3

Reduced-Order H∞ Output-Feedback Control

3.1 Introduction

In the this chapter we address the problem of H∞ reduced-order output-feedback control of state–delayed, continuous-time, state-multiplicative linearsystems via the input–output approach, which enables us to transform theretarded systems to uncertain non-retarded ones [64].

The point of view taken in the present chapter with regard to the timedelay in the system states, is similar to the one taken for the solution of thestate-feedback control and filtering problems in Chapter 2. This point of viewdiffers essentially from the one of Lyapunov–Krasovskii (L–K) [130], [18] aswe extend the input–output approach of [33] to delay-dependent stochasticsolutions of the above problems.

The problem of controlling deterministic linear systems via a reduced-order controller has attracted a lot of interest in the past. In the case wherethe parameters of the system were all known, some iterative methods havebeen suggested [74], [82] that achieve, if they converge, a controller of a pre-scribed order that satisfies prescribed demands on the performance of theclosed-loop system. The solutions achieved by these methods, even if theyconverge to the global minimum of the performance index, cannot guaran-tee the performance in the case where the parameters of the system are notcertain and are only known to reside in a given polytope. To the best ofour knowledge, the problem of reduced-order control for retarded stochasticstate-multiplicative counterpart of the above deterministic systems has notbeen addressed in the literature. Here, we extend the simple design methodwhich has been applied for the solution of the robust SOF control of nonretarded state-multiplicative systems [54] to the case where a reduced-orderoutput-feedback controller is sought for retarded uncertain systems. However,the approach used for the solution of the nondelayed SOF problem in [54] andin Chapter 2 for the retarded case, is substantially different from the approachapplied in this chapter.

E. Gershon & U. Shaked: Advanced Topics in Control & Estimation, LNCIS 439, pp. 61–74.DOI: 10.1007/978-1-4471-5070-1_3 c© Springer-Verlag London 2013

62 3 Reduced-Order H∞ Output-Feedback Control

In [54] (see also Chapter 2, Section 2.8), it has been shown that since,in designing static output-feedback, a constant gain cannot be achieved inpractice and all amplifiers have some finite bandwidth, one can add, in seriesto the measured output of the system, a simple low-pass component witha very high bandwidth. A parameter dependent Lyapunov function is thendescribed for the augmented system that is obtained by incorporating thestates of the additional component into the state space description.

In this chapter we avoid the use of the above low pass filter. We first trans-form the retarded closed-loop system to an augmented one, in such a way thata “static” gain is used. We then transform the latter system to an equivalentdelay free, norm bounded system. Now, in order to avoid the inclusion of thea low pass filter as in [54], we transform, by further augmentation, the lattersystem to a descriptor one. We then apply a special parameter dependent Lya-punov function, to guarantee closed-loop stability and to obtain the requiredcontroller. In our system we allow for a time-varying delay where the uncertainstochastic parameters multiply both the delayed and the nondelayed states inthe state space model of the system. We first derive sufficient condition forobtaining the latter controller for nominal systems and we then extend ourtheory to the robust polytopic case. To demonstrate the tractability of oursolution method, a numerical example taken from the field of aircraft control,is given in Chapter 13 (Example 13.1 ).

3.2 Problem Formulation

We consider the following linear system

dx(t) = [Ax(t) +B1w(t) +A1x(t− τ(t))]dt +B2u(t)dt+Gx(t)dβ(t)+Hx(t− τ(t))dν(t), x(θ) = 0, θ ≤ 0,y(t) = C2x(t) +D21w(t),z(t) = C1x(t) +D12u(t),

(3.1)

where x(t) ∈ Rn is the state vector, w(t) ∈ L2Ft([0,∞);Rq) is an exogenous

disturbance, y(t) ∈ Rm is the measurement vector , z(t) ∈ Rz is the objectivevector, u(t) ∈ R is the control input signal and A, A1, B1, B2, C1, C2 andD12, D21, G, H are constant matrices.

In (3.1a), τ(t) is an unknown continuous time-delay which satisfies:

0 ≤ τ(t) ≤ h, τ(t) ≤ d < 1, (3.2)

where h is the bound on the delay interval. The zero-mean real scalar Wienerprocesses β(t), ν(t) satisfy:

E{β(t)β(s)} =min(t, s), E{ν(t)ν(s)} =min(t, s), E{β(t)ν(s)}= 0.

