H1 Control of Multirate Sampled-Data Systems:

A State-Space Approach

Mats F. S�agfors� and Hannu T. Toivonen

Department of Chemical Engineering,�Abo Akademi University,

FIN{20500 �Abo, FINLAND

msagfors@abo.fi, htoivone@abo.fi

Bengt Lennartson

Control Engineering Laboratory

Chalmers University of Technology

S{412 96 Gothenburg, SWEDEN

bl@control.chalmers.se

Abstract

A state-space solutionof the H1 control problem for periodic multirate sampled-

data systems is presented. The solution is characterized in terms of a pair of discrete

algebraic Riccati equations with a set of associated matrix positive de�niteness condi-

tions and coupling criteria. The solution is derived using two di�erent approaches. In

the �rst approach, the solution is obtained by discretizing two coupled periodic Ric-

cati equations with jumps which characterize the H1-optimal controller. The second

approach is based on the lifting technique. The H1-optimal controller for a lifted rep-

resentation of the multirate system is characterized by a two-Riccati equation solution,

in which the so-called causality constraints are represented by a set of nonstandard

positive de�niteness conditions and coupling criteria.

�Supported by the Academy of Finland

1

1 Introduction

During the last few years there has been an increased interest in the direct design of digital

controllers using continuous-time performance measures. Such systems arise naturally when

a digital controller is applied to a continuous-time plant. In contrast to standard continuous-

time or discrete-time techniques, direct sampled-data methods address both the intersample

behavior of the process, as well as the e�ect the sampling frequency has on the performance.

Multirate systems arise when various measurement sampling functions and control signal

hold elements operate at di�erent rates. The development of a theory for such multirate

systems started in the 1950's with the pioneering work by Kranc [20], Kalman and Bertram

[18] and Friedland [13]. More recent work in the �eld is concerned with e.g. system analysis

and stability [3, 16, 23, 24, 25, 38], optimal control of multirate systems with a quadratic cost

function [1, 2, 6, 8, 11, 21, 22, 26, 39], and H1 control of multirate sampled-data systems

[10, 39].

The H1 problem for single-rate sampled-data systems has recently been studied in a

number of papers [5, 9, 15, 17, 30, 32, 33, 34, 35, 36]. One common approach is to apply

the lifting technique [5] to reduce the hybrid continuous/discrete-time problem to a norm-

equivalent discrete-time H1 problem, to which standard methods can be applied. Another

method is to state a solution directly for the hybrid problem [32, 33]. This leads to a solution

in terms of Riccati equations with jumps. The approaches have recently been shown to give

equivalent solutions [29, 37].

The lifting technique has also been applied to the multirate sampled-data problem [10, 39],

where the periodically time-varying sampled-data system is lifted to give an equivalent time-

invariant discrete system. The main di�culty encountered in this approach is caused by the

causality constraint, which is required for the lifted controller to correspond to a causal

periodic controller. The causality constraint has limited the applicability of standard state-

space solutions on the lifted system description. Previous solutions have therefore been

derived through constrained model matching [10, 39].

In this work, a state-space solution to the periodic multirate sampled-data H1 problem

is derived. The solution is obtained through two di�erent approaches. The �rst approach

is based on a direct Riccati-equation solution, and the second approach is based on lifting.

The direct method has, to our best knowledge, not previously been applied to the multirate

sampled-data H1 problem. The lifting method was used in [10, 39], but no time-domain

solution was derived. The use of both approaches gives insight into the relationship between

the two methods. Both approaches studied here generally give rise to the same synthesis

equations and the same controller.

In the �rst approach, the H1-optimal controller is characterized by a pair of coupled

2

Riccati equations with jumps. Using the results in [29, 37], the Riccati equations can

be discretized and the solution expressed by two periodic Riccati di�erence equations and

discrete-time coupling conditions. The solutions to these periodic equations can then be

expressed in terms of a pair of coupled standard algebraic Riccati equations. This direct

approach allows for a very general problem statement, including non-synchronized sampling

and hold elements.

In the second approach, we adopt the lifting technique to construct a time-invariant

discrete-time system associated with the periodic multirate problem. A solution of the

causally constrained H1-optimal control problem is derived, which consists of two standard

discrete algebraic Riccati equations, a set of associated coupling conditions and a set of

matrix positive de�niteness criteria. An interesting feature of the result is that the alge-

braic Riccati equations are exactly those associated with the unconstrained discrete-time

H1 problem for the lifted time-invariant system description. Thus, standard discrete-time

solution methods can be applied. The additional causality constraint associated with the pe-

riodic multirate system appear in the solution as non-standard positive de�niteness criteria

and coupling conditions, which are easy to check.

The paper is organized as follows. The next section contains the problem de�nition and

a description of the notation used in the paper. In Section 3 a solution to the multirate

sampled-data problem is derived through periodic Riccati equations. A state-space solution

through the lifting technique is presented in Section 4. A simple example is used to illustrate

the results in Section 5, and some concluding remarks are given in Section 6.

2 Problem statement

Consider a linear time-invariant �nite-dimensional continuous-time plant described by

_x(t) = Ax(t) +B1w(t) +B2u(t); x(0) = 0

z(t) = C1x(t) +D12u(t) (1)

where x(t) 2 Rn is the state vector, w(t) 2 Rm1 is the process disturbance, u(t) 2 Rm2 is

the control signal and z(t) 2 Rp1 is the controlled output.

The sampling of the measurements is assumed to be described by a periodic function S

with the period Ty. The outputs may be sampled with di�erent rates or possibly at non-

equidistant time-instants. The discrete signal yd := fyd(iTy + �j); j = 0; 1; : : : ; Ny � 1; i =

0; 1; : : :g available for feedback is given by

yd(iTy + �j) = C2(�j)x(iTy + �j) +D21(�j)vd(iTy + �j); (2)

where vd(�) is a discrete measurement disturbance and iTy + �j (�0 � 0, �j+1 > �j, �Ny=

�0 + Ty) are the times at which a measurement is available. It is assumed that the signal

3

yd(iTy + �j) contains only those measurements obtained at time instant iTy + �j. This

means that the discrete output vector has a periodic and time-varying dimension.

