∞

Measurement-feedback Tracking with Preview

Eli Gershon

Holon Institute of Technology, HIT, Holon, Israel

Keywords:

∞

Control, Discrete-time Systems, Tracking Preview.

Abstract:

Finite-horizon H

∞

output-feedback tracking control for linear discrete time-varying systems is explored along

with the stationary inﬁnite-horizon case. We consider three tracking patterns depending on the nature of the

reference signal i.e : whether it is perfectly known in advance, measured on-line or previewed in a ﬁxed time-

interval ahead. For each of the above three cases a solution is found where, given a speciﬁc reference signal,

the controller plays against nature which chooses the initial condition and the energy-bounded disturbances.

The problems are solved based on a specially devised bounded real lemma for systems with tracking signals.

The ﬁnite-horizon case is extended to the stationary one where similar results are achieved.

1 INTRODUCTION

In the present paper we address the problem of

∞

output-feedback tracking control with preview of

discrete-time linear systems, in both the ﬁnite and the

inﬁnite time settings.

The stability analysis and control design for sys-

tems with stochastic uncertainties have received much

attention in the past (see (Gershon and U. Shaked,

2005) and the references therein), where mainly

continuous-time non retarded systems were conside-

red. In the late 80’s, a renewed interest in the cont-

rol and estimation designs of these systems has been

encountered and solutions to the stochastic control

and ﬁltering problems of both: continuous-time and

discrete-time systems, have been derived that ens-

ure a worst case performance bound in the H

∞

sense

(see (Gershon and U. Shaked, 2005) for an exten-

sive review). Systems whose parameter uncertain-

ties are modeled as white noise processes in a li-

near setting have been treated in (Dragan and Mo-

rozan, 1997a) (Hinriechsen and Pritchard, 1998), for

the continuous-time case and in (Dragan and Moro-

zan, 1997b), (Bouhtouri et al., 1999) for the discrete-

time case. Such models of uncertainties are encoun-

tered in many areas of applications (see (Gershon and

U. Shaked, 2005) and the references therein) such as:

nuclear ﬁssion and heat transfer, population models

and immunology. In control theory such models are

encountered in gain scheduling when the scheduling

parameters are corrupted with measurement noise.

Tracking feedback-control has been a central pro-

blem in both the frequency domain and in the state-

space domain over the last decades. Numerous varia-

tions of this problem, by large, have been published in

both the continuous-time and in the discrete-time set-

tings (Yang and Zhang, 2008), (Wencheng Luo and

Ling, 2005),(Kim and Tao, 2002), (Wei-qian You,

2010) and (Gao and Chen, 2008),(Verriest and Flor-

chinger, 1995). The problem of tracking control with

preview has been solved in the deterministic case by

(Shaked and deSouza, 1995). In the latter work a pre-

viewed tracking signal appears in the system dyna-

mics allowing for three patterns of preview. The ﬁrst

case deals with the inclusion of the tracking signal

where no preview is given [the simplest case] while

in the second case the preview signal is known for a

given ﬁxed ﬁnite time interval in the future which is

smaller the system dynamic horizon. The third pat-

tern deals with the case where the previewed signal is

known along the full ﬁnite-horizon of the system dy-

namics. The latter problems were solved in (Shaked

and deSouza, 1995) by applying a game theoretical

approach where a saddle point strategy is adopted.

The solution of the ﬁnite-horizon, stochastic coun-

terpart, state-feedback tracking control with pre-

view was obtained in (Gershon and U. Shaked,

2005),(Gershon and U. Shaked, 2010), where three

preview patterns of the tracking signal were conside-

red. These include the simple case of on-line measu-

rement of the tracking signal and two patterns where

this signal is either previewed with a ﬁxed time in-

terval ahead or perfectly known in advance along the

system horizon (Gershon and U. Shaked, 2010) (see

Gershon, E.

∞

Measurement-feedback Tracking with Preview.

