- Email: [email protected]
Multi-input uncertain linear systems with terminal sliding-mode control*
Automatica, Vol. 34, No. 3, pp. 389—392, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
PII: S0005–1098(97)00205–7
Technical Communique
Multi-input Uncertain Linear Systems with Terminal Sliding-Mode Control* XINGHUO YU- and MAN ZHIHONG‡ Key Words—Sliding mode; finite time convergence; variable structure; stability; linear systems.
are first outlined, then applied to control the uncertain linear system. Conditions for avoiding singularities are given. Simulations are presented to show the effectiveness of the control design.
Abstract—The design of variable structure control with terminal sliding modes for multi-input uncertain linear systems is discussed. The hierarchical terminal sliding-mode structure is used to enable finite time reachability of the system equilibrium. Conditions for the existence of the terminal sliding modes and implementability of the terminal sliding mode control are given. ( 1998 Elsevier Science Ltd. All rights reserved.
2. THE TERMINAL SLIDING MODE 1. INTRODUCTION
The TSM was first introduced to the control of the dynamic systems based on second-order differential equations such as robotic systems (Man et al., 1994). It is basically a nonlinear switching line,
Sliding-mode control systems have been studied for many years and received many applications. The design of the sliding-mode control involves prescribing switching surfaces with desired dynamic characteristics, and designing the control laws such that the system states reach and remain on the intersection of the switching surfaces. When in sliding, the system exhibits invariance properties such as robustness to certain internal parameter variations and external disturbances. Most commonly used switching surfaces are linear hyperplanes. Such hyperplanes guarantee the asymptotical stability of the sliding mode. It is desirable if certain nonlinear switching surfaces can be chosen which may give rise to a finite time reachability of equilibrium. Such surfaces may greatly improve the system performance. Terminal sliding mode (TSM) is such a finite-time nonlinear sliding mode (Man et al., 1994). For the nth-order single-input linear systems in which the TSMs are reached sequentially, the equilibrium is reached in finite time (Yu and Man, 1996). In this note, we extend the TSM control technique for the single-input systems (Yu and Man, 1996) to multi-input uncertain linear systems. The TSM concept and the hierarchical TSM structure
s"x #bxq@p"0, (1) 2 1 where x "xR , b'0, and p, q are positive integers. 2 1 The system has the following property. For an initial state x at t"t , x (0)"0, and b'0, and 1 0 1 odd positive integers p, q(p'q), it is easily computed that dynamics (1) will reach x "0 in finite 1 time t determined by s p ts" Dx (0)D(p~q)@p . (2) b(p!q) 1 Extension of dynamics (1) can be done by constructing a hierarchical sliding-mode structure such as (Yu and Man, 1996): s "sR #b sq1@p1 , 1 0 1 0 s "sR #b sq2@p2 , 2 1 2 1 s "sR #b sq3@p3 , 3 2 3 2 F F
(3) (4) (5)
s "sR #b sqn~1@pn~1 , (6) n~1 n~2 n~1 n~2 where s "x . The parameters are assumed to be 0 1 b '0, p 'q and p , q are positive odd integers. i i i i i This assumption allows us to achieve higher-order continuous differentiation. With the structure (3)—(6), if s "0 is reached, the stability and n~1 finite-time reachability of the system equilibrium will be guaranteed because it is a concatenation of n dynamics of equation (1) type. So in general, if
