H1 Design of General Multirate Sampled-Data ... - Semantic Scholar
H1 Design of General Multirate Sampled-Data Control Systems
Li Qiu Tongwen Chen Inst. for Math. & Its Appl. Dept. of Elect. & Comp. Engg. University of Minnesota University of Calgary Minneapolis, MN Calgary, Alberta USA 55455 Canada T2N 1N4 Tel: 612-624-1824 Tel: 403-220-8357 Email: [email protected] Email: [email protected] December 10, 1992
Abstract
Direct digital design of general multirate sampled-data systems is considered. To tackle causality constraints, a new and natural framework is proposed using nest operators and nest algebras. Based on this framework explicit solutions to the H1 and H2 multirate control problems are developed in the frequency domain.
Keywords: multirate systems, digital control, sampled-data systems, discrete systems,
H1 optimization, H2 optimization, causality constraint, nest algebra.
1 Introduction There are several reasons to use a multirate sampling scheme in digital control systems: In complex, multivariable control systems, often it is unrealistic, or sometimes impossible, to sample all physical signals uniformly at one single rate. In such situations, one is forced to use multirate sampling. In general one gets better performance if one can sample and hold faster. But faster A/D and D/A conversions mean higher cost in implementation. For signals with di erent bandwidths, better trade-o s between performance and implementation cost can be obtained using A/D and D/A converters at di erent rates. Multirate controllers are in general time-varying. Thus multirate control systems can achieve what single-rate systems cannot; for example, gain margin improvement [25, 16], simultaneous stabilization [25, 31] and decentralized control [2, 41, 36]. The rst author was supported by the Natural Sciences and Engineering Research Council of Canada.
1
Multirate controllers are normally more complex than single-rate ones; but often they
are periodic in a certain sense and hence can be implemented on microprocessors via difference equations with nitely many coecients. Therefore, like single-rate controllers, multirate controllers do not violate the nite memory constraint in microprocessors. The study of multirate systems began in late 1950's [26, 22, 23]; recent interests are re ected in the LQG/LQR designs [7, 1, 9, 29], the parametrization of all stabilizing controllers [27, 34], and the work in [30, 3, 19, 10, 36]. The controller parametrization in [27, 34] provides a basis for designing optimal multirate systems. However, the special structure due to causality in the feedthrough terms of lifted controllers presents a dicult constraint in design; treating this causality constraint is the new feature in multirate optimal design. Causality constraints also arise in discrete-time periodic control [25], where after lifting, the feedthrough terms in controllers must be block lower-triangular. Explicit solutions were obtained for the one-block H1 problem [14, 17] and H2 problem [42]. However, these results do not generalize easily to multirate systems since most multirate designs leads to four-block problems, i.e., the transfer matrices in the associated model-matching problems are in general nonsquare. Our work has been greatly in uenced by the recent trend in sampled-data research, namely, direct digital design based on continuous-time performance specs. Related work on single-rate sampled-data design has been completed in H2 [8, 24, 6] and H1 [20, 40, 5, 38, 39, 37, 21] frameworks. In [33], we performed direct designs for a multirate system with a uniform sampling rate and a uniform hold rate and proposed e ective ways to tackle the causality constraint. Though the setup in [33] captures the essential issue of causality (in a simpli ed form), it also limits the applicability of the results. Our goal in this paper is to treat general multirate systems. In order to do so, a general framework for attacking causality constraints is developed; this is based on ideas from nest spaces and nest algebras. Under this framework, the results on causality [27, 34] become quite transparent; moreover, and more importantly, explicit solutions are obtained for direct multirate designs with H1 and H2 performance criteria.
Setup
To bring in the multirate sampled-data setup, we need to de ne precisely the two basic elements, the periodic sampler S and the (zero-order) hold H (the subscript denotes the period): S maps a continuous signal to a discrete signal and is de ned via = S y () (k) = y (k ):
H maps discrete to continuous via u = H () u(t) = (k);
2
k t < (k + 1) :
Note that the sampler and hold are synchronized at t = 0. The signals may be vector-valued; in this case, for example, = S y simply means
2 1 3 2 S 64 ... 75 = 64 ... m
32 7564
S
y1 3 .. 75; . ym
which corresponds to grouping m samplers with the same rate together. The general multirate sampled-data system is shown in Figure 1. We have used continuous arrows for continuous signals and dotted arrows for discrete signals. Here, G is the continuous-
z
w
G y
- S p p p p
Kd
-
p p p p
u
H
Figure 1: The general multirate sampled-data setup time generalized plant with two inputs, the exogenous input w and the control input u, and two outputs, the signal z to be controlled and the measured signal y . S and H are multirate sampling and hold operators and are de ned as follows:
2S m1 h 6 ... S=4
Sm h
2H 3 nh 75; H = 64 1 . . .
Hn h
3 75:
q
p
These correspond to sampling p channels of y periodically with periods mi h, i = 1; ; p, respectively and holding q channels of with periods nj h, j = 1; ; q respectively. Here mi and nj are di erent integers and h is a real number referred as the base period. If we partition the signals accordingly
2 y1 3 y = 64 ... 75; yp
2 u1 3 2 1 3 2 13 = 64 ... 75; = 64 ... 75; u = 64 ... 75; q
p
uq
then i (k) uj (t)
= yi (kmi h); i = 1; ; p = j (k); knj h t < (k + 1)nj h; j = 1; ; q: 3
We shall allow each channel in y and u to be vector-valued as well; thus without loss of generality we can assume that no two mi are equal and neither are two nj . Kd is the discretetime multirate controller, implemented via a microprocessor; it is synchronized with S and H in the sense that it inputs a value from the i-th channel at times k(mih) and outputs a value to the j -th channel at k(nj h). Figure 1 gives a compact way of describing multirate systems. It is clear that this model captures all multirate systems in which the rates are rationally related, i.e., the ratio of any two rates is rational. Note that any common factor among the mi and nj can be absorbed into h; thus we can assume without loss of generality that the greatest common factor among mi and nj is 1. With this condition, for any multirate system in which rates are rationally related, there exists a unique base period h and a unique set of integers mi and nj so that the system can be put into the framework of Figure 1. Now we introduce a useful notation: Given an operator K and an operator matrix
"
#
P = PP11 PP12 ; 21 22 the associated linear fractional transformation is denoted
F (P; K ) = P11 + P12K (I ? P22K )?1P21: Here we assume that the domains and co-domains of the operators are compatible and the inverse exists. Throughout the paper, G is linear time-invariant (LTI) and nite-dimensional and Kd is linear; additional properties of Kd will be discussed in Section 3. The closed-loop map w 7! z in Figure 1 is F (G; HKdS ). We can now state our main synthesis problem: Design a Kd to give closed-loop stability (to be made precise in Section 4) and achieve the H1 performance requirement kF (G; HKdS )k < , where > 0 is a pre-speci ed performance level and the norm is L2 -induced. Note that the performance requirement is de ned on the continuous-time map and so intersample behaviour is captured in the design spec. Such continuous-time specs are natural since sampled-data systems operate in a continuous-time environment, though controllers are digital. Necessary and sucient conditions will be given under which this multirate H1 control problem is solvable; once solvable, an explicit solution will also be given.