3.2 Problem Formulation 63

We seek a controller of order r ≤ n with the following dynamics:

dxc(t) = [Acxc(t) +Bcy(t)]dt, u(t) = Ccxc(t) +Dcy(t) (3.3)

that, for a prescribed scalar γ > 0 and for all nonzero w(t) ∈ L2Ft([0,∞);Rq),

guarantees JOF < 0 where

JOFΔ= E{

∫ ∞

0

||z(t)||2dt− γ2∫ ∞

0

||w(t)||2dt}. (3.4)

Remark 3.1. Obviously r = n results in a full-order dynamic output-feedbackcontrol while r = 0 yields a static output-feedback gain. We note that inmany practical cases, one seeks a simple controller that, on one hand is easilycomputed and implemented and on the other hand, achieves a disturbanceattenuation bound that is smaller than the one obtained by static gains. Inthese cases, r < n is sought.

In the robust stochastic H∞control problem, we assume that the system pa-rameters lie within the following polytope

ΩΔ=

[A A1 B1 B2 C1 D12 D21 H G

], (3.5)

which is described by its vertices:

Ω = Co{Ω1, Ω2, ..., ΩN}, (3.6)

whereΩi

Δ=

[A(i) A

(i)1 B

(i)1 B

(i)2 C

(i)1 D

(i)12 D

(i)21 H

(i) G(i)], (3.7)

and where N is the number of vertices. In other words:

Ω =

N∑i=1

Ωifi ,

N∑i=1

fi = 1 , fi ≥ 0. (3.8)

Similarly to nominal case, a reduced-order controller of the structure of (3.3)is sought that, for a prescribed positive scalar γ and for all nonzero w(t) ∈L2Ft([0,∞);Rq), guarantees JOF < 0 for all uncertain parameters in the given

polytope, where JOF is given in (3.4).

64 3 Reduced-Order H∞ Output-Feedback Control

3.3 The Delayed Stochastic Reduced-Order H∞Control

Considering the above problem of reduced-order control for nominal systemsand defining the following matrices:

AΔ=

[A 00 0

], B1

Δ=

[B1

0

], B2

Δ=

[0 B2

I 0

], CT

1Δ=

[CT

1

0

], C2

Δ=

[0 IC2 0

],

DT12

Δ=

[0DT

12

], D21

Δ=

[0D21

], A1

Δ=

[A1 00 0

], G

Δ=

[G 00 0

], H

Δ=

[H 00 0

],

KΔ=

[Ac Bc

Cc Dc

], (3.9)

the following theorem provides a sufficient condition for the existence of asolution to the problem:

Theorem 3.1. Consider the system (3.1a–c). There exists a stabilizingreduced-order controller K that achieves negative JOF for all nonzero w ∈L2Ft([0,∞);Rq), for prescribed α > 0, γ > 0 and positive scalar tuning pa-

rameters ε and ε, if there exist matrices Q > 0, Q > 0, RQ > 0 and K thatsatisfy the following LMI:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 Qε

[B1

D21

]Υ15 Ψ16 0 Υ18

∗ −RQ 0 0 Υ25 0 Υ27 0

∗ ∗ −Υ33 0 Υ35 0 0 0

∗ ∗ ∗ −γ2Iq Υ45 0 0 0

∗ ∗ ∗ ∗ −Υ55 0 0 0

∗ ∗ ∗ ∗ ∗ −Iz 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (3.10)

3.3 The Delayed Stochastic Reduced-Order H∞ Control 65

where

Ψ11,a =

[AQ+QAT + B2KC2 + C

T2 K

T BT2 αB2K +QCT

2 − CT2 Q

αKT BT2 + C2Q− QC2 −2αQ

],

Qε = ε

[Q 00 0

],

Υ11 = Ψ11,a + 2Qε +RQ

1−d ,

Υ12 =

[(A1 − εI)Q 0

0 0

],

Υ15 = h

[εQAT + εCT

2 KT BT

2 +QCT2 C2 − CT

2 QC2 αQCT2 − αCT

2 Q

εαKT BT2 − hαQC2 −α2Q

]+ hεQε,

Υ16 =

[QCT

1 + CT2 K

T DT12

αKT DT12

],

Υ18 =

[QGT

0

],

Υ25 = hε

[QAT

1 0

0 0

]− hεQε,

Υ27 =

[QHT

0

],

Υ35 = −hεQε,

Υ45 = h[εBT

1 + DT21C2 αD

T21

],

Υ33 = Υ55 =

[εQ+ CT

2 QC2 αCT2 Q

αQC2 α2Q

].

