A New Representation of Lifted Systems with Applicationsleo.technion.ac.il/publications/newrepresentation/tme439.pdf · systems, which are operators over in nite-dimensional spaces

A New Representation of Lifted Systems with Applications∗

Leonid Mirkin†

Faculty of Mechanical EngineeringTechnion — IIT

Haifa 32000, Israel

Phone: +972 4 8293149Fax: +972 4 8324533

E-mail: [email protected]

Zalman J. PalmorFaculty of Mechanical Engineering

Technion — IITHaifa 32000, Israel

Phone: +972 4 8292086Fax: +972 4 8324533

E-mail: [email protected]

November 26, 1995

Abstract

Lifting, i.e. discretization with built-in intersample behavior, is an emerging techniquefor the analysis and design sampled-data systems. The applicability of the lifting technique,however, is considerably limited due to difficulties in dealing with parameters of the liftedsystems, which are operators over infinite-dimensional spaces rather than finite-dimensionalmatrices. The purpose of this paper is to introduce a new representation for the parame-ters of the lifted systems. This representation simplifies considerably algebraic manipulationover those parameters. It is based on expressing infinite-dimensional operators by dynamicalsystems with two-point boundary conditions operating on a finite time interval. A technicalmachinery, which enables one to reduce algebraic manipulation over the parameters of thelifted systems to simple matrix manipulations in the state-space, is developed in the paperon the basis of the new representation. It is believed that the new representation togetherwith the developed machinery extends considerably the scope of applicability and enhancesthe efficiency of the lifting technique. Several applications considered in the paper (compu-tational issues in sampled-data H2 and H∞ control and Riccati equations for lifted systems)demonstrate the benefits of the proposed representation and its capabilities.

∗Last modified November 17, 1998†Supported by The Center for Absorption in Science, Ministry of Immigrant Absorption State of Israel.

http://leo.technion.ac.il

Technical Report TME– 439; Nov 95 1

1 Introduction

The main motivation for this paper comes from the consideration of the intersample behavior ofsampled-data systems, that is systems consisting of a continuous-time plant and a discrete-time controller, which are connected by a sampling and a hold devices. Dealing with thecontinuous-time behavior of sampled-data systems is complicated by (a) the simultaneous pres-ence of continuous- and discrete-time signals; and (b) inherent periodicity of the interconnectionof a sampler, a discrete-time controller, and a hold. An elegant way to circumvent these diffi-culties is offered by the so-called lifting technique [27, 4, 2] (see also [24, 23]), which allows oneto convert periodic continuous/discrete systems to equivalent, in some sense, pure discrete-timeshift-invariant ones.

Although the reduction of hybrid periodic problems to discrete-time time-invariant onesis clearly advantageous, the lifting method gives rise to another problem: input and outputspaces of the lifted systems become infinite-dimensional. Conceptually, this fact does not leadto considerable difficulties since many LTI system notions have almost one-to-one counterpartsin the lifted domain, see [2, 6]. Moreover, the preservation of the state-space dimensions underlifting guarantees that any observer-based (including H2 and H∞) controller in the lifted domainhas a finite state dimension. Thus, analytic solutions to various sampled-data analysis and designproblems can be obtained relatively straightforwardly in terms of parameters of the lifted models.Yet the latter parameters are no longer finite dimensional matrices but rather operators overinfinite-dimensional input and/or output spaces, such as L2[0; h]. For this reason, the usefulnessof a lifted solution relies essentially upon the ability to treat such infinite-dimensional operators.

Unfortunately, lifted parameters are not readily treatable. Consequently, sampled-data prob-lems solved so far by means of the lifting technique share one key property: the possibility toseparate beforehand the infinite dimensionality from the design by converting the problem toan equivalent finite-dimensional one. If a problem admits such a separation, the complexityof the operations over the infinite-dimensional parameters may be considerably reduced. Yetfor many sampled-data problems the separation is impossible in principle. This is true, forinstance, for any problem where sampling and/or hold devices are to be designed; in this casethe lifted controller is inherently infinite-dimensional. Moreover, even for “separable” problemsthe transparency of a solution might deteriorate rapidly as the controller becomes multirate orincludes generalized sampling and/or hold devices, see e.g. [26]. As a result, many sampled-dataproblems are treated in the literature using other, somehow more problem-oriented, approaches[23, 22].

Thus, the problem of manipulating over the infinite-dimensional parameters of the liftedsystems can be considered as one of the most serious obstacles preventing the use of the liftingtechnique to a fuller extent. It seems, however, that these difficulties are not due to intrinsicdrawbacks of the lifting method but rather are a consequence of an awkward representationof the parameters of the lifted systems. Conventionally [27, 24, 23, 4, 2, 6], these parametersare represented by integral operators, originating in the derivation of the lifted models in thestate-space. Such a representation appears to be unfortunate, since it makes the treatment ofthe lifted parameters quite a complicated and involved problem.

In this respect in this paper we will propose a new representation of the parameters of thelifted systems, which enables one to simplify manipulations over these parameters consider-ably. Instead of considering lifted parameters as integral operators we will treat them as LTIcontinuous-time systems operating over a finite time interval. The advantage of such a rep-resentation lies in the possibility of reducing the operations over these operators to algebraicmanipulations over LTI systems with two-point boundary conditions [14, 8]. The latter ma-

2 Faculty of Mechanical Engineering; Technion—IIT

nipulations, in turn, can be performed by means of the well established state-space machinery.As a result, treating the infinite-dimensional parameters of the lifted systems is simplified con-siderably, enabling a straightforward and unified solution to a wide spectrum of sampled-dataproblems.

In the paper we will consider two applications of the proposed approach. First, we will showthat even for rather general sampling and hold devices the computations arising in sampled-dataH2 and H∞ problems can be handled in a unified and efficient manner. All the computationalformulae are reduced to the computations of the impulse response of LTI continuous-time systemswith two-point boundary conditions, that in turn can be done by computing matrix exponentialsonly. Second, we will consider the discrete-time algebraic Riccati equation, associated with liftedsystems with infinite dimensional parameters. It will be shown that the proposed representationof these parameters makes it possible to reduce such an equation to a finite dimensional discrete-time Riccati equation. In addition, a close connection between the discrete-time ARE associatedwith an LTI system in the lifted domain and the continuous-time ARE associated with the samesystem in continuous time will be established. In particular, we will prove that the stabilizingsolutions to these two Riccati equations coincide.

The paper is organized as follows. Section 2 contains a preliminary material: we briefly reviewthe lifting technique (Subsection 2.1) and LTI systems with two-point boundary conditionsoperating on a finite time interval (Subsection 2.2). The new representation of the parametersof the lifted systems and relevant technical machinery are the subject matters of Section 3. Thenext two sections are devoted to applications of these results: in Section 4 the computationalissues in sampled-data H2 and H∞ problems with generalized sampler and hold are considered;whereas in Section 5 we establish a connection between algebraic Riccati equations, associatedwith continuous-time systems and their lifting. Concluding remarks are given in Section 6.

1.1 Notations

The notations adopted throughout the paper are fairly standard. As usual, Rn denotes then-dimensional Euclidean space and L2n[0; h] denotes the (Hilbert) space of square integrable Rn-valued functions on the interval [0; h]. When the dimensions will be irrelevant we will drop thedimension index and write simply R and L2[0; h]. Cn[0; h] (or simply C[0; h]) denotes the linearspace of bounded Rn-valued continuous functions on the interval [0; h].

