AN APPROXIMATION THEORY OF SOLUTIONS TO OPERATOR Hkmorris/Preprints/ItoMorris.pdf · 2006-09-11 · AN APPROXIMATION THEORY OF SOLUTIONS TO OPERATOR RICCATI EQUATIONS FOR H1CONTROL

AN APPROXIMATION THEORY OF SOLUTIONS TO OPERATORRICCATI EQUATIONS FOR H∞ CONTROL∗

KAZUFUMI ITO† AND K. A. MORRIS‡

SIAM J. CONTROL OPTIM. c© 1998 Society for Industrial and Applied MathematicsVol. 36, No. 1, pp. 82–99, January 1998 004

Abstract. As in the finite-dimensional case, the appropriate state feedback for the infinite-dimensional H∞ disturbance-attenuation problem may be calculated by solving a Riccati equation.This operator Riccati equation can rarely be solved exactly. We approximate the original infinite-dimensional system by a sequence of finite-dimensional systems and consider the correspondingfinite-dimensional disturbance-attenuation problems. We make the same assumptions required inapproximations for the classical linear quadratic regulator problem and show that the sequence ofsolutions to the corresponding finite-dimensional Riccati equations converge strongly to the solu-tion to the infinite-dimensional Riccati equation. Furthermore, the corresponding finite-dimensionalfeedback operators yield performance arbitrarily close to that obtained with the infinite-dimensionalsolution.

Key words. H∞, approximations, partial differential equation, optimal control, infinite dimen-sional

AMS subject classifications. 49N10, 65P05, 93C20

PII. S0363012994274422

1. Introduction. In this paper we discuss H∞ control problems for the linearsystem in a Hilbert space X:

(1.1)d

dtx(t) = Ax(t) +Bu(t) +Dv(t), x(0) = x ∈ X,

where the linear closed operator A generates the C0-semigroup S(t) on X. Let W ,U , and Y be separable Hilbert spaces. The signal v(t) ∈ L2(0,∞;W ) is a W -valueddisturbance and u(t) ∈ L2(0,∞;U) is the control input. We assume that the dis-turbance operator D and the input operator B are bounded, i.e., D ∈ L(W,X) andB ∈ L(U,X). Let C ∈ L(X,Y ) be the reference output operator. For control costε > 0, define the output

(1.2) z(t) = col (Cx(t),√ε u(t))

and the performance index

ρ(u, v;x) = |z|2L2(0,∞;Y ) =∫ ∞

0|Cx(t)|2 + ε|u(t)|2 dt.

Let

U = L2(0,∞;U), W = L2(0,∞;W ).

This paper is concerned with the problem of constructing a stabilizing feedback controllaw u(t) = −Kx(t) such that for each disturbance w ∈ W, the closed-loop solution of

∗Received by the editors September 19, 1994; accepted for publication (in revised form) October8, 1996.

http://www.siam.org/journals/sicon/36-1/27442.html†Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC

27695-8205 ([email protected]).‡Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada

([email protected]).

82

APPROXIMATION OF SOLUTIONS TO RICCATI EQUATIONS 83

(1.1) with x(0) = 0 satisfies

(1.3) ρ(−Kx(t), v; 0) ≤ (γ2 − δ) |v|2L2(0,∞;W )

for given attenuation bound γ > 0 and some δ > 0. The problem described aboveis equivalent to so-called H∞-disturbance attenuation: for given γ > 0 construct anexponentially stabilizing linear feedback u(t) = −Kx(t) such that the attenuationbound

(1.4) supω∈R

∣∣∣∣( C√εK

)(iωI − (A−BK))−1D

∣∣∣∣ < γ

is satisfied. Such problems arise in a variety of contexts; robust stabilization is one ofthe most important.

DEFINITION 1.1. If there is δ > 0 such that for each v ∈ W there exists a controlu ∈ U with

ρ(u, v; 0) ≤ (γ2 − δ) |v|2W ,

the problem is said to be stabilizable with attenuation γ.As in the finite-dimensional case, the H∞ disturbance-attenuation problem is

solvable if and only if the problem is stabilizable with attenuation γ (Theorem 2.2).Furthermore, in this case an appropriate state feedback may be calculated by solv-ing an operator Riccati equation. Unfortunately, this Riccati equation can rarelybe solved exactly. In this paper we approximate the original system (1.1)−(1.2)by a sequence of finite-dimensional systems and consider the corresponding finite-dimensional disturbance-attenuation problems.

The classical linear quadratic regulator (LQR) problem may be regarded as a lim-iting case of the H∞ disturbance-attenuation problem, with the required disturbanceattenuation γ →∞. The approximation theory for the linear quadratic case is fairlycomplete (e.g., see [GI, BK, IT1, IT2]). We make the same assumptions required inthe linear quadratic case and show that a sequence of solutions to finite-dimensionalRiccati equations converges strongly to the solution to the infinite-dimensional Riccatiequation required to solve the H∞ disturbance-attenuation problem. Furthermore,and more importantly, the corresponding finite-dimensional feedback control opera-tors yield performance arbitrarily close to that obtained with the infinite-dimensionalsolution. A key step of the proof of these results is the game-theoretic representation([KE, (2.10) in the proof of Theorem 2.2]) of the solution to the H∞ Riccati equationin terms of the closed-loop solution to the standard LQR problem.