DOI: 10.5220/0006853903250331

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 325-331

ISBN: 978-989-758-321-6

325

also (Gershon and U. Shaked, 2005) for further de-

tails). For all these patterns the solutions were obtai-

ned using a game theory approach where the con-

troller plays against nature which chooses the system

energy-bounded disturbance and the initial condition.

In the present paper, we extend the work of

(Gershon and U. Shaked, 2010) which deals with the

stochastic state-feedback tracking control [zeroing,

of course, the stochastic terms] to the case where

there is no full access to the state-vector and a dyn-

amic output-feedback strategy must be applied. Here,

we treat the case where correlated parameter uncer-

tainties appear in both the system dynamics and the

measurement matrices. An optimal output-feedback

tracking strategy is derived which minimizes the stan-

dard H

∞

performance index, for the three tracking

patterns of the reference signal. In both, the ﬁnite-

horizon horizon case and the stationary one, a min-

max strategy is applied that yields an index of perfor-

mance that is less than or equal to a certain cost. We

address the problem via two approaches: In the ﬁnite-

horizon case we apply the Difference LMI (DLMI)

method (Gershon and U. Shaked, 2005) for the so-

lution of the Riccati inequality obtained, and in the

stationary case we apply a special Lyapunov function

which leads to an LMI based tractable solution.

Notation: Throughout the paper the superscript ‘T ’

stands for matrix transposition, R

denotes the n di-

mensional Euclidean space, R

n×m

is the set of all

n × m real matrices, N is the set of natural num-

bers and the notation P > 0, (respectively, P ≥ 0) for

P ∈ R

n×n

means that P is symmetric and positive

deﬁnite (respectively, semi-deﬁnite). We denote by

(Ω,R

) the space of square-integrable R

− valued

functions on the probability space (Ω,F ,P ), where

Ω is the sample space, F is a σ algebra of a subset

of Ω called events and P is the probability measure

on F . By (F

)

k∈N

we denote an increasing family

of σ-algebras F

⊂ F . We also denote by

(N ; R

)

the n-dimensional space of nonanticipative stochas-

tic processes { f

}

k∈N

with respect to (F

)

k∈N

where

∈ L

(Ω,R

). On the latter space the following l

norm is deﬁned:

||{ f

}||

= E{

∑

∞

|| f

} =

∑

∞

E{|| f

} < ∞,

{ f

} ∈

(N ; R

(1)

where ||·|| is the standard Euclidean norm. We denote

by Tr{·} the trace of a matrix and by δ

i j

the Kronecker

delta function. Throughout the manuscript we refer

to the notation of exponential l

stability, or internal

stability, in the sense of (Bouhtouri et al., 1999) (see

Deﬁnition 2.1, page 927, there). By [Q

]

, [Q

]

−

denote the causal and non causal parts respecti-

vely, of a sequence {Q

, i = 1, 2, ..., N}.

2 PROBLEM FORMULATION

Given the following linear discrete time-varying sy-

stem:

k+1

= A

2,k

1,k

+ B

3,k

= C

2,k

+ D

21,k

(2a-b)

where x

∈ R

is the state vector, y

∈ R

is the mea-

surement vector, w

∈ R

is a deterministic exogenous

disturbance, r

∈ R

is deterministic reference signal

which can be measured on line or previewed, u

∈ R

is the control input signal and x

is an unknown initial

state and where we denote

= C

2,k

3,k

, z

∈ R

, k ∈ [0,N]

(3)

and we assume, for simplicity that:

3,k

2,k

= [0 0

> 0.

Our objective is to ﬁnd a control law {u

} that mini-

mizes the energy of {z

} by using the available know-

ledge on the reference signal, for the worst-case of the

process disturbances {w

},{n

} and the initial condi-

tion x

. We, therefore, consider, for a given scalar

γ > 0, the following performance index:

)

∆



3,N





||z

−γ

[||w

+ ||n

]



− γ

−1

≥0.

(4)

Similarly to (Gershon and U. Shaked, 2010) we con-

sider three tracking problems differing on the infor-

mation pattern over {r

1) H

∞

-tracking with full preview of {r

}: The

tracking signal is perfectly known for the interval

k ∈ [0, N].

