Direct Derivation Method of Noncausal Compensators

for Linear Continuous-Time Systems with Impulsive Effects

Gou Nakura

Abstract— In this paper we study the H∞ tracking problemswith preview for a class of linear continuous-time systemswith impulsive effects. The systems include linear continuous-time systems, linear discrete-time systems and linear systemswith the input realized through a zero-order hold. The nec-essary and sufficient conditions for the solvability of the H∞tracking problem are given by Riccati differential equationswith impulsive effects and terminal conditions. Correspondinglyfeedforward compensator introducing future information isgiven by linear differential equation with impulsive parts andterminal conditions. In this paper we focus on the derivationmethod of noncausal compensator dynamics from the point ofview of dynamics constraint. We derive the pair of noncausalcompensator dynamics and impulsive Riccati equations bycalculating the first variation of the performance index underthe dynamics constraint.

I. INTRODUCTION

It is well known that, for design of tracking control

systems, preview information of reference signals is very

useful for improving performance of the closed-loop systems,

and recently much work has been done for preview control

systems ([2], [3], [4], [8], [9], [10], [11], [13], [14]).

Particularly, U. Shaked et al. have presented the H∞ track-

ing control theory with preview for continuous- and discrete-

time linear time-varying systems by a game theoretic ap-

proach ([2], [14]). The author has extended their theory to

linear impulsive systems ([10], [11]). It is also very important

to consider effects of stochastic noise or uncertainties for

tracking control systems, and so, by Gershon et al, the theory

of stochastic H∞ tracking with preview has been presented

for linear continuous- and discrete-time systems respectively

([3], [4]).

Control and estimation theory for linear systems with

impulsive effects (or linear jump systems), which include

linear continuous- and discrete-time systems, can be widely

applied, for example, to mechanical systems, ecosystems,

chemical processes, financial engineering and so on ([1],

[5], [10], [11], [15], [16]). H∞ control and filtering theory

for linear impulsive systems has been researched in detail

by A. Ichikawa and H. Katayama([5]). Their theory can

be also applied into the sampled-data control systems with

the control input realized through a zero-order hold and the

sampled-observation.

Recently L. A. Montestruque and P. J. Antsaklis ([6],

[7]) have presented their works on model-based networked

systems. Their approach is based on analysis of impulsive

Gou Nakura, Uji, Kyoto, Japan. gg9925 fiesta@ybb.ne.jp

systems. As these examples, impulsive systems and design

theory for them have lately attracted considerable attention.

In this paper we study the H∞ tracking problems with

preview by state feedback for linear continuous-time systems

with impulsive effects on the finite time interval. The author

has already presented the necessary and sufficient conditions

for the solvability of the H∞ tracking problems and given

the control strategies for them ([10], [11]). The necessary

and sufficient conditions for the solvability of the H∞

tracking problem with preview are given by Riccati differ-

ential equation with impulsive parts and terminal conditions.

Correspondingly compensator introducing future information

is described by differential equation with impulsive parts.

However it has not been yet given how the preview or non-

causal feedforward compensators are derived in detail. How

do we derive the form of the compensator dynamics more

directly than the derivation method presented in [2], [10],

[14] and so on? Why does the compensator dynamics have

such form? Even for continuous- or discrete-time systems

the answers to these significant questions have not been fully

given.

In this paper we focus on the design method of noncausal

compensator dynamics from the point of view of dynamics

constraint. We directly derive the pair of Riccati differ-

ential equation and noncausal compensator dynamics with

impulsive parts by variational calculus method. This research

result clarifies the meaning of the form of the noncausal

compensator dynamics and the relashionship of noncausal

tracking control theory and noncausal estimation (smoothing)

theory.

II. PROBLEM FORMULATION

Consider the following linear continuous-time time-

varying system with impulsive effects.

x(t) = A(t)x(t) + B1(t)w(t) + B2(t)uc(t) + B3(t)rc(t)

t �= kτ, x(0) = x0

x(kτ+) = Ad(k)x(kτ) + B1d(k)wd(k)

+B2d(k)ud(k) + B3d(k)rd(k)

zc(t) = C1(t)x(t) + D12(t)uc(t)

+D13(t)rc(t), t �= kτ (1)

zd(k) = C1d(k)x(kτ) + D12d(k)ud(k) + D13d(k)rd(k)

where x ∈ Rn is the state, w ∈ R

p and wd ∈ Rpd are

the exogenous disturbances, uc ∈ Rmc and ud ∈ R

md

are the continuous and impulsive control inputs, zc ∈ Rkc

and zd ∈ Rkd are the controlled outputs, rc(t) ∈ R

rc and

2010 IEEE International Conference on Control ApplicationsPart of 2010 IEEE Multi-Conference on Systems and ControlYokohama, Japan, September 8-10, 2010

978-1-4244-5363-4/10/$26.00 ©2010 IEEE 2350

rd(k) ∈ Rrd are known or mesurable reference signals, and

x0 is an unknown initial state. We assume that all matrices

are of compatible dimensions. Throughout this paper the

dependence of the system matrices on t or k will be omitted

for the sake of notational simplicity.

