PII: S0005–1098(98)00041–7
Automatica, Vol. 34, No. 8, pp. 1021—1024, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00
Technical Communique
On the Localization of Intersample Ripples of Linear Systems Controlled by Generalized Sampled-Data Hold Functions* K. G. ARVANITISKey Words—Sampled-data systems; generalized sampled-data hold functions; exact model matching; intersample ripples; linear nonself-adjoint operators.
has been reported in the literature and several techniques have been developed to settle this important concern (see, for example, Bamieh and Pearson, 1992; Chen and Francis, 1991a, 1995; Chen and Qiu, 1994; Dullerud and Francis, 1991; Kabamba and Hara, 1993; Leung et al., 1991; Tadmor, 1992; Toivonen, 1992; Yamamoto, 1994; Arvanitis and Paraskevopoulos, 1995 and the references cited therein). Nevertheless, it appears that, a thorough study concerning the intersample performance analysis of sampled-data systems controlled on the basis of the GSHF approach does not exist in the relevant literature. In this note, our aim is to investigate the possibility of ameliorating the closed-loop system performance in the intersample time instants, when GSHF are designed in order to achieve the control objective at the sampling instants. Our interest is focused on the case where, the control objective is the exact matching of the closed-loop system to a prespecified discretetime model. In this case, the modulating hold functions are tailored to a given system, in such a way that, for the closed-loop system, a desired discrete-time transfer function matrix to be assigned. In Paraskevopoulos and Arvanitis (1994), a new algebraic technique is presented, for the solution of the discrete exact model matching problem using the GSHF approach. This technique provides necessary and sufficient solvability conditions and the general expressions of the modulating GSHF sought, as functions of arbitrary parameters. In this respect, our purpose here, is to investigate the possibility to improve the ‘‘robustness’’ properties of the aforementioned GSHF technique, by appropriately selecting the values of these arbitrary parameters. The technique proposed to treat this problem, is mainly based on an appropriate error system and on the expansion of the output signal of this system, in suitable basis functions, which are chosen such that some finite rank linear nonself-adjoint operators are represented exactly. It is worth noticed, at this point that, the proposed technique is similar to the well-known lifting technique for continuous-time signals and sampled data systems (see Bamieh et al., 1991; Bamieh and Pearson, 1992; Tadmor, 1992; Yamamoto, 1994; Chen and Francis, 1995). As it is shown in the present note, localization of intersample ripples can be accomplished by a suitable selection of the degrees of freedom incorporated in the general forms of the modulating hold functions. This guarantees that the performance of the closed-loop system is the desired one, even between the sampling instants.
Abstract—In this note, the intersample performance of linear systems, which are controlled on the basis of generalized sampled-data hold functions, in order to achieve exact model matching at the sampling instants, is analyzed. The proposed technique relies on an appropriate error system and of the expansion of its output signal in suitable basis functions, which are selected such that some finite rank linear nonself-adjoint operators are represented exactly. As it is shown, it is plausible to localize the intersample ripples, which may cause a degradation of the control performance, by appropriately selecting the arbitrary elements of the general forms of the modulating hold functions. This guarantees that the performance of the closedloop system is the desired one, not only at the sampling instants, but even between them. ( 1998, Elsevier Science Ltd. All rights reserved. 1. Introduction Generalized sampled-data hold functions (GSHF), first proposed by Kabamba (1987), constitute a powerful tool for the control of linear multivariable systems, alternative to standard dynamic compensation and especially to state observers. The basic idea of the GSHF approach is to periodically sample the plant output and generate the control by means of a hold function applied to the resulting sequence. Until now, the GSHF approach has successfully been applied, in order to solve some very interesting control problems. This approach provides a series of remarkable advantages over other well-established feedback control design techniques (for an overview, see Kabamba, 1987; Arvanitis, 1995 and the references cited therein). From an overview of the so far reported results on the subject, it becomes clear that, although the particular control objective sought can be achieved using the GSHF approach at the sampling instants, little can be said about the performance of the control system between them. Generally speaking, when designing sampled-data controllers for continuous-time linear systems, it is of crucial importance to take into account the intersample behaviour of the system, which may not be the desired one, even though at the sampling instants the system has the desired control performance. For example, it is well known that, in sampled control, a degradation of the control performance may be caused by intersample oscillations, which are usually called intersample ripples (Astrom and Wittenmark, 1984). Recently, much research related to the intersample control performance,
2. Preliminaries Consider the reachable linear time-invariant system of the form x5 (t)"Ax(t)#Bu(t), y(t)"Cx(t), (1)
* Received 19 February 1997; Received in final form 17 February 1998. This paper was recommended for publication in revised form by Editor Peter Dorato. Corresponding author K. G. Arvanitis. Tel. #30 1 7722501; Fax #30 1 7722459; E-mail
[email protected]. -National Technical University of Athens, Department of Electrical and Computer Engineering, Division of Computer Science, Zographou 15773, Athens, Greece.
