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H2 and H∞ Design of Sampled-Data Systems Using Lifting

Part I: General Framework and Solutions

Leonid Mirkin∗

Faculty of Mechanical Eng.Hector Rotstein†

Faculty of Electrical Eng.

Zalman J. Palmor‡

Faculty of Mechanical Eng.

Technion — Israel Institute of TechnologyHaifa 32 000, Israel

September 28, 1997

Abstract

This paper presents a complete solution to the H2 and H∞ problems for sampled-datasystems. As opposed to previous work in the area, it is assumed here that all or some of thesampling function, the discrete-time controller and the hold function are available for design.

The solution is obtained by transforming the problem to discrete-time using the well-known lifting technique. It is then shown that the desired components of the sampled-datacontroller can be “peeled-off” from the inherently infinite dimensional description in thelifted-domain. The procedure for doing this last step is central to the approach in the paper.

Both new and revised solutions are presented in the paper. The solution to the H2 prob-lems is completely new. The solution to the H∞ problems is presented in a unifying frame-work, and is more transparent than the previous existing in the literature. Transparency paysin the form of clearer results. In particular, a separation structure is established between thedesign of (sub)optimal hold and sampler.

1 Introduction

Since the early 90th, much attention has been paid to H2 and H∞ optimal control of continuous-time systems using sampled-data controller, that is a controller implemented by a digital com-puter connected with a plant via an A/D (sampler) and a D/A (hold) converters; see [17, 5, 30,8, 16, 27, 4, 25, 29] for an introduction to the subject and pointers to relevant literature. Ina great majority of these works, however, only the discrete part of the sampled-data controlleris designed, while the sampler and the hold are selected without taking into consideration theplant dynamics or the control objectives. In most cases, an approximate “ideal” sampler isused to obtain the discrete time measurements, while zero or first order holds are the devicesof choice for converting the output of the discrete part of the controller to a continuous timecontrol signal. The advantages of having fixed sampler and hold are numerous: the behavior ofthis devices is fairly well understood and hence can be incorporated (at least heuristically) in the∗Phone: +972 (4) 8293149; Fax: +972 (4) 8324533; E-mail: [email protected]†Phone: +972 (4) 8294642; Fax: +972 (4) 8323041; E-mail: [email protected]‡Phone: +972 (4) 8292086; Fax: +972 (4) 8324533; E-mail: [email protected]

1

2 Faculty of Mechanical Engineering; Technion— IIT

controller design process, the design of hardware is considerably simplified, adequate simulationtools exist, etc. At the same time, fixing the sampler and hold devices irrespectively of controlconsiderations may limit the closed-loop performance, especially when the sampling rate is not“fast” enough with respect to the plant dynamics.

Numerous attempts to incorporate the sampler and, especially, the hold into the designprocess have been made so far basically from the discrete-time performance point of view, see[15, 1]. Loosely speaking, these designs are based on the open-loop compensation of undesirableplant dynamics (e.g., nonminimum-phase zeros) and manage to achieve significant improvementsof discrete-time performance, usually at the expense of a poor inter-sampling behavior. As aconsequence, these works gave rise to criticism [10, 7] and the suspicion that, for instance, ageneralized hold device will generically exhibit undesirable robustness properties. It becomesapparent that the design of sampling and/or hold devices should take into account inter-samplingbehavior, even when sampling is fast.

The design of A/D and D/A converters on the basis of continuous-time performance isconnected with serious technical difficulties. It is not surprising then, that only few results inthis direction are available in the literature. For instance, LQG design of the hold function wasdiscussed by Juan and Kabamba [14] (see also [12]) assuming that the discrete-time part of thecontroller is fixed. The necessary conditions for optimality of the hold function derived in [14] arequite involved and apply only when rather restrictive constraints (fixed monodromy) on the holdfunction are imposed. The design of the H∞ suboptimal hold function is considered by Basar [2]for the state-feedback and by Sun et al. [26] for the output feedback cases. In a remarkable work,Tadmor [27] treated the sampled-data H∞ problem in a general setting, including the design ofsampling and/or hold functions. In a more restrictive setting, [22] proposed a simple approachto the H∞ control when both sampler and hold are design parameters. In all the H∞ resultsabove explicit formulae for the suboptimal sampler and/or hold are obtained. Finally, Bamieh[3] proposed an approximate solution to the ‘1 design of the sampling and hold functions.

An important break-through in the treatment of sampled-data systems was the introductionof “lifting,” an operation that reduces the time-varying sampled data problem into a time-invariant, albeit inherently infinite dimensional, discrete-time one. The idea is conceptuallysimple and involves three steps: i) lift the problem to the discrete-time; ii) solve the resultingformal discrete-time problem in the so-called lifted domain; and iii) “peel-off” the result backto continuous-time. The complexity of dealing with systems in the lifted domain has preventedso-far finding a complete solution to the relevant design problems. The main contribution of thecurrent research is to show how the three design steps can effectively be carried out. In orderto do this, a better understanding of the lifted domain and its connection with continuous-time is required. In order to streamline the presentation, we have divided the paper into twoparts. In the first one, the framework is described and the main solutions are presented, basedon the technical results derived in the second part. The second paper presents some technicaldevelopments which are relevant for the problems under consideration, but have also independentinterest. For clarity, we have tried to make the two parts as self-contained as possible. Webelieve that with this structure we have achieved an adequate tradeoff between readability ofthe material and completeness.

This paper is organized as follows. In Section 2 the problems to be considered throughoutthe paper are formulated. The next two sections are devoted to the solutions of these problemsin the lifted domain. In particular, in Section 3 it is shown how a wide class of sampled-dataproblems can be reduced to a unified “standard problem” in the lifted domain, while in Section 4the lifted solutions to the sampled-data H2 (§4.1) and H∞ (§4.2) problems are presented andsome interesting properties of these solutions are discussed (§4.3). The lifted solutions are

Technical Report TME– 450; Sep 97 3

then “peeled-off” (based on the technical results obtained in the companion paper [23]) inSection 5, which contains complete solutions to the sampled-dataH2 andH∞ problems, includingthe (sub)optimal sampling and hold. In Section 6 various properties and interpretations ofthe optimal sampling and hold functions are discussed. In §6.1 some qualitative conjecturesconcerning the control oriented design of the D/A converters are also presented.