The control signal is assumed to be generated by a zero-order hold function H with the

overall period Tu. The number of control signal updates during one period Tu is Nu. The

individual control signals may be updated at faster rates (with Tu as a common multiple of

the hold times) or the hold times may be periodically time-varying. The multirate zero-order

hold is described by

u(t) = [I �H(�l)]u(kTu + �l) +Hu(�l)ud(kTu + �+l); (3)

t 2 (kTu + �l; kTu + �l+1] ; l = 0; 1; 2; : : :Nu � 1; k = 0; 1; 2; : : ::

where ud := fud(kTu + �+l); l = 0; 1; 2; : : :Nu � 1; k = 0; 1; 2; : : :g is a discrete-time control

signal. The matrixH(�l) := diagf�1(�l); �2(�l); : : : ; �m2(�l)g selects the control signals to be

updated at time instant kTu + �l (�0 � 0, �l+1 > �l, �Nu= �0 + Tu). The functions �m(�l)

take the value 1 if the m:th control signal is updated at time instant kTu + �l, and zero

otherwise. The discrete control signal ud(kTu + �+l) contains only those signals updated at

time kTu + �l. The matrix Hu(�l) is therefore a submatrix of H(�l), which consists of the

nonzero columns of H(�l).

Remark 2.1 The time-varying dimensions of the signals yd and ud guarantee the existence

of matrix inverses which appear in the solution. For an alternative way to guarantee non-

singularity, see [11].

The system de�ned by (1) { (3) is assumed to be periodic with the period T , such that T

is the least common multiple of the period of the sampler Ty and the period of the hold

function Tu. (The solution can, however, with minor modi�cations be extended to cover

the case when the periods of S and H are asynchronously related, cf. [28].) The multirate

sampled-data problem de�ned here is very general. Note that the only assumption on S and

H is their periodicity, otherwise the discrete events may occur at any (non-synchronized)

instants. The solution presented in Section 3 can also be generalized to the case with non-

periodic time-varying sampling. In connection with the lifting technique in Section 4 we

will make some additional technically simplifying assumptions on the synchronization of the

sampling and the hold elements.

The system (1) { (3) can be represented as a mixed continuous/discrete system with

jumps,

�G :

8>><>>:

_xe(t) = Aexe(t) + B1ew(t); t 6= kTu + �lxe(kTu + �+

l) = Ad(�l)xe(kTu + �l) +Bd(�l)ud(kTu + �+

l)

z(t) = C1exe(t)

yd(iTy + �j) = C2d(�j)xe(iTy + �j) +D21(�j)vd(iTy + �j)

(4)

4

where we have extended the state vector, xe(t) := [x0(t) u0(t)]0, and

Ae :=

�A B2

0 0

�; B1e :=

�B1

0

�; C1e := [C1 D12 ] (5)

Ad(�l) :=

�I 0

0 I �H(�l)

�; Bd(�l) :=

�0

Hu(�l)

�; C2d(�j) := [C2(�j) 0 ] : (6)

We need the following de�nitions, cf. [32],

De�nition 2.1 Let �Ad

Ae(�; �) denote the state-transition matrix of the system �G. Then �G

is said to be exponentially stable if there exist c1; c2 > 0 such that

k�(t2; t1)k � c1e�c2(t2�t1); all t2 � t1: (7)

De�nition 2.2 The system �G is said to be stabilizable (respectively, detectable) if there

exists a bounded function Kd(�l) (respectively, Ld(�j)) such that �Ad+BdKd

Ae(respectively,

�I+LdC2

A) satis�es (7).

The following assumptions are made in the paper:

A1 The system �G is stabilizable.

A2 The system �G is detectable.

A3 The matrix D012D12 is non-singular.

A4 The matrices D21(�j)D021(�j) are non-singular.

A5 The continuous-time system (A;B1; C1) is stabilizable and detectable.

The set of admissible controllers is assumed to consist of exponentially stabilizing discrete

causal control laws ud = Kyd such that ud(kTu + �+l) is a function of past measurements

fyd(iTy + �j) : iTy + �j � kTu + �lg only.

The following worst-case performance measure induced by the disturbances w 2 L2[0;1)

and vd 2 l2(0;1) is introduced,

J1(K) := sup(w;vd)6=0

24 kzkL2�kwk2

L2

+ kvdk2l2�1=2

35 : (8)

The multirate sampled-data H1 control problem is de�ned as follows:

Find an admissible controller K, if one exists, which achieves a speci�ed attenuation level

> 0, such that J1(K) < .

5

3 A Riccati equation solution

The following solution to the multirate sampled-dataH1 problem stated in terms of coupled

periodic Riccati equations with jumps stands as a basis for the rest of the treatment in this

section (Theorem 3.1). In Theorem 3.2, the Riccati equations are discretized, and the

solution is given in terms of discrete periodic Riccati equations. Theorem 3.3 shows the

fact that solutions to the periodic equations can be obtained in terms of a pair of coupled

algebraic Riccati equations.

Theorem 3.1 Consider the multirate sampled-data system �G. Suppose that the assump-

tions A1 { A5 hold. Then there exists a discrete admissible controller K which achieves the

performance bound J1(K) < if and only if the following conditions are satis�ed:

(i) There exists a bounded Tu-periodic symmetric positive semide�nite matrix function

Se(t) :=

�S S1S01 S2

�(t); t 2 [0;1) (9)

which satis�es the following periodic Riccati di�erential equation with jumps,

� _Se(t) = A0eSe(t) + Se(t)Ae + �2Se(t)B1eB

01eSe(t) +C01eC1e; t 6= kTu + �l

Se(�l) = A0d(�l)Se(�

+l)[Ad(�l) �Bd(�l)Kd(�l)] (10)

Se(kTu + �l) = Se(�l) l = 0; 1; : : :Nu � 1; k = 0; 1; : : :

where

Kd(�l) := [B0d(�l)Se(�

+l)Bd(�l)]

�1B0d(�l)Se(�

+l)Ad(�l) (11)

such that the Tu-periodic system

_xe(t) = [Ae + �2B1eB01eSe(t)]xe(t); t 6= kTu + �l

xe(kTu + �+l) = [Ad(�l)� Bd(�l)Kd(�l)]xe(kTu + �l) (12)

is exponentially stable.

