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H1 Design of General Multirate Sampled-Data
Control Systems�
Tongwen Chen
Dept. of Elect. & Comp. Engg.
University of Calgary
Calgary, Alberta
Canada T2N 1N4
Tel: 403-220-8357
Email: [email protected]
Li Qiu
Inst. for Math. & Its Appl.
University of Minnesota
Minneapolis, MN
USA 55455
Tel: 612-624-1824
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 impossi-
ble, 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 dif-
ference equations with �nitely many coe�cients. 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 di�cult 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); k� � t < (k + 1)�:
2
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
264 1...
m
375 =
264S�
. ..
S�
375264y1...
ym
375;
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-
G
Kd HS
z
y
w
u
�
� �
�
- p p p p- p p p p-
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:
S =
264Sm1h
. ..
Smph
375; H =
264Hn1h
. ..
Hnqh
375:
These correspond to sampling p channels of y periodically with periods mih, i = 1; � � � ; p,
respectively and holding q channels of � with periods njh, 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
y =
264y1...
yp
375; =
264 1...
p
375; � =
264�1...
�q
375; u =
264u1...
uq
375;
then
i(k) = yi(kmih); i = 1; � � � ; p
uj(t) = �j(k); knjh � t < (k + 1)njh; 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 twomi are equal and neither are two nj . Kd is the discrete-
time 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(njh).
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 =
"P11 P12P21 P22
#;
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 su�cient 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 H1model-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 fXig, 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 fXig and fYig
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
TXi � 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 fXig and fYig. 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, TXi � 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 �Xi: X ! Xi and �Yi : Y ! Yi be
orthogonal projections. Then the condition in (1) is equivalent to
(I ��Yi)T�Xi= 0; i = 0; 1; � � � ; n:
The set of all such operators is denoted N (fXig; fYig) and abbreviated N (fXig) if fXig =
fYig. The following properties are straightforward to verify.
5
Lemma 1
(a) If T1 2 N (fXig; fYig) and T2 2 N (fYig; fZig), then T2T1 2 N (fXig; fZig).
(b) N (fXig) forms an algebra, called nest algebra.
(c) If T 2 N (fXig) and T is invertible, then T�1 2 N (fXig).
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 (fXig).
Lemma 2 Let T be an operator on X .
(a) There exists a unitary operator U1 on X and an operator R1 in N (fXig) such that
T = U1R1.
(b) There exists an operator R2 in N (fXig) 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)?Xi
as the orthogonal complement of Xi+1 in Xi. Decompose X into
X = (X1)?X0� (X2)
?X1� � � � � (Xn)
?Xn�1
:
It follows that under this decomposition any operator R belongs to N (fXig) i� its matrix
is block lower-triangular, all the diagonal blocks being square. Thus it su�ces to show that
for any matrix T on X we can write T = U1R1 where U1 is orthogonal and R1 is block
lower-triangular. This follows from a QR type of factorization for square matrices: T = U1R1
with U1 orthogonal and R1 lower-triangular; partition R1 accordingly to get that R1 is also
block lower-triangular. QED
A Distance Problem
Finally, we look at a distance problem. Let X and Y be �nite-dimensional inner-product
spaces with nests fXig and fYig. Let T be an operator X ! Y . We want to �nd the distance
(via induced norms) of T to N (fXig; fYig), abbreviated N :
dist (T;N ) := infQ2N
kT � Qk: (2)
It is clear that
dist (T;N ) � maxik(I ��Yi)T�Xi
k:
6
Theorem 1
dist (T;N ) = maxik(I ��Yi)T�Xi
k:
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 My.
Lemma 3 Assume that A;B;C are �xed matrices of appropriate dimensions. Then
infXk
"C A
X B
#k = maxfk
hC A
ik; k
"A
B
#kg := �:
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)
?Xn�1
Y = (Y1)?Y0� (Y2)
?Y1� � � � � (Yn)
?Yn�1
:
We get matrix representations for T and Q:
T =
266664T11 T12 � � � T1nT21 T22 � � � T2n...
.
.
.
.
.
.
Tn1 Tn2 � � � Tnn
377775 ; Q =
266664Q11 0 � � � 0
Q21 Q22 � � � 0
.
.
.
.
.
.
.
.
.
Qn1 Qn2 � � � Qnn
377775 ;
Q being block lower-triangular.
Step 2 De�ne Xij = Tij �Qij , i � j, and
P =
266664X11 T12 � � � T1nX21 X22 � � � T2n...
