4

Tracking Control with Preview

4.1 Introduction

In this chapter we address the problem of H∞ state-feedback preview trackingcontrol of state-delayed, continuous-time, state-multiplicative, linear systems,as formulated in Section 4.2 (for a similar deterministic setup see [78], [106],[112]). Similarly to [55],[59], the point of view taken here differs from the“traditional” L-K point of view as we apply the input–output approach.

The problem of tracking control with preview may be treated by the stan-dard theory of non-retarded systems with state-multiplicative noise (see [53]and the references therein). The approach there leads to over-conservative con-ditions which, in the case of retarded systems, also require too many tuningsof scalar parameters. The main source of conservatism stems there from theway stochastic stability is guaranteed via Linear Matrix Inequalities (LMIs)in the standard approach [53].

To reduce this conservatism, a new approach is adopted here where wefirst derive, in Section 4.3, Lemma 4.3.1, a matrix inequality condition forthe stability of the closed-loop system that involves the yet undeterminedcontrol input. Applying the resulting stability condition to the control designproblem in Section 4.4, a matrix inequality condition is obtained that guaran-tees a prescribed H∞ performance. This inequality involves the undeterminedcontrol input too. Realizing that application of Schur complement is equiva-lent to completion to squares, the optimal control strategy is then derived, inTheorem 4.1.

In our system, we allow for a time-varying delay where the uncertainstochastic parameters multiply both the delayed and the non delayed statesin the state space model of the system. We obtain sufficient conditions forstability and H∞ state-feedback tracking control design, in terms of LMIs inCorollaries 4.4.1, 4.4.2 of Section 4.4. We demonstrate our theory, in Section4.5, via an example that demonstrates the impact of the delay length, delayderivative and the preview length on the system performance. An additionalexample is given in Chapter 13 (Example 13.2).

E. Gershon & U. Shaked: Advanced Topics in Control & Estimation, LNCIS 439, pp. 75–93.DOI: 10.1007/978-1-4471-5070-1_4 c© Springer-Verlag London 2013

76 4 Tracking Control with Preview

4.2 Problem Formulation

Given the following linear continuous time-invariant system:

dx(t) = [A0x(t) +A1x(t − τ(t)) +B1w(t) + B2u(t) +B3r(t)]dt+Gx(t− τ(t))dζ(t) + Fx(t)dβ(t), x(θ) = 0, over[−h 0],z(t) = C1x(t) +D12u(t) +D13r(t)

(4.1)

where x(t) ∈ Rn is the system state vector, u(t) ∈ Rl is the input vector,r(t) ∈ L2 is the signal to be tracked, w(t) ∈ L2([0,∞);Rp) is the exogenousdisturbance signal, z(t) ∈ Rq is the signal to be controlled, A, A1, B1, B2,B3, C1, D12, D13, F, and G are constant matrices of the appropriate dimen-sions and τ(t) is the unknown time-delay that satisfies:

0 ≤ τ(t) ≤ h, τ(t) ≤ d < 1. (4.2)

The variables β(t) and ζ(t) are zero-mean real scalar Wiener processes thatsatisfy:

E{dβ(t)} = 0, E{dζ(t)}=0, E{dβ(t)2}=dt, E{dζ(t)2}=dt,

E{dβ(t)dζ(t)}= 0.

We seek a state-feedback control law u(t) that minimizes, for the worst-caseof the process disturbance w(t), the expected value of the energy of z(t) withrespect to the uncertain parameters, by using the available knowledge on thereference signal. We, therefore, consider, for a given scalar γ > 0, the followingperformance index:

JEΔ= E{

∫ ∞

0

(||z(t)||2 − γ2||w(t)||2)dt}. (4.3)

We consider two different tracking problems differing on the informationpattern over r(t) :

1) Stochastic H∞ finite-fixed preview tracking of r(t) : The trackingsignal r(t) is previewed in a known fixed interval, i.e., r(τ ) is known forτ ≤ t+ h where h is a known preview length.2) Stochastic H∞-tracking with zero preview of r(t) : The trackingsignal is measured on line, i.e., at time t, r(τ ) is known for τ ≤ t.

In these two cases we seek a control law u(t) of the formu(t) = Hxx(t) +Hrr(t), where Hx is a causal operator and where thecausality of the operator Hr depends on the information pattern of thereference signal.

For these tracking problems we consider a related linear quadratic min-max game in which the controller plays against nature. We, thus, consider thefollowing game:

4.2 Problem Formulation 77

Find r(t)-dependent strategies w∗ ∈ L2([0,∞);Rp) and u∗(t) ∈ Rl thatsatisfy, ∀r(t) ∈ L2:

JE(r, u∗, w) ≤ JE(r, u

∗, w∗) ≤ JE(r, u, w∗), ∀w(t) ∈ L2([0,∞);Rp)

and ∀u(t) ∈ Rl.

Also find the minimum upper-bound J(r) to the saddle point valueJE(r, u

∗, w∗).In order to solve the above problems, we introduce the following operators:

(Δ1g)(t)Δ= g(t− τ(t)), (Δ2g)(t)

Δ=

∫ t

t−τ(t)

g(s)ds. (4.4)

Considering the above two operators the system (4.1a) can be written as:

dx(t) = [A0 +M ]x(t)dt+ [A1 −M ]w1(t)dt−Mw2(t)dt+B1w(t)dt+B2u(t)dt+B3r(t)dt + Fx(t)dβ(t) +Gw1(t)dζ(t) − Γβdt− Γζdt,y(t) = [A0 +M ]x(t) + [A1 −M ]w1(t)−Mw2(t) +B1w(t) +B2u(t)+B3r(t) − Γβ − Γζ ,

(4.5)

where

Γβ =M

∫ t

t−τ

Fx(s)dβ(s), Γζ =M

∫ t

t−τ

Gw1(s)dζ(s),

w1(t) = Δ1x(t) and w2(t) = Δ2y(t), (4.6)

where the matrix M is an unknown constant matrix to be determined.