Organization
The rest of the paper is organized as follows. In Section 2 we present some concepts and facts about nest operators and nest algebras. These have direct applications in subsequent sections. Section 3 discusses desirable properties for multirate controllers; they are periodicity, causality, and nite-dimensionality. Causality constraints are de ned using operators between appropriate nests. This provides a natural and transparent framework for studying causality constraints in multirate systems. 4
Section 4 deals with internal stability of the setup in Figure 1 and relates it to internal stability of some discrete-time system. Section 5 contains the main contribution of this paper, namely, an explicit solution to the multirate H1 control problem. This is achieved by rst reducing it to a constrained H1 model-matching problem and then solving the latter problem using results in Section 2. A frequency-domain approach is used consistently. In Section 6 we brie y consider the H2 -optimal design of general multirate systems. The techniques developed in this paper also yield an explicit solution to the H2 problem. Finally, Section 6 contains some concluding remarks. The notation is quite standard and will be de ned when introduced. Throughout the paper, we choose to use -transforms instead of z -transforms, where = z ?1 ; in this case, discrete-time spaces such as H2 and H1 are de ned on the open unit disk. Finally, if G is an LTI system, G^ denotes its transfer matrix.
2 Preliminaries In this section we address some topics and computation on nests and nest algebras which are useful in the sequel. We shall restrict our attention to nite-dimensional spaces; more general treatment can be found in [4, 12].
Nests, Nest Operators, and Nest Algebras
Let X be a nite-dimensional space. A nest in X , denoted fXi g, is a chain of subspaces in X , including f0g and X , with the nonincreasing ordering:
X = X0 X1 Xn?1 Xn = f0g: Let X and Y be both nite-dimensional inner-product spaces with nests fXi g and fYi g respectively. Assume the two nests have the same number of subspaces, say, n + 1 as above. A linear map T : X ! Y is a nest operator if
T Xi Yi ;
i = 0; 1; ; n:
(1)
This gives n +1 relations; the rst and the last are trivially satis ed. We shall allow repetitions in fXi g and fYi g. Thus redundancy may occur in (1) and in the results to follow. However, for computation one can eliminate this redundancy as follows: If Xi = Xi+1 , the i-th relation, namely, T Xi Yi , is redundant since Yi Yi+1 and therefore can be eliminated; similarly, if Yi = Yi+1 , we eliminate the (i + 1)-st relation. Let X : X ! Xi and Y : Y ! Yi be orthogonal projections. Then the condition in (1) is equivalent to i
(I ? Y )T X = 0; i
i
i
i = 0; 1; ; n:
The set of all such operators is denoted N (fXi g; fYig) and abbreviated N (fXi g) if fXi g = fYi g. The following properties are straightforward to verify. 5
Lemma 1
(a) If T1 2 N (fXi g; fYig) and T2 2 N (fYig; fZig), then T2T1 2 N (fXi g; fZig). (b) N (fXi g) forms an algebra, called nest algebra. (c) If T 2 N (fXi g) and T is invertible, then T ?1 2 N (fXi g).
Factorization
It is a fact that every operator on X can be factored as the product of a unitary operator and an operator in N (fXi g). Lemma 2 Let T be an operator on X . (a) There exists a unitary operator U1 on X and an operator R1 in N (fXi g) such that T = U 1 R1 . (b) There exists an operator R2 in N (fXi g) and a unitary operator U2 on X such that T = R2U2 . Note that R1 and R2 are invertible if T is invertible. We shall give an elementary proof of this lemma, for it provides a way to compute such factorizations via the well-known QR factorization.
Proof of Lemma 2 We shall look at part (a); part (b) follows similarly. Since Xi Xi+1, we write (Xi+1 )?X as the orthogonal complement of Xi+1 in Xi . Decompose X into X = (X1)?X0 (X2)?X1 (Xn )?X ?1 : It follows that under this decomposition any operator R belongs to N (fXi g) i its matrix is block lower-triangular, all the diagonal blocks being square. Thus it suces to show that for any matrix T on X we can write T = U1 R1 where U1 is orthogonal and R1 is block lower-triangular. This follows from a QR type of factorization for square matrices: T = U1 R1 with U1 orthogonal and R1 lower-triangular; partition R1 accordingly to get that R1 is also block lower-triangular. QED i
n
A Distance Problem
Finally, we look at a distance problem. Let X and Y be nite-dimensional inner-product spaces with nests fXi g and fYi g. Let T be an operator X ! Y . We want to nd the distance (via induced norms) of T to N (fXi g; fYig), abbreviated N : dist (T; N ) := Qinf kT ? Qk: (2) 2N It is clear that
dist (T; N ) max k(I ? Y )T X k: i i
6
i
Theorem 1 dist (T; N ) = max k(I ? Y )T X k: i i
i
This is Corollary 9.2 in [12] specialized to operators on nite-dimensional spaces; it is an extension of a result in [32, 11] on norm-preserving dilation of operators, which is specialized to matrices below. We denote the Moore-Penrose generalized inverse of a matrix M by M y.
Lemma 3 Assume that A; B; C are xed matrices of appropriate dimensions. Then " # h i "A# C A inf k X B k = maxfk C A k; k B kg := : X Moreover, a minimizing X is given by
X = ?BA ( I ? AA)yC: It will be of interest to us how to compute a Q to achieve the in mum in (2); this can be
done by sequentially applying Lemma 3: Step 1 Decompose the spaces X and Y :
X = (X1)?X0 (X2)?X1 (Xn )?X ?1 Y = (Y1)?Y0 (Y2)?Y1 (Yn )?Y ?1 : We get matrix representations for T and Q: 2 3 3 2 T11 T12 T1n Q11 0 0 66 T21 T22 T2n 77 66 Q21 Q22 0 77 6 7 T = 6 .. .. .. 7 ; Q = 66 .. .. 77 ; . 5 4 . . . 5 4 . ... Tn1 Tn2 Tnn Qn1 Qn2 Qnn Q being block lower-triangular. Step 2 De ne Xij = Tij ? Qij , i j , and 3 2 X11 T12 T1n 66 X X T2n 77 P = 66 ..21 ..22 .. 77 : 4 . . . 5 Xn1 Xn2 Xnn n
n
The problem reduces to
min kP k; X ij
where Tij , i < j , are xed. Minimizing Xij can be selected as follows. First, set X11; X21; ; Xn1 and Xn2 ; ; Xnn to zero. Second, choose X22 by Lemma 3 such that 7
the norm of the matrix (I ? Y2 )P X1 (obtained by retaining the rst 2 block rows and the last n ? 1 block columns in P ) is minimized: k(I ? Y2 )P X1 k = maxfk(I ? Y1 )T X1 k; k(I ? Y2 )T X2 kg:
i
h
Fix this X22. Third, choose X32 X33 again by Lemma 3 to minimize
k(I ? Y3 )P X2 k = maxfk(I ? Y2 )T X1 k; k(I ? Y3 )T X3 kg: In this way, we can nd all Xij such that min kP k = max k(I ? Y )T X k: i X i
ij
i
This procedure also gives a constructive proof of the theorem.
3 Multirate Systems In this section we shall examine the multirate controller Kd in Figure 1 as a discrete-time linear operator. To be studied are three basic properties: periodicity, causality, and nite dimensionality.
Periodicity
The sampled-data controller HKd S is in general time-varying. However, the operation at each channel of S and H is periodic. Let l = LCM fm1; ; mp; n1; ; nqg; where LCM means least common multiple. Thus := lh is the least common period for all sampling and hold channels, i.e., is the least time interval in which the sampling and hold schedule repeats itself. Kd can be chosen so that HKd S is -periodic in continuous-time. For this, we need a few de nitions. Let ` be the space of sequences, perhaps vector-valued, de ned on the time set f0; 1; 2; g. Let U be the unit time delay on ` and U the unit time advance. De ne the integers
m i = ml ; i = 1; 2; ; p i l n j = n ; j = 1; 2; ; q: j
We say Kd is (mi ; nj )-periodic if
2 (U )n1 64 ...
(U )n
3 2 U m 1 75Kd64 ...