*Received 29 December 1994; revised 22 July 1996; received in final form 27 October 1998. This paper was recommended for publication in revised form by Editor Professor P. Dorato. Corresponding author Professor X. Yu. Tel. #61-79-309865; Fax #61-79-309729; Email [email protected]. -Department of Mathematics and Computing, Central Queensland University, Rockhampton, Qld 4702, Australia. ‡Department of Electrical Engineering and Computer Science, University of Tasmania, Hobart 7001, Australia. 389
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Technical Communiques
s "0 is reached at ts, i"1, 2 , n!1, then i n~i s will reach s "0 at n~i~1 n~i~1 p n~i ts "ts# i`1 i b (p !q ) n~i n~i n~i ]Ds (ts)D(pn~i~qn~i)@pn~i , (7) n~i~1 i hence the total time reaching x "0 is 1 n~1 p n~i ts"ts # + 1 b (p !q ) n~i i/1 n~i n~i ]Ds (ts)D(pn~i~qn~i)@pn~i . (8) n~i~1 i
tually equivalent to the decomposition *A"BR for R 3Rm]n so is the D in Assumption 2. It is well known (Kailath, 1980; Guzzella and Geering, 1990) that for the uncertain linear system (9) that satisfies the assumptions above, there exist two coordinate transformations ¹ 3Rn]n and P3 Rm]m, such that the time-invariant part of equation (9) can be transformed to the block-companion canonical form A "¹~1A¹, #
3. THE UNCERTAIN LINEAR SYSTEM
Consider
the
multi-input
uncertain
B "¹~1BP, #
A" # 2 —
(13)
where
linear
Q
1 —
(12)
2 —
a# 1
— —
— a# — — m
Qr "l th rowP 1 1 2
B" #
Qr "l #2#l "nth rowP m 1 m
(9)
where x3 Rn is the system state, u3 Rm is the control input, f 3 Rl (l4m) is the disturbance, and A, B are constant known matrices. The following conditions are assumed. Assumption 1. The uncertainty matrix *A satisfies the matching condition rank[*ADB]"rank[B]"m
0
2 0
,
(14)
2 0 2
0
1
where the unspecified entries of A and B are # # zero, and r (i"1, 2 , m) are the partial sums i of the controllability indices l , i.e. r "l , i 1 1 r "l #l , 2 , r "n. The entries of the matrices 2 1 2] m Q 3 R(li~1) li are almost all equal to zero except i for the upper secondary-diagonal elements [Q ] ( j"1, 2 , l !1) are equal to one. The i j,j`1 i controllability indices l denote the number of lini early independent column vectors in the controllability matrix [B , A B , 2 , An~1B ] which are # # # # # associated with the ith column of B . Without loss # of generality, we assume l 5l 525l . Re1 2 m garding construction of the transformations readers are referred to Kailath (1980) and Guzzella and Geering (1990). Because of Assumption 1, the uncertainty term *A can also be transformed into the form (Guzzella and Geering, 1990)
systems defined as xR "[A#*A]x#Bu#Df,
1
(10)
and also is bounded, i.e. D*a D4*aN where *aN is ij ij ij a positive constant. Assumption 2. The disturbance matrix D satisfies the matching condition 2 *A " #
— —
*a# 1 2
— —
— — *a# — — m rank[DDB]"rank[B]"m
(11)
and f is bounded, i.e. D f D4fM , i"1, 2 , l, where i i fM is a positive constant. i Assumption 3. The pair MA, BN is completely controllable. Remark 1. Condition (10) in Assumption 1 is ac-
Qr "l th row 1 1 2
(15)
Qr "l #2#l "nth row, m 1 m
where the unspecified entries in the matrix are zero, and *a# 3 R1]n. From Assumption 1 we can i assume that (16) E*a#E4*aN # i i where the entries of *aN # are positive constants i and E ) E is the Euclidean norm. The same analogy can be applied to the disturbance matrix D which, with the same transformations, can also be
Technical Communiques
391
transformed into — — D" # 2 —
d# 1
— —
— d# — — m
Qr "l th row 1 1 2
(17)
Qr "l #2#l "nth row, m 1 m
where d 3R1]l. Also, it can be assumed that i (18) Ed#E4dM # i i where the entries of dM # are positive constants. i 4. THE TSM CONTROL