Remark 2.1: Note that the system (1) can be reduced to

the system with the control input realized through a zero-

order hold by introducing the auxiliary variable x ∈ Rm as

follows:

First we consider the following system.

xs = Axs + B1w + B2u + B3rc

z(t) =

[

C1xs + D13rc

D12u

]

where u is the input realized through a zero-order hold,

i.e., u(t) = u(k), kτ < t ≤ (k + 1)τ , and τ is a

sampling time. Let xe = col(x′

s, x′). Then the following

system representation is equivalent to the above system

representation for the system with the input realized through

a zero-order hold.

xe =

[

A B2

O O

]

xe +

[

B1

O

]

w +

[

B3

O

]

rc(t), t �= kτ

xe(kτ+) =

[

I O

O O

]

xe(kτ) +

[

B1d

O

]

wd(k)

+

[

O

I

]

ud(k) +

[

B3d

O

]

rd(k)

zc =[

C1 O]

xe + D13rc(t), t �= kτ

zd(k) =[

C1 O]

xe(kτ) +√

τD12u(k) + D13rd(k)

This system is covered by the impulsive system (1) with

B2 ≡ O. Therefore the H∞ tracking control design theory

for the linear impulsive systems we study in this paper can be

applied to the system with the control input realized through

a zero-order hold. Also refer to the references [5], [11] and

so on.

The H∞ tracking problems we address in this paper for

the system (1) are to design control laws uc(t)∈L2[0, T ]and ud(k)∈l2[0, N ] over the finite horizon [0, T ], Nτ <

T < (N + 1)τ using the information available on the

known parts of the reference signals rc(t) and rd(k) and

minimizing the sum of the energy of zc(t) and zd(k), for

the worst case of the initial condition x0, the disturbances

w(t)∈L2([0, T ];Rp) and wd(k)∈l2([0, N ];Rpd). We denote

by L2([0, T ];Rk) and l2([0, N ];Rkd) the space of nonantic-

ipative signals. Considering the average of the performance

index over the statistics of the unknown parts of rc and rd,

we define the following performance index.

JT (x0, uc, ud, w,wd) := −γ2x′

0R−1x0

+

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

ERs{‖zc(s)‖2}ds

+

N∑

k=0

ERk{‖zd(k)‖2} − γ2[‖wd‖2

2 + ‖w‖22] (2)

where Nτ < T < (N + 1)τ , R = R′ > O is a

given weighting matrix for the initial state, ERsand ERk

mean expectations over Rs+hτ and Rk+h, h is the preview

length of rc(t) and rd(k), and Rs and Rj denote the future

information on rc and rd at time s and jτ respectively,

i.e.,Rs := {rc(l); s < l ≤ T } and Rj := {ri; j < i ≤ N}.

We consider two different tracking problems according to

the information structures (preview lengths) of rc and rd as

follows.

Case a) H∞ Fixed-Preview Tracking:

In this case, it is assumed that at the current time t (kτ+ ≤t ≤ (k + 1)τ), rc(s) is known for s ≤ min(T, t + hτ) and

at the time kτ , rd(i) is known for i ≤ min(N, k + h).Case b) H∞ Tracking of Noncausal {rc(·) and rd(·)}:

In this case, the signals {rc(t)} and {rd(k)} are assumed to

be known a priori for the whole time intervals t ∈ [0+, T ]and k ∈ [0, N ].

In order to solve these two problems, we formulate the

following differential game problem for the system (1) and

the performance index (2).

The H∞ Tracking Problem by State Feedback:

Find {u∗

c}, {u∗

d}, {w∗}, {w∗

d} and x∗

0 satisfying the follow-

ing (saddle point) condition:

JT (x0, u∗

c , u∗

d, w,wd)

≤ JT (x∗

0, u∗

c , u∗

d, w∗, w∗

d)

≤ JT (x∗

0, uc, ud, w∗, w∗

d)

where the control strategies u∗

c(t), 0 ≤ t ≤ T and u∗

d(k), 0 ≤k ≤ N , are based on the information Rt+hτ := {rc(l); 0 <

l ≤ t + hτ} and Rk+h := {rd(i); 0 < i ≤ k + h} (0 ≤ h ≤N ), and the current state x(t).

