), p(k)"Sx(k)"0 and the rest one eigenvalue is b. For our multi-input control systems proposed in this paper, similar result can be obtained and is stated in the following theorem. Theorem 3. If the sliding coezcient matrix S has full rank, then n!m eigenvalues of the proposed closed-loop system described in (6) with K "I is determined by the reduced order system x(k#1)"[A!B(SB)\SA)]x(k)OAM x(k), r(k)"Sx(k)"0
(13)
and the rest m eigenvalues are b!1. Proof. If the sliding coe$cient matrix S has full rank, then it is possible to "nd a matrix H3RL\K"L such that the transform matrix T"[H2 S2]2 is invertible, and its inverse matrix is denoted as T\O[R Q ]. L"L\K L"K Hence, it is known that HR"I , HQ"0 , L\K L\K"K SR"0 , and SQ"I . Then we take the transK"L\K K formation w(k)"Tx(k), the closed-loop system (6) becomes
HA R HA Q w(k)#TBv(k) (14) SA R SA Q if K "I, where A "AM #B(SB)\(b!1)S. It is easy to verify that HA R"HAM R, SA R"0, SA Q"(b!1)I . K Therefore, (14) is equivalent to w(k#1)"
HAM R
HA Q w(k)#TBv(k), 0 (b!1)I K which indicates that the n!m eigenvalues of (6) is determined by the reduced order system (13), and the rest m eigenvalues are b!1. 䊐 w(k#1)"
C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211
From the preceding theorem, one can see that the range of b should be set to "b!1"(1 in order to stabilize the control systems, and the decaying rate of r(k) is equal to m eigenvalues of (6) if K "I and v(k)"0 according to (12). On the other hand, in Theorem 3 the assumption that the sliding coe$cient matrix S has full rank will ensure the existence of inverse of SB if B is also full rank, and the design of S for the presetting n!m eigenvalues of reduced order system (13) can refer to the method proposed by El-Ghezawi, Zinober and Billings (1983). Spurgeon (1992) has also presented a discrete-time variable structure control method using controller structure without switching action. In that paper the author proposed a hyperplane design technique which can achieve the property of global uniform asymptotic stability. The main di!erence of the hyperplane design philosophy between Spurgeon's method and ours is that the former's method cannot assign the desired closed-loop eigenvalues directly, whereas in our method one can utilize the methodology proposed by El-Ghezawi et al. (1983) to design the sliding coe$cient matrix S in order to assign the desired closed-loop eigenvalues directly. Then one can apply the controller given by (5) to achieve the property of global asymptotic stability. In other words, one has to use try and error method in order to obtain a desired system dynamic performance if Spurgeon's method is used. However, one of the advantages of Spurgeon's method is that it can deal with a more general class of matched perturbation with the form ""v(k)"" 4a #a ""x"" , where a and a are positive N N constants.
3. Example We consider the following multi-input discrete-time unstable plant
x(k#1)"
1
0.01
0
1
0
0.01
0.01
0
The eigenvalues of closed-loop system (6) are assigned to +0.6, 0.6, 0.8,. Then, according to Theorem 3, b"1.6. On the other hand, the sliding coe$cient matrix S is obtained by using the method proposed by El-Ghezawi et al. (1983) for the assigned eigenvalue 0.8. This sliding coe$cient matrix S is obtained as
S"
2065.2174 2065.2174 !1965.2174
2069.5652 2169.5652 !2069.5652
(u#v)(k)
with nonlinearities, parameter variations and/or disturbances v(k)
Fig. 4. State variable x .
.
1219.4782 1257.7826 !1219.4348 u " x. C 1221.0434 1322.4347 !1284.1304
0.01 !0.01 0
Fig. 3. State variable x .
In accordance with (5), the equivalent control law is designed as
0.01 x(k)
!0.01 0.02 0.99
#
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x x !0.2x #0.1x u !x cos(k)#0.2 sin(k) " . x !x !2x #0.1x #0.01x u #0.5 cos(k)
The simulation results with k "1, i"1, 2 and initial G condition x(0)"[1 !0.5 0.5] are shown in Figs. 3}9. Figs. 3}5 show that the state trajectories are all driven into a small bounded region rapidly. The control inputs u and u are shown in Figs. 6 and 7. The sliding functions p and p also decay rapidly into a small bounded region, as shown in Figs. 8 and 9. Note that the controllers proposed by Furuta (1990), Pan and Furuta (1994), Wang et al. (1992, 1994), Lee and Wang (1995), Chern et al. (1996), Pieper and Surgenor (1994) can be used directly only for single-input plant; the controller
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C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211
Fig. 5. State variable x .
Fig. 8. Sliding surface p .
Fig. 6. Control input u .
Fig. 9. Sliding surface p .
time systems in this paper. When the system contains matching uncertainties, disturbances and nonlinearities, it is shown that in general, unlike the results of controlling continuous system, the state trajectories can only be driven into a small bounded region. The bound of this region is dependent on the magnitude of perturbations. The main reason is that during the sampling time interval the powerful in"nite switching ability of traditional continuously sliding mode control cannot be applied. The idea of the proposed design technique is very easy, and a knowledge of bound of perturbations is not required beforehand. Fig. 7. Control input u .