1.1 Notation

The notation throughout the paper is as follows. As usually, C− and D state for the openleft half plane and the open unit disc, respectively. Rn denotes the n-dimensional Euclideanspace, and L2n[0; h] denotes the (Hilbert) space of square integrable Rn-valued functions on theinterval [0; h]. When the dimensions are irrelevant the dimension index is dropped, by simplywriting R and L2[0; h]. For operators on R ⊕ L2[0; h], ‖·‖2 denotes the induced operator norm,while ‖·‖HS— the Hilbert-Schmidt operator norm. M ′ means the transpose of a matrix M andO∗— the adjoint of a Hilbert space operator O. The notations �(M), �(M) and �(M) statefor the spectrum, the spectral radius and the maximum singular value of a square matrix M,respectively. O1/2means the square root of O = O∗ ≥ 0.

A “bar” above a variable (�) denotes discrete-time signals in Rn, while “breve” (�) — denotesdiscrete-time signals in the lifted domain. Also, we put forward the following operator nota-tion which improves the readability of formulae when both finite and infinite dimensional in-put/output spaces are involved: a bar indicates an operator O with both input and output spacesfinite dimensional; grave accent — O, when the input space is finite dimensional and the outputinfinite dimensional one; acute accent — O, when the input space is infinite dimensional and theoutput finite dimensional one; and finally breve — O, when both input and output spaces areinfinite dimensional.

The compact block notation[A B

C D

]denotes (matrix- or operator-valued) transfer func-

tions either in s or in z domain in terms of their state-space realization. To distinguish lineartime-invariant (LTI) systems in the time domain from the corresponding transfer functions, theformer are denoted by script capital letters, so G(s) implies the transfer function of a continuous-time LTI system G. The lower linear fractional transformation of K over P is denoted as F`

(P;K

).

Finally, RicC− and RicD denote the continuous- and discrete-time Riccati functions. These func-tions are not the standard ones usually found in the literature [32], but rather are defined overproduct spaces:

Ric S : R(2n+m)×(2n+m) × R(2n+m)×(2n+m) → Rn×n× Rm×n

where S = C− or D. There are good reasons for considering this generalized forms, see for

instance [24, 13] and the Appendix in the companion paper [23] for definitions and properties.The functions Ric S are not defined over the whole R(2n+m)×(2n+m) but rather on a subset calleddomRic S. It is worth stressing that the function Ric S has a one–to–one correspondence withthe stabilizing solution of appropriate Riccati equations.

2 Problem statement

Consider the single rate sampled-data control systems illustrated in Fig. 1, where Pc is acontinuous-time generalized plant and w, z, y, and u are (continuous-time) exogenous input,regulated output, measured output, and control input, respectively. The sampled-data controllerconsists of a discrete-time part Kd, a sampler Sh, and a hold Hh, assumed to be synchronized


-w

-

��Pc

-z

y

�

��Sh�

y

��Kd

�u

��Hh

u

Figure 1: General sampled-data setup in time domain.

and with a sampling period h. Throughout this paper the generalized plant Pc is assumed tobe LTI with the following state-space realization:

Pc(s) =

A B1 B2C1 0 D12C2 D21 0

: (1)

The matrices D11 and D22 are both taken to be zero in order to simplify the derivations andobtain more transparent results. Moreover, for the H2 problem D11 = 0 is also a necessarycondition for the cost function to be finite.

The hold and sampler act on the output of the controller u[k] and the measurement y(t)respectively, to generate [15, 1](

Hhu)(kh+ �) = �H(�)u[k]; ∀� ∈ [0; h) (2a)

and

(Shy

)[k] =

∫h−

0�S(�)y(kh

− − �)d� (2b)

for some �H(�) and �S(�) (generalized hold and sampling functions, respectively) defined on theinterval [0; h). During the inter-sample, the hold function shapes the form of the control signalwhile the sampling function is used to weight the measurements. Note, that the integration limitfor Sh is chosen to be h− rather than h. This is motivated by the fact that any implementableKd cannot process information instantaneously. As shown in [22], this assumption considerablysimplifies the treatment of sampled-data systems.

Depending on whether the sampler and hold devices are considered as design variables ornot, there are four possible classes of sampled-data control problems:

Ca: Both sampler and hold are free for design;

Cb: The sampler is fixed but the hold is to be designed;

Cc: The sampler is to be designed but the hold is fixed;

Cd: Both sampler and hold are fixed.

The last case has been extensively treated in the literature, especially when Sh is the idealsampler and Hh is the zero-order hold (see [8] and the references therein). Usually the solutionsin case Cd are based on a conversion of a sampled-data problem to an equivalent pure discreteone which, in turn, can then be solved by known techniques. Such an approach, however,


requires several intermediate steps to be performed. As a result, no closed-form solutions tothe sampled-data H∞ problem is available. Hence, although this paper addresses basically casesCa–c, which involve the design of the sampling and/or hold functions, the results for Cd will alsobe presented.

The control problems to be dealt with in the paper are theH2 andH∞ optimization problems.Under arbitrary choice of the functions �S and �H in (2) and any LTI Kd, the system in Fig. 1is h-periodic in continuous time. For such systems the notions of the H2 and H∞ system normscan be introduced in a natural manner [8]. Thus, the following optimization problems can beposed:

OPH2 : Find an LTI Kd and, possibly, a sampling function �S(�) and/or a hold function �H(�)

so that the resulting sampled-data controller internally stabilizes the system in Fig. 1and minimizes the H2 norm of the closed-loop operator from w to z.

and

OPH∞ : Given a scalar > 0, find (if they exist) an LTI Kd and, possibly, a sampling func-tion �S(�) and/or a hold function �H(�) so that the resulting sampled-data controllerinternally stabilizes the system in Fig. 1 and makes the H∞ norm of the closed-loopoperator from w to z less than .

Remark 2.1. It is worth stressing that the discrete-time part Kd of the sampled-data controller isalways a design parameter. This is in contrast to the approach in [14, 12], where the generalizedhold is designed for a fixed Kd. Nevertheless, the Hh and Sh resulting from the solution to OPH2and OPH∞ are referred to asH2-optimal andH∞-suboptimal hold and sampler, respectively. TheH2-optimality of Hh (Sh) here is understood as the ability to design Kd and, possibly, Sh (Hh)so that the H2 performance achieved by the sampled-data controller HhKdSh supersedes theone achieved using any other hold (sampling) device. Similarly, H∞-suboptimal hold (sampler)refers to the feasibility of designing Kd and, possibly, Sh (Hh) so that the overall sampled-datacontroller be -suboptimal.

The treatment of OPH2 and OPH∞ is complicated owing to their hybrid continuous/discretenature and inherent periodicity. To circumvent these difficulties the so-called lifting techniqueof [31, 6, 4] (see also [28, 27]) can be applied.