The notation that is used in this paper is standard as in [PA]. Backgroundon linear semigroup theory may also be found in [PA]. An outline of the paper is asfollows. In section 2 the solution to the H∞ disturbance-attenuation problem in termsof the operator Riccati equation (2.1) is described, and an approximation theory ofsolutions to (2.1) is developed. Our theoretical results are demonstrated numericallyin section 3 using a Euler–Bernoulli beam example.

2. Approximation theory. In this section we develop an approximation theoryfor the H∞ disturbance-attenuation problem.

DEFINITION 2.1. (1) The pair (A,B) is stabilizable if there exists a bounded linearoperator F : X → U such that A − BF generates an exponentially stable semigroupon X.

84 K. ITO AND K. A. MORRIS

(2) The pair (A,C) is detectable if there exists a bounded linear operator G : Y →X such that A−GC generates an exponentially stable semigroup on X.

(3) The state feedback K ∈ L(X,U) is γ-admissible if it is exponentially stabilizingand the linear feedback u(t) = −Kx(t) is such that the attenuation bound (1.3) isachieved.

A key result in the finite-dimensional and the infinite-dimensional theory is that ifthe problem is stabilizable with attenuation γ, then it is stabilizable by state feedback.

THEOREM 2.2 (see, e.g., [KE, Thm. 4.4]). Assume that (A,B) is stabilizable and(A,C) is detectable. For γ > 0 the following are equivalent:

• there exists a γ-admissible state feedback;• the system is stabilizable with disturbance attenuation γ;• there exists a nonnegative, self-adjoint operator Σ on X satisfying the Riccati

equation

(2.1)(A∗Σ + ΣA− 1

εΣBB∗Σ +

1γ2 ΣDD∗Σ + C∗C

)x = 0

for all x ∈ dom(A), and A− 1εBB

∗Σ + 1γ2DD

∗Σ generates an exponentiallystable semigroup on X.

Moreover, in this case a γ-admissible state feedback is given by K = 1εB∗Σ.

An approximation theory for solutions to (2.1) which numerically approximate thefeedback operator K = B∗Σ/ε is developed below. Let XN be a finite-dimensionalsubspace of X, and let PN be the orthogonal projection of X onto XN . The spaceXN is equipped with the induced norm from X. Consider a sequence of opera-tors AN ∈ L(XN , XN ), BN = PNB ∈ L(U,XN ), DN = PND, and CN =the restriction of C onto XN . The operator AN can be extended to all of X byANPNx.

Approximation Assumptions.(A1) For each x ∈ X we have

(i) eAN tPNx→ S(t)x,

(ii) (eAN t)∗PNx→ S∗(t)x,

uniformly in t on bounded intervals.(A2) (i) The family of pairs (AN , BN ) is uniformly exponentially stabilizable; i.e.,

there exists a uniformly bounded sequence of operators KN ∈ L(XN , U) such that

(2.2)∣∣∣e(AN−BNKN )tPN x

∣∣∣X≤M1 e

−ω1t |x|X

for some positive constants M1 ≥ 1 and ω1.(ii) The family of pairs (AN , CN ) is uniformly exponentially detectable; i.e., there

exists a uniformly bounded sequence of operators GN ∈ L(Y,XN ) such that

(2.3)∣∣∣e(AN−GNCN )tPN x

∣∣∣X≤M2 e

−ω2t, t ≥ 0,

for some positive constants M2 ≥ 1 and ω2.(A3) The input operator B and disturbance operator D are compact.Remarks. (1) Note that (A1) implies that PNx→ x for x ∈ X.(2) Assumption (A1) and the uniform boundedess theorem imply the boundedness

of |eAN tPN |L(X,X) uniformly in t ∈ [0, 1] and N . Then the standard semigroup


theorem, e.g., [PA, Chapter 1, Theorem 2.2], implies that |eAN tPNx|X ≤M0 eω0t|x|X

for some M0 ≥ 1 and ω0 ∈ R.(3) For an important equivalent statement of (A1)(i) we note the Trotter–Kato

theorem.THEOREM 2.3 (Trotter–Kato Theorem; see, e.g., [PA, Chapter 3, Theorem 4.2]).

Assume the stability of approximation

|eAN tPNx|X ≤M0 eω0t|x|X for some M0 ≥ 1 and ω0 ∈ R.

Then the convergence (A1)(i)

eAN tPNx→ S(t)x for every x ∈ X and uniformly on bounded t intervals

is equivalent to the consistency: for some λ ∈ ρ(A)∣∣(λI −AN )−1PNx− (λI −A)−1x∣∣X→ 0

as N →∞ for all x ∈ X.(4) The convergence (A1)(ii) of the adjoint semigroup sequence (eA

N t)∗ is requiredfor the strong convergence of the approximating feedback gain operators. A counter-example may be found in [BIP].