The transpose of a matrix M is denoted as M ′, its spectrum as (M), whereas the trace astr(M). The notation O∗ is adopted for the adjoint of a Hilbert space operator O. The scalarproduct on a Hilbert space H is denoted as 〈·; ·〉H. Given an operator O : L2[0; h] 7→L2[0; h], then‖O‖HS and ‖O‖2 denote the Hilbert-Schmidt and the L2[0; h]-induced norms of O, respectively.Since we will extensively use operator compositions involving operators over infinite dimensionalspaces (such as L2[0; h]), the following notation is aimed to simplify the readability of formulae:bar above a variable denotes an operator O both from and to finite dimensional spaces; graveaccent — O, from a finite dimensional space to an infinite dimensional one; acute accent — O,from an infinite dimensional space to a finite dimensional one; and breve — O, both from andto infinite dimensional spaces.

Continuous-time linear time-invariant (LTI) systems in the time domain is denoted by scriptcapital letters (like G), discrete-time LTI systems — by bar (G), and lifted systems — by breve

(G). The compact block notation[A B

C D

]denotes an LTI system (continuous-, discrete-time,

or lifted) in the time domain in terms of its state-space realization. Finally, the lower linearfractional transformation of K over P is denoted as F`

(P;K

).


2 Preliminaries

This section contains some preliminary results, which will be used in the sequel. In Subsection 2.1we briefly review the lifting technique, while Subsection 2.2 contains some facts about LTIsystems with two-point boundary conditions, operating on a finite time interval. The reader isreferred to [4, 2] and [14, 8, 9], respectively, for more details about these subjects.

2.1 Lifting technique

The notion of lifting is based on a conversion of real valued continuous-time signals into functionalspace valued sequences, that is sequences that take values not on R but rather on some generalBanach space (L2[0; h] in this paper). Formally, let ‘L2[0,h] be the space of sequences, eachelement 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 as follows:

= 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 we restrict the domain of Wh to the Hilbert space L2, then by endowing ‘L2[0,h] with anappropriate 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 essentially thesame system. Indeed, lifting preserves system stability and system induced norms. This allowsone 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 is due to 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.

Now consider the representation of lifted systems in the state-space. Let G be an LTI finitedimensional system with the state-space realization

[A B

C D

]. Then the lifting of G, G, is also

LTI and

G =

[A B

C D

];

A : R 7→R

B : L2[0; h] 7→R

C : R 7→L2[0; h]

D : L2[0; h] 7→L2[0; h]

; (1)

where for arbitrary ∈ R and ∈ L2[0; h]

A = eAh; (2a)

B =

∫h0eA(h−s)B(s)ds; (2b)(

C)() = CeAτ; (2c)(

D)() = D() + C

∫τ0eA(τ−s)B(s)ds: (2d)


It is clearly seen from (1) that the state-space realization of G involves operators over L2[0; h]

rather than finite dimensional matrices only. Fortunately, the dimension of the state vector ispreserved under lifting and hence the operators B and C in (1) have finite rank. This fact makesit possible to reduce the computations in sampled-data H2 and H∞ optimization problems tofinite dimensional ones [13, 3, 23, 2], because the infinite dimensional parts are “hidden” insideoperator compositions like BD∗(I − DD∗)−1C. However the integral representation (2) makesthe process of reducing such expressions to computationally convenient formulae quite difficult.In this respect we will propose a different representation for the operators in (1), which willprovide much simpler way for manipulating these operators.

2.2 LTI systems with two point boundary conditions

An LTI continuous-time system G with two-point boundary conditions on the time interval [0; h]

can in general be described as follows:

x(t) = Ax(t) + Bu(t); x(0) + x(h) = 0; (3a)y(t) = Cx(t) +Du(t) (3b)

where A, B, C, and D are some appropriate dimensioned matrices and and are squarematrices, which shape the boundary conditions of the state vector. The boundary conditionsof (3) are said to be well-posed if x(t) ≡ 0 is the only solution to equation (3a) when u(t) ≡ 0.It can easily be verified that system G given by (3) has well-posed boundary conditions iff thematrix function

G:= + eAh (4)

is nonsingular. In the sequel, the term STPBC will be used for systems with well-posed bound-ary conditions only, unless the opposite is stated explicitly. If (3) has well-posed boundaryconditions, then its response y(t) is uniquely determined by the input u(t) and is as follows:

y(t) = Du(t) +

∫h0KG(t; s)u(s)ds; (5)

where the kernel KG(t; s) has the form

KG(t; s) =

CeAt−1

G e−AsB if 0 ≤ s < t ≤ h

−CeAt−1G eA(h−s)B if 0 ≤ t < s ≤ h

: (6)

It follows that an STPBC is causal if its “” matrix is zero and is anti-causal if so is its “”matrix. Also, it can easily be verified that when t < s, KG(t; s) = CeAt−1

G e−AsB−CeA(t−s)B.Hence, KG(t; s) is continuous at s = t iff CB = 0.

In the sequel we will denote system (3) using the following compact block notation:

G =

(A ΩΥ B

C D

):

The time interval on which the system operates will be usually clear from the context. WhenG is causal we will omit also the boundary condition “window” and write simply

(A B

C D

). We

will say that two STPBC G1 and G2 are equivalent if D1 = D2 and KG1(t; s) = KG2(t; s) almosteverywhere.


Manipulations over STPBC can be performed in terms of matrix parameters of their state-space representation, much like manipulations over the standard LTI causal systems. Below weconsider basic operations, which will be used in the sequel.

• Adjoint system:

G∗ =

(−A ′ eA

′hΥ′ Ξ′G−1Ω′ Ξ′G−1eA

′h C ′

−B ′ D ′

): (7a)

The formula above can be derived from the fact that the kernel of G∗ is just K ′(s; t). Note, thatsince G∗ = I, the system G∗ also has well-posed boundary conditions.

The issues of similarity transformation and order reduction for STPBC are in general morecomplicated than for LTI causal systems, see [9] for details. Yet the following two operations,which are counterparts of those for LTI causal systems, are sufficient for the purposes of thispaper:

• Similarity transformation:

G =

(TAT−1 SΩT−1SΥT−1 TB

CT−1 D

)(7b)

for any nonsingular matrices T and S.

• Dilation:

G =

? ? ?

0 A ?

0 0 ?

? ? ?

0 ?

0 0 ?

? ? ?

0 ?

0 0 ?

?

B

0

0 C ? D

; (7c)

where ? denotes irrelevant blocks.

Equality (7c) actually implies that if some states and boundary conditions of STPBC are either“unobservable” or “uncontrollable”, the state dimension of the system can be reduced.

• Addition:

G1+ G2 =

A10

0

A2

[1 0

0 2

][1 0

0 2

]B1B2

C1 C2 D1+D2

: (7d)

• Multiplication:

G1G2 =

A10

B1C2A2

[1 0

0 2

][1 0

0 2

]B1D2B2

C1 D1C2 D1D2

: (7e)

It can easily be verified that both the sum and the product of STPBC have well-posed boundaryconditions also.

• Inversion (exists iff det(D) 6= 0 and det(+ e(A−BD−1C)h) 6= 0):

G−1 =

(A− BD−1C ΩΥ BD−1

−D−1C D−1

): (7f)

It is seen, that in contrast to causal LTI systems, the nonsingularity of D is not sufficient for theinvertibility of STPBC. It is easily seen that the second condition for the existence of G−1 justconsists on the requirement that G−1 has well-posed boundary conditions. Thus, if an STPBCis invertible then the inversion has well-posed boundary conditions.