where x(t)3Rn, u(t)3Rm, y(t)3Rp and A, B, C are real matrices of appropriate dimensions. Consider now applying to system (1) the following control law: u(t)"F(t)y(k¹)#G(t)w(k¹), (2) 1021
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where t3[k¹, (k#1)¹), k50, and where w(k¹)3Rr is a new control vector. The matrices F(t), G(t), known as generalized sampled-data hold functions, are assumed bounded, integrable and ¹-periodic, i.e.
where l"n(r#p)!rank %, j , i"1, 2, 2, l, are arbitrary 1 parameters and
C D C D
F(t#¹)"F(t), G(t#¹)"G(t), t3[k¹, (k#1)¹), k50 The resulting closed-loop system has the following transfer function matrix:
P
T
exp[A(¹!q)]B(F(q), G(q)) dq. (3) 0 Under the reachability assumption on equation (1), a solution of equation (3) with regard to (F(t), G(t)) is (Kabamba, 1987) (F(t), G(t))"BT exp[AT(¹!t)]W~1(A, B, ¹) (F , F ), (4) ( c where W(A, B, ¹) is the reachability Gramian of (A, B) on [0,¹], having the form W(A, B, ¹) "
P
T
exp[A(¹!q)]BBT exp[AT(¹!q)] dq. 0 In the sequel, our aim is the design of GSHF, in order to achieve at the sampling points t"k¹, exact matching of system (1) to a given discrete reference model. More precisely, consider a linear continuous-time model M, described in state space by x5 (t)"A x (t)#B w(t), y (t) " C x (t), M M M M M M M ] where x 3Rnm nm and y (t)3RP, and its corresponding discreteM M time analogous, obtained by sampling x (t), w(t) and y (t) with M M sampling period ¹ and having the following discrete transfer function matrix: H (z)"C (zI!A )~1B , (5) m m m m ] where H (z)3Cp r(z) and where Cp r(z) denotes the field of the m strictly causal rational discrete transfer function and where in equation (5) the matrix triplet is such that ]
A "exp(A ¹), m M
P
B " m
T
exp(A q)B dq, M M
C "C . m M
0 It is obvious that, system (5) can match system (1) using GSHF if and only if H (z)"C(zI!'!F C)~1F ,H (z)"C (zI!A )~1B . c ( c m m m m After certain algebraic manipulations, the above relation can be reduced to the following algebraic system of equations (Parskevopoulos and Arvanitis, 1994): h%"h,
(6)
h"[( f ) 2 ( f ) ( f ) 2 ( f ) ], c1 cn (1 (n where ( f ) and ( f ) , for i"1, 2, 2, n, are the ith rows of F and ci (i c F , respectively. Note also that in equation (6) ( I ? CT I ? (C')T 2 I ? (C'2n~1)T %" , 2n~1 0 M ? CT 2 + M ? (C'2n~j~1)T 1 j/1 j h"[(k ) 2 (k ) 2 (k ) 2 (k ) ], 11 1p 2n 1 2n p where M "C Aj~1 B , for j"1, 2, 2, 2n, (k ) , for j m m m jk k"1, 2, 2, p is the kth row of M , and where by ? is denoted j the Kronecker product. Equation (6) is solvable if and only if the following rank condition holds:
C
D
CD
rank
% "rank % h
The general solution of equation (6), for F and F is (Parac ( skevopoulous and Arvanitis, 1994) l F "Q # + j Q , c 0 j j j/1
C D
S " 0
l F "S # + j S , ( 0 j j j/1
(7)
(r ) 01 F , (r ) 0n
CD
(p ) (t ) 1j 1j S" F . F , j (p ) (t ) nj nj Note that, (q ) and (r ) for o"1, 2, 2, n are obtained by 0o 0o partitioning the special solution h "h%T[%%T#%T% ]~1 of sp 0 0 equation (6), as follows: Q" j
H (z)"C(zI!'!F C)~1F , # ( c where '"exp(A¹) and where (F , F )" ( c
(q ) 01 F , (q ) 0n
Q " 0