(5) In (A2)(i) if we let KN = KPN , then condition (2.2) becomes the preservationof exponential stability under approximation of the semigroup T (t) generated by A−BK.

(6) Assumption (A3) is equivalent to

lim |PND −D| = 0, lim |PNB −B| = 0 as N →∞

since XN is finite dimensional.Assumptions (A1)–(A2) are identical to those required to show that the solu-

tions to the Riccati equations arising in the approximation theory for linear quadraticproblem converge, e.g., [IT1]. Assumption (A3) is not required in the standard LQRproblem for the existence of solutions to a family of approximating finite-dimensionalRiccati equations. However, this assumption is required to ensure continuity of per-formance measure and to guarantee that the approximating controllers stabilize theinfinite-dimensional system [IT2, MO1, MO2].

Before presenting the approximation result and its proof we state a technicallemma which plays an important role in the proof.

LEMMA 2.4 (Datko lemma; see, e.g., [SA, Theorem 6.2]). Let S(t), t ≥ 0 be alinear C0-semigroup on a Banach space X satisfying the exponential bound

|S(t)| ≤Meωt

for some constants M ≥ 1, ω ≥ 0. Moreover, let 1 ≤ p < ∞, and suppose that thereexists a constant c > 0 such that∫ ∞

0|S(t)x|pX dt ≤ cp |x|

pX , x ∈ X.

Then, for every

α > − 1pcpMp

,


there exists a γ = γ(α, ω,M, c, p) ≥ 1 such that

|S(t)| ≤ γ eαt, t ≥ 0.

Now we state our main approximation result.THEOREM 2.5. Assume that (A,B) is stabilizable, (A,C) is detectable, and (A1)–

(A3) are satisfied. If the original problem is stabilizable with attenuation γ, then so arethe approximating systems for sufficiently large N . For such N , the Riccati equation

(2.4) (AN )∗ΣN +ΣNAN− 1ε

ΣNBN (BN )∗ΣN +1γ2 ΣNDN (DN )∗ΣN +(CN )∗CN = 0

has a nonnegative, self-adjoint solution ΣN and ΣNPN x → Σx strongly in X asN → ∞. Moreover, KN = 1

ε (BN )∗ΣN converges to K = 1εB∗Σ in norm. For N

sufficiently large, KN is γ-admissible for the infinite-dimensional problem.Proof. The proof is given in several steps. First, we give a brief description of the

representation of Σ. This is used to show that for large N the approximating systemsare stabilizable with attenuation γ and so for such N the finite-dimensional Riccatiequation (2.4) has a solution ΣN . We show that ΣN → Σ and KN → K. Finally, weshow that the approximating finite-dimensional feedback KN is γ-admissible for theoriginal system.

Step 1. First we briefly review the representation of Σ. Details may be found in[KE, Theorem 4.4] or [BB]. Since (A,B) is stabilizable and (A,C) is detectable, the(LQR) Riccati equation

(2.5)(A∗Π + ΠA− 1

εΠBB∗Π + C∗C

)x = 0 for all x ∈ dom(A)

has the unique nonnegative, self-adjoint solution Π. Let Sc(t) be the exponentiallystable semigroup generated by A− 1

εBB∗Π.

Consider the quadratic differential game

maxv∈W

minu∈U

ρ(u, v;x)− γ2 |v|2W

subject to (1.1).Define L ∈ L(W, L2(0,∞;X)) by

(2.6) (Lv)(t) =∫ ∞t

S∗c (τ − t)ΠDv(τ) dτ.

For a disturbance v and x ∈ X

(2.7) u∗(t) = −1εB∗ [Πx(t) + (Lv)(t)]

minimizes ρ(u, v;x) over u ∈ U subject to (1.1).For x ∈ X, v ∈ W, write r(t) = (Lv)(t) and define the quadratic form

J(v;x) = ρ(u∗, v;x)− γ2 |v|2W

= (x,Πx+ 2 r(0)) +∫ ∞

02 (Dv(t), r(t))− 1

ε|B∗r(t)|2 − γ2 |v(t)|2 dt.

Defining the self-adjoint operator Q on W by

(2.8) Q = γ2 I +1εL∗BB∗L−D∗L− L∗D,


we have

J(v;x) = −(Qv, v)W + 2 (D∗ΠSc(·)x, v)W + (Πx, x).

If the system (1.1)−(1.2) is stabilizable with attenuation γ, then for some δ > 0

J(v; 0) = ρ(u∗, v; 0)− γ2 |v|2W ≤ −δ |v|2W .

Thus Q ≥ δI and maximization of J(v;x) over v ∈ W is well posed. The solution tothis problem, the worst disturbance v∗, is the unique solution to

(2.9) Qv∗ −D∗ΠSc(·)x = 0, x ∈ X.

We define the self-adjoint operator Σ on X by

(Σx, x) = max J(v, x) = (D∗ΠSc(·)x, v∗)W + (Πx, x)

for x ∈ X. This implies

(2.10) Σx = Πx+∫ ∞

0S∗c (t)ΠDv∗(t) dt.