.
.
.
.
.
.
Xn1 Xn2 � � � Xnn
377775 :
The problem reduces to
minXij
kPk;
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�X1k = maxfk(I � �Y1)T�X1
k; k(I ��Y2)T�X2kg:
Fix this X22. Third, choose
hX32 X33
iagain by Lemma 3 to minimize
k(I ��Y3)P�X2k = maxfk(I � �Y2)T�X1
k; k(I ��Y3)T�X3kg:
In this way, we can �nd all Xij such that
minXij
kPk = maxik(I � �Yi)T�Xi
k:
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 HKdS 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 HKdS 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
�mi =l
mi; i = 1; 2; � � � ; p
�nj =l
nj; j = 1; 2; � � � ; q:
We say Kd is (mi; nj)-periodic if264(U�)�n1
. ..
(U�)�nq
375Kd
264U �m1
. ..
U �mp
375 = Kd:
8
This means shifting the i-th input channel by �mi time units ( �mimih = �) corresponds to
shifting the j-th output channel by �nj units (�njnjh = �). Thus HKdS 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�:
f�(0); �(1); � � �g 7!
8><>:264
�(0)...
�(m� 1)
375;264
�(m)
.
.
.
�(2m� 1)
375; � � �
9>=>; :
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:
Kd :=
264L�n1
.. .
L�nq
375Kd
2664L�1�m1
.. .
L�1�mp
3775:
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, HKdS 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
(0) =
0B@264L �m1
...
L �mp
375264 1...
p
3751CA (0)
=
h 1(0)
0 � � � 1( �m1 � 1)0 � � � p(0)
0 � � � p( �mp � 1)0i0
Note that i(k) is sampled at t = kmih. Similarly,
�(0) =h�1(0)
0 � � � �1(�n1 � 1)0 � � � �p(0)
0 � � � �p(�nq � 1)0i0
and �j(k) occurs at t = knjh. Let � be the set of sampling or hold instants in the interval
[0; �) (modulo the base period h), i.e.,
� :=
[i
f0; mi; 2mi; � � � ; l�mig
This is a �nite set of, say, n + 1 elements (not counting repetitions); order � increasingly
(�r < �r+1):
� = f�r : 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 < �rg
Yr = span f�(0) : �j(k) = 0 if knj < �rg:
Xr and Yr correspond to, respectively, the inputs and outputs occurred after and including
time �rh. It is easily checked that fXrg and fYrg are nests and that the causality condition
on D (the output at time �rh depends only on inputs up to �rh) is exactly
DXr � Yr; r = 0; 1; � � � ; n:
Thus we de�ne D to be (mi; nj)-causal if D 2 N (fXrg; fYrg). 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+1h depends only on inputs up to time �rh.
The following lemma, which is straightforward to prove, justi�es our use of terminology
from a continuous-time viewpoint.
Lemma 4
(a) HKdS is causal in continuous time i� Kd is causal and D is (mi; nj)-causal.
(b) HKdS 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 D2D1 is (mi; nj)-causal; fur-
thermore, if D1 or D2 is strictly causal, then D2D1 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; D
2are 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 �nite-
dimensional di�erence equations for Kd, we further assume Kd is �nite-dimensional. Thus
Kd has a state model
Kd(�) =
266664A B1 � � � Bp
C1 D11 � � � D1p
.
.
.
.
.
.
.
.
.
Cq Dq1 � � � Dqp
377775 :
Let the state for Kd be �. The corresponding equations for Kd (� = Kd ) are
�(k+ 1) = A�(k) +
pXi=1
Bi i(k)
�j(k) = Cj�(k) +
pXi=1
Dji i(k); j = 1; 2; � � � ; q:
Note that i= L �mi
i and �j = L�nj�j . Partitioning the matrices accordingly
Bi =
h(Bi)0 � � � (Bi) �mi�1
i;
Cj =
264
(Cj)0
.
.
.
(Cj)�nj�1
375; Dji =
264
(Dji)00 � � � (Dji)0; �mi�1
.
.
.
.
.
.
