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Frequency-domainperformance boundsfor uncertain structureswith sensor and actuatordynamicsWassim M. Haddad , Emmanuel G. CollinsJr , David C. Hyland & Vijaya-SekharChellaboinaPublished online: 08 Nov 2010.

To cite this article: Wassim M. Haddad , Emmanuel G. Collins Jr ,David C. Hyland & Vijaya-Sekhar Chellaboina (1997) Frequency-domain performance bounds for uncertain structures with sensor andactuator dynamics, International Journal of Control, 66:3, 381-392, DOI:10.1080/002071797224630

To link to this article: http://dx.doi.org/10.1080/002071797224630

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Frequency-domain performance bounds for uncertain structures withsensor and actuator dynamics

WASSIM M. HADDAD² , EMMANUEL G. COLLINS, Jr³ , DAVID C.HYLAND§ and VIJAYA-SEKHAR CHELLABOINA ² ¶

A new majorant robustness analysis test that yields frequency-dependent perfor-mance bounds for closed-loop uncertain vibrational systems with frequency, damp-ing and mode shape uncertainty is developed. Speci® cally, for closed-loop systemsconsisting of uncertain positive real plants in tandem with sensor and actuatordynamics and controlled by strictly positive real compensators, performancebounds are developed by decomposing the equivalent compensator (which includesthe actuator and sensor dynamics) into a generalized positive real part and a non-generalized positive real part, and by using concepts of M-matrices and majorantanalysis.

Nomenclature

R , C , Ip real numbers, complex numbers, and p ´ p identity matrixZ*, Z- * complex conjugate transpose of matrix Z, (Z*)- 1

Z(i,j) (i, j) element of matrix Zdiag (Z1, . . . ,Zn) diagonal matrix with listed diagonal elements

Y £ £ Z Y (i, j) £ Z(i,j) for each i and j, where Y and Z are realmatrices with identical dimensions

|a | absolute value of complex scalar adet (Z) determinant of square matrix Z

i xi 2 euclidean norm of vector x(= (x*x)1 /2)s min(Z), s max(Z) minimum, maximum singular values of matrix Z

i Zi s spectral norm of matrix Z(= s max(Z))i Zi F Frobenius norm of matrix Z(= (tr ZZ*)1 /2)q (Z) spectral radius of a square matrix Z

¸min(Z), ¸max(Z) minimum, maximum eigenvalues of the Hermitian matrix ZZd, Zod diagonal, o� -diagonal part of a square matrix Z

(Zd = diag (Z(1,1), . . . Z(n,n)), Zod = Z - Zd)L [z(t)] Laplace transform of z(t)

He Z, Sh Z 12 (Z + Z*), 1

2 (Z - Z*)

0020-7179/97 $12.00 Ñ 1997 Taylor & Francis Ltd.

INT. J. CONTROL, 1997, VOL. 66, NO. 3, 381± 392

Received 22 August 1995.² School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-

0150, U.S.A. Tel: + 1(404)894 1078: Fax: + 1(404)894 2760; e-mail: wm.haddad@aerospace.-gatech.edu.

³ Department of Mechanical Engineering, Florida A&M/Florida State, Tallahassee, FL32310, U.S.A. Tel: + 1(904)487-6331; fax: + 1(904)487-6337; e-mail: ecollins@evax.eng.fsu.edu.

§ Harris Corporation, Government Aerospace Systems, Division, Melbourne, FL 32902,U.S.A. Tel: + 1(407)729-2138; fax: + 1(407)727-4016.

¶ Tel: + 1(404)894 3000; fax: + 1(404)894 2760; e-mail: gt3582a@acme.gatech.edu.