It is shown in [KE, BB] that Σ satisfies the Riccati equation (2.1) and is uniquewithin the class of nonnegative, self-adjoint solutions to (2.1) such that theclosed-loop semigroup generated by A − 1

εBB∗Σ + 1

γ2DD∗Σ is exponentially

stable on X.Moreover, the optimal pair (u∗, v∗) to the differential game is of feedback form:

u∗(t) = −1εB∗Σx(t), v∗(t) =

1γ2 D

∗Σx(t).

Step 2. Next, we show that the finite-dimensional Riccati equation (2.4) hasa nonnegative solution by showing that the finite-dimensional system is stabilizablewith attenuation γ.

Under assumptions (A1)−(A2) it is shown in [IT1] that the (LQR) Riccati equa-tion on XN

(2.11) (AN )∗Π + ΠNAN − 1ε

ΠNBN (BN )∗ΠN + (CN )∗CN = 0

has the unique nonnegative, self-adjoint solution ΠN and also that ΠNPN x → Πx

strongly in X as N → ∞. Define SNc (t) = e(AN− 1εB

NBN∗ΠN )t. It is also shown in

[IT1] that there exist constants M3 ≥ 1 and ω3 > 0 such that

(2.12) |SNc (t)PNx| ≤M3e−ω3t |x|X .

Let KN = 1εB

N ∗ΠN and K = 1εB∗Π. Then, since B is compact, |KNPN −K| →

0 as N →∞. Assumption (A1) implies∣∣(λI − (AN −BNKN ))−1PNx− (λI − (A−BK))−1x∣∣X→ 0

for all x ∈ X and also the similar statement for the sequence of adjoint operators. Itthus follows from the Trotter–Kato theorem that

(2.13) SNc (t)PNx→ Sc(t)x and (SNc )∗(t)PNx→ S∗c (t)x

for all x ∈ X, uniformly on bounded t-intervals.


Since D is compact,

(2.14) |(SNc )∗(t)ΠNDN − S∗c (t)ΠD| → 0

uniformly in any bounded t-interval. For τ > 0 and p ∈ [1,∞),∫ ∞0|(SNc )∗(t)ΠNDN − S∗c (t)ΠD|p dt

≤∫ τ

0|(SNc )∗(t)ΠNDN − S∗c (t)ΠD|p dt

+∫ ∞τ

(|SNc (t)|p|ΠN |p + |S∗c (t)|p|Π|p)|D|p dt.

Since from (2.12) the second term of the right-hand side is bounded by Me−ω τ forsome positive constants M and ω, it follows from (2.13) that

(2.15)∫ ∞

0|(SNc )∗(t)ΠNDN − S∗c (t)ΠD|p dt→ 0

as N →∞ for all p ∈ [1,∞).Define the linear operators LN and QN on W for the approximate problem that

corresponds to L and Q defined in (2.6) and (2.8), respectively. It then follows from(2.15) that

(2.16) |LN − L| and |QN −Q| → 0

as N →∞.Define

zN (u, v;x) = col (CNxN (t),√εu(t)),

where xN is the state of the approximating system with control u, disturbance v, andinitial condition x. Also define the finite-dimensional cost

ρN (u, v, x) = |zN |2L2(0,∞;Y ).

As in (2.7) define the control

(2.17) uN (t) = −1ε

(BN )∗[ΠNxN (t) + LNv(t)].

Then, with initial condition x = 0,

(2.18) xN (t) =∫ t

0SNc (t− s)

(−1εBN (BN )∗(LNv)(s) +DNv(s)

)ds.

Define the linear operators LN , L ∈ L(L2(0,∞;X), L2(0,∞, X)) by

(LNf)(t) =∫ t

0SNc (t− s)

(1εBN (BN )∗f(s)

)ds


and

(Lf)(t) =∫ t

0Sc(t− s)

(1εBB∗f(s)

)ds

Using the same arguments as above we can show that |LN − L| → 0 as N →∞. Sincefrom (2.7)

x∗(t) =∫ t

0Sc(t− s)

(−1εBB∗(Lv)(s) +Dv(s)

)ds,

it follows from (2.16) and (2.18) that

|xN − x∗|2L2(0,∞;X) ≤ ε1(N) |v|2W ,

where ε1(N)→ 0 as N →∞. Since CN is the restriction of C on XN it follows that

|ρN (uN , v; 0)− ρ(u∗, v; 0)| ≤ ε(N) |v|2W ,

where ε(N) → 0 as N → ∞. Therefore, for sufficiently large N , the approximatingproblems are stabilizable with attenuation γ.

This implies that the finite-dimensional Riccati equation (2.4) has a self-adjointsolution ΣN on XN

(2.19) ΣNPNx = ΠNPNx+∫ ∞

0(SNc )∗(t)ΠNDNvN (t) dt,

where vN (t) ∈ W is the unique solution of

(2.20) QNvN − (DN )∗ΠNSNc (·)PNx = 0.