(Dji)�nj�1;0 � � � (Dji)�nj�1; �mi�1
375
(certain blocks inDji must be zero for the causality constraint), we get the di�erence equations
for Kd (� = Kd ):
�(k+ 1) = A�(k) +
pXi=1
�mi�1Xs=0
(Bi)s i(k �mi + s) (3)
�j(k�nj + r) = (Cj)r�(k) +
pXi=1
�mi�1Xs=0
(Dji)rs i(k �mi + s); (4)
11
where the indices in (4) go as follows: j = 1; 2; � � � ; q and r = 0; 1; � � � ; �nj � 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
y
#=
"G11 G12
G21 G22
# "w
u
#:
G has a state model
G(s) =
264 A B1 B2
C1 D11 D12
C2 D21 0
375:
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
xsd(t) :=
"x(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 = SG22H, the multirate
discretization of G22, we arrive at a multirate discrete-time system. Now lift Kd as before
and G22d by
G22d =
264L �m1
...
L �mp
375G22d
2664L�1�n1
...
L�1�np
3775
12
Kd
G22
S H-
�
p p p p p p-p p p p p p-
�
y u
Figure 2: For stability analysis
Kd
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
p
p
p
p
p
p
p
p
p
pppppppppppppp
�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 � D22dD 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 stabilizesG22d if for any initial
states �(0) and �(0), "�(k)
�(k)
#! 0 as k !1:
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 su�ces 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) +
Z t
k�e(t��)AB2u(�) d�;
13
it follows that x(t)! 0 as t!1. QED
Su�cient 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:
G(s) =
264 A B1 B2
C1 0 D12
C2 0 0
375:
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 discrete-
time problem with time-varying control. The reduction is based on recent advances in H1sampled-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 Smih = SmiSh and Hnjh = HhHnj . So the multirate sampling and
hold operators S and H can be factored as
S =
264Sm1
. ..
Smp
375Sh; H = Hh
264Hn1
. ..
Hnq
375:
De�ning
Kd1 =
264Hn1
. ..
Hnq
375Kd
264Sm1
. ..
Smp
375; (5)
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.
G
Kd1 HhSh
z
y
w
u
� �
�
- p p p p- p p p p-
Figure 4: An equivalent single-rate system
Let D11h : L2[0; h)! L2[0; h) be de�ned by
(D11hw)(t) = C1
Z t
0
e(t��)AB1w(�) 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
Gd
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 p p p p�
p p p p p p p p�p p p p p p p p p
p p p p p p p p p p p p pp p p p p p p p p p p p-p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
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
Gd(�) =
264 Ad B1d B2d
C1d D11d D12d
C2d 0 0
375:
The computation of the matrices in Gd 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:
Gd =
"G11d G12d
G21d G22d
#:
De�ne Kd as before and
Gd =
266664Ll
L �m1Sm1
. ..
L �mpSmp
377775Gd
266664L�1l
Hn1L�1�n1
.. .
HnqL�1�nq
377775
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.
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
Kd
� !p p p p p p p p p p p p p p p p p p� p p p p p p p p p p p p p p p p p p�
p p p p p p p p�p p p p p p p p p
p p p p p p p p p p p p pp p p p p p p p p p p p-p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
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 6: The lifted system
Let P be a discrete-time system with state � and transfer matrix
P (�) =
"A B
C D
#:
Let m;n; �m; �n; l be positive integers such that
m �m = n�n = l:
De�ne
P := L �mSmPHnL�1�n
and the characteristic function on integers
�[p;q)(r) =
(1; p � r < q
0; else:
Lemma 7 A state model for P is
P (�) =
26666664
Al Pn�1r=0 A
l�1�rBP
2n�1r=n Al�1�rB � � �
Pl�1r=l�n A
l�1�rB
C D00 D01 � � � D0;�n�1
CAm D10 D11 � � � D1;�n�1
......
......
CAl�m D �m1;0 D �m1;1 � � � D �m�1;�n�1
37777775;
where
Dij = D�[jn;(j+1)n)(im) +
(j+1)n�1Xr=jn
CAim�1�rB�[0;im)(r):
The corresponding state vector is � = Sl�.
17
The lemma can be proven by manipulating the input-output equations for P . Note that
the transfer matrices for all blocks in Gd can be obtained from this lemma; for example, for
the (1,1) block, namely, G11d = LlG11dL�1l , we take m = n = 1 and �m = �n = l in the lemma
to get
G11d(�) =
26666664
Ald Al�1
d B1d Al�2d B1d � � � B1d
C1d D11d 0 � � � 0
C1dAd C1dB1d D11d � � � 0
.
.
.
.
.
.
.
.
.
.
.
.
C1dAl�1d C1dA
l�2d B1d C1dA
l�3d B1d � � � D11d
37777775:
This realization is also given in, e.g., [16].