III. DESIGN OF TRACKING CONTROLLERS

In this section we present the H∞ tracking theory for linear

impulsive systems by state feedback.

For the system (1), we assume the following condition.

A1: D′

12D12 > O and D′

12dD12d > O

Now we consider the following Riccati differential equa-

tion with impulsive parts.

X + A′X + XA + C′

1C1 +1

γ2XB1B

′

1X

−S′R−1S = O, t �= kτ (3)

X(kτ−) = A′

dX(kτ)Ad + C′

1dC1d

+R′

1T−11 R1(k) − F ′

2V2F2(k) k = 0, 1, · · · (4)

T1(k) > aXI for some aX > 0 (5)

where

R = D′

12D12, S(t) = B′

2X(t) + D′

12C1,

T1(k) = γ2I − B′

1dX(kτ)B1d,

T2(k) = D′

12dD12d + B′

2dX(kτ)B2d,

S(k) = B′

2dX(kτ)B1d,

R1(k) = B′

1dX(kτ)Ad,

R2(k) = D′

12dC1d + B′

2dX(kτ)Ad,

V2(k) = (T2 + ST−11 S′)(k),

2351

V3(k) = (B′

2d + ST−11 B′

1d)(k),

F2(k) = −V −12 (k)(R2 + ST−1

1 R1)(k).

We obtain the following necessary and sufficient condi-

tions for the solvability of the H∞ tracking problem and a

saddle point strategy for it.

Proposition 3.1: Consider the system (1) and suppose A1.

Then the H∞ Tracking Problem is solvable by State

Feedback if and only if there exists a matrix X(t) > O

satisfying the conditions X(0−) < γ2R−1 and X(T ) = O

such that the Riccati equation (3)(4) with (5) holds over

[0, T ]. A saddle point strategy is given by

x∗

0 = [γ2R−1 − X(0−)]−1θ(0) (6)

w∗(t) = γ−2B′

1X(t)x(t) + Cθθ(t), t �= kτ (7)

u∗

c(t) = −R−1S(t)x(t) − Curc(t) − Cθuθc(t), t �= kτ (8)

w∗

d(k) = T−11 (k)[R1(k)x(kτ) + S′(k)ud(k)]

+Dwrd(k) + Dθwθ(kτ+) (9)

u∗

d(k) = F2(k)x(kτ) + Du(k)rd(k) + Dθuθc(kτ+) (10)

where

Cθ = −γ−2B′

1, Cθu = R−1B′

2, Cu = R−1D′

12D13,

Dw(k) = T−11 (k)B′

1dX(kτ)B3d,

Du(k) = −V −12 (k)(B′

2dX(kτ)B3d + D′

12dD13d + SDw),

Dθw(k) = T−11 (k)B′

1d,

Dθu(k) = −V −12 (k)(S(k)Dθw + B′

2d).

θ(t), t ∈ [0, T ], satisfies

θ(t) = −A′

c(t)θ(t) + Bc(t)rc(t), t �= kτ

θ(kτ) = A′

d(k)θ(kτ+) + Bd(k)rd(k)θ(T ) = 0

(11)

where

Ac = A +1

γ2B1B

′

1X − B2R−1S,

Bc = −(XB3 + C′

1D13) + S′Cu,

Ad(k) = Ad + D′

θwR1(k) − D′

θuV2F2(k),

Bd(k) = A′

dX(kτ)B3d + R′

1Dw

−F2V2Du(k) + C′

1dD13d

and θc(t) is the ’causal’ part of θ(·) at time t. This θc is the

expected value of θ over Rs and Rk and given by

θc(s) = −A′

c(s)θc(s) + Bc(s)rc(s), s �= kτ

t ≤ s ≤ tfθc(jτ) = A′

d(j)θc(jτ+) + Bd(j)rd(j),

k < j ≤ kf (kfτ < tf < (kf + 1)τ)θc(tf ) = 0

(12)

where, for kτ+ ≤ t ≤ (k + 1)τ ,{

tf = t + hτ and kf = k + h if (k + h)τ < T

tf = T and kf = N if (k + h)τ ≥ T.(Proof)

Sufficiency: We have already presented the proof of the

sufficiency for the solvability of this H∞ noncausal tracking

problem. Refer to [10].