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1. Introduction

The analysis of uncertain dynamical systems for robust stability and performanceremains one of the most important problems in modern feedback control theory.Hence, considerable e� ort has been devoted to the development of e� cient analysistools that allow a control system to be analysed for robustness with respect tostructured and unstructured uncertainty in the design model. In certain applications,such as the control of ¯ exible structures, if the sensors and actuators are colocatedand also dual, for example, force actuators and velocity sensors or torque actuatorsand angular rate sensors, the plant transfer function is positive real. In practice, theprospect for controlling such systems is quite good because, if sensor and actuatordynamics and time delays are negligible, stability is unconditionally guaranteed solong as the controller is strictly positive real (Joshi 1989, Benhabib et al. 1981). Inpractice, however, sensor and actuator dynamics, as well as system delays, cannot beneglected. Furthermore, many of the positive real robustness analysis and synthesisresults (Joshi 1989, Haddad et al. 1994; and the references therein) consider onlystability issues and ignore performance. However, it is well known that robust per-formance is of paramount importance in practice. Speci® cally, even though stabilityrobustness addresses the qualitative question as to whether or not a system remainsstable for all plant perturbations within a speci® ed class of uncertainties it is impor-tant to quantitatively investigate the performance degradation within the region ofrobust stability.

In papers by Hyland et al. (1994, 1996) the tools of majorant analysis used todevelop robust stability and performance tests by Hyland and Bernstein (1987),Collins and Hyland (1989) and Hyland and Collins (1989, 1991) were extended topositive real plants controlled by strictly positive real compensators. Speci® cally,using M-matrices in the context of majorant analysis, new majorant robustnessanalysis tests were developed that yield frequency-dependent performance boundsfor frequency, damping, and mode shape uncertainty in positive real vibrationalsystems. In particular, whereas singular value performance measures only providebounds on the L 2 norm of the performance variables, the majorant bounds devel-oped by Hyland et al. (1994, 1996) provide frequency-dependent bounds on theindividual elements of the performance variables. For this class of systems the posi-tive real majorant bounds developed by Hyland et al. (1994, 1996) yield lessconservative robustness (stability and performance) predictions over previousnorm-based majorant performance bounds (Hyland and Collins 1989) and the per-formance bound resulting from real structured singular value analysis (Hyland et al.1994, 1996, Fan et al. 1991).

The main purpose of this paper is to extend the results presented by Hyland et al.(1994, 1996) to uncertain positive real structural systems involving sensor andactuator dynamics controlled by strictly positive real compensators. Because theresulting system is no longer positive real the results of Hyland et al. (1994, 1996)can no longer be applied. Extending the framework developed by Hyland et al.(1994, 1996), we develop new frequency-domain performance bounds for thismore general class of uncertain structural systems. Speci® cally, the results are devel-oped by decomposing the equivalent compensator consisting of the original strictlypositive real compensator, along with the sensor and actuator dynamics, into ageneralized positive real part and a non-generalized positive real part, and byusing the concepts of M-matrices and majorant analysis. To demonstrate the

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e� ectiveness of the proposed approach we apply our results to an Euler± Bernoullibeam with closely spaced frequency uncertainty and sensor and actuator dynamics.

2. Mathematical preliminaries

In this section we establish several de® nitions and two key lemmas. A non-nega-tive matrix A is a matrix with nonnegative elements, i.e. A ³ ³ 0. For given matrixnorms a block-norm matrix (Ostrowski 1975) of a given partitioned matrix is a non-negative matrix each of whose elements is the norm of the corresponding partitionedsubblock. The modulus matrix of A Î C

m ´ n is the non-negative matrix

|A|M [|A(i,j )|] (1)Note that the modulus matrix is a special case of a block-norm matrix. Furthermore

|AB|M £ £ |A|M|B|M (2)for B Î C

n´ p.A majorant (Dahlquist 1983) is an element-by-element upper bound for a block-

norm matrix. Speci® cally, A is a majorant of A Î Cm ´ n if

|A|M £ £ A (3)Let A Î C

n´ n. Then ÏA Î Rn´ n is a minorant (Dahlquist 1983) of A if

ÏA(i, i) £ |A(i,i)| (4)ÏA( i,j) £ - |A( i, j)|, i /= j (5)

The following lemma is a direct consequence of the above de® nitions.