Furthermore, AN− 1εB

N (BN )∗ΣN+ 1γ2D

N (DN )∗ΣN generates an exponentially stable

semigroup on XN , and KN = 1ε (BN )∗ΣN is γ-admissible for the approximating

problem.Step 3. We now show that ΣN converges strongly to Σ and KN converges uni-

formly to K.The uniform convergence of QN to Q implies that QN is coercive with QN ≥ δ

2for N sufficiently large. Also, (2.15) implies that∫ ∞

0|(DN )∗ΠNSNc (t)PN −D∗ΠSc(t)|2dt→ 0

as N →∞. Therefore, the solution to (2.20) satisfies

(2.21) |vN |W ≤M |x|X

for some constant M . Note that

Q(vN − v∗) = (Q−QN )vN + (DN )∗ΠNSNc (t)PNx−D∗ΠSc(t)x.

Hence, from (2.16) and (2.21) vN converges strongly to v∗ in W as N →∞ for eachx ∈ X.


It now follows from (2.10), (2.15), and (2.19) that

ΣNPNx→ Σx for all x ∈ X.

Since B is compact, we have that |(BN )∗ΣNPN −B∗Σ| → 0. That is, KN convergesuniformly to K.

Step 4. To prove that for large N the approximating feedback operators KN areγ-admissible for the system (1.1)−(1.2), first note that by the Trotter–Kato theoremthe convergence of KN to K in norm implies that the semigroup SKN generated byA − BKN converges to the semigroup SK generated by A − BK, strongly in X,uniformly in bounded intervals of time. Also, there exists M ≥ 1, ω > 0 such that forsufficiently large N ,

|SKN (t)|X ≤ Me−ωt.

The output (1.2) with a disturbance v, feedback control u(t) = −KNx(t), and initialcondition x(0) = 0 is

zN (t) =[

C

−√εKN

] ∫ t

0SKN (t− s)Dv(s) ds.

The convergence of the semigroups and the compactness of D imply, as in equation(2.15), that for any p ∈ [1,∞),∫ ∞

0|SKN (t)D − SK(t)D|p dt→ 0.

Let z indicate the output (1.2) obtained with the same disturbance v but with feedbackcontrol −Kx(t). Then ∫ ∞

0|zN (t)− z(t)|2Y dt ≤ ε(N) |v|2W ,

where ε(N) → 0 as N → ∞. This implies that for large N , KN is γ-admissible forthe original system.

COROLLARY 2.6. Under the same assumptions as in Theorem 2.3, we have

|vN (t)− v∗(t)|W = 0,

|uN (t)− u∗(t)|U = 0,

and there exist positive constants M4, ω4,M5, and ω5 such that

(2.22) |e(AN− 1εB

N (BN )∗ΣN+ 1γ2D

N (DN )∗ΣN )tPNx| ≤M4e

−ω4 t |x|X ,

(2.23) |e(AN−BN KN )tPNx| ≤M5e−ω5 t |x|X .

Proof. The convergence of the worst disturbance vN follows from the proof ofTheorem 2.3.


We show that for large N , AN − 1εB

N (BN )∗ΣN + 1γ2D

N (DN )∗ΣN generates auniformly stable semigroup TN (t). Note that

vN (t) =1γ2 (DN )∗ΣNxN (t), uN (t) = −1

ε(BN )∗ΣNxN (t)

and

(2.24)∫ ∞

0|CNxN (t)|2 + ε|uN (t)|2 dt = (ΣNPNx, x) + γ2|vN (t)|2W ,

where

xN (t) = TN (t)x = e(AN− 1

εBN (BN )∗ΣN+ 1

γ2DN (DN )∗ΣN )t

PNx, x ∈ X.

Let GN be as in (A2)(ii) so that e(AN−GNCN )t is uniformly exponentially stable. Thenwe have

xN (t) = e(AN−GNCN )tPNx

+∫ t

0e(AN−GNCN )(t−s)(GNCNxN (s) +BNuN (t) +DNvN (t)) dt.

Note that from Holder’s inequality and the Fubini theorem∫ ∞0

∣∣∣∣∫ t

0f(t− s)g(s) ds

∣∣∣∣2 dt ≤ ∫ ∞0

∫ t

0|f(t− s)| ds

∫ t

0|f(t− s)||g(s)|2 ds

≤∫ ∞

0|f(σ)| dσ

∫ ∞0

(∫ ∞s

|f(t− s)| dt)|g(s)|2 ds

≤(∫ ∞

0|f(σ)| dσ

)2 ∫ ∞0|g(t)|2 dt

for f ∈ L1(0,∞) and g ∈ L2(0,∞). It thus follows from (A2)(ii), (2.21), (2.24) that∫ ∞0|xN (t)|2 dt ≤ M2

2

ω2|x|2

+2M2

ω2

∫ ∞0

(|GN |2|CNxN (t)|2 + |BN |2|uN |2 + |DN |2|vN (t)|2) dt

≤ β |x|2X for some β > 0.

Hence, (2.22) follows from the Datko lemma. The proof of (2.23) is identical.Let T (t) be the semigroup generated by A− 1

εBB∗Σ+ 1

γ2DD∗Σ on X. The above

implies that xN (t) converges in L2(0,∞;X) to T (t)x (see the proof of Theorem 2.3)and so uN (t)→ u∗(t) in L2(0,∞;X).