Acknowledgements proposed by Wang et al. (1996) can be used for the plant without the uncertainties in the input matrix, and the plant considered by Fujisaki et al. (1994) does not contain perturbations. Therefore, the controllers proposed in these papers cannot be used directly for this case.
4. Conclusions The designing of sliding mode controllers is successfully proposed for a class of multi-input linear discrete-
The authors would like to thank the Editor, Associate Editor, and the reviewers for their many helpful comments and suggestions that have helped to improve the quality of this paper. References Baily, E., & Arapostathis, A. (1987). Simple sliding mode control scheme applied to robot manipulators. International Journal of Control, 45(4), 1197}1209.
C.-C. Cheng et al. / Automatica 36 (2000) 1205}1211 Chan, C. Y. (1991). Robust discrete-time sliding mode controller. Systems and Control Letters, 23, 371}374. Chern, T. -L., Chuang, C. W., & Jiang, R. -L. (1996). Design of discrete integral variable structure control systems and application to brushless dc motor control. Automatica, 32(5), 773}779. El-Ghezawi, O. M. E., Zinober, A. S. I., & Billings, S. A. (1983). Analysis and design of variable structure systems using a geometric approach. International Journal of Control, 38(3), 657}671. Elmali, H., & Olgac, N. (1992). Sliding mode control with perturbations estimation (SMCPE): A new approach. International Journal of Control, 56(4), 923}941. Fujisaki, Y., Togawa, K., & Hirai, K. (1994). Sliding mode control for multi-input discrete time systems. Proceedings of the 33rd conference on decision and control (pp. 1933}1935). Furuta, K. (1990). Sliding mode control of a discrete system. Systems and Control Letters, 14, 145}152. Lee, R.-C., & Wang, W.-J. (1995). Robust variable structure control synthesis in discrete-time uncertain system.. Control-Theory and Advanced Technology, Part 4, 10(4), 1785}1796. Leon, S.J. (1994). Linear algebra with applications (4th ed.). Macmillan College Publishing Company, Inc. Myszkprowski, P., & Holmberg, U. (1994). Comments on variable structure control design for uncertain discrete-time system. IEEE Transactions on Automatic Control, 39(11), 2366}2367. Pan, Y., & Furuta, K. (1994). Vss controller design for discrete-time system. Control-Theory and Advanced Technology, Part 1, 10(4), 669}687. Pieper, J. K., & Surgenor, B. W. (1994). Discrete time sliding mode control applied to a gantry crane. Proceedings of the 33rd conference on decision and control, 829}834. Sarpturk, S. Z., Istefanopulos, Y., & Kaynak, O. (1987). On the stability of discrete-time sliding mode control systems. IEEE Transactions on Automatic Control, AC-32(10), 930}932. Spurgeon, S. K. (1992). Hyperplane design techniques for discrete-time variable structure control systems. International Journal of Control, 55(2), 445}456. Utkin, V. I. (1977). Variable structure systems with sliding modes. IEEE Transactions on Automatic Control, AC-22(2), 212}222. Utkin, V. I. (1978). Sliding Modes and Their Applications to Variable Structure System. Moscow: MIR Publishers. Vadali, S. R. (1986). Variable structure control of spacecraft large-angle maneuvers. Journal of Guidance and Control of Dynamics, 9(2), 235}239. Wang, W.-J., & Wu, G.-H. (1992). Variable structure control design on discrete-time systems from another viewpoint. Control-Theory and Advanced Technology, 8(1), 1}16.
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Wang, W.-J., Wu, G.-H., & Yang, D.-C. (1994). Variable structure control design for uncertain discrete-time systems. IEEE Transactions on Automatic Control, 39(1), 99}102. Wang, W.-J., Lee, R.-C., & Yang, D.-C. (1996). Sliding mode control design in multi-input perturbed discrete-time systems. Journal of Dynamic Systems, Measurement, and Control, 118, 322}327. Chih-Chiang Cheng was born in Taipei, Taiwan, Republic of China on February 27, 1957. He received the B.S. degree in Electrical Engineering from Chung Yuan Christian University, Chung-Li, Taiwan, in 1981, the M.S. and Ph.D. degree in Electrical Engineering from University of Texas at Arlington, Arlington, Texas, in 1983 and 1991, respectively, as well as Engineer degree in Electrical Engineering from University of Southern California, Los Angeles, in 1985. He is currently an Associate Professor in the department of Electrical Engineering at National Sun Yat-Sen University, Kaohsiung, Taiwan. His current research interests include system and control theory, with emphasis on design of nonlinear control systems such as variable structure control, adaptive control, fuzzy control and digital control. Ming-Hsiung Lin was born on December 12, 1972. He received the B.S. degree in Electronic Engineering from National Taiwan Institutes of Technology, in 1994, and the M.S. degree in Electrical Engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1996. He is currently a product engineer of Asus computer Co.. His main research interests include design of discrete-time variable structure systems and personal computer structure. Jia-Ming Hsiao was born on February 12, 1971. He received the B.S. degree in Electrical Engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1994. Currently he is working toward his Ph.D. at National Sun Yat-Sen University, Kaohsiung, Taiwan. His main research interests include design of variable-structure systems and fuzzy control theory.