(3.11)In the latter case the controller gain is given by:

K = KQ−1. (3.12)

Proof: Augmenting the system (3.1a) to include the states of (3.3a), wedefine the augmented state vector ξ(t) = col{x(t), xc(t)} and obtain thefollowing representation:

dξ(t) = [Aξ(t) + Bw(t)]dt + A1ξ(t− τ(t))dt + Gξ(t)dβ(t) + Hξ(t− τ(t))dν(t),ξ(0) = 0, over [−h 0],

z(t) = Cξ(t) + Dw(t),(3.13)

66 3 Reduced-Order H∞ Output-Feedback Control

where

AΔ= A+ B2KC2,

BΔ= B1 + B2KD21,

CΔ= C1 + D12KC2,

D = D12KD21.

(3.14)

Recalling the definition of (9k), the system (3.13a,b) can be described as aclosed-loop system with

u(t) = Ky(t),

that is applied to the following system:

dξ(t) = [Aξ(t) + B1w(t) + B2u(t)]dt+ A1ξ(t− τ(t))dt + Gξ(t)dβ(t)+Hξ(t− τ(t))dν(t),ξ(0) = 0, over [−h 0],

y(t) = C2ξ(t) + D21w(t),

z = C1ξ(t) + D12u(t),

(3.15)

where ξ(t) ∈ Rn+r, y(t) ∈ Rm+r, u(t) ∈ Rl+r, and where A, B1, B2, C1, C2

and D12, A1, G, H, D21 are given in (3.9). Note that in the latter for-mulation, the problem of finding the dynamic controller parameters (i.eAc, Bc, Cc, and Dc) becomes one of seeking a static output-feedback gainthat achieves the required attenuation level.

In order to solve the above problem, we introduce the following operators:

(D0g)(t)Δ= g(t− τ(t)), (D1g)(t)

Δ=

∫ t

t−τ(t)

g(s)ds, (3.16)

where g(t) is a piecewise continuous-time function. In what follows we use thefact that the induced L2-norm of D0 is bounded by 1√

1−d, and the fact that

the induced L2-norm of D1 is bounded by h (see [77], Lemma 1 for the latternorm). Using the operator notations of (3.16), defining

w1(t) = (D0ξ)(t) = ξ(t− τ(t)), and w2(t) = (D1y)(t) = {∫ t

t−τ

y(s)ds},

where y(t) is given in (3.17b), and introducing a matrix M ∈ R(n+r)×(n+r),we obtain the following system:

dξ(t) = [A+M ]ξ(t)dt+ [B1w(t) + B2u(t)]dt+ (A1 −M)w1(t)dt

−Mw2(t)dt + Gξ(t)dβ(t) + Hw1(t)dν − Γβdt− Γνdt,

y(t) = [A+M ]ξ(t) + (A1 −M)w1(t)−Mw2(t) + B1w(t) + B2u(t)− Γβ − Γν(3.17)

where

3.3 The Delayed Stochastic Reduced-Order H∞ Control 67

Γβ =M

∫ t

t−τ

Gξ(s)dβ(s), and Γν =M

∫ t

t−τ

Hw1(s)dν(s).

We note that the definitions of D1g in (3.16b) is different from the one usedin [77]. The reason is that in the stochastic case the latter definition doesnot lead to the bound of h. We also note that the matrix M is an unknownconstant matrix which is introduced into the dynamics of (3.17a,b) in orderto achieve additional degree of freedom in the design of the reduced-ordercontroller.

Remark 3.2. The system (3.15a) is a special case of the dynamics of (3.17a),similarly to what is shown in Remark 1.1, Chapter 1. This follows by replacingMw2(t)dt in (3.17a) byM{

∫ t

t−τy(t,)dt,}dt, using the above definition of w2(t).