h "[(q ) 2 (q ) (r ) 2 (r ) ], sp 01 on 0n 0n where % is a basis for ker %. It is also noticed that (p ) and (t ) 0 ij ij are the jth rows of the matrices P and T , produced by partitioni i ing % as follows: 0 % "[P 2 P T 2 T ]. 0 1 n i n In view of equation (4), the solution of the exact model matching problem using GSHF, for G(t), F(t) is given by (G(t), F(t))"BT exp[AT(¹!t)]W~1(A, B, ¹)
A
B
l l ] Q #+jQ ,S #+j S . (8) 0 j j 0 j j j/1 j/1 The degrees of freedom, in the general analytical expressions of the modulating hold functions G(t) and F(t), which are represented by the arbitrary parameters j , i"1, 2,2 , v, may 1 be very useful, in order to satisfy other additional design requirements, such as parameter sensitivity reduction, minimum cost controller implementation, etc. In particular, as it is shown in the next section, we can obtain a significant amelioration of the closed-loop system performance, in the sense of the localization of the intersample ripples, by appropriately selecting these arbitrary parameters. 3. Main result In this section, we deal with the problem of the existence of suitable choices of the solutions of equation (6), in order to ameliorate the closed-loop system performance, in the intersample time instants k¹#d. To this end, it is convenient to define, the following extended state-space model, called the error system: x5 (t)"A x (t)#B u(t)#B w(t), y (t)"C x (t), (9) e e e e,1 e,2 e e e where xT (t)"[xT(t) xT (t)]T3Rn`nm and where e M
C C
D
A 0 A" , e 0 A M C " !C e
B
C
D
M
CD
C D
B 0 " , B " , e,1 e,2 0 B M
.
It is pointed out that, in equation (9), y (t)3Rp is the error, e defined by the difference between y (t) and y(t). M Let, in the sequel, w(t)"w(k¹), ∀t3[k¹, (k#1)¹). We shall study the problem of controlling (9) using controllers of the form (2) and (8). In this case, the error y (t) at time k¹#d, is given by e y (k¹#d)"C [exp(A d)#) (d)]x (k¹) e e e 1 e d #C ) (d)# exp[A (d!q)]B dq w(k¹), e 2 e e,2 0 (10)
C
P
D
where ) (d) and ) (d) are defined by 1 2 d ) (d)" exp[A (d!q)]B BT 1 e e,1 0 exp[AT(¹!q)]W~1(A, B, ¹) F [C 0] dq (
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P
d exp[A (d!q)]B BT e e,1 0 exp[AT(¹!q)]W~1(A, B, ¹)F dq c and where F and F are given by equation (7). We shall next c ( show that, it is plausible to drive y (k¹#d) to almost zero e values, by choosing suitably the arbitrary parameters j involved i in the forms of F and F . To this end, it is convenient to c ( introduce the linear operators Q : Rn`nmP¸P[0, ¹), Q : 1 2 2 Rn`nmP¸P[0, ¹) and Q : RrP¸P[0, ¹), defined by 2 3 2 (Q x) (d)"C exp(A d)x, (11a) 1 e e d (Q x)(d)" C exp[A (d!q)]B BT exp[AT(¹!q)]W~1 2 e e e,1 0 l (A, B, ¹) S # + j S [C 0] dqx, (11b) 0 j j j/i d (Q w) (d)" C exp[A (d!q)]B BT exp[AT(¹!q)]W~1 3 e e e,1 0 l (A, B, ¹) Q # + j Q #B dqw, (11c) 0 j j e,2 j/i respectively, where, in general, Lx (I) is the Hilbert space of 2 functions with values in Rx, which are square integrable on 1-R in the sense of the inner product defined by ) (d)" 2
P
A
P
A
B
B
P
(h, t )"