U m
p
q
8
3 75 = Kd:
This means shifting the i-th input channel by m i time units (m i mi h = ) corresponds to shifting the j -th output channel by n j units (nj nj h = ). Thus HKd S is -periodic in continuous time i Kd is (mi; nj )-periodic. Since G is LTI, it follows that the sampled-data system in Figure 1 is -periodic if Kd is (mi ; nj )-periodic. We shall refer as the system period. Now we lift Kd to get an LTI system. For an integer m > 0, de ne the discrete lifting operator Lm via = Lm : 82 (0) 3 2 (m) 3 9 > > = < 7 .. : ; f (0); (1); g 7! >64 ... 75; 64 5 . > ; : (m ? 1) (2m ? 1) Lm maps ` to `m, the external direct sum of m copies of `. If the underlying period for is , then the underlying period for is m . Now extend the input and output spaces of Kd so that the underlying period is ; this corresponds to lifting the controller Kd in the following way: 3 3 2 2
Kd := 64
Ln1
?1
...
Ln
L 75Kd66 m 1 . . . 4
q
L?m1
77 5:
p
It is an easy matter to check, see, e.g., [28], that the lifted controller Kd is LTI i Kd is (mi ; nj )-periodic.
Causality
Figure 1 is a real-time system. So for Kd to be implementable, HKd S must be causal in continuous time. This implies that Kd , as a single-rate system, must be causal; and moreover, the feedthrough term D in Kd must satisfy a certain constraint, that is, some blocks in D must be zero [27, 34]. Now let us characterize this constraint on D using nest operators. Write = Kd ; then (0) = D (0), where by de nitions
02 L 32 m 1 1 6 6 7 . . . . (0) = B @4 54 . . Lm p h 0 = 1 ? 1)0 1(0) 1(m
31 75CA (0)
p
p (0)0
p ? 1)0 p (m
i0
Note that i (k) is sampled at t = kmi h. Similarly, h i0 (0) = 1(0)0 1(n1 ? 1)0 p (0)0 p (nq ? 1)0 and j (k) occurs at t = knj h. Let be the set of sampling or hold instants in the interval [0; ) (modulo the base period h), i.e., :=
[ i
1 ! [ 0[ f0; mi; 2mi; ; l ? mig @ f0; nj ; 2nj ; ; l ? nj gA : j
9
This is a nite set of, say, n + 1 elements (not counting repetitions); order increasingly (r < r+1 ): = fr : r = 0; 1; ; ng: Let (0) and (0) live in the nite-dimensional spaces X and Y respectively. For r = 0; 1; ; n, de ne
Xr = span f (0) : i(k) = 0 if kmi < r g Yr = span f (0) : j (k) = 0 if knj < r g: Xr and Yr correspond to, respectively, the inputs and outputs occurred after and including time r h. It is easily checked that fXr g and fYr g are nests and that the causality condition on D (the output at time r h depends only on inputs up to r h) is exactly D Xr Y r ;
r = 0; 1; ; n:
Thus we de ne D to be (mi; nj )-causal if D 2 N (fXr g; fYr g). This is the same causality constraint in [27, 34] de ned in terms of the elements of D. For later bene t, we de ne D to be (mi ; nj )-strictly causal if
DXr Yr+1 ;
r = 0; 1; ; n ? 1:
This means that the output at time r+1 h depends only on inputs up to time r h. The following lemma, which is straightforward to prove, justi es our use of terminology from a continuous-time viewpoint.
Lemma 4 (a) HKd S is causal in continuous time i Kd is causal and D is (mi; nj )-causal. (b) HKd S is strictly causal in continuous time i Kd is causal and D is (mi ; nj )-strictly causal.
Some conclusions on causality issues [27] are transparent under this new formulation.
Lemma 5 (a) If D1 is (mi ; pk )-causal and D2 is (pk ; nj )-causal, then D2 D1 is (mi; nj )-causal; furthermore, if D1 or D2 is strictly causal, then D2 D1 is also strictly causal. (b) If D is (mi; mi )-causal and invertible, then D?1 is (mi ; mi)-causal. (c) If D is (mi; mi )-strictly causal, then (I ? D)?1 exists and is (mi; mi)-causal.
10
Proof Part (a) follows immediately from Lemma 4:
D1 ; D2 are causal ) HD1S ; HD2S are causal in continuous time ) HD2D1S = HD2SHD1S is causal in continuous time ) D2D1 is causal:
Part (a) also follows from Lemma 1 (a) by some renumbering of the indices. Part (b) follows directly from Lemma 1 (c). For part (c), note that under appropriate decomposition, D is strictly block lower-triangular; so (I ? D)?1 exists and is (mi ; mi)-causal [part (b)]. QED Let us de ne Kd to be (mi; nj )-causal if Kd is causal and D is (mi; nj )-causal.
Finite Dimensionality
We assume Kd is (mi ; nj )-periodic and -causal. Then Kd is LTI and causal. To get nitedimensional di erence equations for Kd , we further assume Kd is nite-dimensional. Thus Kd has a state model 2 3
66 CA1 DB111 K^ d () = 66 .. .. 4 . . Cq Dq1
Bp D1p 777
.. 7 : . 5
Dqp
Let the state for Kd be . The corresponding equations for Kd ( = Kd ) are
(k + 1) = A(k) +
p X
Bi i(k)
i=1 p X
j (k) = Cj (k) +
i=1
Dji i (k); j = 1; 2; ; q:
Note that i = Lm i and j = Ln j . Partitioning the matrices accordingly i h Bi = (Bi )0 (Bi )m ?1 ; 2 (Dji)00 (Dji)0;m ?1 3 2 (Cj )0 3 75 .. .. Cj = 64 ... 75; Dji = 64 . . (Dji )n ?1;0 (Dji)n ?1;m ?1 (Cj )n ?1 (certain blocks in Dji must be zero for the causality constraint), we get the di erence equations for Kd ( = Kd ): j
i
i
i
j
j
(k + 1) = A(k) + j (knj + r) =
p mX ?1 X i
j
(Bi )s i (km i + s) i=1 s=0 p mX i ?1 X (Dji)rs i (km i + s); (Cj )r (k) + i=1 s=0 11
i
(3) (4)
where the indices in (4) go as follows: j = 1; 2; ; q and r = 0; 1; ; n j ? 1. These are the equations for implementing Kd on microprocessors and they require only nite memory. Note that the state vector for Kd is updated every system period . In summary, in this paper we are interested in the class of multirate Kd which are (mi; nj )periodic and -causal and nite-dimensional; this class is called the admissible class of Kd and can be modeled by di erence equations (3-4) with D (mi ; nj )-causal. The corresponding admissible class of Kd is characterized by LTI, causal, and nite-dimensional Kd with the same constraint on D. The causality constraint, namely, that D must be a nest operator, is the new feature in lifted multirate systems, which is the main concern in the subsequent designs. A seemingly easy way out would be to take D = 0, which corresponds to delay the control input u by a system period . However, we would like to argue that this would result in serious performance degradation since for complex multirate systems, the system periods are usually appreciably large.