Noting that y(t,)dt, = dξ(t,)− Gξ(t,)dβ(t,)− Hw1(t,)dν(t,), one obtains that

−Mw2(t)=−M∫ t

t−τ

y(t,)dt,=−M∫ t

t−τ

{dξ(t,)−Gξ(t,)dβ(t,)−Hw1(t,)dν(t,)}

=−Mξ(t) +Mξ(t− τ(t)) + Γβ + Γν = −Mξ(t) +Mw1(t) + Γβ + Γν ,

where Γβ and Γν are defined following (3.17). Substituting the right handside of the latter equation for −Mw2(t) in (3.17a), the dynamics of (3.15a) isrecovered.

Using the fact that ||D0||∞ ≤ 1√1−d

and ||D1||∞ ≤ h [77], (3.17) may be cast

into what is entitled: the norm-bounded uncertain model, by introducing into(3.17) the following new variables:

w1(t) = Δ1ξ(t), and w2(t) = Δ2y(t), (3.18)

where ||Δ1||∞ ≤ 1√1−d

and ||Δ2||∞ ≤ h are diagonal operators having iden-

tical scalar operators on the main diagonal. In order to avoid the use of alow pass filter for the solution of the “static” control problem of (17a,b) [54],

we define ξ(t)Δ= col{ξ(t), y(t)} and obtain the following equivalent descriptor

system representation to the closed-loop system:

Edξ(t) = (A+ M)ξ(t)dt+ Bw(t) + (A1 − M)w1(t)dt+ Gξ(t)dβ(t)

−Mw2(t) + Hw1(t)dν(t) + Γβ,νdt, ξ(0) = 0, over [−h 0],z = [C1 D12K]ξ(t),

(3.19)

where ξ(t) ∈ Rn+m+2r and where

EΔ=

[I 00 0

], w1(t) =

[w1(t)0

], w2(t) =

[w2(t)0

], A

Δ=

[A B2KC2 −I

],

A1Δ=

[A1 00 0

], B

Δ=

[B1

D21

], M =

[M 00 0

], G

Δ=

[G 00 0

], H

Δ=

[H 00 0

],

68 3 Reduced-Order H∞ Output-Feedback Control

Γβ,νΔ=

[−M [Γβ + Γν ]

0

]. (3.20)

In order to derive conditions for the stability of the closed-loop system, weconsider the system of (3.19), with B = 0 and define the following Lyapunovfunction:

V (t, ξ(t)) = ξTEP ξ, PΔ=

[P 0

−α−1C2P α−1P

](3.21)

where P ∈ R(n+r)×(n+r) is a positive definite matrix, P > 0 is a matrix inR(m+r)×(m+r), and α is chosen for simplicity to be a positive scalar.

Noting that E{Γβ,ν} = 0 , we apply the Ito lemma [9], and taking expec-tation we obtain:

E{(LV )(t)} = E{2〈P ξ(t), [(A+ M)ξ(t) + (A1 − M)w1(t)− Mw2(t)]〉}+E{Tr{P [Gξ(t) Hw1(t)]Q[Gξ(t) Hw1(t)]

T }},

where L is the infinitesimal generator associated with the differential equation

of (3.19) [9], QΔ=

[1 00 1

]is the covariance matrix of the augmented Wiener

process vector col{β(t) v(t)}. We also have the following:

Tr{P [Gξ(t) Hw1(t)]Q[Gξ(t) Hw1(t)]T }

= Tr{[ξT (t)GT

wT1 (t)H

T

]P [Gξ(t) Hw1(t)]Q}

= Tr{[ξT (t)GT P Gξ(t) ξT (t)GT P Hw1(t)

wT1 (t)H

T P Gξ(t) wT1 (t)H

T P Hw1(t)

][1 00 1

]}

= ξT (t)GT P Gξ(t) + wT1 (t)H

T P Hw1(t).

In the attempt to establish E{(LV )(t)} ≤ −k||ξ(t)||2, for some k > 0 (uni-formly in t), we have the following requirement:

2ξT (t)PT [(A+ M)ξ(t) + (A1 − M)w1(t)− Mw2(t)] + kξT (t)ξ(t)

+ξT (t)GT P Gξ(t) + wT1 (t)H

T P Hw1(t) < 0.(3.22)

To the latter we add the following term which is nonnegative due to thediagonality of Δ1

ξT (t)(1

1 − dR1 − ΔT1 R1Δ1)ξ(t) = ξT (t)

1

1 − dR1ξ(t)− wT1 (t)R1w1(t).