3 The new representation

3.1 Motivation

Let us return to the lifted system G given by (1) and assume that the feedthrough matrix of itsoriginal G is zero, that is D = 0. Consider the feedthrough term of G, D : L2[0; h] 7→ L2[0; h],which is an infinite rank operator. Inspecting (2d), one can see that D is just the restriction ofG to L2[0; h]. This prompts the representation of D by the following STPBC:

D =

(A B

C 0

):

The benefit of using the latter representation becomes evident when one tries to handle theoperator (I−D∗D)−1, which appears, for instance, in the sampled-dataH∞ problems [2]. Indeed,using the integral representation of D in (2d) one can get that = (I− D∗D)−1! iff

(t) = !(t) −

∫htB ′e−A′(t−s)C ′C

∫s0eA(s−τ)B()dds:

It is not clear, however, how to solve the latter equation. On the other hand, using (7) one caneasily get that:

(I− D∗D)−1 =

(I−

(A B

C 0

)∗(A B

C 0

))−1

=

−A ′

0

C ′CA

[0 0

0 I

][ I 00 0

]0

B

B ′ 0 I

−1

=

−A ′

−BB ′C ′CA

[0 0

0 I

][ I 00 0

]0

B

−B ′ 0 I

and I− D∗D is invertible iff the latter STPBC has well-posed boundary conditions.

Thus, it is seen that the representation of D as an STPBC enables to simplify considerablyalgebraic manipulations over it. This fact has been pointed out by Gohberg and Kaashoek [8]for a rather general class of integral operators L2[0; h] 7→ L2[0; h]. In the context of sampled-data systems, Bamieh and Pearson [2] exploited the STPBC representation of D to computean operator similar to the operator (I − D∗D)−1. In [2], however, this operator has been thenconverted back to the integral form and all other operations, such as the computation of thematrix

Mα:= B(I− D∗D)−1B∗; (8)

have been performed by involved operations over the integral representation. It is worth stressingthat in [2] only the case of zero-order hold and ideal sampler has been treated and the simplifyingassumption D11 = 0 has been made. For more general cases manipulations over the integralrepresentation of operators become even more complicated.

The other parameters of the model (1), that is the operators A, B, and C, are not operatorson L2[0; h] and hence cannot be replaced by STPBC in a straightforward manner. Yet motivatedby the simplicity and efficiency of treating D as the “truncation” of G, we will propose a newrepresentation of all the parameters of (1), which enables one to replace algebraic manipulationsover these parameters with state space manipulations over STPBC.


3.2 Main result

The extension of the idea discussed in the previous subsection requires introduction of two newoperators. Given a number ∈ [0; h], the impulse operator Iθ transforms a vector ∈ Rn intoa modulated -impulse as follows:

= Iθ ⇐⇒ (t) = (t− ):

Define also the operator I∗θ, which transforms a function ∈ Cn[0; h] into a vector from Rn

as follows:

= I∗θ ⇐⇒ = ();

that is I∗θ is in fact the ideal sampler. Note, that the notation I∗θ is chosen as if the operator I∗θwere the adjoint of Iθ. Indeed, it is easy to see that given an h ≥ the equality∫h

0 ′()

(Iθ)()d = 〈I∗θ; 〉R

is satisfied for any ∈ Rn and any ∈ Cn[0; h]. Yet the range of the impulse operator andthe domain of the sampler are not L2[0; h] (they are even not Hilbert spaces) and thus strictlyspeaking we cannot say that I∗θ is the adjoint of Iθ. Nevertheless, we will proceed with thisabuse of notation for the following reason. Throughout the paper we will use the above twooperators just for constructing operators like GIθ : R 7→ L2[0; h] or I∗θG : L2[0; h] 7→ R, whereG is an STPBC with a zero “D” matrix. Since these operators are operators on Hilbert space,the notion of adjoint operator is well defined in this case. Moreover, it will be shown in thenext subsection (Lemma 1) that (GIθ)

∗ = I∗θG∗ and (I∗θG)∗ = G∗Iθ. Therefore, our notation is

justified and one may think of I∗θ as the “adjoint” of the impulse operator.We are now in the position to formulate the main result of the paper:

Theorem 1. For any LTI continuous-time system G =[A B

C D

]its lifting G

:= WhGW−1

h is

also LTI and has the state-space realization (1), where

[A B

C D

]=

[I∗h 0

0 I

] A I B

I 0 0

C 0 D

[ I0 0

0 I

]: (9)

Proof. Follows by a straightforward application of (5).

It is clear that (9) and (2) are equivalent in the sense that these equations define the sameoperators. The advantage of replacing the integral representation in (2) with STPBC in (9) stemsfrom the fact that algebraic manipulations over the latter can be performed efficiently in termsof their state-space realizations. The efficiency of this approach for performing manipulationsover the operator D was demonstrated in the previous subsection. In the next subsection we willshow that the impulse operator Iθ and the sampling operator I∗θ fit nicely into this framework,thus enabling to extend the technique to all other parameters of (1). We will show that Iθ andI∗θ can easily be separated from STPBC when dealing with matrices like (8). This enables oneto reduce the computation of any matrix like (8) to the calculation of a matrix exponential bymeans of straightforward manipulations over state-space realizations. Moreover, in some casesthe impulse and the sampling operators can be “absorbed” into STPBC, that allows one toapply the state-space machinery to more complicated operators, like D∗D+ B∗XB for example.


Finally, note that all operations over the operators in (9) are reduced to operations overSTPBC, the parameters of which are those of the original continuous-time plant. It means thatfinal formulae can be obtained directly in terms of the parameters of the continuous-time plant.This may provide an additional insight into a wide range of sampled-data problem.

3.3 Technical machinery

We start with the following lemma, which constitutes the basis for a “separation” between theoperators Iθ, I∗θ and the STPBC:

Lemma 1. Given an STPBC, G, on the interval [0; h] with D = 0, then

(GIθ)∗ = I∗θG

∗ and (I∗θG)∗ = G∗Iθ

for any ∈ [0; h].

Proof. We prove only the first statement. The second one then follows from the facts that G∗ isalso STPBC and for any Hilbert space operator O, (O∗)∗ = O.

Let KG(t; s) be the kernel of G. Then for any ∈ L2[0; h] and any ∈ R

〈;GIθ〉L2[0,h] =

∫h0 ′(t)

∫h0KG(t; s)(s− )dsdt =

∫h0 ′(s)KG(s; )ds

= 〈I∗θG∗; 〉R;

where the latter equality follows by noticing that G∗ has the kernel K ′(s; t).

To demonstrate the use of Lemma 1, consider the computation of the matrix Mα in (8).It follows from Theorem 1 that B is of the form I∗hB for some B : L2[0; h] 7→ L2[0; h]. Then,Lemma 1 gives that:

Mα = I∗hB(I− D∗D)−1B∗Ih = I∗hO Ih:

Now, the operator O above can easily be computed using the STPBC representations of D and B,while the operators Ih and I∗h get into the picture only at the final stage. Such a “separation” ofthe impulse and the sampling operators from STPBC can efficiently be used in a wide spectrumof sampled-data problems (see Section 4, where some applications are discussed), enabling toreduce computations to the computation of matrices of the form I∗θ1O Iθ2 , where i are either 0or h and O is an STPBC. The latter matrix, in turn, can easily be computed using the following

Lemma 2. Let G =(A ΩΥ B

C 0

), then:

I∗hG I0 = CeAh(+ eAh)−1B; (10a)I∗0G Ih = −C(+ eAh)−1B; (10b)

and if in addition CB = 0, then

I∗0G I0 = C(+ eAh)−1B; (10c)I∗hG Ih = −CeAh(+ eAh)−1B: (10d)

Proof. Formulae (10) can easily be derived using (5). Just note, that if CB = 0 then the kernelKG(t; s) is well defined also when t = s, that in turn guarantees that so are (10c) and (10d).