For the system in the canonical form i.e. equations (13)—(18), xR "(A #*A )x#B u#D f, (19) # # # # we introduce the set of switching surfaces as S(x)"[s (x), s (x), 2 , s (x)]T in order to achieve 1 2 m finite-time reachability as well as the robustness. For each entry of S, s ( j"1, 2 , m), we employ the j following hierarchical structure: s "sR #b sqj,1@pj,1 , j,1 j,0 j,1 j,0 s "sR #b sqj,2@pj,2 , j,2 j,1 j,2 j,1 F F
(20) (21)
"sR #b sqj,lj~1@pj,lj~1 , (22) s j,lj~2 j,lj~1 j,lj~2 j,lj~1 with r "0, i.e. s "x , where s "x 0 1,0 1 j,0 rj~1 s "x . , 2 , s "x and s "s 2,0 r1`1 m,0 rm~1`1 j j,lj~1 Remark 2. It is easily seen that such a structure results in finite-time reachability of the equilibrium. If s "0 is reached, then the system will be j,lj~1 constrained in the sliding-mode structure and reach s "0, 2 , s "0 sequentially in finite time. j,lj~1 j,0 The following control is chosen: u"u #u , (23) %2 /where u is the equivalent control for system (9) %2 without the internal parameter variations and external disturbances, such that S(x)"0 and SQ (x)"0. u is the control that will suppress the /uncertainties and disturbances. Proposition 1. For the system (19), if control u is designed as u"u #u (24) %2 /with the entries of u and u %2 /li~2 dli~k~1 u "!a# x! + b sli,k`1@pi,k`1 , (25) i i,%2 i,k`1 dtli~k~1 k/0
)!dM # fM sgn(s ) u "!*aN # DxD sgn(s i i,li~1 i i,li~1 i,/), (26) !k sgn(s i i,li~1 for i"1, 2 , m, where k '0, sgn is a sign function i defined as sgn(s )"1, 0, !1 for s '0, i,li~1 i,li~1 "0, (0, respectively, then the system will reach the sliding mode S(x)"0 in finite time. Proof. To ensure the reachability of the sliding mode s "s "0, the condition s R s (0 i i,li~1 i,li~1 i,li~1 should be satisfied. Takeing the first-order derivative of s using equation (22), one has i,li~1i d sR "s¨ #b (sqi ,li~1@pi ,li~1) (27) i,li~1 i,li~2 i,li dt i,li~2 since s "sR #b sqi,h@pi,h, for h"l !1, 2 , 1, i,h i,h~1 i,h i,h~1 i and the lth-order derivative of s is i dl`1 dl dl s " s #b (sqi,h@pi,h). i,h dtl i,h~1 dtl i,h dtl`1 i,h~1 Then it can be easily calculated that dli dli~k~1 li~2 " s #+ b (sqi,k`1@pi,k`1). sR i,k`1 dtli~k~1 i,k i,li~1 dtli l,0 k/0 (28) , then Since s "x i,0 ri~1`1r dli dli . s " x dtli i,0 dtli r~1`1 From the canonical form matrices (16) and (17) one can see that dli x "xR "a#x#*a#x#d# f#u . (29) i i i i ri dtli ri~1 The resulting expression is then substituted into equation (29) and multiplied by s as i,li~1 sR "*a#xs #d# f s s i i,li~1 i i,li~1 i,li~1 i,li~1 D!dM # fM Ds D !*aN # DxDDs i i,li~1 i i,li~1 !k Ds D. (30) i i,li~1i Since *a#xs !*aN # DxD Ds D40, i i,li~1 i i,li~1 !dM # fM Ds D40, d# f s i i,li~1 i i,li~1 then s sR D, (31) 4!k Ds l,li~1 i,li~1 i i,li~1
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Technical Communiques
"0 which means that the sliding mode s i,li~1 will be reached in finite time. The TSM s "0 i (i"1, 2 , m) is then reached in finite time. Hence S"0 will be reached in finite time. When S"0, subsequently the system state will reach zero in finite time. K Remark 3. The control given by equation (24) has actually a kind of decoupling effect. Indeed, when the equivalent control component u is activated i,%2 s "0 is reached which means x ( j"1, 2 , l ) i ri~1`j i will reach zero independently. Remark 4. Note that control (24) involves calculation of terms dli~k~1 sqi,k`1@pi,k`1, k"0, 2 , l !2. i dtli~k~1 k It is easy to show that, for any k3 M0, 1, 2 , l !2N, i dli~k~1 sqi,k`1@pi,k`1"f (s , sR , 2 , s(li~1)). i,k i,0 i,0 i,0 dtli~k~1 i,k where f
i,k
. is a nonlinear function, and s "x i,0 ri~1`1
The cases when s P0, u PR, may exist i~1 i which we call the singularity of the control. To avoid the singularity, stringent conditions have to be imposed. Proposition 2. If (l !k!1)p ((l !k)q , then i i,k i i,k when s P0 sequentially from k"l !2 to k"0, i,k i u is bounded. i Proof. Similar to Yu and Man (1996).