(ii) There exists a bounded Ty-periodic symmetric positive semide�nite matrix function Q(�)

which satis�es the following periodic Riccati di�erential equation with jumps,

_Q(t) = AQ(t) + Q(t)A0 + �2Q(t)C01C1Q(t) +B1B01; t 6= iTy + �j

Q(�+j) = [I � Ld(�j)C2(�j)]Q(�j) (13)

Q(iTy + �j) = Q(�j) j = 0; 1; : : :Ny � 1; i = 0; 1; : : :

where

Ld(�j) := Q(�j)C02(�j)[D21(�j)D

021(�j) +C2(�j)Q(�j)C

02(�j)]

�1 (14)

6

such that the Ty-periodic system

_x(t) = (A+ �2Q(t)C01C1)x(t); t 6= iTy + �j

x(iTy + �+j) = [I � Ld(�j)C2(�j)]x(iTy + �j) (15)

is exponentially stable.

(iii)

�[S(t)Q(t)] < 2; all t 2 [0;1) (16)

where �(�) denotes the spectral radius.

Moreover, when these conditions hold, the performance bound J1(K) < is achieved by the

T -periodic controller

_x(t) = �A(t)x(t) + �B2(t)u(t) t 6= iTy + �j;

x(iTy + �j) = x(iTy + �j) + �Ld(�j)[yd(iTy + �j)�C2(�j)x(iTy + ��j)] (17)

ud(kTu + �+l) = �[H0

u(�l)S2(�l)Hu(�l)]�1H0

u(�l) [S01(�l)x(kTu + �l)

+S2(�l)[I �H(�l)]u(kTu + �l)]

where u(t) is given by (3),

�A(t) := A+ �2B1B01S(t)

�B2(t) := B2 + �2B1B01S1(t) (18)

�Ld(�j) = N (�j)C02(�j)[D21(�j)D

021(�j) + C2(�j)N (�j)C

02(�j)]

�1 (19)

and N (t) is given by

N (t) = Q(t)(I � �2S(t)Q(t))�1: (20)

Proof: The multirate sampled-data system de�ned by (4) can be regarded as a periodic

time-varying system. The multirateH1 problem can then be solved by adapting continuous-

time and discrete-time H1 theory of time-varying systems [19, 27] to this case. This leads

to a two-Riccati equation solution of the sampled-data problem. The theorem is a fairly

straightforward generalization of the corresponding single-rate result in [30, 33] and the

dual-rate result in [36].

2

The periodic Riccati di�erential equations with jumps in (10) and (13) can be discretized

to periodic Riccati di�erence equations. Similarly, the coupling condition (16) can be ex-

pressed as a discrete-time criterion. Since the sampling and hold functions may have di�er-

ent periods the discretization points of the Riccati equations involved can occur at di�erent

7

time-instants. The solution can then be expressed in terms of two discrete periodic Riccati

equations with di�erent periods.

Introduce the 2(n +m2)� 2(n+m2) matrix

�(t) :=

��11 �12

�21 �22

�(t) := exp

��

�Ae �2B1eB

01e

�C01eC1e �A0e

�t

�(21)

where Ae, B1e and C1e are de�ned by (5), and de�ne

~Ae(hl; �l) := �11(hl)�1Ad(�l)

~B1e(hl) ~B01e(hl) := � 2�11(hl)

�1�12(hl);

~B2e(hl; �l) := �11(hl)�1Bd(�l) (22)�

~C01e(hl; �l)~D012(hl; �l)

�[ ~C1e(hl; �l) ~D12(hl; �l) ] :=

�A0d(�l)

B0d(�l)

��21(hl)�11(hl)

�1 [Ad(�l) Bd(�l) ]

where hl := �l+1 � �l. Note that the structure of the matrices in (22) associated with the

augmented system (4) allows for the partition

~Ae(�; � ) =:

�~A(�) ~B2(�)0 I

�Ad(� )

~B1e(�) =:

�~B1(�)0

�(23)

~C1e(�; � ) =: [ ~C1(�) � ]

where ~A is n� n, ~B1 is n�m1 and ~C1 is p1 � n. Denote further

~C2(�) := C2(�); ~D21(�) := D21(�): (24)

It is assumed that the matrices �11(�) in (22) are invertible, which is the case if there exist

bounded solutions to (10) and (13), see [37].

Theorem 3.2 Consider the multirate sampled-data system �G. Suppose the assumptions

A1 { A5 hold. Then the conditions in Theorem 3.1 (i){(iii) are equivalent to the following

conditions:

(i) The matrices �11(hl) are invertible and there is an Nu-periodic symmetric positive

semide�nite matrix function Se(�l) satisfying the following periodic Riccati di�erence equa-

tion

Se(�l) = ~A0e(hl; �l)Se(�l+1) ~Ae(hl; �l) + ~C01e(hl; �l)

~C1e(hl; �l)�

E0S(hl; �l)M�1S

(hl; �l)ES(hl; �l) (25)

Se(kTu + �l) = Se(�l); l = Nu � 1 : : : ; 1; 0; k = 0; 1; : : :

8

where

ES(hl; �l) :=

�~B01e(hl)

~B02e(hl; �l)

�Se(�l+1) ~Ae(hl; �l) +

�0

~D012(hl; �l)

�~C1e(hl; �l) (26)

MS (hl; �l) :=

�� 2I 0

0 ~D012(hl; �l)

~D12(hl; �l)

�+

�~B01e(hl)

~B02e(hl; �l)

�Se(�l+1)

�~B01e(hl)

~B02e(hl; �l)

�0

such that

2I � ~B01(hl)S(�l+1) ~B1(hl) > 0; (27)

where S(�l) and ~B1(hl) are given by the partitions in (9) and (23), respectively, and the

discrete Nu-periodic system

xe(kTu + �l+1) = Acl(hl; �l)xe(kTu + �l) (28)

where

Acl(hl; �l) := ~Ae(hl; �l)� [ ~B1e(hl) ~B2e(hl; �l) ]MS(hl; �l)�1ES(hl; �l) (29)

is exponentially stable.