Remark 4.1. Similarly to what is shown in Remark 1.1 in Chapter 1, the dy-namics of (4.1a) is a special case of that of (4.5a) as follows:

Noting (4.6c,d) and applying the operators of (4.4a,b), Equation (4.5a)can be written as:

dx(t) = [A0 +M ]x(t)dt+ [A1 −M ]w1(t)dt −M{∫ t

t−τ

y(t,)dt,}dt+B1w(t)dt

+B2u(t)dt+B3r(t)dt + Fx(t)dβ(t) +Gw1(t)dζ(t) − Γβdt− Γζdt,

w1(t) = x(t − τ(t)).

Now, recalling y(t) of (4.5b) one can write:

dx(t) = y(t)dt+ Fx(t)dβ(t) +Gw1(t)dζ(t)

78 4 Tracking Control with Preview

and therefore y(t,)dt, = dx(t,)− Fx(t,)dβ(t,)−Gw1(t,)dζ(t,). Hence,

−Mw2(t) = −M

∫ t

t−τ

y(t,)dt, = −M

∫ t

t−τ

{dx(t,)− Fx(t,)dβ(t,)−Gw1(t,)dζ(t,)}

= −Mx(t) +Mx(t− τ) + Γβ + Γζ = −Mx(t) +Mw1(t) + Γβ + Γζ ,

where Γβ and Γζ are defined in (4.6a,b) respectively. Replacing the right handside of the latter equation for −Mw2(t) in (4.5a), the dynamics of (4.1a) isrecovered.

Using the facts that ||Δ1||∞ ≤ 1√1−d

and ||Δ2||∞ ≤ h (this follows from

arguments similar to those in [77], see Lemma 1), the system of (5) can bedescribed in the frame work of norm-bounded systems.

4.3 Stability of the Delayed Tracking System

In the sequel, we derive a stability result that is based on the assertion thatthe conditions for the stability of the system (4.5a,b) with the ‘feedback’ thatis described by (4.6c,d), are usually derived using the small gain theorem. It iswell known [15], however, that these conditions are identical to those obtainedby applying the Lyapunov approach to the same system with feedback that isdescribed by the diagonal ‘feedback’ operators Δ1 and Δ2 that, for all x andy in Rn, satisfy the following:

xT (t)ΔT1 Δ1x(t) ≤ (1− d)−1‖x(t)‖2

andyT (t)ΔT

2 Δ2y(t) ≤ h2‖y(t)‖2.In order to derive our stability result, we replace Δi by Δi, i = 1, 2. Not-ing that E{Γβ} = 0 and E{Γζ} = 0, we consider V (t, x(t)) = 〈x(t), Qx(t)〉.Applying Ito formula to V (t, x(t)), defining

A = A0 +M, A1 = A1 −M

and taking expectation we obtain, defining the infinitesimal generator associ-ated with the differential equation of (5a) by L:

E{LV (t, x(t))} = 2E〈Qx(t), Ax(t) + A1Δ1x(t)−MΔ2y(t) +B1w(t)

+B2u(t) +B3r(t)〉 + ETr{Q[Fx(t) Gw1(t)]P [Fx(t) Gw1(t)]T }, (4.7)

where PΔ=

[1 00 1

]is the covariance matrix of the stationary augmentedWiener

process vector col{β(t), ζ(t)} and where the condition for the stability of thesystem is E{LV (t, x(t))} < 0. Now,

4.3 Stability of the Delayed Tracking System 79

Tr{Q[Fx(t) Gw1(t)]P [Fx(t) Gw1(t)]T } = Tr{

[xT (t)F T

wT1 (t)G

T

]Q[Fx(t) Gw1(t)]P}

= xT (t)FTQFx(t) + wT1 (t)G

TQGw1(t).

Adding the following term, which is nonnegative due to the diagonality of Δ1,to (4.7):

xT (t)(1

1 − dR1 − ΔT1 R1Δ1)x(t) = xT (t)

1

1− dR1x(t)− wT1 (t)R1w1(t),

and noting that:

wT2 (t)w2(t) = yT (t)ΔT

2 Δ2y(t) ≤ h2yT (t)R2y(t),

where R1 and R2 are constant positive definite matrices we obtain:

E{LV (t, x(t))} ≤ 2E〈Qx(t), Ax(t)+ A1w1−Mw2+B1w(t)+B2u(t)+B3r(t)〉

−wT1 (t)R1w1(t) + h

2yT (t)R2y(t)− wT2 R2w2 + x

T (t)FTQFx(t)

+ xT (t)1

1− dR1x(t) + wT1 (t)G

TQGw1(t). (4.8)

Taking w(t) = 0 and r(t) = 0 in (4.8) and in y of (4.5b), defining ξ(t) =col{x(t), w1(t), w2(t), u(t)}, we readily obtain from (4.8), that stability is guar-anteed for a feedback control signal u(t) that satisfies the following inequalityE{ξT (t)φξ(t) + yT (t)h2R2y(t)} =

E{ξT (t){

⎡⎢⎢⎢⎢⎢⎢⎣

Υ11 QA1 −QM QB2

∗ −R1 +GTQG 0 0

∗ ∗ −R2 0

∗ ∗ ∗ 0

⎤⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎢⎣

AT0

AT1

−MT

BT2

⎤⎥⎥⎥⎥⎥⎥⎦h2R2

[A0 A1 −M B2

]}ξ(t)} < 0,

where Υ11 = QA+ ATQ+ 11−dR1 + F

TQF, (4.9)

and where φ is the leftmost block matrix in the last inequality. The laterinequality leads, in turn, to the following inequality:

80 4 Tracking Control with Preview

E{ξT (t)φξ(t)} < 0, where φ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 QA1 −QM QB2 AT0 +MT

∗ −R1 +GTQG 0 0 AT

1 −MT

∗ ∗ −R2 0 −MT

∗ ∗ ∗ 0 BT2

∗ ∗ ∗ ∗ −R−12 h−2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

(4.10)

and where ξ(t)Δ=

[ξ(t)y(t)

]. We note that (4.10a) should be satisfied along

the trajectories ξ. This does not mean that φ < 0. Defining QM = QM,taking R2 = ε1Q where ε1 is a positive tuning scalar, and multiplying (4.10a)from the left and the right by diag{I, I, I, I, hR2}, we obtain the followingrequirement for u(t) to achieve closed-loop stability.

E{ξTy (t){

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 −QM QB2 hε1(AT0 Q+QT

M )

∗ −R1 +GTQG 0 0 hε1(AT1 Q−QT

M )

∗ ∗ −ε1Q 0 −hε1QTM

∗ ∗ ∗ 0 hε1Q

∗ ∗ ∗ ∗ −ε1Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦}ξy(t)} < 0, (4.11)

where ξy(t)Δ=

[ξ(t)

h−1R−12 y(t)

]and where Υ11 = QA0 +QM + AT

0 Q +QTM +

11−dR1 + F

TQF, Υ12 = QA1 −QM . We thus obtain the following lemma:

Lemma 4.3.1 The system of (4.1a ) with w = 0 and r = 0 is exponentiallymean square stable for a given feedback control signal u(t) and a given positivescalar ε1, if there exist positive definite matrices 0 < Q ∈ Rn×n, 0 < R1 ∈Rn×n and a matrix QM ∈ Rn×n that satisfy (4.11).

4.4 The State-Feedback Tracking

In the sequel, the solution to the stochastic state-feedback tracking controlproblem for retarded systems is obtained by applying a game theory approachwhich enables us to obtain the strategies of the signals involved, alongside thetwo preview tracking patterns that are applied. We note that the solution tothe non-retarded case in ([53], pages 57–65) has first been obtained for thefinite-horizon case using the game theoretic approach. It has been extendedthere to infinite-horizon where mean square stability of the close-loop systemis obtained in the limit where t→ ∞.

The approach taken in the present manuscript, is somewhat different. Here,we restrict the theory to the infinite-horizon case. We assume, first, that the

4.4 The State-Feedback Tracking 81

closed-loop system is exponentially mean square stable and then derive a LMIcondition that assures the H∞ control performance if the ‘minimizing control’input indeed stabilizes the closed-loop system. We then show, in the Appendix,that the condition of Lemma 4.3.1, for the stability of the closed-loop system,is satisfied as a part of the resulting LMI.

We consider the following inequality:

Ψ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ψ11 Ψ12 A1 −M 0 M PR1√1−d

PCT1 PF

T

∗ Ψ22 hR2(A1 −M) 0 hR2M 0 0 0

∗ ∗ −R1 GT 0 0 0 0

∗ ∗ ∗ −P 0 0 0 0

∗ ∗ ∗ ∗ −R2 0 0 0

∗ ∗ ∗ ∗ ∗ −R1 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Iq 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (4.12)

∀t ∈ [0 ∞), where

Ψ11 = (A0 +M)P + P (A0 +M)T + γ−2B1BT1 −B2R

−1BT2 ,

Ψ12 = hPAT0 R2 + hP R

T2 + hγ−2B1B

T1 − hB2R

−1BT2 ,

Ψ22 = h2[γ−2B1BT1 −B2R

−1BT2 ]−R2,

R = DT12D12.

(4.13)

The cost functional JE of (4.3) cannot be possibly made negative for all w ∈L2([0,∞);Rp) and r(t) ∈ L2. Denoting, therefore, by f(r, t), f(0, t) = 0 theyet unknown positive scalar function that describes the upper-bound on theintegrand of (4.3) and by J(r) =

∫∞0f(r, t)dt, assuming for the moment that∫∞

0 f(r, t)dt exists, the following main result of the paper is obtained for thecase of fixed-finite preview (i.e. h �= 0):

Theorem 4.1. Consider the system of (4.1a,b) and JE of (4.3). Given γ > 0,there exists a control signal u(t) that possesses a solution that minimizes J(r)if there exist a positive definite matrix P ∈ Rn×n, a matrix M ∈ Rn×n andpositive definite matrices R1 ∈ Rn×n, R2 ∈ Rn×n that solve (4.12). When asolution exists, the maximizing and minimizing strategies of Nature and thecontroller are respectively given by:

w∗(t) = −γ−2[BT1 P

−1x(t)+hBT1 R

−12 y(t)], u∗(t) = [Kxx(t)+Krr(t)+Kθθ(t)],

82 4 Tracking Control with Preview

where

Kx = −[h2BT2 φ

−1B2 +DT12D12]

−1[hgφ−1B2 + CT1 D12 + P

−1B2]T ,

Kr = −[h2BT2 φ

−1B2 +DT12D12]

−1(DT12D13 + h

2BT2 φ

−1B3),

Kθ = −[h2BT2 φ

−1B2 +DT12D12]

−1BT2

(4.14)

and where

Q = P−1,R2 = R2M,

φ = −(h2[(R2A1 − R2)[R1 −GTQG]−1(R2A1 − R2)T + R2R

−12 RT

2 ]