K
Remark 5. There is another kind of singularity. There may exist such a case when t "0, for some 0 i, h, where i 3 [1, 2 , m] and h3 [1, 2 , l !1], i s (t )"0. By replacing s (t ) with the following i,h 0 i,h 0 mapping function;
G
z(t ) for z(t )O0, 0 0 (32) d for z(t )"0, 0 where d'0, this singular problem will be avoided. Map(z(t ))" 0
Remark 6. The aforementioned control technique can be extended to model following adaptive control simply by replacing the system state x in the sections above with z"x !x where x is the d d
desired dynamics state and x is the actual system state. Assume that the desired dynamics is xR "A x #B r. d d d d If the matching condition rank[A!A DB]"rank[B]"m d is satisfied, then the design of the model following adaptive control with terminal sliding modes is almost parallel to the design procedure above. Remark 7. It should be noted that for the unmatched perturbation Df to the system where f is a bounded perturbation and DOBR@ for any matrix R@, since the matrix A relies on the trans# formation ¹ in equation (13) ¹~1D cannot be decomposed into the form of cB for any constant c. Hence the condition xR "r (i"1, 2 , m; ri`j ri`j`1 j"1, 2 , r) does not hold and the derivation of u is different. Strong conditions may have to be imposed on parameters q and p . With the i,li~1 i,li~1 presented condition, the boundedness of the system is not guaranteed for an unmatched bounded perturbation. 5. CONCLUSION
The control of multi-input uncertain linear systems with TSMs has been discussed in this paper. It has been shown that if the system uncertainties and disturbances satisfy the matching condition, the TSM control realizes the finite-time reachability of the system equilibrium. Conditions to avoid the singularity of control when the system state enters the TSMs have been given. Acknowledgements—This work is supported in part by a grant from the Australian Research Council. REFERENCES Man, Z., A. P. Paplinski and H. R. Wu (1994). A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE ¹rans. Automat. Control, 39, 2464—2468. Guzzella, L. H. P. Geering (1990). Canonical formulation and general principles of variable structure controllers. In Deterministic Control of ºncertain Systems, ed., A. S. I. Zinober. Peter Peregrinus, London. Kailath, T. (1980). ¸inear Systems. Prentice-Hall, Englewood Cliffs, NJ. Yu, X. Z. Man (1996). Model reference adaptive control systems with terminal sliding modes. Int. J. Control, 64, 1165—1176.
PII: S0005–1098(97)00205–7
Technical Communique
Multi-input Uncertain Linear Systems with Terminal Sliding-Mode Control* XINGHUO YU- and MAN ZHIHONG‡ Key Words—Sliding mode; finite time convergence; variable structure; stability; linear systems.
are first outlined, then applied to control the uncertain linear system. Conditions for avoiding singularities are given. Simulations are presented to show the effectiveness of the control design.
Abstract—The design of variable structure control with terminal sliding modes for multi-input uncertain linear systems is discussed. The hierarchical terminal sliding-mode structure is used to enable finite time reachability of the system equilibrium. Conditions for the existence of the terminal sliding modes and implementability of the terminal sliding mode control are given. ( 1998 Elsevier Science Ltd. All rights reserved.
2. THE TERMINAL SLIDING MODE 1. INTRODUCTION
The TSM was first introduced to the control of the dynamic systems based on second-order differential equations such as robotic systems (Man et al., 1994). It is basically a nonlinear switching line,