4 Internal Stability In this section we look at stability of Figure 1. We assume the continuous G is LTI, causal, and nite-dimensional; partition G as follows:
" # "
z G11 G12 y = G21 G22
G has a state model
#"
#
w : u
3 2 B B A 1 2 G^(s) = 64 C1 D11 D12 75:
C2 D21 0 Let the plant state be x and the controller state be (Kd is admissible). Note that the system in Figure 1 is -periodic. De ne the continuous-time vector " # x ( t ) xsd(t) := (k) ; k t < (k + 1): The (autonomous) system in Figure 1 is internally stable, or Kd internally stabilizes G, if for any initial value xsd (t0), 0 t0 < , xsd (t) ! 0 as t ! 1. In the de nition, w = 0; so Figure 1 reduces to Figure 2, where we assume G22 has the same state as G. Moving S and H around the loop and de ning G22d = S G22H, the multirate discretization of G22, we arrive at a multirate discrete-time system. Now lift Kd as before and G22d by 3 2L 3 2 L?1 m 1 n 1 77 75G22d66 ... ... G22d = 64 5 4 L?n 1 Lm p
12
p
G22
y
- S
-
p p p p p p
u
-
Kd
H
p p p p p p
Figure 2: For stability analysis p p p p p p p p p p p p p p p p p p
G22d
p p p p p p p p p p p p p p p p p p
p
p
p p
p p
p
p
p
p
p
p
p
p
p p
p p p
p p p p p p p
-
p p p p p p p p p p p p p p p
p p
Kd
p p p p p
p
p p
p
p
p
p p
p
p
p
p p
p
Figure 3: The lifted system for stability to get the lifted system of Figure 3. Because G22 is LTI and strictly causal, G22d is (nj ; mi)periodic and -strictly causal. Thus G22d is LTI and causal with D22d (nj ; mi)-strictly causal. So Figure 3 gives an LTI discrete system. In fact, a state model for G22d can be obtained [28]; its state being := S x, or (k) = x(k ). Let us see that Figure 3 is well-posed, i.e., the matrix I ? D22dD is invertible, where D is the feedthrough term of Kd . This follows from Lemma 5: D22dD is (mi ; mi)-strictly causal [Lemma 5 (a)] and so I ? D22d D is invertible [Lemma 5 (c)]. This also implies that the multirate system of Figure 1 is well-posed. The system in Figure 3 is internally stable, or Kd internally stabilizes G22d if for any initial states (0) and (0), " # (k) ! 0 as k ! 1: (k) It is clear that Figure 3 is internally stable if Figure 1 is. Theorem 2 Kd internally stabilizes G i Kd internally stabilizes G22d. Proof Suppose Kd internally stabilizes G22d. It suces to show that x(t) ! 0 as t ! 1. Internal stability of Figure 3 implies that (k) ! 0 as k ! 1 in Figure 3 and hence u(t) ! 0 as t ! 1 in Figure 2. Now since for k t < (k + 1) ,
x(t) = e(t?k)A (k) +
Zt
13
k
e(t? )A B2 u( ) d ;
QED
it follows that x(t) ! 0 as t ! 1.
Sucient conditions for the internal stability to be achievable are that (A; B2) and (C2; A) are stabilizable and detectable respectively and that the system period is non-pathological in a certain sense, see, e.g., [16, 33].
5 H1-Optimal Control With reference to Figure 1, we now study the main synthesis problem: Design an admissible
Kd that internally stabilizes G and achieves the continuous-time H1 performance requirement kF (G; HKdS )k < , where is pre-speci ed and positive. By proper scaling, we can always take = 1.
The general idea in the solution is to reduce the multirate sampled-data problem to a discrete H1 model-matching problem with the causality constraint and then solve the constrained problem explicitly using techniques presented in Section 2 on nest operators and nest algebras. A special case of the reduction process was reported in [33] where a uniform sampling rate and a uniform hold rate are assumed. The solution process is complex enough to require several distinct steps. Appropriate connections to some recent work on H1 control are made in each step. We start with a state model for G:
3 2 A B1 B2 G^ (s) = 64 C1 0 D12 75: C2 0
0
As we saw in the preceding section, the zero block in D22 guarantees the well-posedness of the closed-loop multirate system in Figure 1. For kF (G; HKdS )k to be nite, we must have D21 = 0. The zero block in D11 is for a technical simpli cation, as in the single-rate case [5, 21]. We shall assume that (A; B2) is stabilizable and (C2; A) is detectable. H1
Discretization
The original problem is posed in continuous time; so the rst step is to recast it as a discretetime problem with time-varying control. The reduction is based on recent advances in H1 sampled-data control in the single-rate setting. Introduce the discrete sampling operator Sm : ` ! ` de ned via = Sm () (k) = (km) and the discrete hold operator Hn : ` ! ` via
= Hn () (kn + r) = (k); r = 0; 1; ; n ? 1: 14
It is easily checked that Sm h = Sm Sh and Hn h = Hh Hn . So the multirate sampling and hold operators S and H can be factored as i
i
2 Sm 1 6 ... S=4
j
Sm
j
2 Hn 3 75Sh; H = Hh64 1 . . .
Hn
p
De ning
2 Hn 1 6 ... Kd1 = 4
Hn
3 2 Sm 75Kd64 1 . . . q
Sm
3 75;
3 75:
q
(5)
p
we can view Figure 1 as a ctitious single-rate system but with a time-varying controller Kd1 as in Figure 4. Now the results in, e.g., [5, 21] (there, discrete controllers are not required to be time-invariant), are applicable.
z
G y
-
w u
Sh
- Kd1 -
p p p p
p p p p
Hh
Figure 4: An equivalent single-rate system Let D11h : L2 [0; h) ! L2 [0; h) be de ned by (D11hw)(t) = C1
Zt 0
e(t? )A B1 w( ) d
and assume
kD11hk < 1. Since D11h is the restriction of F (G; HKdS ) on L2 [0; h), this condition is necessary for kF (G; HKdS )k < 1; how to verify this condition was studied in [5]. For the multirate sampled-data H1 problem, invoke the single-rate results to get the equivalent discrete system shown in Figure 5 and the equivalent discrete-time problem: Design Kd1 to give internal stability and achieve kF (Gd; Kd1)k < 1, where the norm now is `2 induced. The H1 discretization Gd (for = 1) is LTI and causal and has a state model 15
p p p p p p p p p p p p p p p p p p
!
p p p p p p p p p p p p p p p p p p
Gd
p p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p
p
p
p
p
p
-
p p p p p
p p
Kd1
p p p p p p p p p p p p p p
p p p p p p p p p p p p p p p p p p
Figure 5: H1 discretized system
3 2 A d B1d B2d G^d () = 64 C1d D11d D12d 75: C2d
0
0
The computation of the matrices in G^ d is given in, e.g., [5, 21]. In this way, we arrive at an equivalent discrete H1 problem; however, Kd1 is constrained to be of the form in (5) with Kd admissible.
Discrete Lifting
The system of Figure 5 is single-rate with the underlying period being h. The next step is to lift to get an LTI con guration with underlying period . Partition Gd as usual:
"
#
G11d G12d : Gd = G 21d G22d De ne Kd as before and
2 66 Ll Lm 1 Sm1 Gd = 66 ... 4
3 2 L?1 77 66 l Hn1 L?n11 77 Gd 66 ... 5 4
3 77 77 5
Hn L?n 1 to get the lifted system of Figure 6, where ! = Ll ! and = Ll . Since Gd is LTI, causal, and nite-dimensional with G22d strictly causal, it is an easy exercise to verify the following properties of Gd . Lm Sm p
p
q
q
Lemma 6 Gd is LTI, causal, and nite-dimensional. Moreover, the feedthrough term D22d of G22d is (nj ; mi )-strictly causal. In fact, a state model for Gd can be obtained using the lemma below. 16
Gd
p p p p p p p p p p p p p p p p p p
p p p p p p p p p p
!
p p p p p p p p p p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p
p
p
p
p p p p p p
p
-
p p p p p p p p p p p p p p
Kd
p p p p p p p p p p p p p p p p p p p p
Figure 6: The lifted system Let P be a discrete-time system with state and transfer matrix
#
"
B : P^ () = CA D Let m; n; m; n ; l be positive integers such that
mm = nn = l: De ne
P := Lm SmPHn L?n 1
and the characteristic function on integers
[p;q) (r) =
(
pr
Li Qiu Tongwen Chen Inst. for Math. & Its Appl. Dept. of Elect. & Comp. Engg. University of Minnesota University of Calgary Minneapolis, MN Calgary, Alberta USA 55455 Canada T2N 1N4 Tel: 612-624-1824 Tel: 403-220-8357 Email: [email protected] Email: [email protected] December 10, 1992
Abstract
Direct digital design of general multirate sampled-data systems is considered. To tackle causality constraints, a new and natural framework is proposed using nest operators and nest algebras. Based on this framework explicit solutions to the H1 and H2 multirate control problems are developed in the frequency domain.