3 Lifted “standard problem”

The notion of lifting is based on a conversion of real valued signals in continuous time intofunctional space valued sequences, that is sequences that take values not from R but ratherfrom some general Banach space (L2[0; h] in this paper). Formally, let ‘L2[0,h] be the space ofsequences, each element of which is a function from L2[0; h], that is

‘L2[0,h] ={� : �[k] ∈ L2[0; h] ∀k ∈ Z+

}:

Then given any h > 0, the lifting operator Wh : L2e 7→ ‘L2[0,h] is defined [6, 4] such that:

� = Wh� ⇐⇒ (�[k]

)(�) = �(kh+ �) ∀� ∈ [0; h]:

It is easy to see that the lifting operator is a linear bijection between L2e and ‘L2[0,h]. Moreover,if the domain of Wh is restricted to the Hilbert space L2, then by endowing ‘L2[0,h] with an


w -��P

z-

u

-

y

�

��K

Figure 2: “Standard problem” in the lifted domain.

appropriate norm the lifting operator can be made an isometry. Hence, treating a system� = G! not as a mapping from ! to � but rather as a mapping from ! to � gives essentiallythe same system (as an input-output mapping). Indeed, lifting preserves system stability andsystem induced norms. This allows one to conclude that G and

G:= WhGW−1

h : ‘L2[0,h] 7→ ‘L2[0,h];

which is called lifting of G, are equivalent. The advantage of treating systems in the liftingdomain stems from the fact that G is time-invariant in discrete time even if G is h-periodic incontinuous time. Hence, any periodic problem in continuous time can be reduced to a time-invariant one in discrete time.

The application of this idea to the sampled-data setup in Fig. 1 is straightforward. In orderto convert this setup to a pure discrete time-invariant one the operators Pc, Sh, and Hh in Fig. 1should be lifted to Pc

:= WhPcW

−1h , Sh

:= ShW

−1h , and Hh

:= WhHh respectively. The lifted

plant Pc is LTI and has a state-space realization

Pc(z) =

A B1 B2

C1 D11 D12

C2 D21 D22

: (3)

Notice that although A is a square matrix with the same dimensions as A, the remaining entriesin this state-space representation are operators acting from or/and to infinite dimensional spaces.The lifted hold Hh is a memoryless gain with transfer function

Hh(z) = �H; (4)

while the lifted sampler includes a backward shift [22]:

Sh(z) = z−1�S: (5)

The expressions for the parameters of the plant Pc, the hold Hh, and the sampler Sh canbe derived using standard lifting arguments [4, 20]. They, however, are not essential for thediscussion in this section and hence are postponed to the companion paper [23].

What is important in the discussion to follow is the fact that lifting puts all blocks in Fig. 1on equal footing, making them discrete-time LTI systems. Consequently, the four cases Ca–d ofthe sampled-data interconnection in Fig. 1 with generalized sampler and hold, can be reduced tothe standard setup of the linear optimal control in Fig. 2, where the generalized plant P is given,while the controller K is to be designed. The lifted sampler Sh and hold Hh can be absorbedeither into the generalized plant or to the controller, depending on whether they are fixed ortreated as free design parameters. The only fixed part that will always be absorbed into the


controller is the backward shift z−1 of Sh. The reason for this is twofold: first, the state-spacedimension of P is preserved, and second, the characterization of K is considerably simplifiedif it is strictly causal [22]. An optimization problem can therefore always be formulated as a“standard problem”, with a strictly proper controller. For the four cases above one gets:

Ca: Here, both �S and �H are absorbed into the controller, giving

P(z) = Pc(z) and K(z) = K(z):= z−1�HKd(z)�S:

Any designed controller can always be factorized as follows:

K(z) =

[AK BK

CK 0

]= �H

[AK BKCK 0

]�S;

where �H and CK are any operators of appropriate dimensions such that �HCK = CK andBK and �S are such that BK�S = BK. This yields Kd, Hh, and Sh.

Cb: In this case �S is absorbed into the lifted plant but �H into the controller, giving

P(z) =

A B1 B2

C1 D11 D12

C2 D21 D22

and K(z) = K(z):= z−1�HKd(z);

where[C2 D21 D22

] := �S

[C2 D21 D22

]:

Any designed controller can always be factorized as follows:

K(z) =

[AK BK

CK 0

]= �H

[AK BKCK 0

];

where �H and CK are any operators of appropriate dimensions such that �HCK = CK.This yields both Kd and Hh.

Cc: Now �H is absorbed into the lifted plant but �S into the controller, giving

P(z) =

A B1 B2

C1 D11 D12

C2 D21 D22

and K(z) = K(z):= z−1Kd(z)�S;

where B2

D12

D22

:=

B2

D12

D22

�H:Any designed controller can always be factorized as follows:

K(z) =

[AK BKCK 0

]=

[AK BKCK 0

]�S;

where BK and �S are any operators of appropriate dimensions such that BK�S = BK. Thisyields both Kd and Sh.


Cd: In this case, both �S and �H are absorbed into the lifted plant, giving

P(z) =

A B1 B2

C1 D11 D12

C2 D21 D22

and K(z) = K(z):= z−1Kd(z);

where

D22:= �SD22�H:

Given a designed K, the discrete-time part of the controller is

Kd(z) = zK(z):

Thus a designed K now contains information not only about Kd, but also possibly about �Hand/or �S. If the hold function is the design parameter, then it is completely characterized bythe “C”-part of the controller. Likewise, if the sampling function is the design parameter, thenit is completely characterized by the “B”-part of the controller.

It is worth stressing, that although for all the cases above the problem fits into the unifiedframework in Fig. 2, it is seen that some of the parameters of P and K might have either finite-or infinite-dimensional input and/or output spaces depending on the situation. In this respect,it is assumed here that the generalized plant takes the form:

P(z) =

A B1 B2

C1 D11 D12C2 D21 D22

=

[A B1 B2

C D•1 D•2

]=

A B

C1 D1•C2 D2•

; (6)

while a designed controller is necessarily of the form

K(z) =

[AK BK

CK 0

]:

The tilde is used to highlight that the corresponding operators may have either finite or infinitedimension depending on whether Sh and Hh are fixed or not.

4 Optimal design in the lifted domain

Having reduced the sampled-data setup to the discrete-time LTI “standard problem” in Fig. 2,the next step is to reformulate and solve OPH∞ and OPH2 in terms of P and K. This is thepurpose of this section.