Sliding-mode control systems have been studied for many years and received many applications. The design of the sliding-mode control involves prescribing switching surfaces with desired dynamic characteristics, and designing the control laws such that the system states reach and remain on the intersection of the switching surfaces. When in sliding, the system exhibits invariance properties such as robustness to certain internal parameter variations and external disturbances. Most commonly used switching surfaces are linear hyperplanes. Such hyperplanes guarantee the asymptotical stability of the sliding mode. It is desirable if certain nonlinear switching surfaces can be chosen which may give rise to a finite time reachability of equilibrium. Such surfaces may greatly improve the system performance. Terminal sliding mode (TSM) is such a finite-time nonlinear sliding mode (Man et al., 1994). For the nth-order single-input linear systems in which the TSMs are reached sequentially, the equilibrium is reached in finite time (Yu and Man, 1996). In this note, we extend the TSM control technique for the single-input systems (Yu and Man, 1996) to multi-input uncertain linear systems. The TSM concept and the hierarchical TSM structure
s"x #bxq@p"0, (1) 2 1 where x "xR , b'0, and p, q are positive integers. 2 1 The system has the following property. For an initial state x at t"t , x (0)"0, and b'0, and 1 0 1 odd positive integers p, q(p'q), it is easily computed that dynamics (1) will reach x "0 in finite 1 time t determined by s p ts" Dx (0)D(p~q)@p . (2) b(p!q) 1 Extension of dynamics (1) can be done by constructing a hierarchical sliding-mode structure such as (Yu and Man, 1996): s "sR #b sq1@p1 , 1 0 1 0 s "sR #b sq2@p2 , 2 1 2 1 s "sR #b sq3@p3 , 3 2 3 2 F F
(3) (4) (5)
s "sR #b sqn~1@pn~1 , (6) n~1 n~2 n~1 n~2 where s "x . The parameters are assumed to be 0 1 b '0, p 'q and p , q are positive odd integers. i i i i i This assumption allows us to achieve higher-order continuous differentiation. With the structure (3)—(6), if s "0 is reached, the stability and n~1 finite-time reachability of the system equilibrium will be guaranteed because it is a concatenation of n dynamics of equation (1) type. So in general, if
*Received 29 December 1994; revised 22 July 1996; received in final form 27 October 1998. This paper was recommended for publication in revised form by Editor Professor P. Dorato. Corresponding author Professor X. Yu. Tel. #61-79-309865; Fax #61-79-309729; Email [email protected]. -Department of Mathematics and Computing, Central Queensland University, Rockhampton, Qld 4702, Australia. ‡Department of Electrical Engineering and Computer Science, University of Tasmania, Hobart 7001, Australia. 389
390
Technical Communiques
s "0 is reached at ts, i"1, 2 , n!1, then i n~i s will reach s "0 at n~i~1 n~i~1 p n~i ts "ts# i`1 i b (p !q ) n~i n~i n~i ]Ds (ts)D(pn~i~qn~i)@pn~i , (7) n~i~1 i hence the total time reaching x "0 is 1 n~1 p n~i ts"ts # + 1 b (p !q ) n~i i/1 n~i n~i ]Ds (ts)D(pn~i~qn~i)@pn~i . (8) n~i~1 i
tually equivalent to the decomposition *A"BR for R 3Rm]n so is the D in Assumption 2. It is well known (Kailath, 1980; Guzzella and Geering, 1990) that for the uncertain linear system (9) that satisfies the assumptions above, there exist two coordinate transformations ¹ 3Rn]n and P3 Rm]m, such that the time-invariant part of equation (9) can be transformed to the block-companion canonical form A "¹~1A¹, #
3. THE UNCERTAIN LINEAR SYSTEM
Consider
the
multi-input
uncertain
B "¹~1BP, #
A" # 2 —
(13)
where
linear
Q
1 —
(12)
2 —
a# 1
— —
— a# — — m
Qr "l th rowP 1 1 2
B" #
Qr "l #2#l "nth rowP m 1 m
(9)
where x3 Rn is the system state, u3 Rm is the control input, f 3 Rl (l4m) is the disturbance, and A, B are constant known matrices. The following conditions are assumed. Assumption 1. The uncertainty matrix *A satisfies the matching condition rank[*ADB]"rank[B]"m
0
2 0
,
(14)
2 0 2
0
1