+h2γ−2B1BT1 −R2),

g = h(AT0 R2 + R

T2 ) + h[Q(A1 −M)[R1 −GTQG]−1(AT

1 R2 − RT2 )

+QMR−12 RT

2 ] + hγ−2QB1B

T1 ,

B2 = B2 + [AwRw,y − hγ−2B1BT1 ]B2,

Aw = ([A1 −M ]− hγ−2B1BT1 [A1 −M ])(I − Rw,y[A1 −M ])−1,

Rw =

[R1 −GTQG 0

0 R2

],

Rw,x = R−1w

[AT

1Q−MTQ

],

Rw,y = R−1w

[hAT

1

−hMT

]R−1

2 ,

B1 = (I + AwRw,y)B1[I + hγ−2BT

1 B1]−1.

(4.15)

The signal θ(τ) is described by

θ(τ) = −AT θ(τ) + Brr(τ), τ ∈ [t t+ h], θ(t+ s) ≡ 0, s ≥ h, (4.16)

with

A = A− γ−2B1[BT1 Q+ hBT

1 A]−B2[h2BT

2 φ−1B2 +D

T12D12]

−1[hgφ−1B2

+CT1 D12 +QB2] + AwRw,x,

Br = B3 − B2Kr − γ−2hB1BT1 B3 + AwRw,yB3,

(4.17)where h is the preview length. The bound on the performance index is thengiven by:

JE(r, u∗, w∗)≤ J(r) (4.18)

4.4 The State-Feedback Tracking 83

where

J(r) = E∫ ∞

0

{rT (α6 + αT5 α

−14 α5)r}dt + E

∫ ∞

0

||BT1 θ||2γ−2dt

+ 2E∫ ∞

0

θT (t){Brr(t)}dt − E∫ ∞

0

||ATwθ||2R−1

w

dt− E∫ ∞

0

||α−1/24 BT

2 θ||2dt(4.19)

and where α4, α5, α6 are given in the following:

Υ3 =

⎡⎣ Υ3,1 hgφ

−1B2 + CT1 D12 +QB2 QB3 + hgφ

−1B3 + CT1 D13

∗ DT12D12 + h

2BT2 φ

−1B2 DT12D13 + h

2BT2 φ

−1B3

∗ ∗ DT13D13 + h

2BT3 φ

−1B3

⎤⎦

Δ=

⎡⎢⎢⎣α1 α2 α3

αT2 α4 α5

αT3 α

T5 α6

⎤⎥⎥⎦ , Υ3,1 = Υ11 + gφ

−1gT + CT1 C1. (4.20)

The proof of the theorem is given in the Appendix where the well-posednessof J(r) is also discussed.

Noting that, unfortunately, the inequality (4.12) is trilinear in the deci-sion variables, two possible solution methods can be applied: the first involvesP, (M, R1) iterations, where we take R2 = εIn and choose ε to be a tuning pa-rameter, similar to the P −K iterations method. This method, if it converges,can only achieve a local minimum. The second method, which we apply in thestatement of Lemma 4.4.1 below, is to assign predetermined values forM, R1

and R2 which entails an over-design while rendering a simple tractable solu-tion. It should be noted here thatM was an additional degree of freedomin our solution method. Substituting in (4.12) M = A1 + εmIn, R1 = εrInand R2 = ε2In, we obtain the following condition:

Υ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 −εmIn 0 A1 + εmInPεr√1−d

PCT1 PF

T

∗ Υ22 −hε2εmIn 0 hε2(A1 + εmIn) 0 0 0

∗ ∗ −εrIn GT 0 0 0 0

∗ ∗ ∗ −P 0 0 0 0

∗ ∗ ∗ ∗ −ε2In 0 0 0

∗ ∗ ∗ ∗ ∗ −εrIn 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Iq 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(4.21)

84 4 Tracking Control with Preview

∀t ∈ [0 ∞), where

Υ11 = (A0 +A1 + εmIn)P + P (A0 +A1 + εmIn)T + γ−2B1B

T1 −B2R

−1BT2 ,

Υ12 = hPAT0 ε2 + hP (A1 + εmIn)

T ε2 + hγ−2B1B

T1 − hB2R

−1BT2 ,

Υ22 = h2[γ−2B1BT1 −B2R

−1BT2 ]− ε2In.

(4.22)

We thus have the following result for the case of finite fixed-preview trackingcontrol:

Lemma 4.4.1 Consider the system of (4.1a,b) and JE of (4.3). Given γ > 0,and three tuning scalars: εm, ε2 > 0 and εr > 0, the state-feedback trackingcontroller minimizes J(r) if there exists a positive definite matrix P ∈ Rn×n

that solve (4.21).

The following preview strategies are obtained:

Corollary 4.4.1 Stochastic H∞-tracking with finite fixed-preview :Since at time t, r(τ ) is known for τ ≤ t + h the following control law isobtained:

u(t) = −[Kxx(t) + Krr(t) + Kθθ(t)], (4.23)

where Kx, Kr and Kθ are defined in (4.14). The latter controller achieves thebound J(r) of (4.19).

Corollary 4.4.2 Stochastic H∞-tracking with no preview : In this casethe control law is given by u(t) = −[Kxx(t) + Krr(t)], where Kx and Kr arethe same as in Corollary 4.4.1. The bound of the performance index is J(r)of (4.19) where all the terms that contain the signal θ(t) are zero.