2) H

∞

-tracking with no preview of {r

}: The

tracking signal measured at time k is known for i ≤ k.

3) H

∞

-tracking with ﬁxed-ﬁnite preview of {r

At time k, r

is known for i ≤ min(N,k + h) where

h is the preview length.

In all the above three cases we seek a control law {u

}

of the form

= H

+ H

where H

is a causal operator and where the causa-

lity of H

depends on the information pattern of the

reference signal. The design objective is to minimize

max

) ∀{w

},{n

} ∈ l

[0,N − 1], x

∈ R

where for all of the three tracking problems we derive

a controller {u

} which plays against it’s adversaries

},{n

} and x

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

326

3 FINITE-HORIZON

OUTPUT-FEEDBACK

TRACKING

We bring ﬁrst the deterministic counterpart of the

state-feedback result of (Gershon and U. Shaked,

2010) which is based on a game theory approach and

which constitutes the ﬁrst step in the solution of the

output-feedback control problem. We consider the sy-

stem of (2a) and (3) and obtain the following result:

Lemma 1: Consider the system of (2a), (3) and

of (4) with the term of n

excluded. Given γ > 0,

the state-feedback tracking game possesses a saddle-

point equilibrium solution iff there exists Q

> 0,∀i ∈

[0,N] that solves the following Riccati-type equation

k+1

− A

k+1

2,k

−1

2,k

k+1

, Q(N)=C

k+1

∆

= Q

k+1

[I − γ

−2

1,k

k+1

]

−1

= B

2,k

k+1

2,k

(5a-c)

and satisﬁes

k+1

> 0, k ∈ [0 N − 1], γ

−1

− Q

> 0,

where R

k+1

∆

= γ

I − B

1,k

k+1

1,k

(6a-c)

When a solution exists, the saddle-point strategies are

given by:

∗

= (γ

−1

− Q

)

−1

∗

= R

−1

k+1

1,k

[θ

k+1

2,k

3,k

)]

∗

= −Φ

−1

2,k

k+1

+ B

3,k

+ Q

−1

k+1

]}

(7a-c)

where the causal part of θ

k+1

= [θ

k+1

]

(8)

and where θ

satisﬁes

k+1

, θ

= C

3,N

= Q

−1

k+1

−1

k+1

, S

k+1

∆

= M

−1

k+1

+ B

2,k

−1

k+1

2,k

k+1

3,k

, T

k+1

∆

(9)

The game value is then given by:

∗

)=kS

−

k+1

−1

k+1

3,k

−kQ

−

k+1

+kD

3,k

+kD

3,N

+θ

(γ

−1

−Q

)

−1

(10)

Proof: (Gershon and U. Shaked, 2010) The proof is

based on adapting the standard completing to squares

arguments to the case. We bring in the Appendix

the Sufﬁciency part of the proof, which is needed

for the derivation of the BRL of the next section.

We note also, at this point, that the solution of the

output-feedback control problem is also based on

these derivations.

Remark 1: It is important to note that the signal of

in (9) is admitted in the above derivation because

of the tracking signal which affects the dynamics of

(2a) (Gershon and U. Shaked, 2010). This signal

accounts for the nature of the tracking pattern, where

it’s causal part (i.e [θ

k+1

]

) appears in the structure of

the controller in accordance with the preview patterns.

Remark 2: Applying the result of Lemma 1 on the

speciﬁc pattern of the reference signal it is shown in

(Gershon and U. Shaked, 2010) that the saddle-point

controller strategy depends on the causal part of θ

k+1

(i.e [θ

k+1

]

), where θ

k+1

is given in (9). The latter

dependency on θ

k+1

appears also in the structure of

the control signal in the output-feedback case.