where the unspecified entries of A and B are # # zero, and r (i"1, 2 , m) are the partial sums i of the controllability indices l , i.e. r "l , i 1 1 r "l #l , 2 , r "n. The entries of the matrices 2 1 2] m Q 3 R(li~1) li are almost all equal to zero except i for the upper secondary-diagonal elements [Q ] ( j"1, 2 , l !1) are equal to one. The i j,j`1 i controllability indices l denote the number of lini early independent column vectors in the controllability matrix [B , A B , 2 , An~1B ] which are # # # # # associated with the ith column of B . Without loss # of generality, we assume l 5l 525l . Re1 2 m garding construction of the transformations readers are referred to Kailath (1980) and Guzzella and Geering (1990). Because of Assumption 1, the uncertainty term *A can also be transformed into the form (Guzzella and Geering, 1990)
systems defined as xR "[A#*A]x#Bu#Df,
1
(10)
and also is bounded, i.e. D*a D4*aN where *aN is ij ij ij a positive constant. Assumption 2. The disturbance matrix D satisfies the matching condition 2 *A " #
— —
*a# 1 2
— —
— — *a# — — m rank[DDB]"rank[B]"m
(11)
and f is bounded, i.e. D f D4fM , i"1, 2 , l, where i i fM is a positive constant. i Assumption 3. The pair MA, BN is completely controllable. Remark 1. Condition (10) in Assumption 1 is ac-
Qr "l th row 1 1 2
(15)
Qr "l #2#l "nth row, m 1 m
where the unspecified entries in the matrix are zero, and *a# 3 R1]n. From Assumption 1 we can i assume that (16) E*a#E4*aN # i i where the entries of *aN # are positive constants i and E ) E is the Euclidean norm. The same analogy can be applied to the disturbance matrix D which, with the same transformations, can also be
Technical Communiques
391
transformed into — — D" # 2 —
d# 1
— —
— d# — — m
Qr "l th row 1 1 2
(17)
Qr "l #2#l "nth row, m 1 m
where d 3R1]l. Also, it can be assumed that i (18) Ed#E4dM # i i where the entries of dM # are positive constants. i 4. THE TSM CONTROL
For the system in the canonical form i.e. equations (13)—(18), xR "(A #*A )x#B u#D f, (19) # # # # we introduce the set of switching surfaces as S(x)"[s (x), s (x), 2 , s (x)]T in order to achieve 1 2 m finite-time reachability as well as the robustness. For each entry of S, s ( j"1, 2 , m), we employ the j following hierarchical structure: s "sR #b sqj,1@pj,1 , j,1 j,0 j,1 j,0 s "sR #b sqj,2@pj,2 , j,2 j,1 j,2 j,1 F F
(20) (21)
"sR #b sqj,lj~1@pj,lj~1 , (22) s j,lj~2 j,lj~1 j,lj~2 j,lj~1 with r "0, i.e. s "x , where s "x 0 1,0 1 j,0 rj~1 s "x . , 2 , s "x and s "s 2,0 r1`1 m,0 rm~1`1 j j,lj~1 Remark 2. It is easily seen that such a structure results in finite-time reachability of the equilibrium. If s "0 is reached, then the system will be j,lj~1 constrained in the sliding-mode structure and reach s "0, 2 , s "0 sequentially in finite time. j,lj~1 j,0 The following control is chosen: u"u #u , (23) %2 /where u is the equivalent control for system (9) %2 without the internal parameter variations and external disturbances, such that S(x)"0 and SQ (x)"0. u is the control that will suppress the /uncertainties and disturbances. Proposition 1. For the system (19), if control u is designed as u"u #u (24) %2 /with the entries of u and u %2 /li~2 dli~k~1 u "!a# x! + b sli,k`1@pi,k`1 , (25) i i,%2 i,k`1 dtli~k~1 k/0
)!dM # fM sgn(s ) u "!*aN # DxD sgn(s i i,li~1 i i,li~1 i,/), (26) !k sgn(s i i,li~1 for i"1, 2 , m, where k '0, sgn is a sign function i defined as sgn(s )"1, 0, !1 for s '0, i,li~1 i,li~1 "0, (0, respectively, then the system will reach the sliding mode S(x)"0 in finite time. Proof. To ensure the reachability of the sliding mode s "s "0, the condition s R s (0 i i,li~1 i,li~1 i,li~1 should be satisfied. Takeing the first-order derivative of s using equation (22), one has i,li~1i d sR "s¨ #b (sqi ,li~1@pi ,li~1) (27) i,li~1 i,li~2 i,li dt i,li~2 since s "sR #b sqi,h@pi,h, for h"l !1, 2 , 1, i,h i,h~1 i,h i,h~1 i and the lth-order derivative of s is i dl`1 dl dl s " s #b (sqi,h@pi,h). i,h dtl i,h~1 dtl i,h dtl`1 i,h~1 Then it can be easily calculated that dli dli~k~1 li~2 " s #+ b (sqi,k`1@pi,k`1). sR i,k`1 dtli~k~1 i,k i,li~1 dtli l,0 k/0 (28) , then Since s "x i,0 ri~1`1r dli dli . s " x dtli i,0 dtli r~1`1 From the canonical form matrices (16) and (17) one can see that dli x "xR "a#x#*a#x#d# f#u . (29) i i i i ri dtli ri~1 The resulting expression is then substituted into equation (29) and multiplied by s as i,li~1 sR "*a#xs #d# f s s i i,li~1 i i,li~1 i,li~1 i,li~1 D!dM # fM Ds D !*aN # DxDDs i i,li~1 i i,li~1 !k Ds D. (30) i i,li~1i Since *a#xs !*aN # DxD Ds D40, i i,li~1 i i,li~1 !dM # fM Ds D40, d# f s i i,li~1 i i,li~1 then s sR D, (31) 4!k Ds l,li~1 i,li~1 i i,li~1