Start by imposing the following assumptions on the generalized plant (6):

(A1): The operator[A− �I B2

]is right invertible ∀ |�| ≥ 1;

(A2): The operator[A− �I

C2

]is left invertible ∀ |�| ≥ 1;

(A3): The operator[A− ejθI B2

C1 D12

]is left invertible ∀� ∈ [0; 2�);


(A4): The operator[A− ejθI B1C2 D21

]is right invertible ∀� ∈ [0; 2�).

These assumptions are the counterparts of the standard assumptions imposed on a discrete-timegeneralized plant, to guarantee input–output stabilizability and non-singularity of the H2 andH∞ problems. Furthermore, the following assumption about P has to be made:

(A5): D21 is a bounded operator.

For the standard discrete-time systems with matrix valued parameters this assumption is obvi-ously redundant. For lifted systems, however, this is not always true since sampling operationsmight be unbounded in the L2 sense [8, Theorem 9.3.1]. Thus the assumption (A5), togetherwith the assumption D22 = 0, guarantees that Sh operates over “proper” signals. Actually, itmeans that pre-filtering by an anti-aliasing filter is provided if necessary.

4.1 H2 optimization

The notion of the H2 system norm can be extended in a natural manner to LTI systems in thelifted domain [5] (see also [17]) although it is not an induced norm. More specifically, the H2

norm of an LTI system G : ‘L2[0,h] 7→ ‘L2[0,h] is defined as follows:

‖G‖2H2:=1

h

∫2π0‖G(ejθ)‖2HSd�:

This definition is consistent with both the deterministic and the stochastic interpretations of theH2 system norm in the time domain, and reduces to the usual definition when G is the lifting ofan LTI continuous-time system [5].

Thus, OPH2 can now be reformulated in the lifted domain as follows:

OPeqH2

: For the plant P given in (6), find a strictly causal K which internally stabilizes P andminimizes the performance index:

JH2:= ‖F`

(P; K

)‖2H2 :

The solution to this H2 problem is presented next without a proof. This is because, in principle,problem OPeq

H2is a discrete-time LTIH2 problem which can be solved by using existing techniques

[8, 30]. Detailed treatment of the H2 problem in the lifted domain in the case when both samplerand hold are fixed and the controller is not constrained to be strictly proper can be found in[21].

The solution of OPeqH2

requires the two H2 DARE’s

X = A ′XA+ C∗1C1− (B∗2XA+ D∗12C1)∗(D∗12D12+ B∗2XB2

)−1(B∗2XA+ D∗12C1); (7a)

and

Y = AYA ′ + B1B∗1− (AYC∗2+ B1D

∗21)(D21D

∗21+ C2YC

∗2

)−1(AYC∗2+ B1D

∗21)∗: (7b)

Notice that for all the four cases described in Section 3, both X and Y of (7) are square matrices(e. g., finite dimensional) with the same dimensions as A. Also, observe that the choice of thehold device affects only the control Riccati equation (7a), while the choice of sampler affectsonly the filtering Riccati equation (7b).

The main result for the OPeqH2

can now be stated.


Theorem 1. Let (A1)–(A5) be satisfied and X2 ≥ 0, Y2 ≥ 0 be the stabilizing solutions to theDARE’s (7). Then the optimal value of the performance index JH2 is

JoptH2

=1

h

(‖D11‖2HS+ tr

{X2B1B

∗1+ C∗1C1Y2+ (A ′X2A− X2)Y2

})and the unique strictly proper controller which achieves the optimal cost Jopt

H2has the state-space

representation

K2(z) =

[A+ B2F2+ L2C2+ L2D22F2 −L2

F2 0

];

where

F2:= −

(D∗12D12+ B∗2X2B2

)−1(B∗2X2A+ D∗12C1); (8a)

L2:= −(AY2C

∗2+ B1D

∗21)(D21D

∗21+ C2Y2C

∗2

)−1: (8b)

Note again that although the parameters of P might be infinite dimensional, the solutionsto the Riccati equations as well as the “A” parameter of the optimal controller are always finitedimensional matrices. The state feedback F2 and the output injection L2 “gains”, however,might operate over infinite dimensional output and input spaces, respectively. Consider, forinstance, the operator F2. When Hh is fixed (cases Cc and Cd), the operator compositionsD∗12C1 = D∗12C1 and D∗12D12= D∗12D12, as well as the operator B2 = B2, are finite dimensionalmatrices, which can easily be computed [5, 20]. Yet when Hh is to be designed (cases Ca andCb) both equation (7a) and F2 involve quite complicated infinite dimensional operators. TheRiccati equations, like (7a), can in principle be dealt with via the associated Hamiltonians [9],that enables to circumvent some problems. Nevertheless, in order to obtain the optimal statefeedback “gain” one has to handle infinite dimensional operators, like D∗12D12+ B∗2XB2. Thetechniques for performing such manipulations over the lifted operators have been developed onlyrecently in [20]. As shown in the companion paper [23], the resulting F2 (and L2) can be obtainedin an elegant form.

4.2 H∞ optimization

Since for continuous-time systems the H∞ norm is the induced L2=L2 norm, the notion of theH∞ system norm can be extended to sampled-data systems by defining it as the ‘2

L2[0,h]=‘2L2[0,h]

induced operator norm. As shown in [4], the H∞ norm can be defined in the frequency domainas follows:

‖G‖H∞ := maxθ∈[0,2π)

‖G(ejθ)‖2:

Correspondingly, the lifted equivalent of OPH∞ for the setup in Fig. 2 takes the form

OPeqH∞ : For the plant P given in (6) and a number > 0, find a strictly causal internallystabilizing controller K such that

‖F`(P; K

)‖H∞ < ;

or show that no such controller exists.


Using the same reasoning as in the OPeqH2

case, the OPeqH∞ problem is a discrete-time LTI H∞problem [6], with the additional constraint that the controller must be strictly proper. Thistype of problems was discussed in detail for the case of finite dimensional parameters in [19].

The solution of the OPeqH∞ requires the following two H∞ DARE’s

X = A ′XA+ C∗1C1− (D∗1•C1+ B∗XA)∗(D∗1•D1• − 2E11+ B∗XB)−1(D∗1•C1+ B∗XA) (9a)

and

Y = AYA ′ + B1B∗1− (B1D

∗•1+ AYC∗)(D•1D

∗•1− 2E11+ CYC∗)−1(B1D

∗•1+ AYC∗)∗; (9b)

where E11:=[I 00 0

]. As in the H2 case, both X and Y solving the H∞ DARE’s (9) are square

matrices of the same dimension as A. Also, the choices of Hh and Sh affect only equations (9a)and (9b), respectively.