The solution of the output-feedback control pro-

blem involves 2 steps where the latter one is a ﬁlte-

ring problem of order n. A second Riccati equation

is thus achieved by applying the BRL to the dynamic

equation of the estimation error. The latter imposes

augmentation of the system to 2n order. This augmen-

ted system contains also a tracking signal component

and therefore one needs to apply a special BRL for

systems with tracking signal. We thus bring ﬁrst the

following lemma:

3.1 BRL for Systems with Tracking

Signal

We consider the following system:

k+1

= A

+ B

1,k

+ B

3,k

= C

+ D

3,k

, z

∈ R

, k ∈ [0, N]

(11a,b)

which is obtained from (2,a) and (3) by setting

2,k

≡ 0 and D

2,k

≡ 0. We consider the following in-

dex of performance:

)

∆



3,N

+||z



−



[||w

]



− γ

−1

, R

−1

≥0.

(12)

We arrive at the following theorem:

Theorem 1: Consider the system of (11a,b) and

of (12). Given γ > 0, J

of (12) satisﬁes J

≤

J(r,ε), ∀{w

} ∈ l

[0,N − 1], x

∈ R

, where

J(r, ε) =

N−1

∑

k=0

k+1

1,k

−1

k+1

1,k

}θ

k+1

N−1

∑

k=0

3,k

+||D

3,N

+ 2

N−1

∑

k=0

k+1

−1

k+1

(

−1

k+1

)

−1

3,k

+ θ

−1

θ,

∞

Measurement-feedback Tracking with Preview

327

if there exists

that solves the following Riccati-

type equation

k+1

I − B

1,k

k+1

1,k

> 0,

Q(0) = γ

−1

− εI,

(13a,c)

for some ε > 0 where

k+1

= C

3,N

−1

k+1

3,k

k+1

3,k

k+1

∆

k+1

[I − γ

−2

1,k

k+1

]

−1

(14a-e)

Proof: Unlike the state-feedback tracking control

in (Gershon and U. Shaked, 2010), the solution of

the BRL does not acquire saddle-point strategies

(Since the input signal u

is no longer an adversary).

It can, however, be readily derived based on the

ﬁrst part of the sufﬁciency of Lemma 1 by setting

2,k

≡ 0 and D

2,k

≡ 0. In the Appendix we bring the

proof of the BRL as a derivation of the proof of the

state-feedback tracking control solution (which is not

included in (Gershon and U. Shaked, 2010). The

latter proof is also essential for the derivation of the

output-feedback tracking control solution.

Remark 3: The choice of ε > 0 in

Q(0) of (13b)

reﬂects on both, the cost value (i.e

J(r,ε)) of (13b)

and the minimum achievable γ. If one chooses

0 < ε << 1 then, the cost of

J(r,ε) increases while

the solution of (13a) is easier to achieve, which

results in a smaller γ. The choice of large ε, on the

other hand, causes the reverse effect, which leads to a

larger γ.

3.2 The Output-feedback Tracking

Control

We consider the system of (2a,b) and (3). Like in the

state-feedback case (Gershon and U. Shaked, 2010)

we seek a control law {u

}, based on the informa-

tion of the reference signal {r

} that minimizes the

tracking error between the system output and the

tracking trajectory, for the worst case of the initial

condition x

, the process disturbances {w

} and {n

We, therefore, consider the performance index of (4)

and we assume that (5a) has a solution Q

k+1

> 0 over

[0,N] where (6a,b) are satisﬁed. The solution of the

output-feedback problem is stated in the following

theorem, for the a priori case, where u

can use the

information on {y

, 0 ≤ i < k} :

Theorem 2: Consider the system of (2a,b), (3) and

of (4). Given γ > 0, the output-feedback tracking

control problem, where {r

} is known a priori for all

k ≤ N (the full preview case) possesses a solution if

there exists

∈ R

2n×2n

> 0, K

o,k

∈ R

n×z

, ∀i ∈ [0,N]

that solves the following Difference LMI (DLMI):







−1

1,k

k+1

−1

1,k

0 γ

−1

1,k

I 0

1,k

0 0 I







≥ 0, (15a,b)

where P

−1

∆

= γ

−2

R R

R R + εI

with a

forward iteration, starting from the above initial

condition of P

, where R is deﬁned in (4), where

1,k

are deﬁned in (23) and where

−1

is given in (24).