From the de�nitions of Kd and Gd, we get after some algebra that the closed-loop map
F(Gd; Kd) in Figure 6 equals LlF(Gd; Kd1)L�1l . So kF(Gd; Kd)k = kF(Gd; Kd1)k since Ll
is norm-preserving. Thus the equivalent LTI problem is now: Design an admissible Kd that
internally stabilizes Gd and achieves kF(Gd; Kd)k1 < 1. Notice that the feedthrough term
Kd(0) must be (mi; nj)-causal; so this is a constrained H1 control problem in discrete time.
Constrained Model-Matching Problem
Now we use the controller parametrization [27, 34] to reduce the problem further to a model-
matching problem. In order to internally stabilize Gd, it su�ces to internally stabilize G22d.
Bring in a doubly-coprime factorization for G22d:
G22d = NM�1= ~M
�1~N2
4 ~X � ~Y
� ~N ~M
35"M Y
N X
#= I
with the conditions:
M(0) = I; ~M (0) = I;
N(0) = ~N(0) = D22d;
X = I; ~X = I;
Y (0) = ~Y (0) = 0:
The standard procedure in [15] yields such a factorization. Since D22d is (nj ; mi)-strictly
causal, it follows from [27, 34] that the set of admissible Kd that provide internal stability is
parametrized by
Kd = (Y � MQ)(X � NQ)�1; Q 2 RH1; Q(0) (mi; nj)-causal:
18
Note that the causality constraint on Kd translates exactly to the same constraint on Q(0).
Now de�ne the three RH1 matrices as in [15]
T1 = G11d + G12dM ~Y G21d
T2 = G12dM
T3 = ~MG21d
to get the closed-loop transfer matrix of Figure 6
F(Gd; Kd) = T1 � T2QT3:
Recall in Section 3 that Q(0) is (mi; nj)-causal i� Q(0) 2 N (fXrg; fYrg), abbreviated
N , where the nests fXrg and fYrg were de�ned in Section 3. In this way we arrive at the
constrained H1 model-matching problem: Find Q 2 RH1 with Q(0) 2 N such that
kT1 � T2QT3k1 < 1:
If such a Q exists, we say the multirate H1 problem is solvable.
An Explicit Solution
Let us focus on the constrained H1 model-matching problem. We write T�(�) for T (��1)0.
For regularity, we need the following assumption:
For every � on the unit circle, T2(�) and T�3(�) are both injective.
Under this assumption, perform an inner-outer factorization T2 = T2iT2o and an co-inner-
outer factorization T3 = T3coT3ci, where T2o and T3co are both invertible over RH1. Apply
unitary transformations to T1 � T2QT3 and de�ne
R =
"R11 R12
R21 R22
#:=
"T�2i
I � T2iT�2i
#T1
hT�3ci I � T�
3ciT3ci
i: (6)
The constrained model-matching problem is equivalent to the following four-block problem
of �nding a Q 2 RH1 with Q(0) 2 N such that
k
"R11 � T2oQT3co R12
R21 R22
#k1 < 1: (7)
Dropping the causality constraint on Q(0) temporarily allows us to parametrize all Q in
RH1 achieving (7). We know from [13] that there exists a Q 2 RH1 such that (7) holds i�
k
"�H?
2
0
0 I
#RjH2�L2k < 1: (8)
19
If (8) is satis�ed, then a procedure in [18] allows us to �nd an RH1 matrix
K =
"K11 K12
K21 K22
#
with K�112; K�1
212 RH1 and kK22k1 < 1 such that all Q 2 RH1 satisfying (7) are charac-
terized by
Q = F(K; Q1); Q1 2 RH1; kQ1k1 < 1: (9)
We refer to [18] for the details of checking inequality (8) and the expression of K. Hereafter,
we shall assume that (8) is true. This is also necessary for the solvability of the multirate H1problem.
In general K22(0) 6= 0, so Q(0) depends on Q1(0) in a linear fractional manner. However,
it is possible to simplify this relation by introducing another linear fractional transformation
[33]:
Q1 = F(V; Q2):
Here V , partitioned as usual, is a constant unitary matrix. It follows that the mapping
Q2 7! Q1 is bijective from the open unit ball of RH1 onto itself [35]. Thus all Q satisfying
(7) can be re-parametrized by
Q = F [K;F(V; Q2)]
= F(L; Q2); Q2 2 RH1; kQ2k1 < 1;
where L, partitioned as usual, can be written in terms of K and V :
L =
"K11 + K12V11(I � K22V11)
�1K21 K12(I � V11K22)�1V12
V21(I � K22V11)�1K21 V21(I � K22V11)
�1K22V12 + V22
#:
For L22(0) = 0, we choose the unitary matrix V to be
V =
264 K0
22(0)
hI � K0
22(0)K22(0)
i1=2
hI � K22(0)K
022(0)
i1=2
�K22(0)
375 :
It can be checked that L12(0) and L21(0) are still nonsingular.