Using (9), necessary and sufficient conditions for the existence of a solution to the OPH∞ ,as well as a particular solution may be established:

Theorem 2. Given plant (6) such that the assumptions (A1)–(A5) hold, then the followingstatements are equivalent:

i) There exists a controller K which solves the OPeqH∞ .

ii) The DARE’s (9) have stabilizing solutions Xγ ≥ 0 and Yγ ≥ 0 such that∥∥∥∥[ X1/2γ 0

0 I

][A B1

C1 D11

][Y1/2γ 0

0 I

]∥∥∥∥2

< : (10)

Given that the conditions of part ii) hold, then the matrix Zγ:= (I− −2YγXγ)

−1 is well definedand one controller which solves OPeqH∞ is

Kγ(z) =

[A+ BFγ+ ZγLγ2(C2+ D2•Fγ) −ZγLγ2

Fγ2 0

];

where

Fγ:= −(D∗1•D1• − 2E11+ B∗XγB)−1(D∗1•C1+ B∗XγA) =

[Fγ1Fγ2

]; (11a)

Lγ:= −(B1D

∗•1+ AYγC

∗)(D•1D∗•1− 2E11+ CYγC

∗)−1 =[Lγ1 Lγ2

]: (11b)

As seen, the direct solution to the H∞ problem involves infinite dimensional operators evenin the case Cd, that is when both sampling and hold functions are fixed. Nevertheless, as shownin the companion paper [23] the techniques developed in [20] enable the reduction of all theseoperator compositions to finite dimensional matrices, which can be easily computed.

4.3 Discussion

Although the lifted solutions of Theorems 1 and 2 are not readily implementable, some insightinto the solutions to OPH2 and OPH∞ can be gained already at this stage. In particular:


Remark 4.1. As noted in Section 3, if the hold (sampling) function is the design parameter,then it is characterized by the “C” (“B”) part of the lifted controller. Hence, the H2-optimalhold is characterized by the operator F2, which becomes F2 in that case, while the H2-optimalsampler — by the operator L2 (L2). On the other hand, by inspecting (8) and (7) one can seethat F2 depends only on the subsystem from u to z, while L2 is completely characterized by theproperties of the subsystem from w to y. Therefore, there is complete separation between thedesign of Hh and Sh in the H2 case. This separation is reminiscent of that between the statefeedback and the state estimation in the standard H2 (LQG) design and thus is not a surprise.

Remark 4.2. Much more surprising is the fact that similar separations arguments apply to thedesign of the H∞ suboptimal hold and sampler as well. This fact is worth stressing, giventhe coupling which exists between the full information and the output estimation problemsin the H∞ output feedback control [32]. Although the “B” part of the lifted H∞ suboptimalcontroller contains the coupling term Zγ, the latter is finite-dimensional and hence can alwaysbe absorbed into the discrete-time part of the controller. Consequently, the H∞ suboptimal holdand sampler are characterized by the operators Fγ2 and Lγ2, respectively. It is seen from (11)and (9) that these operators are not completely independent, since both of them are affectedby the subsystem from w to z. Nevertheless, Fγ2 does not depend on the measurement y andLγ2 does not depend on the control u. Hence, both the hold and the sampling functions can bedesigned independently one from the other in the H∞ case also. This is in contrast to previousworks in the literature [28, 22], where the H∞ suboptimal sampler depended on the hold and itis not clear how to recover the separation.

Remark 4.3. It is not difficult to verify that as → ∞ the H∞ Riccati equations (9) reduceto the corresponding H2 ones (7). Moreover, as → ∞, the conditions of statement ii) ofTheorem 2 hold automatically, Zγ = I (since both Xγ and Yγ are uniformly bounded),

limγ→∞ Fγ =

[0

F2

];

and

limγ→∞ Lγ =

[0 L2

]:

Thus in the limit case → ∞ the H∞-suboptimal controller Kγ(z) approaches K2(z), the H2-optimal one.

5 Main results

The solutions to the OPH2 and OPH∞ can now be presented. Again, these solutions are stronglybased on the technical machinery developed in the companion paper. To keep the presentationclear, it is assumed that a zero-order hold is used in cases Cc and Cd, while an ideal sampler isused in cases Cb and Cd. From Remarks 4.1 and 4.2 it follows that such simplifications do notaffect the results concerning the (sub)optimal sampler and hold. More general Sh and Hh canbe treated in a similar fashion. Also, as follows from Remark 4.3, the solution of Theorem 2approaches that of Theorem 1 as → ∞. For that reason, only the H∞ results are presentedbellow. The H2 results can be obtained from the H∞ ones by the simple substitution −1 = 0.The only part of the H2 solution which needs to be treated independently is the calculation ofthe optimal performance Jopt

H2. This issue is touched upon in §5.5.


Let

�X:=

−D ′12C1 −B ′2 −D ′12D12A −2B1B

′1 B2

−C ′1C1 −A ′ −C ′1D12

;�Y:=

−D21B′1 −C2 −D21D

′21

A ′ −2C ′1C1 C ′2−B1B

′1 −A −B1D

′21

;and

�:= exp

0 −D ′12C1 −B ′2 −D ′12D120 A −2B1B

′1 B2

0 −C ′1C1 −A ′ −C ′1D120 0 0 0

h =

I �12 �13 �140 �22 �23 �240 �32 �33 �340 0 0 I

:Define the matrices

�X:=

�12 �13 �14�22 �23 �24�32 �33 �34

;�Y

:=

0 C2 0

� ′22 − −2� ′32 � ′22C′2

− 2� ′23 � ′33 − 2� ′23C′2

;and the matrix function of an argument � ∈ Z+

�ν:=

0 0 0

In 0 0

0 In 0

∈ R(ν+2n)×(2n+ν):

The following quantity is also required:

0:= ‖Pc,11‖L2[0,h]; (12)

where Pc,11(s) = C1(sI − A)−1B1 (in other words, 0 is the L2[0; h]-induced norm of the sub-system from w to z). This quantity can be computed as described in [8, Section 13.5] or [11].Since 0 = ‖D11‖2, it is clear that 0 is the lower bound for the achievable H∞ performance insampled-data systems under any choice of Sh and Hh. Hence it is natural to consider only thecases where > 0.