Proof: Using the expression that is achieved in the

Appendix for

) in the state-feedback

case, the index of performance is now given by:

) =

)

−γ

||n

= −γ

|| ¯w

− γ

||x

− x

∗

−1

−γ

−1

∑

N−1

k=0

[{|| ¯u

1,k

}]

− γ

||n

J(r).

(16)

We note that in the full preview case [θ

k+1

]

= θ

k+1

Using the following deﬁnitions:

¯u

= Φ

1/2

k+1

+ Φ

−1/2

k+1

2,k

k+1

3,k

+ Q

−1

k+1

)

¯w

= γ

−1

1/2

k+1

− γ

−1

−1/2

k+1

1,k

k+1

+ B

2,k

3,k

) + θ

k+1

(17)

we note that

¯w

= γ

−1

1/2

k+1

− w

∗

), ¯u

= Φ

1/2

k+1

− u

∗

where w

∗

, u

∗

are deﬁned in (7b,c) respectively. Note

also that the terms that are not accessed by the con-

troller (i.e the terms with x

), are exclude from u

∗

Considering the above ¯w

and ¯u

we seek a controller

of the form

¯u

= −

1,k

ˆx

We, therefore, re-formulate the state equation of (2a)

adding the additional terms to recover the original

equation of (2a). Considering the above, we obtain

the following new state equation:

k+1

1,k

¯w

2,k

¯u

3,k

4,k

k+1

(18)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

328

where

= Q

−1

k+1

1,k

= γB

1,k

−1/2

k+1

2,k

= Q

−1

k+1

2,k

−1/2

k+1

3,k

= B

3,k

+ B

1,k

−1

k+1

1,k

k+1

3,k

−

2,k

−1/2

k+1

2,k

k+1

3,k

4,k

1,k

−1

k+1

1,k

−

2,k

−1/2

k+1

2,k

k+1

−1

k+1

(19a-e)

Replacing for ¯w

and ¯u

we obtain:

k+1

= (Q

−1

k+1

1,k

(γ

−1

1/2

k+1

− w

∗

)) +

2,k

(Φ

1/2

k+1

− u

∗

)) +

3,k

4,k

k+1

(20)

We consider the following Luenberger-type state

observer:

ˆx

k+1

ˆx

+ K

o,k

−C

2,k

ˆx

) + d

, ˆx

= 0,

ˆz

1,k

ˆx

(21)

where

2,k

¯u

3,k

4,k

k+1

Denoting e

= x

− ˆx

and using the latter we obtain:

k+1

= (

− K

o,k

2,k

1,k

ˆw

where we deﬁne

ˆw

∆

= [ ¯w

]

1,k

= [

1,k

− K

o,k

21,k

Deﬁning also ξ

= [x

]

, ¯r

∆

= [r

k+1

]

, we

obtain

k+1

1,k

ˆw

3,k

¯r

˜z

1,k

(22a-b)

where

11,k

2,k

1,k

22,k

1,k

−K

o,k

21,k

3,k

4,k

0 0

11,k

−

2,k

1,k

22,k

− K

o,k

2,k

1,k

= Φ

−1/2

k+1

2,k

k+1

1,k

= [0

1,k

(23a-g)

Applying the results of Theorem 2 to the system of

(22) we obtain the following Riccati-type equation:

[

−1

k+1

− γ

−2

1,k

]

−1

(24)

where

is given in (15b). Denoting

−1

and

using Schur complement we obtain the DLMI of

(15a). The latter DLMI is initiated with the initial

condition of (15b) which corresponds to the case

where a weighting γ

−1

is applied to ˆx

in order

to force nature to select ˆx

= 0 in the corresponding

differential game (see (Gershon and U. Shaked,

2005), Chapter 9, for details).