Using the facts:

− 2ξT (t)PT Mw2(t) ≤ h2yT (t)R2y(t) + ξT (t)PT MR−1

2 MT P ξ(t) (3.23)

3.3 The Delayed Stochastic Reduced-Order H∞ Control 69

where R1 and R2 are constant positive definite matrices, and

wT2 (t)w2(t) = [yT (t)ΔT

2 0]

[Δ2y(t)

0

]≤ h2yT (t)R2y(t),

we find that (3.22) holds if the following inequality is satisfied:

2ξT PT [(A+ M)ξ(t) + (A1 − M)w1(t)] + kξT (t)ξ(t) + ξT (t)GT P Gξ(t)

+wT1 (t)H

T P Hw1(t) + ξT (t) 1

1−dR1ξ(t)− wT1 (t)R1w1(t) + h

2yT (t)R2y(t)

+ξT (t)PT MR−12 MT P ξ(t) < 0.

(3.24)Denoting ζ(t) = col{ξ(t), w1(t)}, (3.24) is equivalent to the following:

ζT (t)

[Ψ11 P

T (A1 − M)+

∗ −R1 + HT P H

]ζ(t)+h2yT (t)R2y(t)<0,

whereΨ11 = PT (A+ M) + (AT + MT )P + 1

1−dR1 + GT P G+ PT MR−1

2 MT P .Substituting for y from (3.17b), the following inequality is then obtained:

Ψ=

⎡⎢⎢⎢⎣

Ψ11 PT A1 − Y Y h(ATR2 + YT )

∗ −R1 + HT P H 0 h(AT

1 R2 − Y T )∗ ∗ −R2 −hY T

∗ ∗ ∗ −R2

⎤⎥⎥⎥⎦<0,

where

Ψ11 = PT A+ AT P +Y +Y T +1

1− dR1+ GT P G, Y = PT M, Y = R2M.

(3.25)

In order to avoid using BMI for the solution of (3.25a) (since M appears inboth Y and Y ), we chooseM = εI in M of (3.20g) where ε is a positive tuningscalar. We thus obtain:

Y = PT M = PT

[εI 00 0

], Y = R2M = R2

[εI 00 0

]. (3.26)

This choice transforms (3.25a) into a LMI, for a given ε. The stability of theclosed-loop system is therefore guaranteed if there exist P > 0, K, R1 >0, R2 > 0, and a positive tuning scalar ε that satisfy (3.25a). To satisfyJOF < 0 where JOF is given in (3.4) it is required that:

E{∫ ∞

0

[LV + zT (t)z(t)− γ2wT (t)w(t)]dt} < 0, (3.27)

where in the expression for E{LV } the operators Δ1 and Δ2 are used. Usingarguments similar to those that guarantee stability, the latter requirement isdescribed by the following inequality:

70 3 Reduced-Order H∞ Output-Feedback Control

Γ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ψ11 PT A1 − Y Y PT B hATR2 + hYT [C1 D12K]T

∗ −R1+ HT P H 0 0 hAT

1 R2 − hY T 0

∗ ∗ −R2 0 −hY T 0

∗ ∗ ∗ −γ2Iq hBTR2 0

∗ ∗ ∗ ∗ −R2 0∗ ∗ ∗ ∗ ∗ −Iz

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦< 0,

(3.28)

where Ψ11 is given in (3.25b).The latter inequality is not affine in the decision variables P and K. De-

noting,

Q = P−1 Δ=

[Q 0

QC2 αQ

],

we multiply (3.28) by diag{QT , QT , QT , I, QT , I} and diag{Q, Q, Q, I, Q, I}from the left and the right, respectively. Carrying out the various multiplica-tions we obtain the following inequality: Ψ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ψ11 Ψ12 QTY Q B hQT [ATR2 + YT ]Q Ψ16 0 Ψ18