Remark 4.2. Considering (4.12), deleting the second to the fifth column androw blocks of Ψ and taking M = A1, R1 = O(h) and R2 = O( 1

h), one obtains

the result of Theorem 4.2, page 63 of [53] for the nondelayed stochastic state-feedback tracking case. We note also that (4.12) and therefore (4.21), do notdepend on B3 and D13, similarly to the result in Theorem 4.2 of [53].

Remark 4.3. We note that a simple strategy is applied for the numerical so-lution of (4.21). The solution of (4.21) involves a search for three scalar vari-ables: εm, ε2 and εr. One may start by taking εm = ε2 = εr and seek, usingline searching, values for these tuning parameters that leads to a stabilizingcontroller of minimum γ. Once such a controller is obtained, standard opti-mization techniques can be used, say Matlab function “fminsearch”, whichseek the combination of the three scalar parameters that bring γ to a localminimum.

4.6 Conclusions 85

4.5 Example

We consider the system of (4.1a,b) taken from [53] with the objective functionof (3) where :

A =

[0 1−1 −0.4

], F =

[0 00 −0.1

], B1=

[1

−1

], B2=

[01

], B3=

[10

]

C1 =[−0.50.4

], D12 = 0.1, D13 = −1 and G = 0 in (4.1a).

The time varying delay τ(t) is assumed to satisfy |τ(t)| ≤ h = 0.20sec and|τ (t)| ≤ 0.1 and we consider a preview length of h = 4sec. Seeking a state-feedback controller, we apply the result of Corollary 4.4.1 and obtain a nearminimum attenuation level of γ = 1.77, which is found for ε2 = 40, εm = 0.013and εr = 1.6. The control law is given by (4.14a-c) where:

Kx =[−249.4425 −48.2705

],

Kr =[1.6735e− 004

],

Kθ =[−5.8024 −85.0952

].

The corresponding closed-loop poles are [−5.8484, −42.8221].In order to demonstrate the effect of the delay and its derivative bounds

on the attenuation level in this example we consider, for simplicity, the caseof zero preview and obtain, applying Corollary 4.4.2 and taking ε2 = 40, thefollowing near minimum attenuation levels in Table 4.1.

Table 4.1 The minimal attenuation levels obtained as a function of the delay lengthand the delay derivative bound

h d εr εm γmin

0 0 1 0.012 1.06

0.25 0 1.6 0.035 1.97

0.4 0 1.4 0.087 2.78

0.6 0 1.45 0.066 3.92

0.6 0.4 1.5 0.067 4.15

0.6 0.9 1.14 0.095 4.60

We note that the result for h = 0 (i.e., zero delay) (given in the first line ofTable 4.1) is in line with Remark 4.2: the minimal attenuation level of γ = 1.06was also obtained in [53] for the non-delayed stochastic state-feedback trackingcontrol.

4.6 Conclusions

In this chapter the problem of tracking signals with preview in presence ofWiener-type stochastic parameter uncertainties is solved for retarded LTI sys-

86 4 Tracking Control with Preview

tems. The retarded systems have been transformed to norm-bounded uncer-tain delay-free systems via the input-output approach. The state-feedbacktracking control problem has then been solved for the infinite horizon case.Applying a min–max approach to the equivalent norm-bounded system, min-imizing control and maximizing disturbance strategies have been obtained,which are based on the measurement of the system states and the previewedreference signal. The performance index in the tracking problem includes aver-aging over the statistics of the stochastic parameters in the system state-spacemodel. An upper-bound on this index of performance has been obtained forthe two patterns of preview control tracking problems treated in the paper.

The solution method used in the present chapter is different from the oneapplied to the non delayed case. In the latter case, a finite-horizon solution hasbeen obtained at first and then it has been extended to the infinite-horizoncase where stability was guaranteed in the limit of t → ∞. An attempt to“combine” the already solved preview tracking result with the input–outputapproach along the approach used for the non-delayed case yields, unfortu-nately, a complicated and a most conservative solution to the delayed trackingproblem. This is why an alternative approach has been adopted in this prob-lem. Inequality conditions for stability and performance are first derived interms of the yet undetermined control signal u(t) and only in the last stepof the solution the optimal control signal is derived. This approach yieldsa less conservative result that is most amenable for calculating a stabilizingcontroller.

It is shown that when the delay in the system tends to zero, theresults of our theory coincide with those of the non delayed stochastic state-feedback tracking control design. The tractability and solvability of our the-ory is demonstrated via two examples, where the effect of the time delay andit’s derivative on the performance index is shown in Example 4.5 and wherethe applicability of our theory to real practical control engineering is demon-strated in Chapter 13, Section 13.2 . Extension of these results to the casewhere there is no access to the system states and the controller has to rely onnoisy measurements of the output, is not treated here – it is left for a futurework.

4.7 Appendix

Proof of Theorem 4.1: The proof is based on minimizing a positive upper-bound on the index of performance. By applying several completions to squareoperations and using Schur complement formula, the optimal strategy forw(t) is first obtained. We then introduce a new signal θ(t), which allows thederivation of the control strategy for two preview patterns. We first treat thefixed-finite preview case and then the zero-preview control case.