hq(t)l(t) dt with h, t3Lx (I). 2 I Also defined by S the shift operator of the form a (S x) (t)"x(t#a) a and let the sequence of operators MP N, k50, be defined *kT,(k`1)T) as the sequence of orthogonal projections from L [0, R) (see 2 Naylor and Sell, 1982 for details). In this respect, the error signal y , whose elements belong to L [0, R) , can be uniquely decome 2 posed as follows: = y"+ P y . e *kT,(k`1)T) e k/0 Using definitions (11a)—(11c), the components of the error y in e the above decomposition are given by P Y "S Q x (k¹)#S Q x (k¹) *kT,(k`1)T) e ~kT 1 e ~kT 2 e #S Q w(k¹), k50. (12) ~kT 3 Relation (12) defines a discrete representation of relation (10), with the error y as element of the infinite-dimensional space e LP [0, ¹). It is worth noticed, at this point, that the operators Q , 2 1 Q and Q , have finite ranks. The approach proposed here, to 2 3 study the intersample behavior of the closed-loop system, relies on the expansion of the error signal y in suitable basis functions. e These functions, will be selected such that the finite rank operators Q , Q and Q , are represented exactly. 1 2 3 In order to develop a more useful discrete-time representation of equation (10), it is convenient to characterize the operator Q , 1 in terms of its Schmidt pair decomposition (Gohberg and Krein, 1969; Bettayeb et al., 1980). This decomposition is summarized in the following Proposition.
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where R(A , C , ¹) is the positive-semidefinite matrix e e T R(A , C , ¹)" exp(ATq)CTC exp(A q) dq. e e e e e e 0 The left singular vectors n , j"1, 2, 2, r , are given by j 1 n (d)"p~1C exp(A d)t . j j e e j
P
(15)
Proof. it is well known that the Schmidt pair decomposition of the operator Q is given by equation (13). It is also easy to show 1 that the singular values p and the singular vectors t satisfy the j j eigenvalue problem Q* Q t "p2t , j"1, 2, 2, r , (16) 1 1 j j j 1 where Q* is the adjoint operator of Q . The operator Q* has the 1 1 1 following form:
P
T
exp(ATq) CTt(q) dq, e e 0 so that equation (16) reduces to the matrix eigenvalue problem (14). The expression (15) for the left singular vectors follows from the definition of Q and the expansion (13). h 1 Let us now introduce the finite-dimensional subspace Y of T L [0, ¹), defined as 2 Y "VI n #Q Rn`nm#Q Rr, T j/1 j 2 3 where by VI n is denoted the linear span of the vectors n , j/1 j j j"1, 2, 2 , r . Also, introduce the closed subspace Y of 1 e L [O, ¹) , defined by 2 = Y " y 3Lp [0, R): y " + P y e e 2 e *kT,(k`1)T) e k/0 Q*" 1
G
H
and S P y 3Y . kT *kT,(k`1)T) e T Next, denote by q the dimension of Y and let Mh N, Y T j j"1, 2, 2 , q , be the orthonormal basis of Y constructed by y T setting h "n , for j"1, 2, 2 , r and by constructing h , for j j 1 j j"r #1, r #2, 2 , q by Gram—Schmidt orthogonalization. 1 1 Y The error signal y 3Y can be uniquely expanded as e e = qY + (y*) h , y"+S e jk j e ~kT j/1 k/0 where the sequence My* N, with y* 3 RqY defined as ek ek y* "[(y*) (y*) 2 (y*) Yk ] ek e ik e 2k e q is in lqY, i.e. the Hilbert space of sequences z*"Mz*, z*, 2 N, 0 1 z*3Rq2Y, with inner product defined as k = Sx*, z*T" + x*Tz*, x*, z* 3 lqY. k k 2 k/0 A discrete-time representation of relation (10) is then obtained by introducing matrix representations of the operators Q , Q 1 2 and Q involved in Eq. (12). These representations are associated 3 with the matrices E 3RqY](n`nm), E 3RqY](n`nm) and E 3RqY]r, 1 2 3 which are defined as (see also Gohberg and Krein, 1969; Bettayeb et al., 1980 for analogous results).