∗ −QTR1Q 0 0 hQT [AT1 R2 − Y T ]Q 0 Ψ27 0

∗ ∗ −QTR2Q 0 −hQT Y T Q 0 0 0

∗ ∗ ∗ −γ2Iq hBTR2Q 0 0 0

∗ ∗ ∗ ∗ −QTR2Q 0 0 0

∗ ∗ ∗ ∗ ∗ −Iz 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(3.29)

where we define KΔ= KQ and where

3.3 The Delayed Stochastic Reduced-Order H∞ Control 71

Ψ11 = Ψ11,a + QT [Y + Y T ]Q+ QTR1Q

1−d ,

Ψ11,a =

[AQ+QAT + B2KC2 + C

T2 K

T BT2 αB2K +QCT

2 − CT2 Q

αKT BT2 + C2Q− QC2 −α[Q+ Q]

],

Ψ12 = A1Q− QTY Q,

Ψ16 = QT [C1 D12K]T =

[QT CT

1 + CT2 K

T DT12

αKT DT12

],

Ψ18 = QT GT

[I

0

],

Ψ2,7 = QT HT

[I

0

].

(3.30)

Taking R2 =

[εP 0

0 Q−1

], where ε is a positive scalar tuning parameter, de-

noting RQ = QTR1Q, R = R2Q =

[εI 0C2 αI

]and carrying further the various

multiplications, we obtain the following LMI:

Υ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 A1Q−Qε Qε B h[QT AT R+ εQε] Ψ16 0 QT GT

[I

0

]

∗ −RQ 0 0 h[QT AT1 R− εQε] 0 Ψ2,7 0

∗ ∗ −Υ33 0 −hεQε 0 0 0

∗ ∗ ∗ −γ2Iq hBT R 0 0 0

∗ ∗ ∗ ∗ −Υ55 0 0 0

∗ ∗ ∗ ∗ ∗ −Iz 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(3.31)where Qε is defined in (3.11b),

Υ11 = Ψ11,a +Qε +QTε +

RQ

1− d,

Υ33 = Υ55 = QTR2Q =

[εQ+ CT

2 QC2 αCT2 Q

αQC2 α2Q

].

Solving for A, A1, B and H, G and carrying out the various multiplications,the LMI of (3.10) is obtained.

72 3 Reduced-Order H∞ Output-Feedback Control

Remark 3.3. Considering the inequality of (3.29) for the case where h tends tozero, one would expect a condition that is equivalent to the one obtained forthe stochastic nonretarded case [53] . Indeed, letting h and R1 tend to zeroand R2 tend to infinity in (3.29) and choosing in (3.26a,b) M = A1 in Y andY , the result of [53] is recovered.

Theorem 3.1 provides a sufficient condition for the existence of the gain matrixK that guarantees the required performance for a single plant (3.15) in thepolytope Ω. As such, this gain matrix provides by (3.9k) the reduced-ordercontroller for this single plant that satisfies (3.4). Since the elements in (3.31)are affine in the systemmatrices, the condition of Theorem 3.1 can be extendedto cope with all the parameters in Ω.

Denoting the matrices of (3.15) that correspond to the i-th vertex of Ωi

(and thus also to the vertices of Ω) by: A(i), A(i)1 ,B

(i)1 , B

(i)2 , C

(i)1 , D

(i)12 and

D(i)21 , H

(i), G(i), i = 1, 2, ..., N , we obtain the following result:

Corollary 3.3.1 There exists a reduced-order controller K that achievesnegative JOF of (3.4) over the entire polytope Ω, for all nonzero w ∈L2Ft([0,∞);Rq), for prescribed γ > 0, α > 0 and positive scalar tuning pa-

rameters ε and ε, if there exist matrices Q > 0, Q > 0, RQ > 0 and K thatsatisfy the following set of LMIs:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ(i)11 Υ

(i)12 Qε

[B

(i)1

D(i)21

]Υ

(i)15 Υ

(i)16 0 Υ

(i)18

∗ −RQ 0 0 Υ(i)25 0 Υ

(i)27 0

∗ ∗ −Υ (i)33 0 Υ

(i)35 0 0 0

∗ ∗ ∗ −γ2Iq Υ(i)45 0 0 0

∗ ∗ ∗ ∗ −Υ (i)33 0 0 0

∗ ∗ ∗ ∗ ∗ −Iz 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, i = 1, 2, ..., N

(3.32)

where Υ(i)11 , Υ

(i)12 , Υ

(i)15 , Υ

(i)16 , Υ

(i)18 , Υ

(i)25 , Υ

(i)27 , Υ

(i)33 , Υ

(i)35 and Υ

(i)45 are given in

(3.11) and where the nominal matrices are replaced by those corresponding tothe vertex i of the polytope.