(ii) The matrices �11(sj), where sj := �j+1� �j, are invertible and there is an Ny-periodic

symmetric positive semide�nite matrix function Q(�j) satisfying the following periodic Ric-

cati di�erence equation

Q(�j+1) = ~A(sj)Q(�j) ~A0(sj)+ ~B1(sj) ~B

01(sj)�EQ(sj ; �j)M

�1Q

(sj ; �j)E0Q(sj ; �j)

Q(iTy + �j) = Q(�j); j = 0; 1; : : :Ny � 1; i = 0; 1; : : : (30)

where

EQ(sj ; �j) := ~A(sj)Q(�j) [ ~C01(sj)

~C02(�j) ] (31)

MQ(sj ; �j) :=

�� 2I 0

0 ~D21(�j) ~D021(�j)

�+

�~C1(sj)~C2(�j)

�Q(�j)

�~C1(sj)~C2(�j)

�0

such that

2I � ~C1(sj)[I � Ld(�j) ~C2(�j)]Q(�j) ~C01(sj) > 0 (32)

where Ld(�) is given by (14) and the discrete Ny-periodic system

x(iTy + �j+1) = [ ~A(�j) �EQ(sj ; �j)M�1Q

(sj ; �j)

�~C1(sj)~C2(�j)

�]x(iTy + �j) (33)

is exponentially stable.

(iii)

�[S(t)Q(t)] < 2; for t = iTy + �j; such that t � T;

j = 0; 1; : : : ; Ny � 1; i = 0; 1; : : : (34)

9

where S(t) is given by (10) and T is the period of the multirate system �G. Moreover, when

these conditions hold, the performance bound J1(K) < is achieved by the controller in

(17).

Proof: For the single-rate case, a similar discretization of the equations corresponding to

(10) and (13) was done in [29, 37] and the proof is therefore not given in its full length here.

The most obvious di�erence is the periodic time-varying nature of the resulting discrete-

time equations (25) and (30). By the analysis in [29, 37], the existence of bounded solutions

to (10) and (13) between the discretization points is equivalent to the conditions (27) and

(32), respectively, and the stability criteria (12) and (15) are equivalent to the discrete-time

criteria in (28) and (33), respectively.

An important step is the result that the coupling condition (16) can be discretized too.

The discrete-time coupling condition is obtained from the observation that the quantity

�[S(t)Q(t)] � 2 can change sign at the discontinuous jumps only, i.e. at the instants f�lg

and f�jg [29, 37]. From the equations describing the discontinuities in (10) and (13) we know

that S(�+l) � S(�l) and Q(�j) � Q(�+

j). By the property of the spectral radius function,

�(UV ) � �(ZV ) i� U � Z, where U;Z; V are positive de�nite matrices of compatible

dimensions, it then follows that if �(S(�j)Q(�j)) � 2 then �(S(t)Q(t)) � 2 is true in the

whole interval t 2 (�j�1; �j]. Thus, also the coupling condition can be discretized. The

periodicity of the Riccati equations then give the set of coupling conditions (34) . 2

Remark 3.1 Observe that S(t) in (34) is obtained from the matrix exponential (21)

Se(t) = [�21(�l � t) + �22(�l � t)Se(�l)] [�11(�l � t) + �12(�l � t)Se(�l)]�1

;

�l�1 < t � �l (35)

and the partition in (9).

Remark 3.2 An equivalent expression for the coupling condition (34) can be stated in terms

of the periodicity of Se(�):

�[S(t)Q(t)] < 2; for t = kTu + �+l; such that t � T;

k = 0; 1; : : : ; Nu � 1; l = 0; 1; : : : (36)

where S(t) and Q(t) are given by (10) and (13), respectively.

Remark 3.3 The controller (17) can be discretized by observing that the state transition

matrix � �A(�; �) associated with the matrix �A(�) and the matrix �(t2; t1) :=R t2t1�(t2; �) �B2(�)d�

can be obtained through the matrix exponential in equation (21) [36].

10

Remark 3.4 Observe that no separate causality constraint problem arises in this solution,

as opposed to the lifting solution [10, 39]. The controller (17) is periodically time-varying. A

lifted time-invariant controller description can be obtained by lifting the periodic controller

(17).

The periodic Riccati equations in (10) and (13) are discretized at those points only where

a discrete event actually occurs. An alternative, and perhaps more common formulation, is

to de�ne a small time interval, say h, which is a fraction of all sampling and hold intervals.

This way of discretizing would result in two discrete Riccati equations with the same period.

Further, the coupling condition would be expressed in terms of the discrete periodic solutions

instead of the expressions (34) or (36). We would also need one matrix exponentiation only

in order to obtain the matrices involved in the discrete Riccati equations. This way of

discretizing may, however, lead to a very long period of the periodic equations, since they

may be discretized at times where no action actually occurs. In particular, a common

sampling-time fraction may lead to a very long period for systems with non-synchronized

sampling, i.e. when there are discrete time actions both at times ft+kTg and ft+kT+�g for

some � < T . The present approach is also applicable to the case when � is irrational. This

case is not possible to solve using a constant discretization step. Our problem formulation

is stated in order to achieve the shortest possible period of the periodic Riccati di�erence

equations in (25) and (30), with the possible expense of more matrix exponentiations. The

alternative formulation with one single sampling-time fraction can be regarded as a special

case of this description.

The following theorem shows that the periodic Riccati di�erence equations in (25) and

(30) can be solved via standard discrete algebraic Riccati equations. See also [21] for an

alternative generation of an algebraic Riccati equation.

Theorem 3.3 Assume that there exists a bounded stabilizing positive semide�nite matrix

function P (�) satisfying the following discrete N -periodic Riccati equation

P (tk+1) = A(tk)P (tk)A0(tk) +R1(tk)� [A(tk)P (tk)C

0(tk) + R12(tk)]

� [C(tk)P (tk)C0(tk) +R2(tk)]

�1[C(tk)P (tk)A

0(tk) +R012(tk)] (37)

A(tk) = A(tk+N ); C(tk) = C(tk+N ); Ri(tk) = Ri(tk+N )

k = 0; 1; 2; : : :

Then there exists a solution P := P (tN ) to the following algebraic Riccati equation

P = �AP �A0 + �R1 ���AP �C0 + �R12

� ��CP �C0 + �R2

��1 � �CP �A0 + �R012�

(38)

where�A := AN ; �C := CN

�R1 := R1;N ; �R12 := R12;N ; �R2 := R2;N ;(39)

11

where the matrices are de�ned by the following recursive equations

Ak := A(tk)Ak�1; A0 := A(t0);

Ck :=

�Ck�1

C(tk)Ak�1

�; C0 := C(t0);

R1;k := A(tk)R1;k�1A0(tk) +R1(tk); R1;0 := R1(t0); (40)

R12;k := [A(tk)R12;k�1; A(tk)R1;k�1C0(tk) + R12(tk)] ; R12;0 := R12(t0)