We start by seeking a finite positive scalar J(r) =∫∞0 f(r, t)dt such that

4.7 Appendix 87

E∫ ∞

0

[LV + zT (t)z(t)− γ2wT (t)w(t)]dt < J(r), ∀r(t) ∈ L2,

∀w(t) ∈ L2([0,∞);Rp), (4.24)

while minimizing J(r) and requiring that the inequality 4.24 will also guar-antee that E{LV } < 0. We note that since x0 ≡ 0 and V is a Lyapunovfunction, 4.24 will guarantee that JE < J(r). Considering (4.8), we defineξ(t) = col{x(t), w1(t), w2(t), w(t), u(t), r(t)}, and require f(r, t) to satisfy thefollowing inequality:

E{ξT (t){

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 −QM QB1 QB2 QB3

∗ −R1 +GTQG 0 0 0 0

∗ ∗ −R2 0 0 0

∗ ∗ ∗ −γ2I 0 0

∗ ∗ ∗ ∗ 0 0

∗ ∗ ∗ ∗ ∗ 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+Γ T1 h

2R2Γ1 + ΓT2 Γ2}ξ(t)} < f(r, t),

where

Γ T1 = col{(A0 +M)T , (A1 −M)T , −MT , BT

1 , BT2 , B

T3 }

Γ T2 = col{CT

1 , 0, 0, 0, DT12, D

T13},

and where

Υ11 = QA+ ATQ+1

1− dR1 + FTQF, Υ12 = QA1.

We thus obtain the requirement of E{ξTy (t)Γ1ξy(t)} < f(r, t), where ξTy (t) =[ξT (t) hξT (t)Γ T

1 ξT (t)Γ T2 ], and where

Γ1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 −QM QB1 QB2 QB3 h(AT0 +MT ) CT

1

∗ −R1 +GTQG 0 0 0 0 h(AT

1 −MT ) 0

∗ ∗ −R2 0 0 0 −hmT 0

∗ ∗ ∗ −γ2I 0 0 hBT1 0

∗ ∗ ∗ ∗ 0 0 hBT2 DT

12

∗ ∗ ∗ ∗ ∗ 0 hBT3 DT

13

∗ ∗ ∗ ∗ ∗ ∗ −R−12 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

88 4 Tracking Control with Preview

Multiplying the above Γ1 from both sides by diag{I, I, I, I, I, I, R2, I} anddenoting R2 = R2M, the following requirement is obtained:

E{ξTy (t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 −QM QB1 QB2 QB3 h(AT0 R2 + RT

2 ) CT1

∗ −R1 +GTQG 0 0 0 0 h(AT1 R2 − RT

2 ) 0

∗ ∗ −R2 0 0 0 −hRT2 0

∗ ∗ ∗ −γ2I 0 0 hBT1 R2 0

∗ ∗ ∗ ∗ 0 0 hBT2 R2 DT

12

∗ ∗ ∗ ∗ ∗ 0 hBT3 R2 DT

13

∗ ∗ ∗ ∗ ∗ ∗ −R2 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ξy(t)}

(4.25)< f(r, t),

where ξTy (t) = [ξT (t) hξT (t)Γ T1 R

−12 ξT (t)Γ T

2 ].

Remark 4.4. The requirement of (4.25), if satisfied for a given u(t), guaranteesthe stability of the closed-loop system by Lemma 4.3.1. The latter stems fromthe fact that for r ≡ 0, f(r, t) = 0 (by definition) and the five columns androws of (12) are the first, second, third, fifth, and seventh columns and rowsof (4.25), respectively. The latter fact leads to E{LV } < 0.

Next, we apply Schur complement to the second and third columns and rows

of (4.25) and we obtain, defining w(t)Δ= col{w1(t), w2(t)} the following re-

quirement:

E{ξT (t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 QB1 QB2 QB3 Υ15 CT1

∗ −γ2I 0 0 hBT1 0

∗ ∗ 0 0 hBT2 DT

12

∗ ∗ ∗ 0 hBT3 DT

13

∗ ∗ ∗ ∗ Υ55 0

∗ ∗ ∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ξ(t)}−E{(w−w∗)T Rw(w−w∗)} < f(r, t),

(4.26)

where ξ(t) = col{x(t), w(t), u(t), r(t), hR−12 Γ1ξ(t), Γ2ξ(t)},

Υ11 = Υ11 +QA1[R1 −GTQG]−1AT1Q +QMR−1

2 MTQ,

Υ15 = h(AT0 R2 + R

T2 ) + h[Q(A1 −M)[R1 −GTQG]−1(AT

1 R2 − RT2 )

+QMR−12 RT

2 ],

Υ55 = −R2 + h2[(R2A1 − R2)[R1 −GTQG]−1(R2A1 − R2)

T + R2R−12 RT

2 ],

w∗ = R−1w [

[[AT

1 −MT ]Q−MTQ

]x(t) +

[h(A1 −M)T

−hMT

]R−1

2 y(t)],

4.7 Appendix 89

and where Rw is given in (4.15g). The latter operation is equivalent to com-pletion to squares for w that makes the w∗ the optimal strategy. We notethat the second term of the left side of (4.26) is not omitted (by choosing theoptimal strategy) but is rather left for the latter part of the proof. Completingto squares for w, we apply Schur complement to the second column and rowblocks of (4.26) and we obtain:

E{ξT (t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 QB2 QB3 Υ14 CT1

∗ 0 0 hBT2 D

T12

∗ ∗ 0 hBT3 D

T13

∗ ∗ ∗ Υ44 0

∗ ∗ ∗ ∗ −I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ξ(t)} − E{(w − w∗)T Rw(w − w∗)}

− E{(w − w∗)T γ2(w − w∗)} < f(r, t), (4.27)

where ξ(t) = col{x(t), u(t), r(t), hR−12 Γ1ξ(t), Γ2ξ(t)}.

The signal w∗(t) is now the optimal strategy for w where,

Υ11 = Υ11 + γ−2QB1B

T1 Q,

Υ14 = Υ15 + hγ−2QB1B

T1 ,

Υ44 = Υ55 + h2γ−2B1B

T1 ,

w∗ = −γ−2[BT1 Qx+ hB

T1 R

−12 y].