Lemma 1: L et Z Î Cn n. If ÏZd is a minorant of Zd and Zod is a majorant of Zod,

then ÏZd - Zod is a minorant of Z.

A matrix Z Î Rn´ n is a non-singular M-matrix (Fiedler and Ptak 1962, Seneta

1973, Berman and Plemmons 1979) if it has nonpositive o� -diagonal elements (i.e.,Z( i,j) £ 0 for i /= j ) and positive principal minors. Recall that the inverse of a non-singular M-matrix is a nonnegative matrix (Fiedler and Ptak, 1962; Seneta, 1973;and Berman and Plemmons, 1979).

Lemma 2 (Dahlquist 1983): Assume that Z Î Cn´ n and let ÏZ Î R

n n be aminorant of Z. If ÏZ is a non-singular M-matrix, then Z is non-singular and

|Z- 1|M £ £ ÏZ- 1 (6)Finally, a square transfer function G(s) is called generalized positive real (Ander-

son and Moore 1968) if G(s) has no imaginary poles and He G(jx ) is non-negativede® nite for all real x . Note that a generalized positive real transfer function need notbe asymptotically stable.

3. Robust stability and performance for uncertain systems with sensor and actuatordynamics

We begin by considering the following nth-order, uncertain, matrix second-ordervibrational system with proportional damping and rate measurements

Èh (t) + 2ZX Çh (t) + X2 h (t) = Bu(t) + Dw(t) (7)

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y(t) = C Çh (t) (8)z(t) = E Çh (t) (9)

whereX = diag ( X 1, . . . , X n) (10)Z = diag ( z 1, . . . , z n) (11)

and where X i > 0, z i > 0, i = 1, . . . ,n, u Î Rnu is the control signal, w Î R

nw is thedisturbance variable or reference signal, y Î R

ny represents the rate measurements,and z Î R

nz represents the performance variables, restricted to be a linear function ofthe modal rates. The parameters X i and z i, i = 1, . . . ,n, denote the modal frequenciesand damping ratios, respectively. It is assumed that

X Î ­ {X 0 + ¢ X : |¢ X |M £ £ ¢ X } (12)

Z Î Z {Z0 + ¢Z : |¢Z|M £ £ ¢Z} (13)

B Î B {B0 + ¢B : |¢B|M £ £ ¢B} (14)

C Î C {C0 + ¢C : |¢C|M £ £ ¢C} (15)

D Î D {D0 + ¢D : |¢D|M £ £ ¢D} (16)

E Î E {E0 + ¢E : |¢E|M £ £ ¢E} (17)where X 0 Î R

n´ n is a given nominal matrix, ¢ X Î Rn´ n denotes the perturbation

from the nominal matrix X 0, and ¢ X Î Rn´ n is the matrix majorant of ¢ X (and

similarly for Z0 Î Rn´ n, B0 Î R

n´ nu , C0 Î Rny ´ n, D0 Î R

n´ nw and E0 Î Rnz ´ n). Note

that because X 0 and Z0 are diagonal ¢ X , ¢ X , ¢Z, and ¢Z are assumed to bediagonal.

Next, de® ne the ordered pairs H1 ( X , Z), H2 (B, C) and H3 (D,E ), andde® ne H1, H2 and H3 to be the corresponding uncertainty sets; that is,H1 Î H1 ­ ´ Z, H2 Î H2 B ´ C, and H3 Î H3 D ´ E, Additionally, de® neH (H1,H2,H3) and H H1 ´ H2 ´ H3 so that H Î H. Note that H1 is the uncer-tainty set that corresponds to errors in the frequencies and damping ratios, and H2

and H3 are uncertainty sets that correspond to errors in the mode shapes. It followsfrom (12) ± (17) that H1, H2 and H3 are arcwise-connected.