Lemmas 1 and 2 together with formulae (7) constitute a powerful machinery for dealingwith sampled-data systems in the lifted domain. Although these results are relatively simpleand straightforward, to the best of the authors’ knowledge they have not been exploited in theliterature before. Consequently, some available results in sampled-data control literature werederived using unnecessarily cumbersome techniques and significant simplifications have beenoverlooked (see Remark 4.1).

The impulse and the sampling operators can be not only easily separated from, but can alsobe “absorbed” into the STPBC:

Lemma 3. Let

G1:=(A1 Ω1 Υ1 B1C1 D1

); GC1

:=(A1 Ω1 Υ1 I

C1 0

);

G2:=(A2 Ω2 Υ2 B2C2 D2

); GB2

:=(A2 Ω2 Υ2 B2I 0

);

and i, i = 1; 2, be either 0 or 1. Then for any appropriately dimensioned matrix M

G1G2+ GC1Iλ1hMI∗λ2hGB2 =

A10

B1C2A2

[Ω1 (1− λ2)M10 Ω2

][Υ1 λ2M10 Υ2

]B1D2B2

C1 D1C2 D1D2

; (11)

where M1 =(11− (1− 1)1

)M.

Proof. Let Gα denote the STPBC in the right-hand side of (11) and[11() 12()

0 22()

]:= exp

([A1 B1C20 A2

]

):

Using (4) one can get that

−1Gα

=

[G1 112(h) + (1− 2)M1+ 2M122(h)

0 G2

]−1

= −1G1G2

−

[I

0

]−1

G1M1

((1− 2)I+ 222(h)

)−1

G2

[0 I

]:

Now, using (6) it is a matter of a simple algebra to verify that

KGα(t; s) = KG1G2(t; s) + C111(t)−1G1M1

((1− 2)

−1G2222− 222(h)−1

G22)22(−s)B2;

which leads to (11) and, thus, completes the proof.

Although the operator in the left-hand side of (11) is quite complicated, the matrixM affectsthe STPBC in the right-hand side through “reshaping” its boundary conditions only. This,perhaps surprising, result makes the representation proposed in Theorem 1 a really powerfultool for the analysis and design of sampled-data systems in the lifted domain. Roughly speaking,while Lemmas 1 and 2 cover the cases when a sampler and a hold are fixed, Lemma 3 enablesto extend the scope of the applicability of the lifting technique also to the design of A/D andD/A converters. Relation (11) can be used to compute infinite-dimensional operators, like(D∗D + B∗XB)−1, which appear in various sampled-data control and filtering problems (seeSection 5, where Lemma 3 is used in the solution of a lifted Riccati equation; [20], where it isused to design H2 and H∞ optimal sampling and hold functions; and [18], where the sampled-data H∞ filtering problem is treated).


w

u

P

z

y

-

Sh -y

K -u

Hh

Figure 1: Sampled-data setup

4 Computational issues in sampled-data optimal control

The purpose of this section is to demonstrate the capabilities of the proposed new representationof Theorem 1 in handling the computations in sampled-data H2 and H∞ optimization problems.We will consider these problems in a rather general setting, without any simplifying assumptionson the plant parameters and with generalized sampling and hold functions. We will show thatall the computations involving the infinite dimensional parameters of the lifted systems in theabove general setup can easily be reduced to the computations of matrix exponentials.

It is worth noting that since the purpose of this section is to treat the relevant computationalissues only, we will not be concerned with the detailed statement of the sampled-data H2 andH∞ problems. The reader is referred to the book by Chen and Francis [6] and the referencestherein for details. For the same reason, we will not present complete solutions to theH2 and H∞problems. Rather, we will exploit the well known fact [13, 3, 2] that both of these sampled-dataproblems can be reduced to equivalent pure discrete-time finite-dimensional ones. The latterproblems, in turn, should then be solved using standard methods. The “discretization” in thefirst stage of such an approach is connected with the computation of matrices involving infinite-dimensional parameters of the lifted plant. Although such an approach leads to an unnecessarilycumbersome solution for the controller, it requires minimum preliminaries to be presented andat the same time captures all the essential computational issues. Just note that conceptuallyclearer and more efficient approach to the solutions of both the H2 and the H∞ sampled-dataproblems is to solve these problems directly in the lifted domain, thus avoiding the two-stageprocedure mentioned above. Moreover, the computations required in the direct approach aresignificantly reduced, see [17, 21] for details.

4.1 General sampled-data setup

The general sampled-data control system setup we intend to deal with in this section is depictedin Fig. 1, where P is a continuous-time generalized plant, K is a discrete-time controller, Sh isa sampler with a period h, and Hh is a hold device synchronized with Sh.

We assume that the plant P is LTI and has the following state-space realization:

P =

A B1 B2C1 D11 D12C2 D21 D22

;where the partitioning is compatible with that in Fig. 1.

As a hold device Hh we consider the following zero-order generalized hold:

u = Hhu ⇐⇒ u(kh+ ) = H()u[k] ∀ ∈ [0; h);


w

u

P

z

y

-

K

Figure 2: Lifted sampled-data setup

where the generalized hold function H is assumed to be of the form

H() = CHeAHτBH (12)

for some finite dimensional matrices AH, BH, and CH. Although this choice of H is slightly lessgeneral than is conventionally assumed [12, 1], any function from L2[0; h] can be approximatedby (12) with arbitrary accuracy. Thus (12) does not imply much loss of generality. Moreover,the majority of single rate generalized hold devices designed in the literature for LTI systemsare indeed of the form (12) (see [12, 28, 23, 22] for example).

The sampler Sh is also assumed to belong to the class of zero-order generalized samplers:

y = Shy ⇐⇒ y[k] =

∫h0S(h− )y(kh+ )d; (13)

where

S() = DS() + CSeASτBS

for some finite dimensional matrices AS, BS, CS, and DS. Note that strictly speaking the upperlimit in (13) has to be h− rather than h. This point, however, is only relevant in the case wheny(t) is discontinuous at t = kh whenever DS 6= 0. To avoid this situation we will make thefollowing assumption:

(A1): DS[D21 D22

]= 0.

Assumption (A1) states that the sampler Sh operates over proper signals. Also, (A1) impliesin fact that prefiltering by an antialiasing filter is provided if necessary. Hence, it guaranteesboundedness of the sampling operations also [6].

The sampler Sh in (13) is in some sense “non-causal” since y[k] depends on y(t) in theinterval [kh; (k+ 1)h) rather than in [(k− 1)h; kh) as is the usual convention [1, 23]. However,actually we are interested in the causality of the continuous-time controller K

:= HhKSh and

non-causality of Sh is just a matter of synchronizing the time scales for y and u. Under the“causal” sampler the causality of K is equivalent to the causality of K. In our choice of Sh, K iscausal iff K is strictly causal [19]. Although the discrete controller can no longer be consideredas arbitrary, in some cases the strict causality constraint does not complicate the design1. Onthe other hand, the choice of the sampler in (13) avoids the increase in the state-space dimensionof the plant, which takes place when the conventional generalized sampler is considered [1] (see[19] for a detailed discussion of this issue in a more general setting).