Applying Schur complement to the fourth and fifth columns and rows of theleftmost block matrix of (4.27) the following requirement is obtained:

JE(t)Δ= E{ξT3 Υ3ξ3}−E{(w− w∗)T Rw(w− w∗)}−E{(w−w∗)T γ2(w−w∗)} <f(r, t),

(4.28)

where Υ3 is defined in (4.20), ξ3(t) = col{x(t), u(t), r(t)} and where φ and gof (4.15c,d) are obtained by:

φ = −Υ44 = −(Υ55 + h2γ−2B1B

T1 ) =

−(h2[(R2A1 − R2)[R1 −GTQG]−1(R2A1 − R2)T + R2R

−12 RT

2 ]

+h2γ−2B1BT1 −R2), g = Υ14 = Υ15 + hγ

−2QB1BT1 =

h(AT0 R2 + R

T2 ) + h[Q(A1 −M)[R1 −GTQG]−1(AT

1 R2 − RT2 ) +QMR

−12 RT

2 ]+hγ−2QB1B

T1 .

In order to find the minimizing strategy for the controller, we seek to minimize

the expression of E{Φ(r)} Δ= E{ξT3 (t)Υ3ξ3(t)} in JE(t) of (4.28). The latter

minimization will also hold for the integral form of (4.28). We obtain:

E{Φ(r)} Δ= E{xT (t)α1x(t) + 2xT (t)α2u(t) + 2xT (t)α3r(t) + u

T (t)α4u(t)

+2uT (t)α5r(t) + rT (t)α6r(t)},

90 4 Tracking Control with Preview

where α1, α2, ..α6 are defined in (4.20). Completing to squares for u(t), wefind that

E{Φ(r)} = E{xT (t)(α1−α2α−14 αT

2 )x(t)+(uT (t)+(rT (t)αT5 +xT (t)α2)α

−14 )α4

(u(t)+α−14 (αT

2 x(t)+α5r(t)))+2xT (t)α2α−14 α5r(t)+r

T (t)(α6+αT5 α

−14 α5)r(t)}.

In order to allow for the different preview tracking patterns, we add the fol-lowing identically zero-term to JE(t) of (4.28) following integration, where weapply the Ito lemma [9]:

0 = 2E∫ ∞

0

d{θT (t)x(t)}+ 2θT (0)x(0) = 2EθT (∞)x(∞)− 2θT (0)x(0) + 2θT (0)x(0)

= 2E∫ ∞

0

θT (t)x(t)dt+ 2θT (0)x(0)

+2E∫ ∞

0

θT (t){[Ax(t) + B1w(t) + B2u(t) + Brr(t) + Aww0(t)]dt

+ Fx(t)dβ(t) +Gw1(t)dζ − Γβdt− Γζdt} (4.29)

where θ(t) is defined in (4.16) and where we denote:

w0(t) = w − w∗, w = w − w∗.

The result of (4.29) makes use of the fact

dx(t) = [Ax(t) + B1w(t) + B2u(t) + Brr(t) + Aww0(t)]dt+ Fx(t)dβ(t)

+Gw1(t)dζ − Γβdt− Γζdt

= [Ax(t) +B1w(t) +B2u(t) + B3r(t) + A1w1(t)−Mw2(t)]dt+ Fx(t)dβ(t)

+Gw1(t)dζ − Γβdt− Γζdt,

where B1, B2, Aw and A, Br are given in (4.15) and (4.17) respectively.

Remark 4.5. The signal θ(t) plays a central roll only in the case of fixed-finitepreview pattern of Corollary 4.4.1. In this case, the signal r(τ) is known fort < τ < t + h. Note that, following (4.16), θ(t) is retrieved by backwardsintegration from t + h towards t. In the zero preview case (Corollary 4.4.2),θ(τ) ≡ 0.

We note that due to the expectation operator in (4.29), the noise terms in dβand dζ vanish. Completing to squares for w(t) and then for w0(t) we obtain,for JE(t) =

∫∞0 JEdt, noting that x(0) = 0:

JE(t) = E∫ ∞

0

{xT Υxx}dt+ E∫ ∞

0

{uTα4u}dt+ 2E∫ ∞

0

{xTα2α−14 α5r}dt

4.7 Appendix 91

+E∫ ∞

0

{rT (α6 + αT5 α

−14 α5)r}dt− E

∫ ∞

0

||[w − γ−2BT1 θ]||2γ2

+E∫ ∞

0

||[w0 + R−1w AT

wθ]||2Rw− E

∫ ∞

0

||ATwθ||2R−1

w

dt

+2E∫ ∞

0

θT (t)x(t)dt + 2E∫ ∞

0

θT (t){Ax(t) + B2u(t) + Brr(t)}dt

+E∫ ∞

0

||BT1 θ||2γ−2dt

whereΥx = α1 − α2α

−14 αT

2 ,u(t) = u(t) + α−1

4 [α2x(t) + α5r(t)].(4.30)

Next, completing to squares for u(t) we obtain:

JE(t) = E∫ ∞

0

{xT Υxx}dt+ E∫ ∞

0

{(u+ α−14 BT

2 θ)Tα4(u+ α−14 BT

2 θ)}dt

+E∫ ∞

0

{rT (α6 + αT5 α

−14 α5)r}dt− E

∫ ∞

0

||[w − γ−2BT1 θ]||2γ2 + E

∫ ∞

0

||BT1 θ||2γ−2dt

−E∫ ∞

0

||ATwθ||2R−1

w

dt− E∫ ∞

0

||α−1/24 BT

2 θ||2dt+ 2E∫ ∞

0

θT (t){Brr(t)}dt

+E∫ ∞

0

||[w0 + R−1w AT

wθ]||2Rw+ 2E

∫ ∞

0

[θ + AT θ + [α2α−14 α5]r]

Txdt.