r Q " +Ç p n tT , (13) 1 j j j j/1 where r 4n#n is the rank of Q , t 3Rn`nm and 1 m 1 j n 3Lp [0, ¹), j"1, 2, 2, r , are orthonormal right and left sinj 2 1 gular vectors, and p , j"1, 2, 2, r are the singular values of j 1 the operator Q . The singular values and the right singular 1 vectors of Q are given by the following eigenvalue problem 1 (Bettayeb et al., 1980):
E "[p t 2 p t 2 0]T, (17a) 1 1 1 r1 r1 2, n`nm j/1,2, E "MSh , Q e TN , 2 i 2 j i/1,2, 2,qY (17b) 2 E "MSh , Q e TNj/1,2,2 , rY, 3 i 3 j i/1,2, ,q where e , j"1, 2, 2 , n#n and e , j"1, 2, 2 , r are the jth j m j unit vectors in Rn`nm and Rr, respectively. Note that the matrices E and E depend on the arbitrary parameters j , j"1, 2, 2, v. 2 3 j We emphasize this fact by setting in the sequel, E ,E (j) and 2 2 E ,E (j), where j"[j j 2j ]. 3 3 1 2 v On the basis of equations (12), (17a) and (17b), the discretetime representation of equation (10) is given by
R(A , C , ¹)t "p2 t , i"1, 2, 2, r , e e j j j 1
y* "[E #E (j)]x (k¹)#E (j)w(k¹). ek 1 2 e 3
Proposition 1. The operator Q : Rn`nmPLP[0, ¹) defined by 1 2 equation (11a) has the representation
(14)
(18)
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From the above analysis it becomes clear that, intersample ripples can be localized, if y (k¹#d)"0, ∀d3[0, ¹). It is also e clear that, in order to annihilate y (k¹#d), it suffices to annihie late y* and then (y*) . This can be attained, for every x(k¹) and ek e jk w(k¹), by choosing the arbitrary parameters j , j"1, 2, 2, v, j involved in equaion (18) in the form of the vector j, as suggested by the following theorem. ¹heorem 1. The intersample ripples which occur in the closedloop system output, can be localized, if the arbitrary parameters j , j"1, 2, v, are chosen such that the following two relationj ships simultaneously hold: E (j)"!E , E (j)"0. (19) 2 1 3 To summarize, in order to guarantee that the closed-loop system performance is the desired one, not only at the sampling instants, but even between them, the additional conditions (19) must be satisfied. Remark. From the previous analysis, it is easily recognized that, the technique based on the discrete time representation (12) (or Equation (18)) of the continuous-time signal y (k¹#d), which is e used here, in order to treat the problem of the localization of intersample ripples, is remarkably similar to the well-known lifting technique for continuous-time signals and sampled data systems. On the basis of this lifting technique, continuous-time systems can be viewed as linear, time-invariant, discrete-time systems with infinite-dimensional input and output spaces and with either finite- or infinite-dimensional state space (for a detailed analysis of this technique, see Bamieh et al., 1991; Bamieh and Pearson, 1992; Tadmor, 1992; Yamamoto, 1994; Chen and Francis, 1995). 4. Conclusions The intersample performance of linear continuous-time systems, controlled by generalized sampled-data hold functions, in order to achieve exact model matching at the sample instants, has been analyzed in the present note. The proposed technique is based on the expansion of an error signal in suitable basis functions. As it is shown, intersample ripple-free exact model matching control can be achieved, by appropriately selecting the arbitrary elements, incorporated in the general forms of the modulating hold functions. Although, the present note treats the standard situation, with a linear time-invariant continuous-time system and a control law based on GSHF, it is fairly straightforward to include more general situations, such as linear periodic systems, or the use of periodic multirate-input controllers and multirate-output controllers. References Arvanitis, K. G. (1995). Adaptive decoupling of linear systems using multirate generalized sampled-data hold functions. IMA J. Math. Control Inform., 12, 157—177.