Keywords: multirate systems, digital control, sampled-data systems, discrete systems,
H1 optimization, H2 optimization, causality constraint, nest algebra.
1 Introduction There are several reasons to use a multirate sampling scheme in digital control systems: In complex, multivariable control systems, often it is unrealistic, or sometimes impossible, to sample all physical signals uniformly at one single rate. In such situations, one is forced to use multirate sampling. In general one gets better performance if one can sample and hold faster. But faster A/D and D/A conversions mean higher cost in implementation. For signals with di erent bandwidths, better trade-o s between performance and implementation cost can be obtained using A/D and D/A converters at di erent rates. Multirate controllers are in general time-varying. Thus multirate control systems can achieve what single-rate systems cannot; for example, gain margin improvement [25, 16], simultaneous stabilization [25, 31] and decentralized control [2, 41, 36]. The rst author was supported by the Natural Sciences and Engineering Research Council of Canada.
1
Multirate controllers are normally more complex than single-rate ones; but often they
are periodic in a certain sense and hence can be implemented on microprocessors via difference equations with nitely many coecients. Therefore, like single-rate controllers, multirate controllers do not violate the nite memory constraint in microprocessors. The study of multirate systems began in late 1950's [26, 22, 23]; recent interests are re ected in the LQG/LQR designs [7, 1, 9, 29], the parametrization of all stabilizing controllers [27, 34], and the work in [30, 3, 19, 10, 36]. The controller parametrization in [27, 34] provides a basis for designing optimal multirate systems. However, the special structure due to causality in the feedthrough terms of lifted controllers presents a dicult constraint in design; treating this causality constraint is the new feature in multirate optimal design. Causality constraints also arise in discrete-time periodic control [25], where after lifting, the feedthrough terms in controllers must be block lower-triangular. Explicit solutions were obtained for the one-block H1 problem [14, 17] and H2 problem [42]. However, these results do not generalize easily to multirate systems since most multirate designs leads to four-block problems, i.e., the transfer matrices in the associated model-matching problems are in general nonsquare. Our work has been greatly in uenced by the recent trend in sampled-data research, namely, direct digital design based on continuous-time performance specs. Related work on single-rate sampled-data design has been completed in H2 [8, 24, 6] and H1 [20, 40, 5, 38, 39, 37, 21] frameworks. In [33], we performed direct designs for a multirate system with a uniform sampling rate and a uniform hold rate and proposed e ective ways to tackle the causality constraint. Though the setup in [33] captures the essential issue of causality (in a simpli ed form), it also limits the applicability of the results. Our goal in this paper is to treat general multirate systems. In order to do so, a general framework for attacking causality constraints is developed; this is based on ideas from nest spaces and nest algebras. Under this framework, the results on causality [27, 34] become quite transparent; moreover, and more importantly, explicit solutions are obtained for direct multirate designs with H1 and H2 performance criteria.
Setup
To bring in the multirate sampled-data setup, we need to de ne precisely the two basic elements, the periodic sampler S and the (zero-order) hold H (the subscript denotes the period): S maps a continuous signal to a discrete signal and is de ned via = S y () (k) = y (k ):
H maps discrete to continuous via u = H () u(t) = (k);
2
k t < (k + 1) :
Note that the sampler and hold are synchronized at t = 0. The signals may be vector-valued; in this case, for example, = S y simply means
2 1 3 2 S 64 ... 75 = 64 ... m
32 7564
S
y1 3 .. 75; . ym
which corresponds to grouping m samplers with the same rate together. The general multirate sampled-data system is shown in Figure 1. We have used continuous arrows for continuous signals and dotted arrows for discrete signals. Here, G is the continuous-
z
w
G y
- S p p p p
Kd
-
p p p p
u
H
Figure 1: The general multirate sampled-data setup time generalized plant with two inputs, the exogenous input w and the control input u, and two outputs, the signal z to be controlled and the measured signal y . S and H are multirate sampling and hold operators and are de ned as follows:
2S m1 h 6 ... S=4
Sm h
2H 3 nh 75; H = 64 1 . . .
Hn h
3 75:
q
p
These correspond to sampling p channels of y periodically with periods mi h, i = 1; ; p, respectively and holding q channels of with periods nj h, j = 1; ; q respectively. Here mi and nj are di erent integers and h is a real number referred as the base period. If we partition the signals accordingly
2 y1 3 y = 64 ... 75; yp
2 u1 3 2 1 3 2 13 = 64 ... 75; = 64 ... 75; u = 64 ... 75; q
p
uq
then i (k) uj (t)
= yi (kmi h); i = 1; ; p = j (k); knj h t < (k + 1)nj h; j = 1; ; q: 3
We shall allow each channel in y and u to be vector-valued as well; thus without loss of generality we can assume that no two mi are equal and neither are two nj . Kd is the discretetime multirate controller, implemented via a microprocessor; it is synchronized with S and H in the sense that it inputs a value from the i-th channel at times k(mih) and outputs a value to the j -th channel at k(nj h). Figure 1 gives a compact way of describing multirate systems. It is clear that this model captures all multirate systems in which the rates are rationally related, i.e., the ratio of any two rates is rational. Note that any common factor among the mi and nj can be absorbed into h; thus we can assume without loss of generality that the greatest common factor among mi and nj is 1. With this condition, for any multirate system in which rates are rationally related, there exists a unique base period h and a unique set of integers mi and nj so that the system can be put into the framework of Figure 1. Now we introduce a useful notation: Given an operator K and an operator matrix
"
#
P = PP11 PP12 ; 21 22 the associated linear fractional transformation is denoted
F (P; K ) = P11 + P12K (I ? P22K )?1P21: Here we assume that the domains and co-domains of the operators are compatible and the inverse exists. Throughout the paper, G is linear time-invariant (LTI) and nite-dimensional and Kd is linear; additional properties of Kd will be discussed in Section 3. The closed-loop map w 7! z in Figure 1 is F (G; HKdS ). We can now state our main synthesis problem: Design a Kd to give closed-loop stability (to be made precise in Section 4) and achieve the H1 performance requirement kF (G; HKdS )k < , where > 0 is a pre-speci ed performance level and the norm is L2 -induced. Note that the performance requirement is de ned on the continuous-time map and so intersample behaviour is captured in the design spec. Such continuous-time specs are natural since sampled-data systems operate in a continuous-time environment, though controllers are digital. Necessary and sucient conditions will be given under which this multirate H1 control problem is solvable; once solvable, an explicit solution will also be given.