Taking w0 = −R−1w AT

wθ, θ = −AT θ− [α2α−14 α5]r, the above equation can be

expressed by:

JE(t) = E∫ ∞

0

{xT Υxx}dt+ E∫ ∞

0

{(u+ α−14 BT

2 θ)Tα4(u + α

−14 BT

2 θ)}dt

−E∫ ∞

0

||[w − γ−2BT1 θ]||2γ2 + J(r),

where

J(r) = E∫ ∞

0

{rT (t)(α6 + αT5 α

−14 α5)r(t)}dt + E

∫ ∞

0

||BT1 θ(t)||2γ−2dt

−E∫ ∞

0

||ATwθ(t)||2R−1

w(t)

dt− E∫ ∞

0

||α−1/24 BT

2 θ(t)||2dt

+2E∫ ∞

0

θT (t){Brr(t)}dt,

is independent of u(t) and w(t) and is given in explicit form in (4.19).

92 4 Tracking Control with Preview

In order to minimize JE(t) for all possible r, namely to achieve a minimumJ(r) and thus find the minimal possible value of f(r, t), we seek a conditionwhich will assure Υx < 0. Considering (4.30), assuming for simplicity thatCT

1 D12 = 0 and recalling that R = DT12D12, we require:

Υ11 + gφ−1gT + CT

1 C1 − (hgφ−1B2 +QB2)Ψ−13 (hgφ−1B2 +QB2)

T < 0,

where Ψ3 = R+ h2BT2 φ

−1B2. Defining then:

Γ1 = gφ−1gT − h2gφ−1B2Ψ−13 BT

2 φ−1gT ,

Γ2 = −2hgφ−1B2Ψ−13 BT

2 Q,

Γ3 = −QB2Ψ−13 BT

2 Q,

it is thus required that:

Υ11 + CT1 C1 + Γ1 + Γ2 + Γ3 < 0.

Using the matrix inversion lemma we find:

Γ1 = g(φ+ h2B2R−1BT

2 )−1gT ,

Γ2 = −2hg(φ+ h2B2R−1BT

2 )−1B2R

−1BT2 Q,

Γ3 = −Qφ(φ+ h2B2R−1BT

2 )−1B2R

−1BT2 Q,

and have

Γ1 + Γ2 + Γ3 = (g − hQB2R−1BT

2 )(φ+ h2B2R−1BT

2 )−1(gT − hB2R

−1BT2 Q)

−QB2R−1BT

2 Q,

where we note that

−h2QB2R−1BT

2 (φ+ h2B2R−1BT

2 )−1B2R

−1BT2 Q−Qφ(φ+h2B2R

−1BT2 )

−1

B2R−1BT

2 Q=−QB2R−1BT

2 Q.

Namely, we require:

Υx = Υ11 +CT1 C1 −QB2R

−1BT2 Q+ (g − hQB2R

−1BT2 )(φ+ h2B2R

−1BT2 )

−1

(gT − hB2R−1BT

2 Q) < 0.

The latter inequality can be written as :

Γ =

[Υ11 + C

T1 C1 −QB2R

−1BT2 Q g − hQB2R

−1BT2

∗ −(φ+ h2B2R−1BT

2 )

]< 0.

Multiplying the above Γ by diag{Q−1, I} from the left and the right, re-placing for φ and g and applying Schur complement, we readily obtain thefollowing inequality:

4.7 Appendix 93

⎡⎢⎢⎢⎢⎢⎢⎣

Ψ11 Ψ12 A1 −M M

∗ Ψ22 hR2(A1 −M) hR2M

∗ ∗ −[R1 −GTP−1G] 0

∗ ∗ ∗ −R2

⎤⎥⎥⎥⎥⎥⎥⎦< 0,

where we denote P = Q−1 and where

Ψ11 = (A0 +M)P + P (A0 +M)T + γ−2B1BT1 −B2R

−1BT2 + PR1P

11−d

+PCT1 C1P + PFTP−1FP,

Ψ12 = hPAT0 R2 + hP R

T2 + hγ−2B1B

T1 − hB2R

−1BT2 ,

Ψ22 = h2[γ−2B1BT1 −B2R

−1BT2 ]−R2, R = DT

12D12.

Applying Schur complement again, the inequality of (4.12) then follows.

Remark 4.6. In order to prove that∫∞0f(r, t)dt exists (and therefore that J(r)

of (4.19) is well defined) we consider (4.16) and obtain that

θ(t) =

∫ t+h

t

e−AT (t−τ)Brr(τ)dτ =

∫ 0

−h

e−AT τ Brr(t− τ )dτ .

Since J(r) includes integral terms that contain the signal θ(t) multiplied bysome constants, the L2 norm of θ(t) should be derived. We readily find that

∫ ∞

0

θT (t)θ(t)dt = Tr{∫ ∞

0

θ(t)θT(t)dt}

= Tr{∫ ∞

0

{[∫ 0

−h

e−AT τ Brr(t − τ)dτ ][∫ 0

−h

e−AT τ Brr(t − τ)dτ ]T }dt}

= Tr{∫ 0

−h

∫ 0

−h

e−AT τ BrΨ(r, τ, τ)BTr e

−Aτdτdτ}

where Ψ(r, τ, τ )Δ=

∫∞0r(t−τ )rT (t−τ)dt and where by the Cauchy-Shwartz in-

equality Ψ(r, τ, τ ) is finite for all r(t) ∈ L2. Obviously the additional quadraticterm of r(t) in J(r) of (4.19) is finite too.