Furthermore, let µ(H,s) L [Çh (t)] for all H Î H so that, with h (0) = Çh (0) = 0,(7) ± (9) has the frequency-domain representation

µ(H,s) = U (H1,s)(Bu(s) + Dw(s)) (18)y(H,s) = Cµ(H,s) (19)z(H,s) = Eµ(H, s) (20)

whereU (H1,s) diag ( u 1(s, z 1, X 1), . . . , u n(s, z n, X n)), H1 Î H1 (21)

and

u i(s, z i, X i)s

s2 + 2z i X is + X2i

(22)

Note that for all H1 Î H1, U (H1,s) is positive real, and

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He U (H1, jx ) > 0, H1 Î H1, x Î R , x /= 0 (23)To make the model more realistic we now include sensor and actuator dynamics.

Because these dynamics can be experimentally determined, we assume that thesedynamics are known. The matrix transfer function W a of actuator dynamics andthe matrix transfer function W s of sensor dynamics are given by

W a(s) diag ( W a,1(s), . . . , W a,n(s)) (24)W s(s) diag ( W s,1(s), . . . , W s,n(s)) (25)

Appending these dynamics to the system (18) ± (20) yields

µ(H, s) = U (H1,s)(BW a(s)u(s) + Dw(s)) (26)y(H, s) = W s(s)Cµ(H,s) (27)z(H, s) = Eµ(H,s) (28)

Remark 1: Note that even though (8) restricts sensor measurements to ratemeasurements, the sensor model given by (27) allows for position and accelerationmeasurements. Speci® cally, integrators and di� erentiators can be appended as partof the sensor dynamics to obtain position and acceleration measurements from therate measurements. Hence, the results of this paper can be applied to the case ofposition and acceleration measurements.

Next, assume that the linear feedback law

u(s) = - K(s)y(s) (29)stabilizes the nominal system, i.e. the system (26) ± (28) with H1 = ( X 0, K 0) andH2 = (B0,C0). Furtheremore, assume colocated velocity feedback so that B = CT

(Joshi 1989). Substituting (29) into (26) gives

[I + U (H1,s)F(H2,s)]µ(H,s) = U (H1,s)Dw(s) (30)where F(H2,s) B W (s)BT and W (s) W a(s)K(s) W s(s).

Now, for x /= 0 and H2 Î H2, de® ne Fpr(H2, jx ) to be the generalized positivereal part and Fnpr(H2, jx ) to be the non-generalized positive real part F(H2, x ),respectively so that

F(H2, jx ) = Fpr(H2, jx ) + Fnpr(H2, jx ), x /= 0 (31)where

Fpr(H2, jx ) F(H2, j x ), ¸min(He W (jx )) ³ 00, otherwise

(32)

and

Fnpr(H2, jx ) F(H2, jx ) - Fpr(H2, jx ) (33)

for all x /= 0. Similarly, de® ne

W pr( jx ) W (jx ), ¸min(He W (j x )) ³ 0, x Î R

0, otherwise

andW npr(jx ) W (j x ) - W pr( jx ), x /= 0

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The following three lemmas are important to the development of the robust stabilityand performance bounds presented in this paper.

Lemma 3: L et H1 Î H1, H2 Î H2, and x Î R , x /= 0. Then det U (H1, jx ) /= 0 andHe[U - 1(H1, jx ) + Fpr(H2, jx )]> 0. Hence

det[U - 1(H1, jx ) + Fpr(H2, jx )] /= 0 (34)

Proof: For the proof, see Haddad and Bernstein (1991). u

For simplicity of exposition we de® ne

C (H, j x ) [U - 1(H1, jx ) + Fpr(H2, j x )]- 1, x /= 00 x = 0

(35)

for all H Î H, H1 Î H1, H2 Î H2, and x Î R . Furthermore, de® ne S : R ® R as

S ( a ) a , a ³ 00, a < 0

The next two lemmas are a direct generalization of Theorems 6.1 and 6.2 of Hylandet al. (1994, 1996). The proofs follow from majorant analysis and standard singularvalue inequalities, and hence are omitted. For details of similar proofs see Hyland etal. (1996). For the statement of the ® rst result de® ne