R2;k :=

�R2;k�1; R12;k�1C

0(tk)

C(tk)R12;k�1 C(tk)R1;k�1C0(tk) +R2(tk)

�; R2;0 := R2(t0):

Proof: The result is obtained by induction, cf. [7, 12]. 2

By Theorem 3.3, we can obtain the solutions to the periodic Riccati equations (25) and

(30) at one point during their periods by solving algebraic Riccati-equations, to which there

exist algorithms with known convergence properties. The other solution points to (25) and

(30) at the remaining time instants can then be determined using the original equations, or

from the recursion

P (tk+1) = AkPA0k +R1;k � [AkPC

0k + R12;k][CkPC

0k +R2;k]

�1[CkPA0k + R012;k]: (41)

The solution to the multirate sampled-data H1 problem can thus be stated in terms of a

pair of standard discrete algebraic Riccati equations together with the set of non-standard

positive de�niteness criteria (27), (32) and the set of coupling conditions (34).

4 A lifting based solution

Periodic multirate control problems can be solved by the lifting technique, in which the

system is represented by a time-invariant, lifted system. The lifted system has as its inputs

and outputs the inputs and outputs during one period of the periodic system. The controller

can then be designed for the time-invariant lifted system. However, the requirement that

the controller should represent a causal periodic controller imposes conditions on the direct

coupling term of the lifted controller. Due to these causality constraints, standard H2 and

H1 synthesis methods are not directly applicable to the problem of designing controllers

for periodic systems via lifting. Various special methods have therefore been studied for the

problem of designing optimal controllers which satisfy a causality constraint [10, 26, 39, 40].

In this section a two-Riccati equation solution is presented to the H1 problem for lifted

systems with controller causality constraints. The result is closely related to the Riccati-

equation based solution given in Section 3, and thus it also gives insight into the connection

between the two solution procedures. A particular feature of the approach presented here

is that lifting is applied separately to the periodic state feedback and estimation problems,

which need not have the same periods.

12

In this section, it is assumed that all hold elements are synchronized, i.e. there exist times

kTu such that the function H(kTu) = I in (3). This enables us to formulate a particularly

simple solution of the associated state-feedback problem.

4.1 A multirate state-feedback problem

Introduce the �nite-dimensionalNu-periodic discrete system associated with the multirate

system (4), cf. [10, 39],

~xe(kNu + l + 1) = ~Ae(hl; �l)~xe(kNu + l) + ~B1e(hl) ~wu(kNu + l)

+ ~B2e(hl; �l)~ud(kNu + l)

~zu(kNu + l) = ~C1e(hl; �l)~xe(kNu + l) + ~D12(hl; �l)~ud(kNu + l); (42)

l = 0; 1; : : :Nu � 1; k = 0; 1; : : :

where hl = �l+1 � �l, ~ud(kNu + l) := ud(kTu + �+l), ~xe(kNu + l) := [~x0u(kNu + l) u0(kTu +

�l)]0 := xe(kTu + �l) and ~wu(�) denotes a discrete-time disturbance in l2. The system

matrices are given by (22). The system (42) is the equivalent discrete representation of (1)

and (3) obtained by applying lifting to describe the sampled-data system in the intervals

[kTu + �l; kTu + �l+1] [5, 15, 36]. Thus, a sampled state-feedback controller K stabilizes the

sampled-data system (4) and achieves the H1-type performance bound J1(K) < if and

and only if it stabilizes (42) and the l2-induced norm from ~wu to ~zu is less than .

In this section we apply the lifting technique to solve the problem of �nding a causal

stabilizing Nu-periodic state-feedback controller for the periodic system (42) such that the

closed-loop system has the l2-induced norm from ~wu to ~zu less than . Motivated by the

multirate sampled-data problem, we consider the periodic discrete state-feedback problem

only. However, the multirate full-information problem can be solved in a similar way.

The Nu-periodic system (42) has the state ~xe(�). However, as the lower part of the state

~xe(�) consists of the control inputs, which are known to the controller, any Nu-periodic

state-feedback law ~ud = Ke~xe can be written as

~ud = K~xu; (43)

where K is a dynamic Nu-periodic controller, which includes the lower part of ~xe(�) as part

of its state.

Introduce the lifted signals

�wu(k) := [ ~w0u(kNu) ~w0

u(kNu + 1) � � � ~w0

u(kNu + Nu � 1) ]

0

�ud(k) := [ ~u0d(kNu) ~u0

d(kNu + 1) � � � ~u0

d(kNu + Nu � 1) ]

0(44)

�zu(k) := [ ~z0u(kNu) ~z0u(kNu + 1) � � � ~z0u(kNu +Nu � 1) ]0:

13

Assuming that H(kTu) = I, the associated time-invariant lifted system is given by

�xu(k + 1) = Au�xu(k) + B1;u �wu(k) + B2;u�ud(k)

�zu(k) = C1;u�xu(k) +D11;u �wu(k) +D12;u�ud(k) (45)

where �xu(k) := ~x(kNu) and

Au := ~A(Tu)

B1;u := [ ~A(�Nu� �1) ~B1(h0) ~A(�Nu

� �2) ~B1(h1) � � � ~B1(hNu�1) ]

B2;u := [ ~A(�Nu� �1) ~B2(h0)Hu(�0) ~A(�Nu

� �2) ~B2(h1)Hu(�1) � � � (46)

� � � ~B2(hNu�1)Hu(�Nu�1) ]

C1;u :=

2664

~C1(h0)~C1(h1) ~A(�1 � �0)

...~C1(hNu�1)

~A(�Nu�1 � �0)

3775

D11;u := [D11;u(p; q)]; p; q = 0; 1; : : : ; Nu � 1

D12;u := [D12;u(p; q)]; p; q = 0; 1; : : : ; Nu � 1

where

D11;u(p; q) :=

�0; if p � q~C1(hp) ~A(�p � �q+1) ~B1(hq); if p > q

D12;u(p; q) :=

8<:0; if p < q~D12(hp; �p); if p = q~C1(hp) ~A(�p � �q+1) ~B2(hq)Hu(�q); if p > q

(47)

and ~A(�), ~B1(�), ~B2(�), ~C1(�) and ~D12(�; �) are de�ned by (22) and (23). Note that for p > q,

~A(�p � �q) = ~A(hp�1) ~A(hp�2) � � � ~A(hq): (48)

Notice that the lifted system (45) has an n-dimensional state vector. This is due to the

assumption H(kTu) = I, i.e., that all inputs are updated at time instants kTu. If the state

of the lifted systems were de�ned at time instants where only part of the inputs are subject

to change, then the state of the lifted system would include those elements of u which are

not updated at these time instants, cf. the augmented state in (4).