In the case where {r

} is measured on line, or with

preview h > 0, we note that nature strategy which is

not restricted to causality constraints, will be the same

as in the case of full preview of {r

}, meaning that ¯w

of (17) is unchanged. We obtain the following:

Lemma 2: H

∞

Output-feedback Tracking with full

preview of {r

}: We obtain

¯u

= Φ

1/2

k+1

+Φ

−1/2

k+1

2,k

k+1

3,k

−1

k+1

[θ

k+1

]

)

where we note that in this case [θ

k+1

]

= θ

k+1

. Sol-

ving (9) we obtain:

[θ

k+1

]

k+1

∑

N−k−1

j=1

k+1, j

N− j

where

k+1

∆

k+1

k+2

...

N−1

k+1, j

∆







k+1

k+2

...

N− j−1

j < N − k − 1

I j = N − k − 1







(25a-b)

Proof: Considering (9) and taking k + 1 = N we

obtain:

N−1

where θ

is given in (9). Similarly we obtain for N −2

N−2

N−1

N−2

[

N−1

] +

N−2

N−1

N−2

N−1

The above procedure is thus easily iterated to yield

(25a,b). Taking, for example N = 3 one obtains from

(9) the following equation for θ

The same result is recovered by taking k = 0 in (25a,b)

where j = 1,2.

Lemma 3: H

∞

Output-feedback Tracking with no

preview of {r

}: In this case [θ

k+1

]

= 0 since at time

k, r

is known only for i ≤ k. We obtain:

¯u

= Φ

1/2

k+1

+ Φ

−1/2

k+1

2,k

k+1

3,k

Lemma 4: H

∞

Output-feedback tracking with

ﬁxed-ﬁnite preview of {r

}: In this case we obtain:

¯u

= Φ

1/2

k+1

+Φ

−1/2

k+1

2,k

k+1

3,k

−1

k+1

[θ

k+1

]

)

and

2,k

¯u +

3,k

4,k

[θ

k+1

]

where [θ

k+1

]

satisﬁes :

[θ

k+1

]







∑

j=1

k+1, j

k+h+1− j

k + h ≤ N −1

k+1

∑

h−1

j=1

k+1, j

N− j

k + h = N







where

k+1, j

is obtained from (25b) by replacing N

by k + h + 1.

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329

4 STATIONARY

OUTPUT-FEEDBACK

TRACKING CONTROL

We treat the case where the matrices of the system in

(2) and (3) are all time-invariant, N tends to inﬁnity

and the system of (2) is exponential l

stable. In this

case, the solution

of (13a,b), if it exists, will tend to

the mean square stabilizing solution of the following

equation:

Q = A

MA +C

C, γ

I − B

Q(0) = γ

−1

− εI.

(26a,c)

We Introduce the following Lyapunov function:

= ξ

Qξ

, with

Q =

Q α

, (27)

where ξ

is the state vector of (22), Q and

Q are n × n

matrices and α ia a scalar. Considering (26) and the

above result, we obtain the following theorem:

Theorem 3: Consider the system of (2), (3) where

the matrices A,B

,C,D

and D

are

all constant and T → ∞. Given γ > 0, there exists

a controller that minimizes max

of (4) if there exist

Q = Q

∈ R

n×n

Q =

∈ R

n×n

, Y ∈ R

n×q

and a

tuning scalar parameter α that satisfy

ϒ > 0 where

is the following LMI:







Q α

Q ϒ(1,3) ϒ(1,4) 0 0 0

∗

Q ϒ(2,3)

ϒ(2,4) 0 0

∗ ∗ Q α

Q ϒ(3,5) ϒ(3,6) 0

∗ ∗ ∗

Q ϒ(4,5) 0 0

∗ ∗ ∗ ∗ I 0 0

∗ ∗ ∗ ∗ ∗ I 0

∗ ∗ ∗ ∗ ∗ ∗ I







(28)

where

∆

= K

Q,ϒ(1,3) =

Q −

ϒ(1,4) = (

−

)α

ϒ(2,3) =

Q + α

Q − αC

ϒ(2,4) = (α

)

Q −C

ϒ(3,5) = γ

−1

(Q + α

, ϒ(3,6) = −γ

−1

αY

ϒ(4,5) = γ

−1

(1 + α).