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Technical Communiques
"0 which means that the sliding mode s i,li~1 will be reached in finite time. The TSM s "0 i (i"1, 2 , m) is then reached in finite time. Hence S"0 will be reached in finite time. When S"0, subsequently the system state will reach zero in finite time. K Remark 3. The control given by equation (24) has actually a kind of decoupling effect. Indeed, when the equivalent control component u is activated i,%2 s "0 is reached which means x ( j"1, 2 , l ) i ri~1`j i will reach zero independently. Remark 4. Note that control (24) involves calculation of terms dli~k~1 sqi,k`1@pi,k`1, k"0, 2 , l !2. i dtli~k~1 k It is easy to show that, for any k3 M0, 1, 2 , l !2N, i dli~k~1 sqi,k`1@pi,k`1"f (s , sR , 2 , s(li~1)). i,k i,0 i,0 i,0 dtli~k~1 i,k where f
i,k
. is a nonlinear function, and s "x i,0 ri~1`1
The cases when s P0, u PR, may exist i~1 i which we call the singularity of the control. To avoid the singularity, stringent conditions have to be imposed. Proposition 2. If (l !k!1)p ((l !k)q , then i i,k i i,k when s P0 sequentially from k"l !2 to k"0, i,k i u is bounded. i Proof. Similar to Yu and Man (1996).
K
Remark 5. There is another kind of singularity. There may exist such a case when t "0, for some 0 i, h, where i 3 [1, 2 , m] and h3 [1, 2 , l !1], i s (t )"0. By replacing s (t ) with the following i,h 0 i,h 0 mapping function;
G
z(t ) for z(t )O0, 0 0 (32) d for z(t )"0, 0 where d'0, this singular problem will be avoided. Map(z(t ))" 0
Remark 6. The aforementioned control technique can be extended to model following adaptive control simply by replacing the system state x in the sections above with z"x !x where x is the d d
desired dynamics state and x is the actual system state. Assume that the desired dynamics is xR "A x #B r. d d d d If the matching condition rank[A!A DB]"rank[B]"m d is satisfied, then the design of the model following adaptive control with terminal sliding modes is almost parallel to the design procedure above. Remark 7. It should be noted that for the unmatched perturbation Df to the system where f is a bounded perturbation and DOBR@ for any matrix R@, since the matrix A relies on the trans# formation ¹ in equation (13) ¹~1D cannot be decomposed into the form of cB for any constant c. Hence the condition xR "r (i"1, 2 , m; ri`j ri`j`1 j"1, 2 , r) does not hold and the derivation of u is different. Strong conditions may have to be imposed on parameters q and p . With the i,li~1 i,li~1 presented condition, the boundedness of the system is not guaranteed for an unmatched bounded perturbation. 5. CONCLUSION
The control of multi-input uncertain linear systems with TSMs has been discussed in this paper. It has been shown that if the system uncertainties and disturbances satisfy the matching condition, the TSM control realizes the finite-time reachability of the system equilibrium. Conditions to avoid the singularity of control when the system state enters the TSMs have been given. Acknowledgements—This work is supported in part by a grant from the Australian Research Council. REFERENCES Man, Z., A. P. Paplinski and H. R. Wu (1994). A robust MIMO terminal sliding mode control for rigid robotic manipulators. IEEE ¹rans. Automat. Control, 39, 2464—2468. Guzzella, L. H. P. Geering (1990). Canonical formulation and general principles of variable structure controllers. In Deterministic Control of ºncertain Systems, ed., A. S. I. Zinober. Peter Peregrinus, London. Kailath, T. (1980). ¸inear Systems. Prentice-Hall, Englewood Cliffs, NJ. Yu, X. Z. Man (1996). Model reference adaptive control systems with terminal sliding modes. Int. J. Control, 64, 1165—1176.