Assumptions (A1)–(A5) are formulated in terms of the system description in the lifted do-main. Results in the companion paper [23] allow replacing them for more readily checkableconditions. In particular, when Hh is the design parameter, assumptions (A1) and (A3) can beequivalently formulated as

(A1 ′): The pair (A;B2) is stabilizable;

(A3 ′): The matrix[A− j!I B2C1 D12

]is left invertible ∀! ∈ R and D ′12D12> 0;

while when Hh is the zero-order hold (�H(�) = I) they can be replaced with


(A1 ′′): The pair (�22; �24) is stabilizable for any > 0;

(A3 ′′): The matrix

�12 �14�22− ejθI �24�32 �34

is left invertible ∀� ∈ [0; 2�) and any > 0.

Note, that although (A1 ′′) is equivalent to (A1) only if −1 = 0, OPH∞ possesses no solutionswhenever (A1 ′′) is violated. Hence, the replacement of (A1) with (A1 ′′) does not affect thesolution.

Analogously, when Sh is is the design parameter, assumptions (A2) and (A4) can be equiva-lently expressed as

(A2 ′): The pair (C2; A) is detectable;

(A4 ′): The matrix[A− j!I B1C2 D21

]is right invertible ∀! ∈ R and D21D ′21> 0

and (A5) is redundant, while when it is the ideal sampler (�S(�) = �(�)), (A2) and (A4) can bereplaced with

(A2 ′′): The pair (C2; �22) is detectable for any > 0;

(A4 ′′): The matrix[�22− ejθI 2�23

C2 0

]is right invertible ∀� ∈ [0; 2�) and any > 0;

(A5 ′′): D21= 0.

5.1 Both hold and sampler are free

In case Ca the solution to OPH∞ is as follows:

Theorem 3 (Ca). Given plant (1) such that assumptions (A1 ′)–(A4 ′) hold, then for any > 0the matrix �33 is nonsingular and the following statements are equivalent:

i) There exist Kd, Hh, and Sh which solve OPH∞ .

ii) (�X; �m) ∈ dom(RicC−) and (�Y; �r) ∈ dom(RicC−) and the following conditions hold:

(a) Xγ ≥ 0 and �(Xγ�23�−133) < 1;

(b) Yγ ≥ 0 and �(�−133�32Yγ) <

2;

(c) �(Yγ(�33+ −2�32Yγ)

−1Xγ(�′33− � ′23Xγ)

−1)< 2;

where (Xγ; Fγ) = RicC−(�X; �m) and (Yγ; L′γ) = RicC−(�Y; �r).

Furthermore, if the conditions of part ii) hold, then the matrix Zγ:= (I− −2YγXγ)

−1 is welldefined and one possible choice for Kd, Hh, and Sh is:

Kd(z) = z

[Zγ�12+�22 Zγ

I 0

];

where[�11 �120 �22

]:= exp

([A+ −2YγC

′1C1+ LγC2 Lγ(C2+ −2D21B

′1Xγ)

0 A+ −2B1B′1Xγ+ B2Fγ

]h

);


and

�H(�) = Fγe(A+γ−2B1B

′1Xγ+B2Fγ)τ; (13a)

�S(�) = −e(A+γ−2YγC′1C1+LγC2)τLγ; (13b)

Proof (outline). Actually, it suffices to prove that the solution given in the Theorem is equivalentto the one in Theorem 2. To this end note, that by [23, Lemma 3] and its dual, the DARE’s (9)have stabilizing solutions iff (�X; �m) ∈ dom(RicC−) and (�Y; �r) ∈ dom(RicC−). Then, items(a)-(c) are equivalent to (10) by [23, Lemma 8].

Now, consider the lifted controller Kγ(z) (= Kγ(z)) in Theorem 2. It is clear that �H = Fγ2

and �S = Lγ2 can be chosen. Then (13) follow directly from [23, Lemma 3] and its dual. It alsofollows from that Lemma that

A+ BFγ = �22 and L2(C2+ D2•Fγ) = �12;

from which the formula for Kd(z) follows.

5.2 Free hold and fixed sampler

In case Cb the solution to OPH∞ is as follows:

Theorem 4 (Cb). Given plant (1) such that assumptions (A1 ′), (A2 ′′), (A3 ′), (A4 ′′), (A5 ′′) holdand let the sampling function be �S(�) = �(�). Then for any > 0 the following statementsare equivalent:

i) There exist Kd and Hh which solve OPH∞ .

ii) (�X; �m) ∈ dom(RicC−) and (�Y; �r) ∈ dom(RicD) and the following conditions hold:

(a) Xγ ≥ 0 and �(Xγ�23�−133) < 1;

(b) Yγ ≥ 0 and �(�−133�32Yγ) <

2;

(c) �(Yγ(�33+ −2�32Yγ)

−1Xγ(�′33− � ′23Xγ)

−1)< 2;

where (Xγ; Fγ) = RicC−(�X; �m) and (Yγ; L′γ) = RicD(�Y; �r).


−1 is welldefined and one possible choice for Kd and Hh is:

Kd(z) = z

[(I+ ZγLγC2)e

(A+γ−2B1B′1Xγ+B2Fγ)h −ZγLγ

I 0

]and �H(�) as in (13a).

Proof (outline). The first part can be proven in a similar fashion as the first part of Theorem 3,except that the DARE (9b) is solved by using [23, Lemma 7]. To get Kd(z) just note that if Shis the ideal sampler and D21= 0, then[

C2 D2•]

= C2[A B

]; (14)

which completes that proof.


5.3 Fixed hold and free sampler

In case Cc the solution to OPH∞ is as follows:

Theorem 5 (Cc). Given plant (1) such that assumptions (A1 ′′), (A2 ′), (A3 ′′), (A4 ′) hold and letthe hold function be �H(�) = I. Then for any > 0 the following statements are equivalent:

i) There exist Kd and Sh which solve OPH∞ .

ii) (�X; �m) ∈ dom(RicD) and (�Y; �r) ∈ dom(RicC−) and the following conditions hold:

(a) Xγ ≥ 0 and �(Xγ�23�−133) < 1;

(b) Yγ ≥ 0 and �(�−133�32Yγ) <

2;

(c) �(Yγ(�33+ −2�32Yγ)

−1Xγ(�′33− � ′23Xγ)

−1)< 2;

where (Xγ; Fγ) = RicD(�X; �m) and (Yγ; L′γ) = RicC−(�Y; �r).


−1 is welldefined and one possible choice for Kd and Sh is:

Kd(z) = z

[11+ 12FγZγ I

FγZγ 0

];

where[11 120 I

]:= exp

([A+ −2YγC

′1C1+ LγC2 B2+ −2YγC

′1D12

0 0

]h

);

and �S(�) as in (13b).

Proof (outline). The first part can be proven similar to the first part of Theorem 3, except thatthe DARE (9a) is solved by [23, Lemma 7]. In order to obtain the formula for Kd(z) it is moreconvenient to use the dual form of the H∞ suboptimal controller, i.e.