Proof: The proof outline for the above stationary

case resembles the one of the ﬁnite-horizon case.

Considering the stationary version of (2), (3) the sta-

tionary state-feedback control problem is solved to

obtain the optimal stationary strategies of both w

∗

s,k

and u

∗

s,k

(Gershon and U. Shaked, 2010). Thus we

obtain:

∗

s,k

= R

−1

k+1

[θ

k+1

+P(Ax

)],

∗

s,k

= −Φ

−1

M[Ax

+ B

+ Pθ

k+1

]},

∆

= P

−1

[I − γ

−2

−1

]

−1

where P

−1

is the stationry version of the solution Q

of (5).

Using the above optimal strategies we transform

the problem to an estimation one, thus arriving to

the stationary counterpart of the augmented system of

(22). Applying the result of (26) to the latter system

the algebraic counterpart of (24) is obtained which,

similarly to the ﬁnite-horizon horizon case, becomes

the following stationary version of (15):







−1

P γ

−1

0 γ

−1

I 0

0 0 I







≥ 0. (29)

Multiplying the above LMI by diag{I

−1

}

from the left and the right, denoting

Q =

−1

and

using the matrix partition of

Q of (27), the result of

(28) is obtained.

5 CONCLUSIONS

In this paper we solve the problem of deterministic

output-feedback tracking control with preview. Un-

like the state-feedback case the solution is not obtai-

ned by applying a game theory approach, (where a

saddle-point tracking strategy is derived) but rather

as a min-max optimization problem. The output-

feedback tracking problem is solved by applying

a special form of the BRL to a ﬁltering problem

which is formulated once the state-feedback solution

is obtained. The parameters of the a priori type state

observer used in our proof are recovered by iterative

solution of the DLMI of Theorem 2 which can be ea-

sily implemented. The result of the ﬁnite horizon case

were extended to the stationary case, where a simple

LMI condition is formulated for all the three preview

patterns. The theory is demonstrated by a numerical

example.

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Appendix

Proof Sketch of Theorem 1: The proof of Theorem

1 is based on the proof of Lemma 1 which concerns

the state-feedback control problem (for a detailed

proof see (Gershon and U. Shaked, 2010), (Gershon

and U. Shaked, 2005)). We ﬁrst bring in Part I, a

proof sketch of Lemma 1 which is also needed for

the solution of the output-feedback control. We then

bring, in Part II, the derivation of the sufﬁciency part

of Theorem 1, which is derived from Lemma 1, and

the necessity part of the proof of Theorem 1.

Part I: The proof of Lemma 1 follows the standard

line of applying a Lyapunov-type quadratic function

in order to comply with the index of performance.

This is usually done by using two successive com-

pleting to squares operations however, since the

reference signal of r

is introduced in the dynamics

of (2a), we apply a third completing to squares

operation with the aid of the ﬁctitious signal of θ

k+1

This latter signal ﬁnally affects the controller design

through it’s causal part [θ

k+1

]

(for a detailed proof

see (Gershon and U. Shaked, 2010), (Gershon and

U. Shaked, 2005)).

Part II: The sufﬁcient part of the proof of Theorem 1

stems from the above proof of Lemma 1 where B

2,k

0, D

2,k

= 0. Analogously to the proof of Lemma 1,

we obtain the following: J

) =

−

N−1

∑

k=0

|| ˆw

− R

−1

k+1

1,k

k+1

−γ

||x

− (γ

−1

−

)

−1

−γ

−2

where we replace θ

k+1

, Q

k+1

and

, respecti-

vely and where

= [R

−1

−γ

−2

]

−1

, x

= γ

−2

= [γ

−1

−

]

−1

The necessity follows from the fact that for r

≡ 0, one

gets

J(r,ε) = 0 (noting that in this case

≡ 0 in (14)

and therefore the last 3 terms in

J(r,ε) of Theorem

1 are set to zero) and J

< 0. Thus the existence of

Q > 0 that solves (13) is the necessary condition in

the BRL.

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