Organization
The rest of the paper is organized as follows. In Section 2 we present some concepts and facts about nest operators and nest algebras. These have direct applications in subsequent sections. Section 3 discusses desirable properties for multirate controllers; they are periodicity, causality, and nite-dimensionality. Causality constraints are de ned using operators between appropriate nests. This provides a natural and transparent framework for studying causality constraints in multirate systems. 4
Section 4 deals with internal stability of the setup in Figure 1 and relates it to internal stability of some discrete-time system. Section 5 contains the main contribution of this paper, namely, an explicit solution to the multirate H1 control problem. This is achieved by rst reducing it to a constrained H1 model-matching problem and then solving the latter problem using results in Section 2. A frequency-domain approach is used consistently. In Section 6 we brie y consider the H2 -optimal design of general multirate systems. The techniques developed in this paper also yield an explicit solution to the H2 problem. Finally, Section 6 contains some concluding remarks. The notation is quite standard and will be de ned when introduced. Throughout the paper, we choose to use -transforms instead of z -transforms, where = z ?1 ; in this case, discrete-time spaces such as H2 and H1 are de ned on the open unit disk. Finally, if G is an LTI system, G^ denotes its transfer matrix.
2 Preliminaries In this section we address some topics and computation on nests and nest algebras which are useful in the sequel. We shall restrict our attention to nite-dimensional spaces; more general treatment can be found in [4, 12].
Nests, Nest Operators, and Nest Algebras
Let X be a nite-dimensional space. A nest in X , denoted fXi g, is a chain of subspaces in X , including f0g and X , with the nonincreasing ordering:
X = X0 X1 Xn?1 Xn = f0g: Let X and Y be both nite-dimensional inner-product spaces with nests fXi g and fYi g respectively. Assume the two nests have the same number of subspaces, say, n + 1 as above. A linear map T : X ! Y is a nest operator if
T Xi Yi ;
i = 0; 1; ; n:
(1)
This gives n +1 relations; the rst and the last are trivially satis ed. We shall allow repetitions in fXi g and fYi g. Thus redundancy may occur in (1) and in the results to follow. However, for computation one can eliminate this redundancy as follows: If Xi = Xi+1 , the i-th relation, namely, T Xi Yi , is redundant since Yi Yi+1 and therefore can be eliminated; similarly, if Yi = Yi+1 , we eliminate the (i + 1)-st relation. Let X : X ! Xi and Y : Y ! Yi be orthogonal projections. Then the condition in (1) is equivalent to i
(I ? Y )T X = 0; i
i
i
i = 0; 1; ; n:
The set of all such operators is denoted N (fXi g; fYig) and abbreviated N (fXi g) if fXi g = fYi g. The following properties are straightforward to verify. 5
Lemma 1
(a) If T1 2 N (fXi g; fYig) and T2 2 N (fYig; fZig), then T2T1 2 N (fXi g; fZig). (b) N (fXi g) forms an algebra, called nest algebra. (c) If T 2 N (fXi g) and T is invertible, then T ?1 2 N (fXi g).
Factorization
It is a fact that every operator on X can be factored as the product of a unitary operator and an operator in N (fXi g). Lemma 2 Let T be an operator on X . (a) There exists a unitary operator U1 on X and an operator R1 in N (fXi g) such that T = U 1 R1 . (b) There exists an operator R2 in N (fXi g) and a unitary operator U2 on X such that T = R2U2 . Note that R1 and R2 are invertible if T is invertible. We shall give an elementary proof of this lemma, for it provides a way to compute such factorizations via the well-known QR factorization.
Proof of Lemma 2 We shall look at part (a); part (b) follows similarly. Since Xi Xi+1, we write (Xi+1 )?X as the orthogonal complement of Xi+1 in Xi . Decompose X into X = (X1)?X0 (X2)?X1 (Xn )?X ?1 : It follows that under this decomposition any operator R belongs to N (fXi g) i its matrix is block lower-triangular, all the diagonal blocks being square. Thus it suces to show that for any matrix T on X we can write T = U1 R1 where U1 is orthogonal and R1 is block lower-triangular. This follows from a QR type of factorization for square matrices: T = U1 R1 with U1 orthogonal and R1 lower-triangular; partition R1 accordingly to get that R1 is also block lower-triangular. QED i
n
A Distance Problem
Finally, we look at a distance problem. Let X and Y be nite-dimensional inner-product spaces with nests fXi g and fYi g. Let T be an operator X ! Y . We want to nd the distance (via induced norms) of T to N (fXi g; fYig), abbreviated N : dist (T; N ) := Qinf kT ? Qk: (2) 2N It is clear that
dist (T; N ) max k(I ? Y )T X k: i i
6
i
Theorem 1 dist (T; N ) = max k(I ? Y )T X k: i i
i
This is Corollary 9.2 in [12] specialized to operators on nite-dimensional spaces; it is an extension of a result in [32, 11] on norm-preserving dilation of operators, which is specialized to matrices below. We denote the Moore-Penrose generalized inverse of a matrix M by M y.
Lemma 3 Assume that A; B; C are xed matrices of appropriate dimensions. Then " # h i "A# C A inf k X B k = maxfk C A k; k B kg := : X Moreover, a minimizing X is given by
X = ?BA ( I ? AA)yC: It will be of interest to us how to compute a Q to achieve the in mum in (2); this can be
done by sequentially applying Lemma 3: Step 1 Decompose the spaces X and Y :
X = (X1)?X0 (X2)?X1 (Xn )?X ?1 Y = (Y1)?Y0 (Y2)?Y1 (Yn )?Y ?1 : We get matrix representations for T and Q: 2 3 3 2 T11 T12 T1n Q11 0 0 66 T21 T22 T2n 77 66 Q21 Q22 0 77 6 7 T = 6 .. .. .. 7 ; Q = 66 .. .. 77 ; . 5 4 . . . 5 4 . ... Tn1 Tn2 Tnn Qn1 Qn2 Qnn Q being block lower-triangular. Step 2 De ne Xij = Tij ? Qij , i j , and 3 2 X11 T12 T1n 66 X X T2n 77 P = 66 ..21 ..22 .. 77 : 4 . . . 5 Xn1 Xn2 Xnn n
n
The problem reduces to
min kP k; X ij
where Tij , i < j , are xed. Minimizing Xij can be selected as follows. First, set X11; X21; ; Xn1 and Xn2 ; ; Xnn to zero. Second, choose X22 by Lemma 3 such that 7
the norm of the matrix (I ? Y2 )P X1 (obtained by retaining the rst 2 block rows and the last n ? 1 block columns in P ) is minimized: k(I ? Y2 )P X1 k = maxfk(I ? Y1 )T X1 k; k(I ? Y2 )T X2 kg:
i
h
Fix this X22. Third, choose X32 X33 again by Lemma 3 to minimize
k(I ? Y3 )P X2 k = maxfk(I ? Y2 )T X1 k; k(I ? Y3 )T X3 kg: In this way, we can nd all Xij such that min kP k = max k(I ? Y )T X k: i X i
ij
i
This procedure also gives a constructive proof of the theorem.
3 Multirate Systems In this section we shall examine the multirate controller Kd in Figure 1 as a discrete-time linear operator. To be studied are three basic properties: periodicity, causality, and nite dimensionality.
Periodicity
The sampled-data controller HKd S is in general time-varying. However, the operation at each channel of S and H is periodic. Let l = LCM fm1; ; mp; n1; ; nqg; where LCM means least common multiple. Thus := lh is the least common period for all sampling and hold channels, i.e., is the least time interval in which the sampling and hold schedule repeats itself. Kd can be chosen so that HKd S is -periodic in continuous-time. For this, we need a few de nitions. Let ` be the space of sequences, perhaps vector-valued, de ned on the time set f0; 1; 2; g. Let U be the unit time delay on ` and U the unit time advance. De ne the integers
m i = ml ; i = 1; 2; ; p i l n j = n ; j = 1; 2; ; q: j
We say Kd is (mi ; nj )-periodic if
2 (U )n1 64 ...
(U )n
3 2 U m 1 75Kd64 ...