Conversely, we have the following theorem.THEOREM 2.7. Suppose the Riccati equation (2.4) has uniformly bounded nonneg-

ative self-adjoint solutions ΣN on XN for N sufficiently large with

(2.25) |vN (t)|W ≤M |x|2X for some M > 0,


where

vN (t) =1γ2 (DN )∗ΣNe(AN− 1

εBN (BN )∗ΣN+ 1


PNx.

Assume that (A1), (A2)(ii), and (A3) hold. Then, the system (1.1) is stabilizable withattenuation γ.

Proof. It follows from [GI] that there exist a nonnegative self-adjoint operatorΣ on X and a subsequence of ΣN such that ΣNPNx converges weakly to Σx in Xfor x ∈ X. It follows from (A3) that (BN )∗ΣNPNx and (DN )∗ΣNPNx convergestrongly to B∗Σx and D∗Σx, respectively. Let

TN (t) = e(AN− 1

εBN (BN )∗ΣN+ 1


and T (t) be the semigroup generated by A− 1εBB

∗Σ+ 1γ2DD

∗Σ on X. It then followsfrom (A1) and the Trotter–Kato theorem that

TN (t)PNx→ T (t)x for each x ∈ X,

uniformly on bounded t-intervals.Note that

ΣNPNx = e(AN )∗tΣNTN (t)PNx+∫ t

0e(AN )∗(t−s)(CN )∗CNTN (t− s)PNx ds, x ∈ X.

Taking the limit as N →∞, we obtain

Σx = S∗(t)ΣT (t)x+∫ t

0S∗(t− s)C∗CT (t− s)x ds,

which implies that Σ is a solution to (2.4).Moreover, it follows from (2.25) and the proof of Corollary 2.6 that

|TN (t)PNx| ≤M4e−ω4 t |x|X

for some positive constants M4, ω4, provided that (A2)(ii) holds. Hence, the semi-group T (t) is exponentially stable.

It follows from Theorem 2.2 that the system (1.1) is stabilizable with attenu-ation γ.

The optimal disturbance attenuation problem for the infinite-dimensional system(1.1) is to find

γ = inf γ

over all γ such that (1.2) is stabilizable with attenuation γ. Let {γN} indicate thecorresponding optimal disturbance attenuation for the approximating problems. The-orem 2.3 implies that

(2.26) lim supN→∞

γN ≤ γ.

However, we have a stronger result.THEOREM 2.8. Assume that (A1)–(A3) hold, (A,B) is stabilizable, and (A,C) is

detectable. Then

limN→∞

γN = γ.


Proof. Because of (2.26), it is sufficient to show that

lim infN→∞

γN ≥ γ.

Assume that this statement is false. Then there is an δ > 0 such that for all N thereis M > N with γM < γ− δ. In this way we can construct a subsequence {γM} of thesequence {γN} with

γM < γ − δ.

Thus, the approximating system is stabilizable with attenuation γ − δ/2 and

ρM (uM , v; 0) ≤ (γ − δ/2) |v|2W ,

where for any v ∈ W, uM (t) is defined by (2.17). Moreover, we have

|ρM (uM , v; 0)− ρ(u∗, v; 0)| ≤ ε(M) |v|2W ,

where ε(M) → 0 as M → ∞ and u∗(t) ∈ U is given by (2.7). Hence the originalproblem is stabilizable with attenuation γ − δ/2. This contradicts the optimality ofγ and thus (2.26) holds.

The above theorem implies that if a sequence of approximating problems thatsatisfy assumptions (A1)–(A3) are stabilizable with attenuation γ, then so is theinfinite-dimensional problem. Thus, Theorem 2.7 can be regarded as a partial converseof Theorem 2.3. The difference between this theorem and Theorem 2.5 is that theassumption of uniform stabilizability (A2)(i) in Theorem 2.5 is replaced by uniformboundedness of ΣN and vN .

3. Example. Consider a Euler–Bernoulli beam clamped at one end and letw(r, t) denote the deflection of the beam from its rigid body motion at time t andposition r. The deflection can be controlled by applying a torque at the clamped end(r = 0). We assume that the hub inertia Ih is much larger than the beam inertia, sothat, letting θ(t) indicate the rotation angle, u(t) = Ihθ(t) is a reasonable approxima-tion to the applied torque. The disturbance v(t) induces a uniformly distributed loadρdv(t). Use of the Kelvin–Voigt damping model leads to the following description ofthe beam vibrations:

ρ∂2w

∂t2+ Cv

∂w

∂t+

∂2

∂r2

[EI

∂2w

∂r2 + CdI∂3w

∂r2∂t

]=ρr

Ihu(t) + ρ d v(t), 0 < r < L.

The boundary conditions are

w(0, t) = 0,∂w

∂r(0, t) = 0,

[EI

∂2w

∂r2 + CdI∂3w

∂r2∂t

]r=L

= 0,[EI

∂3w

∂r3 + CdI∂4w

∂r3∂t

]r=L

= 0.