From the construction of the lifted system (45) it follows that an Nu-periodic state-

feedback controller K for (42) can be characterized as a time-invariant full-information con-

troller KL := [Kx Kw] : l2 � l2 ! l2 for (45), such that

�ud(k) = ([Kx Kw ]

��xu�wu

�)(k) :=

264

(K~xu)(kNu)...

(K~xu)(kNu + Nu � 1)

375 (49)

14

By construction, the controller (43) stabilizes the periodic system (42) if and only if the

controller (49) stabilizes (45), and the l2-induced norm of the closed-loop system (42), (43)

is equal to the H1 norm of the closed loop (45), (49).

The problem of constructing a stabilizing periodic state-feedback controller for (42) which

achieves a speci�ed l2-induced norm bound can thus be reduced to an H1-optimal full-

information control problem for the time-invariant lifted system (45). However, the equiva-

lent H1 problem de�ned for (45) is not a standard H1 problem, because the requirement

that (43) should be causal imposes corresponding causality constraints on the structure of

the lifted controller in (49). In particular, partitioning Kw in compliance with �ud and �wd,

the direct coupling term of Kw should have zeros on and above the main block diagonal.

For future reference, it is convenient to introduce a connection between the Riccati-

equation solution in Section 3 and the lifted time-invariant system (45).

Theorem 4.1 Consider the multirate system (4) and the associated discrete Nu-periodic

Riccati equation (25). Let Hu(kTu) = I. Then

Se(kTu) =

�S 0

0 0

�(50)

where S is the symmetric positive semide�nite solution of the discrete algebraic Riccati

equation

S = A0uSAu + C01;uC1;u � E0

SM�1SES (51)

where

ES :=

�B01;uB02;u

�SAu +

�D0

11;u

D012;u

�C1;u

MS :=

�B01;uB02;u

�S

�B01;uB02;u

�0+

�� 2I +D0

11;uD11;u D011;uD12;u

D012;uD11;u D0

12;uD12;u

�(52)

Proof: The result follows from (25), Theorem 3.3 and the structures of the matrices in (22).

2

The following theorem gives a state-space solution to the causally constrained lifted full-

informationH1 problem associated with the multirateH1-optimal state-feedback problem.

Theorem 4.2 Consider the lifted system (44), (45) associated with the Nu-periodic system

(42). There exists a stabilizing full-information controller KL described by (49) such that

the associated Nu-periodic controller K is causal and such that the closed-loop system (45),

(49) has H1-norm less if and only if there exists a stabilizing solution S to the discrete

algebraic Riccati equation (51) such that

2I � ~B01(hl)S(�l+1) ~B1(hl) > 0; l = Nu � 1; : : : ; 0 (53)

15

where the matrices fS(�l)g are de�ned recursively according to (25) with the partition in (9),

and

Se(�Nu) :=

�S 0

0 0

�: (54)

In this case, a stabilizing causal Nu-periodic state-feedback controller K which achieves the

norm bound is given by

~ud(kNu + l) = �V �1S

(hl; �l)[ ~B02e(hl; �l)S

�e(�l) ~Ae(hl; �l)

+ ~D012(hl; �l)

~C1e(hl; �l)]~xe(kNu + l) (55)

l = 0; 1; : : :Nu � 1; k = 0; 1; : : :

where ~xe(kNu + l) = [ ~x0u(kNu + l) u0(kNu + l) ],

S�e (�l) := Se(�l)[I � �2 ~B1e(hl) ~B01e(hl)Se(�l)]

�1; (56)

and

VS(hl; �l) := ~B02e(hl; �l)S�e (�l)

~B2e(hl; �l) + ~D012(hl; �l)

~D12(hl; �l): (57)

Proof: By standard discrete H1 control theory, a necessary condition for the existence of

a full-information controller KL which achieves the H1 performance bound for the system

(45) is that the discrete algebraic Riccati equation (51) has a stabilizing symmetric positive

semide�nite solution. In this case, with �xu(0) = 0, we have the expansion [14]

k�zuk2l2� 2k �wuk

2l2=

1Xk=0

Lk( �wu(k); �ud(k)) (58)

where

Lk( �wu(k); �ud(k)) :=

��wu � �w0

u

�ud � �u0d

�0(k)MS

��wu � �w0

u

�ud � �u0d

�(k) (59)

and ��w0u

�u0d

�(k) := �M�1

SES�xu(k) (60)

Moreover, the contribution to the quadratic cost (58) from the discrete time instant k can

be expressed as

�z0u(k)�zu(k) � 2 �w0u(k) �wu(k) = Lk( �wu(k); �ud(k)) + �x0u(k)S�xu(k)

��x0u(k + 1)S�xu(k + 1) (61)

From the de�nitions of the signals �zu(k) and �wu(k), and the system equation (42), which

describes the system behavior between the discrete time instants kNu and kNu + Nu, we

have similarly the expansion

16

�z0u(k)�zu(k) � 2 �w0u(k) �wu(k) =

Nu�1Xl=0

[~z0u(kNu + l)~zu(kNu + l)� 2 ~w0u(kNu + l) ~wu(kNu + l)]

=

Nu�1Xl=0

Il( ~wu(kNu + l); ~ud(kNu + l)) + ~x0u(kNu)S~xu(kNu)

�~x0u(kNu + Nu)S~xu(kNu + Nu) (62)

where

Il( ~wu; ~ud) :=

�~wu � ~w0

u

~ud � ~u0d

�0MS (hl; �l)

�~wu � ~w0

u

~ud � ~u0d

�(63)

and �~w0u

~u0d

�(kNu + l) := �MS (hl; �l)

�1ES(hl; �l)~xe(kNu + l) (64)

The result of the theorem then follows from (58),(61) and (62) by standard discrete H1

control theory [4, 14, 31]. 2

Remark 4.1 Notice that the solution of the causally constrained H1 state-feedback problem

is obtained in terms of the same algebraic Riccati equation (51) which is associated with the

lifted system (45) and H1 problem without causality constraints. The constraints appear

only in the nonstandard form of the positive de�niteness conditions (53) and the controller

structure (55).