U m
p
q
8
3 75 = Kd:
This means shifting the i-th input channel by m i time units (m i mi h = ) corresponds to shifting the j -th output channel by n j units (nj nj h = ). Thus HKd S is -periodic in continuous time i Kd is (mi; nj )-periodic. Since G is LTI, it follows that the sampled-data system in Figure 1 is -periodic if Kd is (mi ; nj )-periodic. We shall refer as the system period. Now we lift Kd to get an LTI system. For an integer m > 0, de ne the discrete lifting operator Lm via = Lm : 82 (0) 3 2 (m) 3 9 > > = < 7 .. : ; f (0); (1); g 7! >64 ... 75; 64 5 . > ; : (m ? 1) (2m ? 1) Lm maps ` to `m, the external direct sum of m copies of `. If the underlying period for is , then the underlying period for is m . Now extend the input and output spaces of Kd so that the underlying period is ; this corresponds to lifting the controller Kd in the following way: 3 3 2 2
Kd := 64
Ln1
?1
...
Ln
L 75Kd66 m 1 . . . 4
q
L?m1
77 5:
p
It is an easy matter to check, see, e.g., [28], that the lifted controller Kd is LTI i Kd is (mi ; nj )-periodic.
Causality
Figure 1 is a real-time system. So for Kd to be implementable, HKd S must be causal in continuous time. This implies that Kd , as a single-rate system, must be causal; and moreover, the feedthrough term D in Kd must satisfy a certain constraint, that is, some blocks in D must be zero [27, 34]. Now let us characterize this constraint on D using nest operators. Write = Kd ; then (0) = D (0), where by de nitions
02 L 32 m 1 1 6 6 7 . . . . (0) = B @4 54 . . Lm p h 0 = 1 ? 1)0 1(0) 1(m
31 75CA (0)
p
p (0)0
p ? 1)0 p (m
i0
Note that i (k) is sampled at t = kmi h. Similarly, h i0 (0) = 1(0)0 1(n1 ? 1)0 p (0)0 p (nq ? 1)0 and j (k) occurs at t = knj h. Let be the set of sampling or hold instants in the interval [0; ) (modulo the base period h), i.e., :=
[ i
1 ! [ 0[ f0; mi; 2mi; ; l ? mig @ f0; nj ; 2nj ; ; l ? nj gA : j
9
This is a nite set of, say, n + 1 elements (not counting repetitions); order increasingly (r < r+1 ): = fr : r = 0; 1; ; ng: Let (0) and (0) live in the nite-dimensional spaces X and Y respectively. For r = 0; 1; ; n, de ne
Xr = span f (0) : i(k) = 0 if kmi < r g Yr = span f (0) : j (k) = 0 if knj < r g: Xr and Yr correspond to, respectively, the inputs and outputs occurred after and including time r h. It is easily checked that fXr g and fYr g are nests and that the causality condition on D (the output at time r h depends only on inputs up to r h) is exactly D Xr Y r ;
r = 0; 1; ; n:
Thus we de ne D to be (mi; nj )-causal if D 2 N (fXr g; fYr g). This is the same causality constraint in [27, 34] de ned in terms of the elements of D. For later bene t, we de ne D to be (mi ; nj )-strictly causal if
DXr Yr+1 ;
r = 0; 1; ; n ? 1:
This means that the output at time r+1 h depends only on inputs up to time r h. The following lemma, which is straightforward to prove, justi es our use of terminology from a continuous-time viewpoint.
Lemma 4 (a) HKd S is causal in continuous time i Kd is causal and D is (mi; nj )-causal. (b) HKd S is strictly causal in continuous time i Kd is causal and D is (mi ; nj )-strictly causal.
Some conclusions on causality issues [27] are transparent under this new formulation.
Lemma 5 (a) If D1 is (mi ; pk )-causal and D2 is (pk ; nj )-causal, then D2 D1 is (mi; nj )-causal; furthermore, if D1 or D2 is strictly causal, then D2 D1 is also strictly causal. (b) If D is (mi; mi )-causal and invertible, then D?1 is (mi ; mi)-causal. (c) If D is (mi; mi )-strictly causal, then (I ? D)?1 exists and is (mi; mi)-causal.
10
Proof Part (a) follows immediately from Lemma 4:
D1 ; D2 are causal ) HD1S ; HD2S are causal in continuous time ) HD2D1S = HD2SHD1S is causal in continuous time ) D2D1 is causal:
Part (a) also follows from Lemma 1 (a) by some renumbering of the indices. Part (b) follows directly from Lemma 1 (c). For part (c), note that under appropriate decomposition, D is strictly block lower-triangular; so (I ? D)?1 exists and is (mi ; mi)-causal [part (b)]. QED Let us de ne Kd to be (mi; nj )-causal if Kd is causal and D is (mi; nj )-causal.
Finite Dimensionality
We assume Kd is (mi ; nj )-periodic and -causal. Then Kd is LTI and causal. To get nitedimensional di erence equations for Kd , we further assume Kd is nite-dimensional. Thus Kd has a state model 2 3
66 CA1 DB111 K^ d () = 66 .. .. 4 . . Cq Dq1
Bp D1p 777
.. 7 : . 5
Dqp
Let the state for Kd be . The corresponding equations for Kd ( = Kd ) are
(k + 1) = A(k) +
p X
Bi i(k)
i=1 p X
j (k) = Cj (k) +
i=1
Dji i (k); j = 1; 2; ; q:
Note that i = Lm i and j = Ln j . Partitioning the matrices accordingly i h Bi = (Bi )0 (Bi )m ?1 ; 2 (Dji)00 (Dji)0;m ?1 3 2 (Cj )0 3 75 .. .. Cj = 64 ... 75; Dji = 64 . . (Dji )n ?1;0 (Dji)n ?1;m ?1 (Cj )n ?1 (certain blocks in Dji must be zero for the causality constraint), we get the di erence equations for Kd ( = Kd ): j
i
i
i
j
j
(k + 1) = A(k) + j (knj + r) =
p mX ?1 X i
j
(Bi )s i (km i + s) i=1 s=0 p mX i ?1 X (Dji)rs i (km i + s); (Cj )r (k) + i=1 s=0 11
i
(3) (4)
where the indices in (4) go as follows: j = 1; 2; ; q and r = 0; 1; ; n j ? 1. These are the equations for implementing Kd on microprocessors and they require only nite memory. Note that the state vector for Kd is updated every system period . In summary, in this paper we are interested in the class of multirate Kd which are (mi; nj )periodic and -causal and nite-dimensional; this class is called the admissible class of Kd and can be modeled by di erence equations (3-4) with D (mi ; nj )-causal. The corresponding admissible class of Kd is characterized by LTI, causal, and nite-dimensional Kd with the same constraint on D. The causality constraint, namely, that D must be a nest operator, is the new feature in lifted multirate systems, which is the main concern in the subsequent designs. A seemingly easy way out would be to take D = 0, which corresponds to delay the control input u by a system period . However, we would like to argue that this would result in serious performance degradation since for complex multirate systems, the system periods are usually appreciably large.