K(z) =

[A+ LγC+ (B2+ LγD•2)Fγ2Zγ −Lγ2

Fγ2Zγ 0

](which becomes K(z) in this case). Then, using the dual result to [23, Lemma 3] one can getthat

A+ LγC = 11 and B2+ LγD•2 = 12;

from which the formula for Kd(z) follows.

5.4 Both hold and sampler are fixed

In case Cd the solution to OPH∞ is as follows:

Theorem 6 (Cd). Given plant (1) such that assumptions (A1 ′′)–(A4 ′′), (A5 ′′) hold and let thehold function be �H(�) = I and the sampling function be �S(�) = �(�). Then for any > 0the following statements are equivalent:

i) There exists Kd which solves OPH∞ .


ii) (�X; �m) ∈ dom(RicD) and (�Y; �r) ∈ dom(RicD) and the following conditions hold:

(a) Xγ ≥ 0 and �(Xγ�23�−133) < 1;

(b) Yγ ≥ 0 and �(�−133�32Yγ) <

2;(c) �

(Yγ(�33+ −2�32Yγ)

−1Xγ(�′33− � ′23Xγ)

−1)< 2;

where (Xγ; Fγ) = RicD(�X; �m) and (Yγ; L′γ) = RicD(�Y; �r).


−1 is welldefined and one possible choice for Kd is:

Kd(z) = z

[(I+ ZγLγC2)(�22+ �23Xγ+ �24Fγ) −ZγLγ

Fγ 0

]:

Proof (outline). The first part can be proven similar to the first part of Theorem 3, except thatinstead of [23, Lemma 3] one has to use [23, Lemma 7]. The formula for Kd(z) then follows by(14) and the equality

A+ BFγ = �22+ �23Xγ+ �24Fγ;

which, in turn, follows from the fact that (Xγ; Fγ) = RicD(�X; �m).

5.5 The optimal H2 performance

In this subsection the formula for the H2 optimal performance index JoptH2

given in Theorem 1,is expressed in terms of the matrices A, B1, and C1. This result is not new (see, e.g., [5] or [20,Theorem 2]) and is presented here for completeness.

Form the matrix exponential

�:= exp

−A ′ C ′1C1 0

0 A B1B′1

0 0 −A ′

h =

�11 �12 �130 �22 �230 0 �11

:Then

Lemma 1. Let X2 and Y2 be the stabilizing solutions to the Riccati equations (7), then theoptimal value of the performance index JH2 is

JoptH2

= 1h tr(� ′22(�13+ X2�23+ �12Y2+ X2�22Y2) − X2Y2

):

6 Discussion

Theorems 3–5 yield not only the discrete-time part Kd of the sampled-data controller, but alsothe H∞ suboptimal (or the H2 optimal) sampling and/or hold functions1. In this section, genericproperties of the (sub)optimal sampler and hold functions are presented, together with theirinterpretation. The first property is the separation between the design of the sampler and thehold, which has already been discussed in §4.3. The separation property has a clear significance:both sampler and hold are, in a sense, open-loop devices. Consequently, the design of the holddoes not depend on the measurement y(t) and the design of the sampler does not depend on thecontrol action u(t). The remaining properties refer to the Sh and Hh, are discussed separatelybelow.

1See Remark 2.1 for what is meant by optimal sampling and hold functions.


6.1 Optimal hold

Consider the H∞ control CARE

A ′X+ XA+ C ′1C1+ 1γ2XB1B

′1X− (B ′2X+D ′12C1)

′(D ′12D12)−1(B ′2X+D ′12C1) = 0; (15)

which reduces to the H2 one as →∞. The H∞ state feedback gain is then:

F2:= −(D ′12D12)

−1(B ′2X+D ′12C1):

The following corollary can be derived from Theorems 3 and 4.

Corollary 1. Let X = X ′ ≥ 0 be the stabilizing solution to the CARE (15). Then one possibleH∞ suboptimal hold function is

�H(�) = F2e(A+γ−2B1B

′1X+B2F2)τ; ∀� ∈ [0; h) (16)

independently of the choice of the sampler and the measured output y(t). When −2 = 0, thisbecomes the unique H2 optimal solution.

The following remarks are in order:

Remark 6.1. Notice that the optimal hold for the nonsingular H2 and H∞ problems is always“asymptotically stable,” in the sense that its “A” matrix is Hurwitz. In other words, thecontinuation of the hold in (16) over the whole interval [0;∞) belongs to L2. The “stability”property is intuitively reasonable: being the generalized hold an “open-loop” device, it preventsany sources of instability during the inter-sample.

Remark 6.2. As a curious corollary to the previous remark, the standard zero-order hold (ZOH)can never appear as part of an optimal hold for nonsingular problems; this is because the “A”matrix of a ZOH is the zero-matrix. The ZOH does appear as part of an optimal hold forsingular H2 and H∞ problems, for which A+ −2B1B

′1X+B2F2 has an eigenvalue at the origin.

Such problems may arise when integral control is required, see [32, §17.3]. In the sampled-datacase, the integral control action is actually “re-distributed” between the discrete-time part ofthe controller Kd and the hold Hh.

Remark 6.3. It is worth mentioning that the design of the H2 and H∞ (sub)optimal holds doesnot depend on the sampling period h. The optimal hold function remains the same for all hand only the interval changes on which �H(�) is defined. Also, it can easily be verified that withthe optimal Hh the system is stabilizable for all h. In other words, if Hh is either H2 or H∞optimal, then no sampling period can lead to the loss of controllability (absence of pathologicalsampling).

The “stability” property discussed in Remarks 6.1 and 6.2 becomes particularly interestingwhen compared with previous works on the design of generalized hold functions. Indeed, formost hold devices designed in the literature from the discrete-time performance point of view(see [1] and the references therein) the continuation of �H on the whole R+ does not necessarilybelong to L2 (and even L∞). For example, if the hold function of the form �H(�) = CHe

AHτBHis designed, then the typical choice is AH = −A ′, where A is the “A” matrix of a plant. Hence,the “stability” of such a hold depends on the open-loop plant dynamics. One is then led to ask,what is the underlying reason that rules out the use of “unstable” hold functions.