Remark 4.2 Theorem 4.2 gives a causal Nu-periodic controller which achieves the spec-

i�ed l2-induced norm bound for the system (42). The corresponding lifted time-invariant

controller KL in (49) can be recovered by applying lifting to the periodic controller (55). It

is easy to see that the lifted version of (55) is a static full-information controller of the form

KL = [Kx Kw], where Kw has zeros on and above the main block diagonal.

4.2 A multirate estimation problem

An analogous dual result is obtained for the multirate estimation problem. De�ne the �nite-

dimensional Ny-periodic discrete-time system associated with the multirate system (4),

~xy(iNy + j + 1) = ~A(sj)~xy(iNy + j) + ~B1(sj) ~wy(iNy + j)

~zy(iNy + j) = ~C1(sj)~xy(iNy + j) (65)

~yd(iNy + j) = ~C2(�j)~xy(iNy + j) + ~D21(�j)~vd(iNy + j)

j = 0; 1; : : :Ny � 1 i = 0; 1; : : :

17

where sj := �j+1��j, ~xy(iNy+j) := x(iTy+�j), ~yd(iNy+j) := yd(iTy+�j), ~vd(iNy+j) :=

vd(iTy+�j) and ~wy(�) denotes a discrete disturbance in l2. The system matrices are de�ned

by (23) and (24). The multirate H1-optimal estimation problem consists of �nding an

Ny-periodic stable causal estimator

zy = F~yd (66)

which achieves the performance bound

sup( ~wy;~vd)6=0

k~zy � zykl2[k ~wyk2l2 + k~vdk

2l2]1=2

< (67)

The time-invariant system obtained by applying lifting to (65) over the period Ny of the

sampling function is given by

�xy(i + 1) = Ay�xy(i) +B1;y �w(i)

�zy(i) = C1;y�xy(i) +D11;y �w(i) (68)

�yd(i) = C2;y�xy(i) +D21;y �w(i)

where we have de�ned �xy(i) := ~xy(iNy), ~w0 := [ ~w0y ~v0

d]0; and the lifted signals

�w(i) := [ ~w0(iNy) ~w0(iNy + 1) � � � ~w0(iNy + Ny � 1) ]0

�zy(i) := [ ~z0y(iNy) ~z0

y(iNy + 1) � � � ~z0

y(iNy +Ny � 1) ]

0(69)

�yd(i) := [ ~y0d(iNy) ~y0

d(iNy + 1) � � � ~y0

d(iNy +Ny � 1) ]

0:

The system-matrices of the lifted system are given by

Ay := ~A(Ty)

B1;y := [ ~A(�Ny� �1) �B1(s0) ~A(�Ny

� �2) �B1(s1) � � � �B1(sNy�1) ]

C1;y :=

2664

~C1(s0)~C1(s1) ~A(�1 � �0)

...~C1(sNy�1)

~A(�Ny�1 � �0)

3775

C2;y :=

2664

~C2(s0)~C2(s1) ~A(�1 � �0)

...~C2(sNy�1)

~A(�Ny�1 � �0)

3775 (70)

D11;y := [D11;y(p; q)]; p; q = 0; 1; : : :; Ny � 1

D21;y := [D21;y(p; q)]; p; q = 0; 1; : : :; Ny � 1

where

D11;y(p; q) :=

�0; if p � q~C1(sp) ~A(�p � �q+1) �B1(sq); if p > q

D21;y(p; q) :=

8<:0; if p < q�D21(sp); if p = q~C2(sp) ~A(�p � �q+1) �B1(sq); if p > q

(71)

18

where we have introduced ��B1(sj)�D21(�j)

�:=

�~B1(sj) 0

0 ~D21(�j)

�: (72)

The Ny-periodic estimator (66) is equivalent to a time-invariant lifted estimator FL : l2 !

l2 described by

�zy(i) = (FL�yd)(i) :=

264

(F ~yd)(iNy)...

(F ~yd)(iNy +Ny � 1)

375 : (73)

Thus, FL achieves the performance bound

sup�w6=0

k�zy � �zykl2k �wkl2

< (74)

for the system (68) if and only if F achieves the bound (67) for the system (65). The

multirateH1-optimal estimation problem can thus be described in terms of a time-invariant

H1-optimal estimation problem for the lifted system (68) subject to a causality constraint

imposed by the causality of (66) and the structure of (73). More speci�cally, the direct

coupling term of FL is required to have a lower block-triangular structure.

The following auxiliary result gives the solution to the periodic Riccati equation (30) in

terms of an algebraic Riccati equation associated with the lifted system (68).

Theorem 4.3 Consider the multirate system (4) and the associated discrete Ny-periodic

Riccati equation (30). Then Q := Q(iTy+�0) is the symmetric positive semide�nite solution

of the discrete algebraic Riccati equation

Q = AyQA0y+ B1;yB

01;y �EQM

�1QE0Q

(75)

where

EQ := AyQ

�C1;y

C2;y

�0+B1;y

�D11;y

D12;y

�0

MQ :=

�C1;y

C2;y

�Q [C01;y C02;y ] +

�� 2I +D11;yD

011;y D11;yD

021;y

D21;yD011;y D21;yD

021;y

�: (76)

The following theorem gives a state-space solution to the causally constrained liftedH1 es-

timation problem associated with the multirate H1-optimal estimation problem.

Theorem 4.4 Consider the lifted system described by equations (68) and (69) associated

with the Ny-periodic system (65). There exists a stable estimator FL described by (73) such

that the associated Ny-periodic estimator F is causal and such that the H1 performance

19

bound (74) is achieved if and only if the there exists a stabilizing solution Q to the discrete

algebraic Riccati equation (75) such that

2I � ~C1(sj)[I � Ld(�j) ~C2(�j)]Q(�j) ~C01(sj) > 0 j = 0; 1; : : : ; Ny � 1 (77)

where Ld is given by (14) and the matrices fQ(�j)g are de�ned recursively according to (30)

with

Q(�0) = Q: (78)

In this case, a stable causal Ny-periodic estimator F which achieves the norm bound (67) is

given by

xy(iNy + j + 1�) = ~A(sj)xy(iNy + j) (79)

xy(iNy + j) = xy(iNy + j�) + Ld(�j)[yd(iTy + sj) � ~C2(�j)xy(iNy + j�)]

zy(iNy + j) = ~C1(sj )xy(iNy + j)

Proof: The proof is analogous to the proof of Theorem 4.2.