4 Internal Stability In this section we look at stability of Figure 1. We assume the continuous G is LTI, causal, and nite-dimensional; partition G as follows:
" # "
z G11 G12 y = G21 G22
G has a state model
#"
#
w : u
3 2 B B A 1 2 G^(s) = 64 C1 D11 D12 75:
C2 D21 0 Let the plant state be x and the controller state be (Kd is admissible). Note that the system in Figure 1 is -periodic. De ne the continuous-time vector " # x ( t ) xsd(t) := (k) ; k t < (k + 1): The (autonomous) system in Figure 1 is internally stable, or Kd internally stabilizes G, if for any initial value xsd (t0), 0 t0 < , xsd (t) ! 0 as t ! 1. In the de nition, w = 0; so Figure 1 reduces to Figure 2, where we assume G22 has the same state as G. Moving S and H around the loop and de ning G22d = S G22H, the multirate discretization of G22, we arrive at a multirate discrete-time system. Now lift Kd as before and G22d by 3 2L 3 2 L?1 m 1 n 1 77 75G22d66 ... ... G22d = 64 5 4 L?n 1 Lm p
12
p
G22
y
- S
-
p p p p p p
u
-
Kd
H
p p p p p p
Figure 2: For stability analysis p p p p p p p p p p p p p p p p p p
G22d
p p p p p p p p p p p p p p p p p p
p
p
p p
p p
p
p
p
p
p
p
p
p
p p
p p p
p p p p p p p
-
p p p p p p p p p p p p p p p
p p
Kd
p p p p p
p
p p
p
p
p
p p
p
p
p
p p
p
Figure 3: The lifted system for stability to get the lifted system of Figure 3. Because G22 is LTI and strictly causal, G22d is (nj ; mi)periodic and -strictly causal. Thus G22d is LTI and causal with D22d (nj ; mi)-strictly causal. So Figure 3 gives an LTI discrete system. In fact, a state model for G22d can be obtained [28]; its state being := S x, or (k) = x(k ). Let us see that Figure 3 is well-posed, i.e., the matrix I ? D22dD is invertible, where D is the feedthrough term of Kd . This follows from Lemma 5: D22dD is (mi ; mi)-strictly causal [Lemma 5 (a)] and so I ? D22d D is invertible [Lemma 5 (c)]. This also implies that the multirate system of Figure 1 is well-posed. The system in Figure 3 is internally stable, or Kd internally stabilizes G22d if for any initial states (0) and (0), " # (k) ! 0 as k ! 1: (k) It is clear that Figure 3 is internally stable if Figure 1 is. Theorem 2 Kd internally stabilizes G i Kd internally stabilizes G22d. Proof Suppose Kd internally stabilizes G22d. It suces to show that x(t) ! 0 as t ! 1. Internal stability of Figure 3 implies that (k) ! 0 as k ! 1 in Figure 3 and hence u(t) ! 0 as t ! 1 in Figure 2. Now since for k t < (k + 1) ,
x(t) = e(t?k)A (k) +
Zt
13
k
e(t? )A B2 u( ) d ;
QED
it follows that x(t) ! 0 as t ! 1.
Sucient conditions for the internal stability to be achievable are that (A; B2) and (C2; A) are stabilizable and detectable respectively and that the system period is non-pathological in a certain sense, see, e.g., [16, 33].
5 H1-Optimal Control With reference to Figure 1, we now study the main synthesis problem: Design an admissible
Kd that internally stabilizes G and achieves the continuous-time H1 performance requirement kF (G; HKdS )k < , where is pre-speci ed and positive. By proper scaling, we can always take = 1.
The general idea in the solution is to reduce the multirate sampled-data problem to a discrete H1 model-matching problem with the causality constraint and then solve the constrained problem explicitly using techniques presented in Section 2 on nest operators and nest algebras. A special case of the reduction process was reported in [33] where a uniform sampling rate and a uniform hold rate are assumed. The solution process is complex enough to require several distinct steps. Appropriate connections to some recent work on H1 control are made in each step. We start with a state model for G:
3 2 A B1 B2 G^ (s) = 64 C1 0 D12 75: C2 0
0
As we saw in the preceding section, the zero block in D22 guarantees the well-posedness of the closed-loop multirate system in Figure 1. For kF (G; HKdS )k to be nite, we must have D21 = 0. The zero block in D11 is for a technical simpli cation, as in the single-rate case [5, 21]. We shall assume that (A; B2) is stabilizable and (C2; A) is detectable. H1
Discretization
The original problem is posed in continuous time; so the rst step is to recast it as a discretetime problem with time-varying control. The reduction is based on recent advances in H1 sampled-data control in the single-rate setting. Introduce the discrete sampling operator Sm : ` ! ` de ned via = Sm () (k) = (km) and the discrete hold operator Hn : ` ! ` via
= Hn () (kn + r) = (k); r = 0; 1; ; n ? 1: 14
It is easily checked that Sm h = Sm Sh and Hn h = Hh Hn . So the multirate sampling and hold operators S and H can be factored as i
i
2 Sm 1 6 ... S=4
j
Sm
j
2 Hn 3 75Sh; H = Hh64 1 . . .
Hn
p
De ning
2 Hn 1 6 ... Kd1 = 4
Hn
3 2 Sm 75Kd64 1 . . . q
Sm
3 75;
3 75:
q
(5)
p
we can view Figure 1 as a ctitious single-rate system but with a time-varying controller Kd1 as in Figure 4. Now the results in, e.g., [5, 21] (there, discrete controllers are not required to be time-invariant), are applicable.
z
G y
-
w u
Sh
- Kd1 -
p p p p
p p p p
Hh
Figure 4: An equivalent single-rate system Let D11h : L2 [0; h) ! L2 [0; h) be de ned by (D11hw)(t) = C1
Zt 0
e(t? )A B1 w( ) d
and assume
kD11hk < 1. Since D11h is the restriction of F (G; HKdS ) on L2 [0; h), this condition is necessary for kF (G; HKdS )k < 1; how to verify this condition was studied in [5]. For the multirate sampled-data H1 problem, invoke the single-rate results to get the equivalent discrete system shown in Figure 5 and the equivalent discrete-time problem: Design Kd1 to give internal stability and achieve kF (Gd; Kd1)k < 1, where the norm now is `2 induced. The H1 discretization Gd (for = 1) is LTI and causal and has a state model 15
p p p p p p p p p p p p p p p p p p
!
p p p p p p p p p p p p p p p p p p
Gd
p p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p
p
p
p
p
p
-
p p p p p
p p
Kd1
p p p p p p p p p p p p p p
p p p p p p p p p p p p p p p p p p
Figure 5: H1 discretized system
3 2 A d B1d B2d G^d () = 64 C1d D11d D12d 75: C2d
0
0
The computation of the matrices in G^ d is given in, e.g., [5, 21]. In this way, we arrive at an equivalent discrete H1 problem; however, Kd1 is constrained to be of the form in (5) with Kd admissible.
Discrete Lifting
The system of Figure 5 is single-rate with the underlying period being h. The next step is to lift to get an LTI con guration with underlying period . Partition Gd as usual:
"
#
G11d G12d : Gd = G 21d G22d De ne Kd as before and
2 66 Ll Lm 1 Sm1 Gd = 66 ... 4
3 2 L?1 77 66 l Hn1 L?n11 77 Gd 66 ... 5 4
3 77 77 5
Hn L?n 1 to get the lifted system of Figure 6, where ! = Ll ! and = Ll . Since Gd is LTI, causal, and nite-dimensional with G22d strictly causal, it is an easy exercise to verify the following properties of Gd . Lm Sm p
p
q
q
Lemma 6 Gd is LTI, causal, and nite-dimensional. Moreover, the feedthrough term D22d of G22d is (nj ; mi )-strictly causal. In fact, a state model for Gd can be obtained using the lemma below. 16
Gd
p p p p p p p p p p p p p p p p p p
p p p p p p p p p p
!
p p p p p p p p p p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p p p p p p p p p
p
p
p
p
p p p p p p
p
-
p p p p p p p p p p p p p p
Kd
p p p p p p p p p p p p p p p p p p p p
Figure 6: The lifted system Let P be a discrete-time system with state and transfer matrix
#
"
B : P^ () = CA D Let m; n; m; n ; l be positive integers such that
mm = nn = l: De ne
P := Lm SmPHn L?n 1
and the characteristic function on integers
[p;q) (r) =
(
pr