If a solution to the latter set of LMIs exists, the matrices of the controller(3.3) that stabilizes the system (3.1) and achieves the required performanceover the polytope Ω is given by

KΔ=

[Ac Bc

Cc Dc

]= KQ−1. (3.33)

3.3 The Delayed Stochastic Reduced-Order H∞ Control 73

Remark 3.4. We note that a simple strategy is applied for the numerical solu-tion of (3.32), which also applies to the solution of the LMI of (3.10) for thenominal case. The solution of (3.32) involves a search for two scalar variables:ε and ε. One may start by taking ε = ε and seek, using line searching, valuesfor both tuning parameters that leads to a stabilizing controller of minimumγ. Once such a controller is obtained, standard optimization techniques canbe used, say Matlab function “fminsearch”, which seeks the combination ofthe two scalar parameters that bring γ to a local minimum.

The above theorem seeks a single set of Q and Q that will satisfy the LMIsof (3.32). As such it provides what is called the quadratic stabilizing solution.This solution which may be sometimes quite conservative can be replaced byanother, less conservative, still quadratic, solution as follows: It turns out,however, that the special structure of P may sometimes lead to zero Bc andCc, thus resulting in suboptimal values of γ and will lead to static, ratherthan dynamic, output-feedback controllers. To circumvent this difficulty, wehave the following remark:

Remark 3.5. An efficient way for computing the reduced-order controllerwhich avoids the occurrence of Bc = 0 and Cc = 0, on the expense of re-stricting the structure of the controller, is one where we fix Dc a priori (sayDc = 0) and take Cc = BT

2 F where F ∈ Rn×r is a free tuning matrix. Thematrices Ac and Bc are then sought which lead to the required performance.In this case, the closed-loop system is described by:

dξ(t) = Aξ(t)dt + Bw(t)dt+ A1ξ(t− τ(t))dt + Gξ(t)dβ(t) + Hξ(t− τ(t))dν(t),z(t) = Cξ(t)dt+Dw(t)dt

(3.34)

where ξΔ= col{x, xc} and where

A =

[A+B2DcC2 B2B

T2 F

BcC2 Ac

],

B =

[B1 +B2DcD21

BcD21

],

C =[C1 +D12DcC2 D12BT

2 F],

and D = D12DcD21. Defining KΔ=

[Ac Bc

]we obtain the following:

A = A+ B2KC2,

B = B1 +KD21,

C = C1 + D12KC2,

D = D11 + D12KD21

74 3 Reduced-Order H∞ Output-Feedback Control

where

A =

[A+B2DcC2 B2B

T2 F

0 0

],

D11 = D12DcD21,

B1 =

[B1 +B2DcD21

0

],

B2 =

[0I

],

C1 =[C1 +D12DcC2 D12B

T2 F

],

C2 =

[0 IC2 0

],

D21 =

[0D21

],

D12 = 0.

Obviously, this approach is suitable for the case where B2 is without uncer-tainty. A similar approach can be applied for the case where C2 is withoutuncertainty, in which case we assign similar structure to Bc.

3.4 Conclusions

In this chapter the problem of linear H∞ reduced-order output-feedback con-trol of state-multiplicative retarded systems has been presented and solvedfor both nominal and uncertain polytopic systems. The stochastic uncertain-ties have been encountered in both the delayed and the nondelayed statesin the state space model of the system. The delay has been assumed to beunknown and time-varying where only the bounds on its length and rate ofchange are given. Delay dependent synthesis method has been applied whichis based on the input–output approach. The resulting nonretarded system isthen augmented to obtain a descriptor system which avoids the inclusion ofan artificial low pass filter, in series to the output. A stability condition isderived for the closed-loop system, which subsequently leads to the solutionof the reduced-order output-feedback control problem.

Some over-design is entailed in our solution due to the special structurechosen for the Lyapunov function and due to the special structure imposedon R2 in the resulting LMIs. In the uncertain case, an additional over-design stems from the fact that a quadratically stable solution is sought.The efficiency of the proposed results is demonstrated via example 13.1 inChapter 13.