To recap, the set of all Q 2 RH1 achieving (7) is parametrized by
Q = F(L; Q2); Q2 2 RH1; kQ2k1 < 1:
Here L has the desirable properties that L22(0) = 0, L12(0) and L21(0) are nonsingular. Thus
Q(0) = L11(0) + L12(0)Q2(0)L21(0): (10)
This is an a�ne function Q2(0) 7! Q(0).
20
Now we bring in the causality constraint on Q(0). Our goal is to �nd a Q2 2 RH1 with
kQ2k1 < 1 such that Q(0) in (10) lies in N (fXrg; fYrg). Since Q(0) depends only on Q2(0)
and in general kQ2k1 � kQ2(0)k, the equivalent problem is to �nd a constant matrix Q2(0)
with kQ2(0)k < 1 such that Q(0) 2 N .
Now we use Lemma 2 to reduce the problem to a distance problem. Introduce matrix
factorizations (Lemma 2)
L12(0) = R1U1; L21(0) = �U2R2;
where R1; R2; U1; U2 are all invertible, U1; U2 are orthogonal, and R1; R2 belongs to the nest
algebras N (fYrg);N (fXrg) respectively.
The computation of such factorizations follow from the proof of Lemma 2: First, change
coordinates in X and Y so that
X = (X1)?X0� (X2)
?X1� � � � � (Xn)
?Xn�1
Y = (Y1)?Y0� (Y2)
?Y1� � � � � (Yn)
?Yn�1
:
This corresponds to permute the components in X and Y according to the order of times when
they occur. Second, do standard QR type factorizations to get the matrices under the new
coordinates, see the proof of Lemma 2. Finally, change coordinates back via permutations to
get the desired matrices.
Substitute the factorizations into (10) and pre- and post-multiply by R�11
and R�12
re-
spectively to get
R�11Q(0)R�1
2= R�1
1L11(0)R
�12� U1Q2(0)U2:
De�ne
W := R�11Q(0)R�1
2; T := R�1
1L11(0)R
�12; P := U1Q2(0)U2:
It follows that Q(0) 2 N (fXrg; fYrg) i� W 2 N (fXrg; fYrg) (Lemma 1) and kQ2(0)k < 1 i�
kPk < 1. Therefore, we arrive at the following equivalent matrix problem: Given T , �nd P
with kPk < 1 such thatW = T�P 2 N ; or equivalently, �nd W 2 N such that kT�Wk < 1.
This can be solved via the distance problem studied in Theorem 1:
dist (T;N ) = maxrfk(I ��Yr )T�Xr
kg =: �:
Let Wopt 2 N achieves the distance, i.e., kT �Woptk = �. The following result summarizes
what we have derived.
Theorem 3 The matrix problem is solvable, i.e., there exists a matrix P with kPk < 1 such
that T � P 2 N , i� � < 1. Moreover, if � < 1, P := T � Wopt solves the problem with
kPk = �.
How to compute � and Wopt were discussed in the procedure given at the end of section 2:
After a change of coordinates in X and Y , which corresponds to permuting their components
chronologically, � can be found via computing the spectral norms of several matrices and
21
Wopt via sequentially applying Lemma 3. If � < 1, we can backtrack to get P (Theorem 3),
Q2(�) = Q2(0), and �nally Q which solves the multirate H1 control problem.
To summarize, let us list the solvability conditions for the multirate H1 control problem
kF(G;HKdS)k < 1:
(a) kD11hk < 1;
(b) k
"PH?
2
0
0 I
#RjH2�L2k < 1;
(c) � < 1.
Condition (a) was studied in detail in [5] and would usually be satis�ed for a reasonable design
since normally the base period h is much smaller than the system period �. Condition (b)
is the solvability condition for a standard four-block H1 problem, see, e.g., [18] for checking
this condition. When conditions (a-b) hold, condition (c) amounts to computing the norms
of several constant matrices.