^C 0( jx ) p- 1(jx )Un, x /= 0

where

p(jx ) max mink

2( z 0,k - ¢ z k)( X 0,k - ¢ X k)

+ 12[S ( s min(B0M(jx )) - s max(M(jx ))||¢B||F)]2,

mink

1x

( X 0,k - ¢ X k)2 - x - s max (Sh W pr(jx ))( s max(B0) + ||¢B||F)2,

mink

x - 1x

( X 0,k - ¢ X k)2 - s max (Sh W pr(jx ))( s max(B0) + ||¢B||F)2

W pr(jx ) + W *pr(jx ) = M(jx )M*(jx )

and Un denotes the n ´ n matrix with all unity elements.

Lemma 4: L et U (H1,s) and Fpr(H2, jx ), x Î R , x /= 0, be given by (21) and (32),respectively. Then

|C (H, jx )|M £ £ ^C 0(jx ), H Î H

For the statement of the next lemma, de® ne

P(jx ) diag (P1(jx ) . . . ,Pn(j x ))where

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Pk(jx ) max 2( z 0,k - ¢z k)( X 0,k - ¢X k) + min (He W pr(jx ))m

l= 1[S (B0(k,l) - ¢B(k,l))]2,

minX Î ­

X20,kx

- x - s max(Sh W pr(jx ))m

l= 1[|B0(k,l)| + ¢B(k,l)]2

and let Fod(jx ) be given by

[Fod(jx )](i,j) = s max( W pr(j x ))m

k= 1

(|B0( i,k) | + ¢B(i,k))21 /2

´m

k= 1

(|B0( j,k)| + ¢B( j,k))21 /2

, i /= j

Lemma 5: L et U (H1,s) and Fpr(H2, jx ), x Î R , x /= 0, be given by (21) and (32),respectively, and let Fd(H2, j x ) and Fod(H2, j x ), respectively, denote the diagonaland o� -diagonal matrices corresponding to Fpr(H2, jx ). If P(jx ) - ÇFod(jx ) is a non-singular M-matrix, then

|C (H, j x )|M £ £ [P(jx ) - Fod(jx )]- 1, H Î H

Next, de® ne ^C (jx ) by

[C (jx )]( i,j) min ([C 0(jx )](i, j ),[(P(jx ) - Fod(jx ))- 1]( i, j )), x Î R , x /= 0

0, x = 0(36)

and let Fnpr(jx ) be a majorant of Fnpr(H2, jx ) for all H2 Î H2, that is,

|Fnpr(H2, jx )|M £ £ Fnpr(jx ), H2 Î H2, x /= 0 (37)

Now, note that Fnpr(H2, j x ), H2 Î H2 and x /= 0, is given by

[Fnpr(H2, jx )](i, j) =m

l= 1

m

k= 1B(i,l) W npr(l,k)B( j,k)

and |[Fnpr(H2, j x )](i,j)| £ [Fnpr(jx )](i, j) , where [Fnpr( jx )](i,j) is given by

[Fnpr(j x )](i,j) =

s max( W npr(jx ))m

k= 1

(|B0(i,k)| + ¢B(i,k))21 /2 m

k= 1

(|B0( j,k)| + ¢B( j,k))21 /2

Next we present the main result of this paper, which gives robust stability andperformance bounds for the uncertain vibrational system described by (26)± (29).