1In H2 and H∞ optimizations the design is even simplified, see e.g. [16].


The system in Fig. 1 is h-periodic in the continuous-time domain. Hence we can apply thelifting operator Wh and get the equivalent discrete-time shift invariant system, shown in Fig. 2,where

P:=[

Wh 0

0 Sh

]P[

W−1h 0

0 Hh

]:

To derive the state-space realization of the LTI system P note, that in the lifted domain thesampler ShW

−1h and the hold WhHh are both memoryless gains I∗hGS and GHI0, respectively,

where

GH =

(AH BHCH 0

);

GS =

(AS BSCS DS

):

Then, using Theorem 1 it is not difficult to verify that

P =

A B1 B2

C1 D11 D12

C2 D21 D22

;A = I∗hGA I0;

B1 = I∗hGB1 ;

B2 = I∗hGB2GH I0;

C1 = GC1 I0;

C2 = I∗hGSGC2 I0;

D11= GD11 ;

D12= GD12GH I0;

D21= I∗hGSGD21 ;

D22= I∗hGSGD22GH I0;

where:[GA GBjGCi GDij

]:=

A I BjI 0 0

Ci 0 Dij

i; j = 1; 2:

Having the state-space model of P we are now in the position to consider the sampled-dataH2 and H∞ control problems for the setup in Fig. 1.

4.2 Sampled-data H2 problem

The H2 problem for the sampled-data setup in Fig. 1 consists on minimizing the H2 norm ofthe closed-loop operator from w to z, that is ‖F`

(P;HhKSh

)‖H2 , by means of a strictly causal

K (for the definition and interpretations of the H2 norm for sampled-data systems see [13, 3]).The following lemma, which is completely in the spirit of [3], establishes that the solution to thesampled-data H2 problem can be reduced to an equivalent pure discrete-time one:

Lemma 4. Form the discrete-time system

P2 =

A2 B2,1 B2,2

C2,1 0 D2,12C2,2 D2,21 D2,22

;where B2,1, D2,21, C2,1, and D2,12 are any matrices such that

M2,12:=

[B2,1D2,21

][B ′2,1 D ′2,21

]=

[B1

D21

][B∗1 D∗21

]; (14a)

M2,21:=

[C ′2,1D ′2,12

][C2,1 D2,12

]=

[C∗1D∗12

][C1 D12

]; (14b)


and

M2,22:=

[A2 B2,2C2,2 D2,22

]=

[A B2C2 D22

]: (14c)

Then the controller HhKSh internally stabilizes P iff K internally stabilizes P2 and for anystrictly causal stabilizing K

‖F`(P;HhKSh

)‖2H2 =

1

h

(‖D11‖2HS+ ‖F`

(P2; K

)‖2H2)

and ‖D11‖HS is finite iff D11= 0.

Proof. Although the H2 problem considered here is more general than the one considered in[3], the Lemma can be proven following the same arguments as in [3]. Just note that the keypoint in [3] was the fact that the feedthrough term of the lifted closed-loop operator is D11,which means that it is not affected by the controller. Although in our case

[D21 D22

]6= 0,

we consider strictly proper controllers only. Hence, the same arguments apply.

Lemma 4 states that any sampled-data H2 problem can be reduced to a standard finite-dimensional discrete H2 problem. In order to formulate the latter problem, however, one hasto compute the matrices M2,12 and M2,21, which involve infinite-dimensional parameters ofthe lifted plant P. If the parameters of P were represented via integrals like in (1), then thecomputation of the matrices in (14a) and (14b) would be reduced to the computation of multipleintegrals involving matrix exponentials [13, 3]. Then for the simple case where Sh is the idealsampler and Hh is the zero-order hold the formulae of Van Loan [25] could be used to reduce itto the computations of matrix exponentials only [3]. Yet for the generalized sampling and holddevices this approach might be too involved.

On the other hand, the use of Theorem 1 together with the machinery of Lemmas 1 and 2makes it possible to get computationally efficient formulae in a direct and simple manner. Tothis end one just needs to substitute the STPBC representation of the parameters of P into theformulae for M2,ij, multiply STPBC, and then compute the impulse responses of the resultingsystems. Formally, define the following STPBC:

G12:=[

GC1 GD12GH]

=

A B2CH I 0

0 AH 0 BHC1 D12CH 0 0

; (15a)

G21:=

[GB1

GSGD21

]=

AS BSC2 BSD210 A B10 I 0

CS DSC2 0

; (15b)

and

G22:=

[GA GB2GH

GSGC2 GSGD22GH

]=

AS BSC2 BSD22CH 0 0

0 A B2CH I 0

0 0 AH 0 BH0 I 0 0 0

CS DSC2 0 0 0

: (15c)


Then it is straightforward to show that

M2,12= I∗0G∗12G12I0; (16a)M2,21= I∗hG21G

∗21Ih; (16b)

and

M2,22= I∗hG22I0: (16c)

One can see now that the computations of the matrices M2,12and M2,21as well as the matrixM2,22 are reduced to the computations of the impulse response of the respective STPBC. Toperform the latter one just needs to get the realizations of the LTI systems G∗12G12 and G21G

∗21

and then apply Lemma 2.To this end form the matrix:

2 =

AS 0 0 BSC2 BSD21B′1 BSD22CH BSD21D

′21B

′S

0 −A ′H −C ′HB′2 C ′HD

′12C1 0 C ′HD

′12D12CH 0

0 0 −A ′ C ′1C1 0 C ′1D12CH 0

0 0 0 A B1B′1 B2CH B1D

′21B

′S

0 0 0 0 −A ′ 0 −C ′2B′S

0 0 0 0 0 AH 0

0 0 0 0 0 0 −A ′S

and partition eΨ2h in a fashion compatible with the partition of 2 as follows:

eΨ2h =

2,11 0 0 2,14 2,15 2,16 2,170 2,22 2,23 2,24 2,25 2,26 2,270 0 2,33 2,34 2,35 2,36 2,370 0 0 2,44 2,45 2,46 2,470 0 0 0 2,55 0 2,570 0 0 0 0 2,66 0

0 0 0 0 0 0 2,77

:

Also, denote the following two matrices:

MS:=

[I 0

DSC2 CS

]; (17a)

MH:=

[I 0

0 BH

]: (17b)

Then we can formulate the main result of this subsection as follows:

Theorem 2. Form the matrix eΨ2h and partition it as above. Then

M2,12= M ′H

[2,44 2,460 2,66

] ′[2,34 2,362,24 2,26

]MH;

M2,21= MS

[2,45 2,472,15 2,17

][2,44 0

2,14 2,11

] ′M ′S;

M2,22= MS

[2,44 2,462,14 2,16

]MH;

and, in case where D11= 0,

‖D11‖2HS= tr ′2,442,35

:


Proof. The formulae for M2,ij can easily be derived from (16) using Lemma 2 and taking intoaccount the equalities[

2,33 0

2,23 2,22

]−1

=

[2,44 2,460 2,66

] ′and [

2,55 2,570 2,77

]−1

=

[2,44 0

2,14 2,11

] ′:

Now, using the integral expression for the Hilbert-Schmidt norm [3] we can write:

‖D11‖2HS= tr(∫h0

∫t0C1e

A(t−s)B1B′1eA′(t−s)C ′1dsdt

)= tr

(eA′h

∫h0e−A′(h−t)C ′1

∫t0C1e

A(t−s)B1B′1e

−A′sdsdt

);

from which the formula for ‖D11‖HS follows by noting that the integral in the last expression is

just I∗h

(−A ′ C ′

I 0

)(A B

C 0

)(−A ′ I

B ′ 0

)I0.