Arvanitis, K. G. and P. N. Paraskevopoulos. (1995). Sampleddata minimum H=-norm regulation of continuous-time linear systems using multirate-output controllers. J. Opt. ¹heor. Appl., 87, 235—267. Astrom, K. J. and B. Wittenmark. (1984). Computer Controlled Systems: ¹heory and Design. Prentice-Hall, Englewood Cliffs, NJ. Bamieh, B., J. B. Pearson, B. A. Francis and A. Tannenbaum. (1991). A lifting technique for linear periodic systems with applications to sampled-data control. Systems Control ¸ett., 17, 79—88. Bamieh, B. and J. B. Pearson. (1992). A general framework for linear periodic systems with application to H sampled-data = control. IEEE ¹rans. Automat. Control, AC-37, 418— 435. Bettayeb, M., L. M. Silvermann and M. G. Safonov. (1980). Optimal approximation of continuous-time systems. Proc. 19th IEEE C.D.C.. Albuquerque, NM., Vol 1, pp. 195—198. Chen, T. and B. A. Francis. (1991a). H -optimal sampled-data 2 control. IEEE ¹rans. Automat. Control., AC-36, 387—397. Chen, T. and B. A. Francis. (1991b). Linear time-varying H 2 optimal control of sampled-data systems. Automatica, 27, 963—974. Chen, T. and B. A. Francis. (1995). Optimal Sampled-Data Control Systems. Springer, London, UK. Chen, T. and L. Qiu. (1994). H design of general multirate = sampled-data control systems. Automatica, 30, 1139 —1152. Dullerud, G. E. and B. A. Francis. (1991). L analysis and design 1 of sampled-data systems. IEEE ¹rans. Automat. Control., AC-37, 436—446. Gohberg, I. C. and M. G. Krein (1969). Introduction to the ¹heory of ¸inear Nonselfadjoint Operators. American Mathematical Society, Providence, RI. Kabamba, P. T. (1987). Control of linear systems using generalized sampled-data hold functions. IEEE ¹rans. Automat. Control., AC-32, 772—783. Kabamba, P. T. and S. Hara (1993). Worst case analysis and design of sampled-data control systems. IEEE ¹rans. Automat. Control., AC-38, 1337—1357. Leung, G. M. H., T. P. Perry and B. A. Francis (1991). Performance analysis of sampled-data control systems. Automatica, 27, 699—704. Naylor, A. W. and G. R. Sell (1982). ¸inear Operator ¹heory in Engineering and Science, Springer, New York. Paraskevopoulos, P. N. and K. G. Arvanitis. (1994). Exact model matching of linear systems using generalized sampleddata hold functions. Automatica, 30, 503—506. Tadmor, G. (1992). H optimal sampled-data control in con= tinuous time systems Int. J. Control, 56, 99—141. Toivonen, H. T. (1992). Sampled-data control of continuoustime systems with an H optimality criterion. Automatica, 28, = 45—54. Yamamoto, Y. (1994). A function space approach to sampleddata control systems and tracking problems. IEEE ¹rans. Automat. Control, AC-39, 703—713, 1994.