Theorem 1: L et H Î H. If

q ( ^C (j x ) Fnpr(jx )) < 1, x Î R (38)

then the closed-loop system (26) ± (29) is asymptotically stable. Furthermore, the outputz(H, jx ) satis® es the bound

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maxH Î H

|z(H, jx )|M £ £ |E|M[In - ^C (j x ) Fnpr(jx )]- 1^C (jx )|Dw(jx )|M, x Î R (39)

Proof: It follows from the multivariable Nyquist criterion that to establishasymptotic stability of the closed-loop uncertain system given by (30) it su� ces toshow that

det[I + U (H1, jx )F(H2, jx )] /= 0, for all x Î R , H1 Î H1, H2 Î H2

Using the de® nition of a minorant, it follows that I - ^C (j x ) Fnpr(jx ) is a minorant ofI + C (H, jx )Fnpr(H2, jx ) for all H Î H and H2 Î H2. Now (38) implies thatI - ^C (jx )Fnpr(j x ) is a non-singular M-matrix. Hence, it follows from Lemma 2that I + C (H, jx )Fnpr(H2, jx ) is invertible for all H Î H, H2 Î H2 and x Î R .Furthermore, since by Lemma 3 U (H1, jx ) and C (H, j x ) are invertible for allH1 Î H1, H Î H and x /= 0, and by noting that

det[I + U (H1, jx )F(H2, jx )]= det[I + C (H, j x )Fnpr(H2, jx )]´ det[C - 1(H1, jx )]det[U (H1, jx )]

it follows that

det[I + U (H1, jx )F(H2, jx )] /= 0, for all H1 Î H2, H2 Î H2, x /= 0

Furthermore

det[I + U (H1,0)F(H2,0)]= 1 for all H1 Î H1, H2 Î H2

since U (H1,0) = 0. Now the performance bound (39) is a direct consequence of (2),(6) and (20). u

Remark 2: Note that if F(H2,s) is strictly positive real for all H2 Î H2, then thespectral radius condition (38) is always satis® ed, since Fnpr(H2, jx ) = 0 for allx Î R . Hence, Theorem 1 predicts unconditional stability for all uncertain positivereal plants controlled by strictly positive real compensators. Furthermore, in thiscase the performance bound given by (39) specializes to the performance boundobtained in Hyland et al. (1994, 1996).

It is important to note that the results presented in this section need not berestricted to positive real plants (without the appended sensor and actuatordynamics) and positive real compensators. Speci® cally, if we assume that U (H1,s)is not positive real, for all H1 Î H1, in (21) and de® ne Gµw,pr(H, jx ) and Gµw,npr(H, jx )to be the generalized positive real and non-generalized positive real parts, respec-tively, of

Gµw(H, j x ) [I + U (H1, j x )F(H2, j x )]- 1U (H1, j x )

such that¸min(HeGµw,pr(H, jx )) > 0

Gµw,npr(H, j x ) Gµw(H, jx ) - Gµw,pr(H, jx )

for all x Î R and H Î H, then Theorem 1 holds with minor modi® cations. Note thatin this case no positivity assumption on either the plant (without the appendedsensor and actuator dynamics) or the compensator is required. However, it should

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be noted that the bounds on C (H, jx ) for all H Î H and x Î R are signi® cantly morecomplex. This case will be addressed in a future paper.

4. Illustrative numerical example

To demonstrate the e� ectiveness of the proposed approach, we present anillustrative example. Speci® cally, consider the simply supported Euler± Bernoullibeam with governing partial di� erential equation for the transverse de¯ ectionw(x, t) given by

m(x) ¶ 2w(x, t)¶ t2 +

¶ 2

¶ x2 EI(x) ¶ 2w(x, t)¶ x2 = f (x, t)

with boundary conditions

w(x, t)|x= 0,L = 0, EI(x) ¶ 2w(x, t)¶ x2

x= 0,L= 0

where m(x) is mass per unit length and EI(x) is the ¯ exural rigidity, with E denotingYoung’s modulus of elasticity and I(x) denoting the cross-sectional area moment ofinertia about an axis normal to the plane of vibration and passing through the centerof the cross-sectional area. Finally, f (x, t) is the force distribution due to controlactuation. Assuming uniform beam properties, the modal decomposition of thissystem has the form

w(x, t) =¥

r= 1W r(x)qr(t)

L

0mW 2

r (X) dx = 1, W r(x) = (2/mL )1 /2 sinrp xL , r = 1,2, . . .

where, assuming uniform proportional damping, the modal coordinates qr satisfy

Èqr(t) + 2z X r Çqr(t) + X2rqr(t) =

L

0f (x, t)W r(x) dx, r = 1,2, . . .