4.3 Sampled-data H∞ problem

The H∞ problem for the sampled-data setup in Fig. 1 consists on finding a strictly causal K

(if it exists) such that ‖F`(P;HhKSh

)‖H∞ < for a given constant > 0 (for the definition

and interpretation of the H∞ norm for sampled-data systems see [23, 2]). The following lemmaestablishes that the sampled-data H∞ problem is equivalent to a pure discrete-time H∞ problem:

Lemma 5. Let P be such that ‖D11‖2 < . Form the discrete-time system

P∞ =

A∞ B∞,1 B∞,2C∞,1 0 D∞,12C∞,2 D∞,21 D∞,22

;where B∞,1, D∞,21, C∞,1, and D∞,12 are any matrices such that

M∞,21 :=[B∞,1D∞,21

][B ′∞,1 D ′∞,21 ] = 2

[B1

D21

]( 2I− D∗11D11

)−1[B∗1 D∗21

];

M∞,12 :=[C ′∞,1D ′∞,12

][C∞,1 D∞,12 ] =

[C∗1D∗12

]( 2I− D11D

∗11

)−1[C1 D12

];

and

M∞,22 :=[A∞ B∞,2C∞,2 D∞,22

]=

[A B2C2 D22

]+

[B1

D21

]D∗11

( 2I− D11D

∗11

)−1[C1 D12

]:

Then the following two statements are equivalent:

i) The controller HhKSh internally stabilizes P and ‖F`(P;HhKSh

)‖H∞ < .


ii) The controller K internally stabilizes P∞ and ‖F`(P∞; K)‖H∞ < .

Proof. The Lemma can be easily proven by the same loop shifting arguments as in [2].

The condition ‖D11‖2 < of Lemma 5 does not depend on the sampling and hold devices.Hence, it can be checked in the fashion proposed in [2, 6]. All other matrices involving infinitedimensional operators, that is M∞,12, M∞,21, and M∞,22, do depend on Sh and Hh. Forgeneralized S and H the application of the technique of [2] might be too difficult. Yet theproposed approach makes the calculations for the generalized Sh and Hh not more involved thanfor the ideal sampler and the zero-order hold.

As for the H2 case, it is easy to see that all the matrices M∞,ij can be treated as impulseresponses of STPBC. Although the expressions for M∞,ij look more cumbersome then for M2,ij

in Lemma 4, we will show that the computational formulae for the H∞ case are not morecomplicated then those for the H2 one. More involved derivation is required in this case. Itwill be shown, however, that all the manipulations are rather straightforward and the insight,gained by the fact that all manipulations are over LTI systems, can be exploited to simplify thefinal formulae.

First, denote

Rγ:= (I− 1

γ2D11D

′11)

−1;

Sγ:= (I− 1

γ2D ′11D11)

−1;

[Aγ BγCγ Dγ

]:=

[A B2C2 D22

]+1

2

[B1D21

]D ′11Rγ

[C1 D12

]:

Then, assume at first that = 1. This assumption considerably simplifies the derivation andcan always be assured by an appropriate scaling of the parameters of P, for example[

C1 D11 D12]→ −1

[C1 D11 D12

]: (18)

Using (7e) and (7f) it is not difficult to show that

(I− GD11G

∗D11

)−1=

Aγ−C ′1RγC1

B1SγB′1

−A ′γ

[I 0

0 0

][ 0 0

0 I

]B1D

′11Rγ

−C ′1RγRγC1 RγD11B

′1 Rγ

and

(I− G∗D11GD11

)−1=

Aγ−C ′1RγC1

B1SγB′1

−A ′γ

[I 0

0 0

][ 0 0

0 I

]B1Sγ

−C ′1D11SγSγD

′11C1 SγB

′1 Sγ

and if ‖D11‖2 < 1 then the two STPBC above have well-posed boundary conditions. Also, G12and G21 defined by (15) can be factorized as follows:

G12=

(A I B2C1 0 D12

) AH 0 BH0 I 0

CH 0 0

:= QCQH;

G21=

AS 0 BS0 I 0

CS 0 DS

A B1I 0

C2 D21

:= QSQB:


Now, since the “transmission zeros” of both (I − GD11G∗D11

)−1 and (I − G∗D11GD11)−1 coincide

with the “poles” of Q∗CQC, G∗D11QC, and QBQ∗B, “pole-zero cancelations” in the systems

Q12:= Q∗C

(I− GD11G

∗D11

)−1QC;

Q21:= QB

(I− G∗D11GD11

)−1Q∗B;

and in

Q22:= P∗11

(I− GD11G

∗D11

)−1QC

can be expected. Although Qij are STPBC rather than LTI causal systems and hence the initialconditions have to be taken into account, the “pole-zero cancelation” arguments above applyhere as well. Indeed, using (7b) and (7c) it is possible to show by a simple algebra that

Q12=

Aγ

−C ′1RγC1

B1SγB′1

−A ′γ

[I 0

0 0

][ 0 0

0 I

]I

0

Bγ−C ′1RγD12

0

D ′12RγC1

I

B ′γ

0

0

0

D ′12RγD12

;

Q21=

Aγ

−C ′1RγC1

B1SγB′1

−A ′γ

[I 0

0 0

][ 0 0

0 I

]0

−I

B1SγD′21

−C ′γI

Cγ

0

D21SγB′1

0

0

0

D21SγD′21

;and

Q22=

Aγ−C ′1RγC1

B1SγB′1

−A ′γ

[I 0

0 0

][ 0 0

0 I

]I

0

Bγ−C ′1RγD12

D ′11RγC1 SγB′1 0 D ′11RγD12

:Proceeding further, we can get that

Q22:=

[GA GB2GC2 GD22

]+ QBQ22

=

Aγ

−C ′1RγC1

B1SγB′1

−A ′γ

[I 0

0 0

][ 0 0

0 I

]I

0

Bγ−C ′1RγD12

I

Cγ

0

D21SγB′1

0

0

0

Dγ

:Now, it is easy to see that[

M∞,12 M ′∞,22M∞,22 M∞,21

]=

[I∗0Q∗H 0

0 I∗hQS

][Q12 Q∗22Q22 Q21

][QHI0 0

0 Q∗SIh

]: (19)

Thus, as in the H2 case, the calculations are reduced to the calculation of impulse responses ofan STPBC, that in turn allows to use Lemma 2.

Form the following matrix:

∞ =

AS 0 BSCγ BSD21SγB′1 BSDγCH BSD21SγD

′21B

′S

0 −A ′H − 1γ2C ′HD

′12RγC1 −C ′HB

′γ − 1

γ2C ′HD

′12RγD12CH −C ′HD

′γB′S

0 0 Aγ B1SγB′1 BγCH B1SγD

′21B

′S

0 0 − 1γ2C ′1RγC1 −A ′γ − 1

γ2C ′1RγD12CH −C ′γB

′S

0 0 0 0 AH 0

0 0 0 0 0 −A ′S

:


Then, partitioning eΨ∞h according to the partition of ∞ as follows:

eΨ∞h =

∞,11 0 ∞,13 ∞,14 ∞,15 ∞,160 ∞,22 ∞,23 ∞,24 ∞,25 ∞,260 0 ∞,33 ∞,34 ∞,35 ∞,360 0 ∞,43 ∞,44 ∞,45 ∞,460 0 0 0 ∞,55 0

0 0 0 0 0 ∞,66

we can formulate the main result of this subsection:

Theorem 3. Form the matrix eΨ∞h and partition it as above. Then if ‖D11‖2 < the matrix∞,44 is invertible and

M∞,12= −M ′H

[0 −1∞,44

′∞,55 − ′∞,55∞,24−1∞,44][∞,23 ∞,25∞,43 ∞,45

]MH;

M∞,21= MS

[∞,34 ∞,36∞,14 ∞,16

][−1∞,44 −−1∞,44∞,46 ′∞,110 ′∞,11

]M ′S;

and

M∞,22= MS

([∞,33 ∞,35∞,13 ∞,15

]−

[∞,34∞,14

]−1∞,44[ ∞,43 ∞,45 ])MH;

where MS and MH are defined by (17).