In analogy with the state-feedback problem, the time-invariant lifted estimator FL asso-

ciated with (79) can be determined by applying lifting to the periodic estimator (79).

4.3 The multirate output-feedback problem

The causally constrained lifted state-feedback controller and estimator results of Theorems

4.2 and 4.4 can be combined to provide a lifting-based solution of the multirate output

feedback problem. Although the result can be stated for general periodic sampling and hold

functions, it will for simplicity be assumed that the sampling and hold functions operate

synchronously.

Theorem 4.5 Consider the multirate sampled-data system described by (1){(3). Assume

that the sampling and hold functions operate synchronously, i.e. Nu = Ny =: N and �l = �l,

l = 0; 1; : : : ; N . There exists a causal N -periodic controller K which stabilizes the sampled-

data system and which achieves the performance bound J (K) < if and only if the following

conditions are satis�ed:

(a) The conditions of Theorem 4.2 are satis�ed,

(b) the conditions of Theorem 4.4 are satis�ed, and

(c) �[S(�l)Q(�l)] < , l = 0; 1; : : : ; N � 1.

20

Moreover, when conditions (a){(c) hold, a stabilizing multirate controller which achieves

the H1 performance bound is given by

xe(kN + l + 1�) = Acl(hl; �l)xe(kN + l)

xe(kN + l) = xe(kN + l�)

+

�I

0

��Ld(�l)[~yd(kN + l)� C2d(�l)xe(kN + l�)]

~ud(kN + l) = �V �1S

(hl; �l)[ ~B02e(hl; �l)S

�e(�l+1) ~Ae(hl; �l)

+ ~D012(hl; �l)

~C1e(hl; �l)]xe(kNu + l) (80)

where Acl(�; �) is de�ned in (29), VS (�; �) is given by (57), and �Ld(�) is given by (19).

Proof: The �rst part of the proof follows directly from Theorems 4.1 { 4.4 and standard

H1 control theory. The equivalence of the controllers (17) and (80) can be shown in analogy

with the single-rate case [28, 37]. 2

Remark 4.3 Notice that Theorem 4.5 gives the multirateH1-optimal controller in terms of

the discrete algebraic Riccati equations of Theorems 4.1 and 4.3, which are the Riccati equa-

tions of the standard H1 problem associated with the lifted system. The causality relations

of the multirate system appear in the solution in the form of the nonstandard positive de�-

niteness conditions (53) and (77) and the coupling conditions in Theorem 4.5 (c). Theorem

4.5 gives a periodic controller which achieves the H1 performance bound for the multirate

system. The corresponding time-invariant lifted controller can be determined by lifting the

periodic controller (80) over the period N .

5 Example

The following simple numerical example is used to illustrate some of the results in this paper.

Consider the system

_x(t) = �0:8x(t) + w(t) + 2u(t)

z(t) =

�1

0

�x(t) +

�0

1

�u(t): (81)

Assume that there are two measurement devices that give sampled data with two di�erent

rates. Both devices measure the state, but the more frequently available measurement is

corrupted by a larger disturbance

yd1(is) = x(is) + 0:7vd1(is); i = 0; 1; 2; : : :

y2d(js) = x(js) + 0:1v2d(js); j = 0; 2; 4; : : : (82)

21

where s = 1:5, such that yd(is) = [yd1(is) y2d(is)]0 for i = 0; 2; 4; : : : and yd(is) = yd1(is) for

i = 1; 3; 5; : : :. The control signal

u(t) = ud(kh+); t 2 (kh; kh+ h] ; k = 0; 1; 2; : : : (83)

is updated with the period h = 2. The sampling function has the period Ty = 3 and the

single-rate hold function has the period Tu = 2. The period of the hybrid multirate system

de�ned by (81){(83) is T = 6.

By separate discretization of the feedback and the estimation problems we obtain a

time-invariant discrete Riccati-equation (25) associated with the feedback problem and a

2-periodic discrete Riccati-equation (30) associated with state estimation, cf. Theorem 4.3.

The latter one can further be lifted to a time-invariant equation. Direct application of the

lifting technique in order to construct a time-invariant description of the hybrid system

(81){(83) with a constant discretization interval would require the interval to be 0:5, and

the associated periodic discrete Riccati-equations would have the period 12.

The solutions S(t) and Q(t) to (10) and (13), respectively, together with the spectral radius

condition (16) are illustrated in Figure 1. Note that although the spectral radius condition

must be satis�ed for all times, it is su�cient to check the coupling condition at three points

during the period T , namely at fkh+; k = 0; 1; 2g, cf. Remark 3.2. Alternatively, we

may apply (34) which gives four coupling conditions at fis; i = 0; 1; 2; 3g. The optimal

cost is inf = 0:971 and the solutions to the discrete Riccati equations (25) and (30) are

S(kTu) = 0:507, Q(iTy + s) = 0:810 and Q(iTy) = 0:971.

6 Conclusion

A state-space solution to the multirate sampled-data H1 problem has been derived. The

solution is expressed in terms of a pair of discrete algebraic Riccati-equations together with a

set of matrix positive-de�niteness conditions and a set of coupling constraints. The result has

been derived both through a direct Riccati-equation approach and via the lifting technique.

Both methods generally lead to the same equations and the same controller. Although this

paper has focused on the H1 problem, a similar relationship between lifting and a two-

Riccati solution can be shown to hold in the multirate Linear Quadratic/H2 problem as

well.

The results of this paper provide a closed state-space solution to the lifted time-invariant

H1 problem with causality constraints. A particular feature of the solution is that the

algebraic Riccati equations are the same as those associated with the unconstrained lifted

H1 problem. The causality constraint of the multirate problem appear here as a set of

additional, nonstandard positive-de�niteness constraints and a set of coupling conditions.

22

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25

0 1 2 3 4 5 60

1

2

t

S(t)

0 1 2 3 4 5 60

0.5

1

t

Q(t)

0 1 2 3 4 5 60

0.5

1

t

S(t)Q(t)

γinf2

Figure 1: The solutions S(t), Q(t), and the coupling condition S(t)Q(t) for = inf over

one period [0; T ] of the multirate system (81) { (83).

26