For simplicity, assume that L = p and m = EI = 2/ p so that (2/mL )1 /2 = 1.Furthermore, we place a colocated velocity/force actuator pair at x = 0.55L . Finally,modelling the ® rst two modes, de® ning the plant state as x = [q1 Çq1 q2 Çq2]T, andde® ning the performance of the beam in terms of the velocity at x = 0.7L , theresulting state-space model and problem data are

Çx(t) = Ax(t) + Bu(t) + D1w(t)

y(t) = Cx(t) + D2w(t)where

A = block-diagi= 1,2

0 1- X

2i - 2z X i

, X i = i2, z = 0.01

B = CT = [0 0.9877 0 - 0.309]T, D1 = [B 04´ 1], D2 = [0 1.9]with the performance variables

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z(t) = E1x(t) + E2u(t)

where

E1 =0 0.809 0 - 0.9510 0 0 0

, E2 = [0 1.9]T

Using Theorem 3.2 of Haddad et al. (1994) (see also Lozano-Leal and Joshi(1988)) we design a strictly positive real compensator K(s). Next we assumefrequency uncertainty in both X 1 and X 2 with ¢ X 1 = 0.5 and ¢ X 2 = 0.4. To re¯ ecta more realistic setting, we include actuator and sensor dynamics described by

W a(s) =20

0.01s2 + s + 20, W s(s) =

200.01s2 + s + 20

Because of the actuator and sensor dynamics, W a(s)K(s) C s(s) is positive real only upto x = 2.5 rad/s as seen in Fig. 1. Hence the techniques developed by Hyland et al.(1994, 1996) for generating frequency-domain performance bounds cannot beapplied here. For the assumed uncertainty range the real structured singular valuebound (mixed-¹ bound) (Hyland et al. 1996, Fan et al. 1991) and the complexblock-structured majorant bound (Hyland and Collins 1991) are in® nite since bothmethods predict instability. Speci® cally, real structured singular value analysispredicts stability for the range ¢ X 1 < 0.09 and ¢ X 2 < 0.36, whereas complexblock-structured majorant analysis predicts stability for the range ¢ X 1 < 0.06 and¢ X 2 < 0.123. The proposed majorant analysis test guarantees robust stability for theassumed uncertainty range and gives a ® nite performance bound (Fig. 2). It is

390 W. M. Haddad et al.

Figure 1. Compensator with sensor and actuator dynamics.

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important to note that in this case the proposed majorant bounds provide asigni® cant improvement over the mixed-¹ bound.

5. Conclusion

This paper has developed frequency-domain performance bounds for closed-loopuncertain positive real vibrational systems involving sensor and actuator dynamicscontrolled by strictly positive real compensators. These results are developed byusing properties of M-matrices in conjunction with majorant analysis. The resultspresented here can also be used to analyse robust stability and performance forsystems with time delay. Speci® cally, a Pade approximation model can be appendedto the compensator dynamics to account for phase lag in the bandwidth of interest.The e� ectiveness of the proposed approach was demonstrated on a vibrationaluncertain system with sensor and actuator dynamics.

ACKNOWLEDGMENTS

This research was supported in part by the National Science Foundation underGrant ECS-9496249, and the Air Force O� ce of Scienti® c Research under ContractF49620-92-C-0019 and Grant F49620-96-1-0125.

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Frequency-domain performance bounds 391

Figure 2. Performance bound for the Euler± Bernoulli beam.

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