Necessity: As explained in [10], because of arbitrariness

of the reference signals rc(·) and rd(·), by considering the

case of rc(·) ≡ 0 and rd(·) ≡ 0, one can also easily deduce

the necessity for the solvability of the H∞ tracking problem.

The direct derivation method of the pair of the noncausal

compensator dynamics and impulsive Riccati equation by

variational calculus will be explained in detail in the next

section. (Q.E.D.)

Next, utilizing the above saddle point strategy for the

game problem, we present a solution to each of the two

H∞ tracking problems by state feedback.

Theorem 3.1: Consider the system (1) and suppose A1.

Then each of the H∞ tracking problems is solvable by

state feedback if and only if there exists a matrix X(t) > O

satisfying the conditions X(0−) < γ2R−1 and X(T ) = O

such that the Riccati equation (3)(4) and the condition (5)

hold over [0, T ]. Moreover, the following results hold using

Kx,c = −R−1S, Krc= −R−1D′

12D13,

Kθ,c = −R−1B′

2, Kx,d = F2(k),

Krd= −V −1

2 (k)(D′

12dD13d + V3(k)X(kτ)B3d),

Kθ,d = −V −12 V3(k).

Case a) The control law for the H∞ fixed-preview tracking

is

us,a,c(t) = Kx,cx(t) + Krcrc(t) + Kθ,cθc(t), t �= kτ

us,a,d(k) = Kx,dx(kτ) + Krdrd(k) + Kθ,dθc(kτ+)

with θc(·) given by (12).

Case b) The control law for the H∞ tracking of noncausal

rc(·) and rd(·) is

us,b,c(t) = Kx,cx(t) + Krcrc(t) + Kθ,cθ(t), t �= kτ

us,b,d(k) = Kx,dx(kτ) + Krdrd(k) + Kθ,dθ(kτ+)

with θ(·) given by (11) since θ(t) = θc(t), ∀ t ∈ [0, T ].Remark 3.1: The compensator dynamics (12) in the fixed

preview case has the same form as the compensator dynamics

(11) in the noncausal case, while the terminal conditions in

these two cases are different.

Remark 3.2: With regard to the value of the performance

index and how to calculate it for each control strategy, refer

to [10].

IV. DIRECT DERIVATION METHOD OF

NONCAUSAL COMPENSATOR DYNAMICS

In this section we consider the direct derivation method

of noncausal compensator dynamics (11) and Riccati dif-

ferential equation (3)-(5) with impulsive parts, which give

the proof of the necessity for the solvability of this H∞

noncausal tracking problem, by the variational calculus.

We assume that the signals {rc(t)} and {rd(k)} are known

a priori for the whole time interval [0, T ]. Notice that

this situation is the same as the noncausal H∞ tracking

problem considered by U. Shaked et. al.([2], [14]) Also

notice that in this case the expectations ERsand ERk

are

not necessary.

2352

In this case we can regard the dynamics in the system (1)

x(t) = Ax(t) + B1w(t) + B2uc(t) + B3rc(t), t �= kτ,

x(kτ+) = Adx(kτ) + B1dwd(k)

+B2dud(k) + B3drd(k)

as the constraint over the time interval [0, T ]. Then we can

define the following Lagrangian

JT (x0, uc, ud, w,wd) := JcT + JdT ,

JcT

:= −γ2‖w‖22 +

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

‖zc(s)‖2ds

+‖C1x(s) + D13rc(s)‖2QT

+

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

2λ′(s){Ax(s) + B1w(s)

+B2uc(s) + B3rc(s) − x(s)}ds

and

JdT

:= −γ2x′

0R−1x0 − γ2‖wd‖2

2

+

N∑

k=0

‖C1dx(kτ) + D12dud(k) + D13drd(k)‖2

+N

∑

k=0

2λ′(kτ+){Adx(k) + B1dwd(k) + B2dud(k)

+B3drd(k) − x(kτ+)}where QT ≥ O and λ(·) is a Lagrange multiplier. We

calculate the first variation of JT with regard to x, uc, ud,

w, wd and λ, and obtain

δJT := δJcT + δJdT

where

δJcT

= −{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

2γ2δw′wds

+

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

2δx′(s)C′

1{C1x(s) + D12uc(s) + D13rc(s)}+2δu′

c(s)D′

12{C1x(s) + D12uc(s) + D13rc(s)}ds

+δx′(T ){C ′

1QT C1x(T ) + 2C ′

1QT D13rc(T )}

+

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

2δλ′(s){Ax(s) + B1w(s)