The values of the physical parameters in this example are listed in Table 1. Letx(t) = (w(·, t), ∂∂tw(·, t)), H be the closed linear subspace of H2(0, 1) defined by

H ={w ∈ H2(0, 1) : w(0) =

dw

dr(0) = 0

},


TABLE 1Physical constants.

E 2.1 ∗ 1011 N/m2

I 1.167× 10−10 m4

ρ 2.975 kg/mCv .001Ns/m2

Cd .01Ns/m2

L 7mIh 121.9748 kgm2

d .04 1/kg

and X = H × L2(0, 1). Here H2(0, 1) is the Hilbert space defined by

H2(0, 1) ={φ ∈ C1(0, 1) :

d

drφ is absolutely continuous and

d2

dr2φ ∈ L2(0, 1)

}.

If the tip deflection is measured, a state-space formulation of the above partial differ-ential equation problem is

d

dtx(t) = Ax(t) +Bu(t) +Dv(t),

y(t) = Cx(t) = w(L, t),

where

A =

0 I

−EIρ

d4

dr4 −CdIρ

d4

dr4 −Cvρ

, B =

0

r

Ih

, D =

0

d

with domain

dom (A) ={

(φ, ψ) ∈ X : ψ ∈ H and

M = EId2

dr2φ+ CdId2

dr2ψ ∈ H2(0, 1) with M(L) =

d

drM(L) = 0

}.

Define V = H ×H. Then A can be defined by

〈Ax, z〉V ∗×V = −a(x, z) for x, z ∈ V,

where the sesquilinear form a(·, ·) on V × V is given by

a((φ1, ψ1), (φ2, ψ2)) = −σ(ψ1, φ2) + σ

(φ1 +

CdEψ1, ψ2

)+(Cvρψ1, ψ2

)L2

for (φi, ψi) ∈ V, i = 1, 2. Here,

σ(φ, ψ) =∫ L

0

EI

ρ

d2

dr2φ(r)d2

dr2ψ(r) dr .


0 50 100 150 200-3

-2

-1

0

1

2

3

time (s)

(m)

FIG. 1. Response of open (..) and closed loop (–) to v(t) = 1, t ≤ 100s: 10 elements.

The sesquilinear form a is continuous on V × V and coercive; i.e.,

Re a(z, z) ≥ ω |z|2V − β |z|2X for z ∈ V

for appropriately chosen positive constants ω, β. It thus follows that A generatesan exponentially stable analytic semigroup on X [SH, section 4]. The operators Band D are clearly bounded operators from R to X. Sobolev’s inequality implies thatevaluation at a point is bounded on H, and so the output operator C is bounded fromX to R.

Let HN ⊂ H be a sequence of finite-dimensional subspaces. The approximatinggenerator AN on XN = HN ×HN is defined by

〈−ANxN , zN 〉 = a(xN , zN ) ∀xN , zN ∈ XN ,

and PN , BN , CN are as defined at the beginning of section 2. This type of ap-proximation is generally referred to as a Galerkin approximation. Suppose that theapproximating subspaces HN satisfy the H-approximation property: for all φ ∈ Hthere exists a sequence φN ∈ HN with

(H1) limN→∞

|φN − φ|H = 0.

It is shown in [IT2, MO2] that as long as the approximating spaces satisfy the H-approximation property, assumptions (A1)–(A3) are satisfied. The standard finite-element cubic B-spline approximations [OP] do satisfy the H-approximation property,and so all the assumptions of Theorem 2.3 are satisfied by this problem.


0 50 100 150 200-30

-20

-10

0

10

20

30

time (s)

(m)

FIG. 2. Response of open (..) and closed loop (–) to v(t) = sin(ωt) : 10 elements.

Our numerical calculations were carried out using a series of cubic B-spline ap-proximations for HN , and the corresponding series of finite-dimensional Riccati equa-tions were solved with ε = .1 and γ = 2.3. Figure 1 compares the open- and closed-loopresponses Cx(t) = w(L, t) with a temporary step disturbance for the approximationwith 10 elements. The feedback controller leads to a closed loop which is able toalmost entirely reject this disturbance. Figure 2 compares the open- and closed-loopresponses to the periodic disturbance sin(ωt) where ω is the first resonant frequency:ω = mini |Im(λi(A10)|. The resonance in the open loop is not present in the closedloop.

Since the input space U = R, the feedback operator KN is a bounded linearfunctional on XN and hence can be uniquely identified with an element of XN ,usually called the gain. Figure 3 displays the convergence of the feedback gainspredicted by Theorem 2.3. Since XN is a product space, the first and second com-ponents of the gains are displayed separately as displacement and velocity gains,respectively.

The sequence of operators AN −BNKN is uniformly exponentially stable, and so

max1≤i≤N

Reλi(AN −BNKN )

converges to a nonzero number as N → ∞, which can be verified theoretically. Thisconvergence is displayed in Figure 4 for several different values of ε and γ = 2.3.Notice that as ε is decreased, the convergence becomes slower. A robust stabilitytheorem provides some insight into this behavior.