Finally, we remark that multirate H1 controllers which are arbitrarily close to optimality
can be computed based on the solvability conditions (a-c) (with proper scaling) and a standard
bisection search.
6 H2-Optimal Control
In this section we use the techniques developed to solve explicitly a general multirate H2-
optimal model-matching problem. The model-matching problem arises in multirate control
problems from either a sampled-data point of view [33] or from a discrete-time LQG point of
view [29].
The problem is as follows: Design a Q 2 RH1 with Q(0) being (mi; nj)-causal such that
the H2-norm of the transfer matrix T1 � T2QT3 is minimized, i.e.,
min
Q2RH1;Q(0)2N
kT1 � T2QT3k2; (11)
where Ti are all in RH1. Here we shall make the same regularity assumption as in Section 5:
For every � on the unit circle, T2(�) and T�3 (�) are injective.
Bring in an inner-outer factorization T2 = T2iT2o and a co-inner-outer factorization T3 =
T3coT3ci with T2o and T3co both invertible over RH1. De�ning R as in (6), we get
kT1 � T2QT3k2
2 = k
"T�2i
I � T2iT�2i
#(T1 � T2QT3)
hT�3ci I � T�
3ciT3ci
ik22
k
"R11 � T2oQT3co R12
R21 R22
#k22
= kR11� T2oQT3cok2
2 + kR12k2
2 + kR21k2
2 + kR22k2
2:
22
The last three terms are independent of Q; so the problem in (11) reduces to minimizing only
the �rst term. Writing
Q = Q0 + �Q1; Q0 2 N ; Q1 2 RH1;
we have
inf
Q2RH1;Q02N
kR11 � T2oQT3cok2
2
= infQ02N
inf
Q12RH1
kR11� T2oQ0T3co � �T2oQ1T3cok2
2
= infQ02N
inf
Q12RH1
k��1[R11 � T2oQ0T3co]� T2oQ1T3cok2
2
� infQ02N
k�H?2
n��1[R11� T2oQ0T3co]
ok22
= k�H?2
R11k2
2 + infQ02N
kR110� T2o(0)Q0T3co(0)k2
2;
where R110 is the constant term in R11. Note that the equality is achieved by setting
Q1 = T�12o �H2
h��1(R11 � T2oQ0T3co)
iT�13co: (12)
Thus the optimal Q can be obtained in two steps: Solve the matrix 2-norm optimization
infQ02N
kR110 � T2o(0)Q0T3co(0)k2
2
for Q0 and then substitute Q0 into (12) to get the optimal Q1.
The matrix 2-norm optimization problem can be solved via matrix factorization as well.
Note that the matrix 2-norm is induced by the inner product:
hA;Bi := trace (A0B):
Thus the set of matrices in N can be regarded as a subspace in the space of matrices mapping
X to Y . So the orthogonal projections �N and �N? are well-de�ned. In fact, �N amounts
to simply retaining the blocks corresponding to the unconstrained blocks in N and zeroing
the other blocks.
For square and nonsingular matrices T2o(0) and T3co(0), bring in useful factorizations
(Lemma 2)
T2o(0) = U2R2; T3co(0) = R3U3;
where U2; R2; U3; R3 are all square, U2; U3 are orthogonal, and R2; R3 are invertible and belong
to the nest algebras N (fYrg) and N (fXrg) respectively. Then
minQ02N
kR110� U2R2Q0R3U3k2
= minQ02N
kU 02R110U03 � R2Q0R3k2
� k�N?[U02R110U
03]k2
23
The inequality follows from the fact that R2Q0R3 2 N i� Q0 2 N (Lemma 1) and becomes
equality if we select
Q0 = R�12�N [U
02R110U
03]R�1
3:
This optimal Q0 is in N also by Lemma 1.
On summarizing, we have derived the following result.
Theorem 4 The optimal Q in (11) is given by
Qopt = Q0;opt + �T�12o �H2
h��1(R11 � T2oQ0;optT3co)
iT�13co;
where the optimal constant term Q0;opt is
Q0;opt = R�12�N [U
02R110U
03]R
�13:
7 Conclusions
In this paper we introduced a new framework based on nest operators for handling causality
constraints in multirate systems. This framework allows us to develop explicit solutions to
syntheses of general multirate control systems with H2 and H1 performance criteria.
The results in this paper are presented in an operator setting; for example, the solution
techniques in Sections 5 and 6 are developed in the frequency domain. For computational
e�ciency, it would be useful to develop a time-domain approach via state-space methods; this
would make a good project for future research.
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