To answer this question, consider the continuous-time LQR problem for the process

x(t) = Ax(t) + B2u(t)


and the cost function

J =

∫∞0

(C1x(t) +D12u(t)

) ′(C1x(t) +D12u(t)

)dt:

The solution to the problem of minimizing J is given by the feedback law [18] u(t) = F2x(t), sothat for a given tk and any � > 0, the closed-loop state vector satisfies the equation

x(tk+ �) = e(A+B2F2)τx(tk):

Substituting this equation into the formula for the control signal, gives

u(tk+ �) = F2e(A+B2F2)τx(tk):

On the other hand, it follows from (2a) and (16) that the H2 optimal hold produces the controlsignal

u(kh+ �) = F2e(A+B2F2)τuk; ∀� ∈ [0; h):

The comparison of the latter two expressions prompts the following interpretation of the H2

optimal hold:

Interpretation 1. The H2 optimal hold, given by (16) subject to −1 = 0, attempts to “recon-struct” the LQR feedback control law, assuming that Kd produces at the kth sampling instantan estimation of state vector of the plant at t = kh.

Similarly to the H2 case, the interpretation of the H∞ suboptimal hold can be obtained fromthe continuous-time H∞ state feedback problem as follows:

Interpretation 2. The H∞ suboptimal hold, given by (16), attempts to “reconstruct” the H∞suboptimal state feedback control law assuming that i) Kd produces at the kth sampling instantan estimate of the state vector of the plant at t = kh; and ii) the disturbance w is the worst-caseone (in an H∞ sense), i.e., w(t) = 1

γ2B ′1Xx(t).

These interpretations explain the “stability” of the optimal hold. Indeed, optimal continuous-time control must guarantee the internal stability of the closed-loop system. Consequently, thereconstruction of the continuous-time control signal in an open-loop fashion lead to the “stabledynamics” of the hold. When the integral control results from the continuous-time design, the“A” matrix of the hold function has an eigenvalue at the origin. This is consistent with thediscussion in Remark 6.2.

The main conclusion of the interpretations above is that the optimal hold attempts to recon-struct a “good” LTI continuous-time control law. Notice that this differs from some previousapproaches to the design of Hh which, while seeking to circumvent basic limitations of linearcontinuous-time control, attempted to outperform a continuous-time controller. The results ofthis paper suggest that the design of the hold should be based on the understanding that theoptimal continuous-time control is the “best” possible choice; consequently, the sampled-datacontroller should mimic it as good as possible. We believe that this idea can be extended fordesigning controllers in problems with no known analytic solution but with computable optimalcontinuous time controller. For example, the H2 or H∞ problems when Hh or Sh are constrainedto be scalar.


6.2 Optimal sampler

Consider now the H∞ filtering CARE

AY + YA ′ + B1B′1+ 1

γ2YC ′1C1Y − (YC ′2+ B1D

′21)(D21D

′21)

−1(YC ′2+ B1D′21)′ = 0; (17)

which again reduces to the H2 one when →∞. The H∞ filter gain is then as follows:

L2:= −(YC ′2+ B1D

′21)(D21D

′21)

−1:

The following corollary can be derived from Theorems 3 and 5:

Corollary 2. Let Y = Y ′ ≥ 0 be the stabilizing solution of the CARE (17). Then one possibleH∞ suboptimal sampling function is

�S(�) = −e(A+γ−2YC′1C1+L2C2)τL2; ∀� ∈ [0; h) (18)

independently of the choice of the hold and the control input u(t). When −2 = 0, this becomesthe unique H2 optimal solution.

The following remarks are in order:

Remark 6.4. Analogously with the case of the optimal hold, the optimal sampling function (18)is asymptotically stable in the sense defined in Remark 6.1. Moreover, the design of Sh does notdepend on the sampling period and it produces a detectable system for all h. In other words,if Sh is either H2 or H∞ optimal, then no sampling period can lead to the loss of observability(absence of pathological sampling).

Remark 6.5. Another interesting property of the optimal sampler is that is does not requirepre-filtering of the measurement output y(t) by an anti-aliasing filter. This is because, looselyspeaking, the sampler itself serves as an anti-aliasing filter. Mathematically, this means that thegeneralized sampler Sh with sampling function (18) is bounded as an operator L2 7→ ‘2.

Remark 6.6. The anti-aliasing capability of the optimal sampler is actually a consequence of thefact that the ideal sampler (the sampling function �S(�) = �(�)) can never appear as a partof an optimal sampler for nonsingular H2 and H∞ problems. The crucial assumption here isthe full row rank of the matrix D21. The latter means that all of the measurement channelsare corrupted by noise. For such a y(t) the instantaneous sampling is an illegal operation, thataccounts for the absence of the impulse component in the optimal sampling function. In thecase when an anti-aliasing filter is present, the matrix D21 necessarily looses row rank and theproblem becomes singular. It can be shown that in such a case the optimal sampler is of theform DS�(�) + CSe

ASτ for some AS, CS, and DS 6= 0 (but still, DSD21 = 0). This result givesrise to an interesting observation. It is well known that whenever Sh contains the ideal sampler,the measured output must be pre-filtered by an anti-aliasing filter. The discussion above impliesthat the opposite is also true: whenever the measured output is pre-filtered by an anti-aliasingfilter, an optimal sampler contains the ideal sampler.

7 Concluding remarks

In this paper a framework for the treatment of a rather general class of sampled-data controlproblems has been proposed. The cases when the sampler, the hold, or both, are available fordesign has been considered. Necessary and sufficient conditions for the existence of controllers


as well as state-space formulas for all the blocks involved have been provided for all these cases.Conditions and the formulas for state-space matrices can be readily implement in a computer.

The essence of this framework is to convert hybrid periodically time-varying problems to aunified discrete-time LTI “standard problem” using the lifting transformation with the subse-quent solution of a problem directly in the lifted domain. The lifting procedure is then carriedone step forward, by peeling-off the representations in the lifted domain back to continuous,or more specifically sampled-data, time. Specifically, the following sampled-data problems havebeen completely solved:

• The sampled-data H2 problems when sampling and/or hold functions are the design pa-rameters. The solution to this problem is, to the best of our knowledge, completely new.

• The sampled-data H∞ problems under similar conditions. These solutions, although notnew, are considerable more transparent than the previously existing in the literature.In particular, (a) a separation structure has been established between the design of H∞suboptimal sampler and hold; and (b) when both sampler and hold are fixed, the firstclosed-form solution to the H∞ problem, which avoids involved intermediate steps, hasbeen derived.

In the final section, several comments which reconcile and compare the present result with someprevious approaches to the design of generalized hold functions, are discussed.

This paper is complemented with [23], where all the technical results required to establishthe main results in this paper are developed. The reader is therefore referred to this work fordetails and interesting additional results concerning sampled-data systems in the lifted domain.

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