Proof. The formulae can be derived from (19) noting that all Qij share the same state space. Onejust has to take into account scaling (18) and the equalities −1∞,22= ′∞,55and −1∞,66= ′∞,11.Remark 4.1. It is of interest to compare the formulae in Theorem 3 with those derived in theliterature for similar problems by means of the lifting technique. The book [6, §13.6] containsthe full derivation of the scheme proposed in [2] for the case of the ideal sampler (AS = 0,BS = 0, CS = 0, and DS = I) and the zero-order hold (AH = 0, BH = I, and CH = I). Thecomputations there are also reduced to the computations of matrix exponentials, however thedimension of these exponentials is larger then the dimension of the matrix ∞ in our result.It seems that the reason is that the manipulations with the integral representation (2) doesnot provide sufficient insight into the problem and hence some “cancelations” may have beenoverlooked. The approach in [10] is based on an “H∞ discretization,” which is different from theone given in Lemma 5. The motivation there was that for such a discretization it happens thatA∞ = A, B∞,2 = B2, C∞,2 = C2, and D∞,22= D22. However, the computational formulae in[10] (which are presented only for the ideal sampler and the zero-order hold) again involve matrixexponentials of higher dimension than in our case. Finally, the recent report [5] argues that oneof the advantages of using the so-called frequency-domain lifting over the time domain liftingis due to the possibility of deriving easily implementable formulae for the computation of theL2-induced norm of the sampled-data sensitivity operator for systems involving the generalizedhold function. Nevertheless, we have shown in this subsection that Theorem 1 allows one tohandle computations for even more general sampled-data H∞ problems (the problem treated in[5] corresponds to the case of the ideal sampler and B1 = 0) in a straightforward and efficientmanner.


5 The algebraic Riccati equations in the lifted domain

The results presented in the previous section requited the use of formulae (7) and Lemmas 1 and2 only. In this section we intend to demonstrate the capabilities of Lemma 3. We will investigatea general discrete-time algebraic Riccati equation (DTARE) associated with the lifting G of anLTI system G. Perhaps surprisingly, we will show that the DTARE associated with G mighthave more solution than the continuous-time algebraic Riccati equation (CTARE) associatedwith G. At the same time, however, the unique stabilizing solutions to these equations, as wellas the conditions of their existence, coincide.

Let G be a finite dimensional LTI continuous-time system with the realization

G =

[A B

C D

]and J = J ′ be a square matrix of the same dimension as the output dimension of G such thatthe matrix D ′JD is nonsingular. We will associate with G and J the following operator:

RG,J(X):= A ′X+ XA+ C ′JC− (XB+ C ′JD)(D ′JD)−1(D ′JC+ B ′X);

which is said to be the continuous-time Riccati operator. Then the equation

RG,J(X) = 0 (20)

is the well known CTARE, which plays an important role in various continuous-time controlproblems, such as H2 (J = I) and H∞ (J =

[I 00 −I

]) optimizations, robust control and so on and

is extensively investigated in the literature (see [15] and the references therein).Now consider the lifting of G,

G:= WhGW−1

h =

[A B

C D

]: (21)

The Riccati operator associated with G and J is the discrete-time Riccati operator

RG,J(X):= A ′XA− X+ C∗JC−

(A ′XB+ C∗JD

)(D∗JD+ B∗XB

)−1(D∗JC+ B∗XA

)and the corresponding DTARE is

RG,J(X) = 0: (22)

DTAREs are well understood when associated with standard finite-dimensional discrete systemsG. In the latter case all the parameters of the DTARE RG,J(X) = 0 are finite-dimensionalmatrices. The case when the discrete-time Riccati operator is associated with “semi-lifted”systems of the form G is not much more complicated. Such a case may arise for example in thesolution of the sampled-data H2 problem, like that considered in Subsection 4.2. If instead ofconverting the problem there to an equivalent finite-dimensional one we tried to solve it directlyin the lifted domain, we would have to solve the control DTARE RP12,I

(X) = 0 and its dual.Although these equations involve infinite-dimensional parameters of the lifted plant, they arehidden inside finite-dimensional matrices, like D∗12D12, and thus do not present any difficulties.

The discrete-time Riccati operator associated with the lifted system G seems to be muchmore difficult to analyze. In this case the infinite dimensionality in the last term cannot becircumvented in a direct fashion. A possible way to deal with (22) is to consider the associated


simplectic pencil, which is finite dimensional (see [7], where such an approach is outlined). Inthe case of the DTAREs, however, the associated simplectic pencil can in general give onlythe sufficient conditions for the existence of solutions to DTAREs. Moreover, it may also beimportant to calculate the lifted state feedback “gain”

F:= −(D∗JD+ B∗XB)−1(D∗JC+ B∗XA):

That calculation appears to be a nontrivial problem. A comprehensive treatment of DTAREsis possible by deflating subspaces of extended simplectic pencils [11], but the extended pencilassociated with (22) is infinite-dimensional again.

On the other hand, it is easy to see that the factors in the last term of RG,J(X) fall underthe scope of Lemma 3. This observation suggests that the Riccati operator can be calculateddirectly, term by term. To this end introduce the matrix

H =

[H11 H12H21 H22

]:=

[A 0

−C ′JC −A ′

]−

[B

−C ′JD

](D ′JD)−1

[D ′JC B ′

];

which is the Hamiltonian matrix associated with the CTARE (20), and denote

eHh=

[11 1221 22

];

where the partitioning corresponds to that of H. Then we can formulate the following lemma:

Lemma 6. Given system (21) and a matrix J = J ′, then the operator D∗JD+B∗XB is invertibleon L2[0; h] if and only if det(D ′JD) 6= 0 and det(22− X12) 6= 0. Moreover, if these twoconditions hold true, then

RG,J(X) = I∗0

H11H21

H12H22

[I 00 0

][ 0 0−X I

]I

X

−X I 0

I0 (23a)

and also

F = −(D ′JD)−1

H11H21

H12H22

[I 00 0

][ 0 0−X I

]I

X

D ′JC B ′ 0

I0: (23b)

Proof. Using Lemma 3 one can easily obtain that

D∗JD+ B∗XB =

−A ′

0

−C ′JCA

[0 00 I

][ I −X0 0

]−C ′JDB

B ′ D ′JC D ′JD

;D∗JC+ B∗XA =

−A ′

0

−C ′JCA

[0 00 I

][ I −X0 0

]0

I

B ′ D ′JC 0

I0;

and

C∗JC+ A ′XA = I∗0

−A ′

0

−C ′JCA

[0 00 I

][ I −X0 0

]0

I

I 0 0

I0:


Now, from (7f) it follows that

(D∗JD+ B∗XB

)−1= (D ′JD)−1

H11H21

H12H22

[I 00 0

][ 0 0−X I

]−B

C ′JDD ′JC B ′ D ′JD

(D ′JD)−1

and the inversion exists iff the matrices D ′JD and[I 0

0 0

]+

[0 0

−X I

]eHh=

[I 0

21− X11 22− X12

]are nonsingular. This immediately yields the existence conditions of the Lemma.