0 1 2 3 4 5 6 70

1

2

3

4

r (m)

(m)

Displacement Gains

0 1 2 3 4 5 6 70

2

4

6

8

10

r (m)

(m)

Velocity Gains

FIG. 3. Convergence of the feedback gains: 2 elements *, 4 elements ..., 6 elements . .,8 elements , 10 elements, .

Let the functionsH,Go, andG in the following theorem indicate transfer functionsof systems with a finite number of inputs and outputs. For a given nominal plant Goand weighting function r ∈ H∞, the class A(Go, r) consists of all plants G that havethe same number of right-hand-plane poles as Go and that satisfy

|G(iω)−Go(iω)| < |r(iω)|.

That is, A(Go, r) contains the systems whose frequency response is within r(iω) ateach frequency ω of that of the nominal system Go.

THEOREM 3.1 (see [CD]). Suppose that a controller H stabilizes the system Go.For any r ∈ H∞ the controller H stabilizes all G ∈ A(Go, r) if and only if, for all ω,

(3.1) |H(1 +GoH)−1(iω)| |r(iω)| ≤ 1.

For any approximation N , we have the nominal plant

Go = (sIN −AN )−1[BN DN ]

and the controller

H =[KN

0

].

Thus, we have

H(I +GoH)−1 = KN (sIN −AN +BNKN )−1(sIN −AN )


1 2 3 4 5 6 7 8 9 10-10

-1

-10-2

-10-3

-10-4

-10-5

Number of elements

FIG. 4. Convergence of max1≤i≤N Re(λi): ε = .1 (—), .001 (- -), .00001.(...).

and

H(I +GoH)−1Go

[0v

]= KN (sIN −AN +BNKN )−1DNv.

The second row in the attenuation bound (1.4) is commonly interpreted as a con-straint on the control effort. However, it can also be interpreted as a robust stabilityconstraint. If the inequality (1.4) is satisfied, then the robustness criterion (3.1) issatisfied with

r(iω) =√ε

γGo(iω)

[0I

].

This interpretation explains why convergence is slower for smaller ε. The stabilitymargin for the approximation N is affected by the constraint that the higher-orderapproximations also be stabilized. Increasing ε while holding all other parameterssuch as γ constant means that the computed controller must stabilize a larger familyof systems. Hence the stability margin is less sensitive to change in the approximationindex N .

REFERENCES

[BB] A. BENSOUSSAN AND P. BERNHARD, On the standard problem of H∞-optimal control forinfinite dimensional systems, in Identification and Control in Systems Governed byPartial Differential Equations, H. T. Banks, R. H. Fabiano, and K. Ito, eds., SIAM,Philadelphia, 1993, pp. 117–140.


[BK] H. T. BANKS AND K. KUNISCH, The linear regulator problem for parabolic systems, SIAMJ. Control Optim., 22 (1984), pp. 684–698.

[BIP] J. A. BURNS, K. ITO, AND G. PROPST, On nonconvergence of adjoint semigroups for controlsystems with delays, SIAM J. Control Optim., 26 (1988), pp. 1442–1454.

[CD] M. J. CHEN AND C. A. DESOER, Necessary and sufficient conditions for robust stability oflinear distributed feedback systems, Internat. J. Control, 35 (1984), pp. 255–267.

[GI] J. S. GIBSON, Linear-quadratic optimal control of hereditary differential systems: Infinitedimensional Riccati equations and numerical approximations, SIAM J. Control Optim.,21 (1983), pp. 95–139.

[IT1] K. ITO, Strong convergence and convergence rates of approximating solutions for algebraicRiccati equations in Hilbert spaces, in Distributed Parameter Systems, F. Kappel, K.Kunisch, and W. Schappacher, eds., Springer-Verlag, Berlin, New York, 1987, pp. 151–166.

[IT2] K. ITO, Finite-dimensional compensators for infinite-dimensional systems via Galerkin-typeapproximation, SIAM J. Control Optim., 28 (1990), pp. 1251–1269.

[KE] B. VAN KEULEN, H∞-Control for Distributed Parameter Systems: A State-Space Ap-proach, Birkhauser Boston, Cambridge, MA, 1993.

[MO1] K. A. MORRIS, Convergence of controllers designed using state-space methods, IEEE Trans.Automat. Control, 39 (1994), pp. 2100–2104.

[MO2] K. A. MORRIS, Design of finite-dimensional controllers for infinite-dimensional systems byapproximation, J. Math. Systems, Estim. Control, 4 (1994), pp. 1–30.

[OP] N. OTTOSEN AND H. PETERSSON, Introduction to the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1992.

[PA] A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equa-tions, Springer-Verlag, New York, 1983.

[SA] D. SALAMON, Structure and stability of finite dimensional approximations for functionaldifferential equations, SIAM J. Control Optim., 23 (1985), pp. 928–951.

[SH] R. E. SHOWALTER, Hilbert Space Methods for Partial Differential Equations, Pitman, Lon-don, 1977.

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