Using (7e), (7b), (7c), and (7d) it is straightforward to show that (23b) holds true and that

RG,J(X) + X = I∗0

H11H21

H12H22

[I 00 0

][ 0 0−X I

]I

0

0 I 0

I0;

from which (23a) can be verified using Lemma 2 (just note that the X part in the “B” matricesof the STPBC in (23) does not affect the result and serves to guarantee that the condition ofLemma 2 holds).

Remark 5.1. By a simple basis change formulae (23) can be rewritten as

RG,J(X) = I∗0

A+ BF

−RG,J(X)

−B(D ′JD)−1B ′

−(A+ BF) ′

[I 00 0

][ 0 00 I

]I

0

0 I 0

I0 (23a ′)

and

F =

A+ BF

−RG,J(X)

−B(D ′JD)−1B ′

−(A+ BF) ′

[I 00 0

][ 0 00 I

]I

0

F −(D ′JD)−1B ′ 0

I0; (23b ′)

where F := −(D ′JD)−1(B ′X+D ′JC) is the continuous-time state feedback gain.

Lemma 6 establishes the equivalence between the Riccati operator for lifted systems and theimpulse response of an STPBC. The latter, in turn, can easily be computed using Lemma 2.More precisely, one can get that for any X such that det(22− X12) 6= 0

RG,J(X) = −(22− X12)−1(21− X11) − X:

It follows from this equation that the DTARE (22) is equivalent to the following quadraticmatrix equation:

22X− X11− X12X+ 21= 0: (24)

Following the reasoning in [15, Proposition 7.1.1], it can be shown that a matrix X is a solutionto this equation iff Im

[IX

]is an invariant subspace of eHh. Since the latter matrix is simplectic,

(24) is a DTARE. On the other hand, it is well known [15, §7.1] that a matrix X is a solutionto CTARE (20) iff Im

[IX

]is an invariant subspace of H. These facts suggest that the Riccati

equations (20) and (24) are closely connected. Indeed, since any invariant subspace of H is alsoan invariant subspace of its exponential eHh, any solution to (20) is also a solution to (24). The


opposite, however, is not true in general [15, Theorem 1.7.5]. It implies that (24), and hencethe lifted DTARE (22), might have more solutions than (20), depending on the eigenvalues ofH and the sampling period h.

The following lemma characterizes the relationship between solutions to the continuous-timeRiccati equation (20) and its lifted counterpart (22):

Lemma 7. Any solution X to CTARE (20) is also a solution to DTARE (22). If none of thepoints + j2πh k, k 6= 0, belongs to (H) whenever ∈ (H), then the converse is also true.

Proof. First, prove that if det(D ′JD) 6= 0 then X is a solution to (22) iff it is a solution to (24).It follows from the reasoning above that to this end suffices to prove that det(22− X12) 6= 0

whenever X is a solution to (24). Suppose that the converse holds true. Then there exists anonzero ∈ ker(22− X12)

′. Premultiplying (24) by ′ we get that ∈ ker(21− X11)′ too.

But this means that[

−XI

] ∈ ker eH

′h = 0 and, hence, = 0, which is a contradiction.Now, we can replace (22) with (24). Then the first claim of the Lemma follows immediately

from the discussion preceding the Lemma. In order to prove the second claim note that theeigenvalue test of the Lemma is just the condition that eλ1h 6= eλ2h for any 1 6= 2 from (H).Then the proof is completed by using [15, Theorem 1.7.5].

Remark 5.2. The condition in Lemma 7 for (22) to have no additional solution relative to (20)is only sufficient. If the eigenvalue test of Lemma 7 is violated, then eHh has additional in-variant subspaces, which are not H-invariant. These subspaces, however, are not necessarilycomplementary to Im

[0I

]and, hence, might not correspond to solutions to (22).

The result of Lemma 7 is quite surprising; it implies, in fact, that G might have some addi-tional properties relative to G. In most cases, however, one is interested in the stabilizing solutionto an algebraic Riccati equation, that is the solution, which corresponds to the unique stable2

invariant subspace of the Hamiltonian or the simplectic matrix associated with the equation.Recall, that the stabilizing solution X to CTARE (20) is its unique real symmetric solution suchthat A+BF (= H11+H12X) is Hurwitz. Analogously, the stabilizing solution X to DTARE (22)is its unique real symmetric solution such that A + BF is Schur and D∗JD + B∗XB is invertibleon L2[0; h]. The next theorem states that with respect to their stabilizing solutions the Riccatiequations (20) and (22) are equivalent.

Theorem 4. A matrix X = X ′ is the stabilizing solution to CTARE (20) iff it is the stabilizingsolution to DTARE (22).

Proof. It can easily be shown that the matrices H and eHh share the same stable invariantsubspace. It means that the stabilizing solutions to (20) and (24) as well as the conditions fortheir existence coincide. Then, the reasoning in the proof of Lemma 7 implies that in order toprove the Theorem suffices to show that A+ BF = 11+ 12X whenever X is a solution to (24).To this end note that

BF = I∗h

H11H21

H12H22

[I 00 0

][ 0 0−X I

]I

X

I 0 0

I0− A;

from which we have:

A+ BF = 11− 12(22− X12)−1(21− X11) = 11+ 12X;

where the latter equality is obtained by (24). This completes the proof.2“Stable” here is understood as corresponding to eigenvalues in the open left half plane C− in the continuous-

time or the open unit disk D in the discrete-time cases.


Theorem 4 implies that if one is concerned with the stabilizing solution to (22) only, thenthe DTARE (22) is equivalent to the standard finite-dimensional CTARE (20). Moreover, us-

ing Theorem 4 and (7c) it can easily be shown that that F =(A + BF I

F 0

)I0, where F is the

continuous-time optimal gain. Theorem 4 and the above expression play, actually, the key rolein the solution of the sampled-data H2 and H∞ control problems when sampling and/or holddevices are not fixed but rather are the design parameters, see [21].

6 Concluding remarks

In this paper we have proposed a new representation of the parameters of LTI continuous-timesystems in the lifted domain. This representation allows one to reduce algebraic operationsover infinite-dimensional parameters of the lifted systems to finite-dimensional matrix manip-ulations in the state space. Consequently, the well understood state-space machinery can beused, enabling unified and intuitively clear solutions for a wide spectrum of sampled-data con-trol problems. The examples, which have been considered in Sections 4 and 5, demonstrate thebenefits of the proposed approach. In particular, we have shown that:

• The computations in the sampled-data H2 and H∞ problems with generalized sampler andhold can be reduced to the computations of matrix exponentials and elementary algebraicmatrix manipulations in a unified and efficient fashion.

• Infinite-dimensional lifted DTARE can be reduced to a finite-dimensional DTARE. More-over, the stabilizing solution to the discrete-time ARE associated with the lifting G ofan LTI system G is equivalent to the stabilizing solution to the continuous-time AREassociated with G.

It is believed that the new representation together with the machinery proposed in the paperenhances considerably the capability and efficiency of the lifting technique. In particular, theresults of Section 5 combined with the results of [19] pave the way to solve sampled-data H2

and H∞ problems, when the sampler or/and the hold are not fixed a priory but rather are thedesign parameters. This is the topic of a current research (see [21, 20]).

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