+B2uc(s) + B3rc(s) − x(s)}ds

+

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

2{δx′(s)A′ + δw′(s)B′

1 + δu′

c(s)B′

2}λ(s)ds

−2λ′(T )δx(T ) + 2λ′(0+)δx(0+)

+

{

N−1∑

k=0

∫ (k+1)τ

kτ+

+

∫ T

Nτ+

}

2λ′δx(s)ds

and

δJdT

= −2γ2δx′

0R−1x0 − 2γ2

N∑

k=0

δwd(k)wd(k)

+

N∑

k=0

{

2δx′(k)C′

1d{C1dx(k) + D12dud(k) + D13drd(k)}

+2δu′

d(k)D′

12d{D12dud(k) + D13drd(k)}}

+

N∑

k=0

2δλ′(kτ+){Adx(k) + B1dwd(k) + B2dud(k)

+B3drd(k) − x(kτ+)}

+N

∑

k=0

2λ′(kτ+){Adδx(k) + B1dδwd(k) + B2dδud(k)

−δx(kτ+)}Let δJT = 0 and then we obtain the following conditions:

(i) the conditions of optimality

w∗(s) = γ−2B′

1λ(s), s �= kτ, (13)

w∗

d(k) = γ−2B′

1dλ(kτ+), (14)

u∗

c(s) = −R−1(D′

12C1x(s)

+D′

12D13rc(s) + B2λ(s)), s �= kτ, (15)

D′

12d{D12du∗

d(k) + D13drd(k)} = −B′

2dλ(kτ+). (16)

(ii) the canonical equations

x(s) = Ax(s) + B1w∗(s) + B2u

∗

c(s) + B3rc(s), (17)

s �= kτ,

x(kτ+) = Adx(kτ) + B1dw∗

d(k)

+B2du∗

d(k) + B3drd(k),(18)

λ(s) = −A′λ(s) − C′

1{C1x(s) + D12u∗(s)

+D13rc(s)}, s �= kτ, (19)

λ(kτ) = A′

dλ(kτ+) + C′

1d{C1dx(kτ)

+D12du∗

d(k) + D13drd(k)}.(20)

(iii) the boundary conditions

x∗

0 = γ−2Rλ(0), (21)

λ(T ) = C′

1QT C1x(T ) + 2C ′

1QT D13rc(T ). (22)

From the boundary condition (22), we set

λ(t) = X(t)x(t) + θ(t),

λ(kτ+) = X(kτ)x(kτ+) + θ(kτ+)

and

λ(kτ) = X(kτ−)x(kτ) + θ(kτ).

2353

Then from (21) we have

x∗

0 = γ−2R[X(0)x∗(0) + θ(0)],

which is equivalent to the worst case (6) of the initial state

x0. We have also the continuous part (7) of the worst case

disturbance and the continuous part (8) of the control strategy

immediately. Notice that

λ(t) = θ(t) + X(t)x(t) + X(t)x(t)

= θ(t) + X(t)x(t) + X(t){Ax(t) + B1w∗(t)

+B2u∗(t) + B3rc(t)}.

Substituting this and λ(t) = θ(t) + X(t)x(t) into the

canonical equation (19), we obtain the following equality.

λ(t) = θ(t) + X(t)x(t) + X(s){Ax(t) + B1w∗(t)

+B2u∗(t) + B3rc(t)}

= −A′{θ(t) + X(t)x(t)}−C ′

1{C1x(t) + D12u∗(t) + D13rc(t)}

By the arbitrariness of x, with regard to the first order

terms of x, we obtain the following continuous parts of

the impulsive Riccati equation with the terminal condition,

which give the necessary conditions for the solvability of the

H∞ noncausal tracking problem.

X + A′X + XA + C′

1C1 +1

γ2XB1B

′

1X

−S′R−1S = O, X(T ) = C′

1QT C1 (23)

where

R = D′

12D12, S(t) = B′

2X(t) + D′

12C1.

With regard to the terms without depending on x, we obtain{

θ(t) = −A′(t)θ(t) + B(t)rc(t)θ(T ) = 2C ′

1QT D13rc(T )(24)

where

A = A +1

γ2B1B

′

1X − B2R−1S,

B = −(XB3 + C′

1D13) + S′Cu, Cu = R−1D′

12D13,

which gives the preview compensator dynamics with the

terminal condition.

Next we consider the impulsive parts. Using (14) and (18),

we obtain

w∗

d(k) = γ−2B′

1dX(kτ){Adx(kτ) + B1dw∗

d(k)

+B2du∗

d(k) + B3drd(k)} + γ−2B′

1dθ(kτ+),

which is equivalent to

[I − γ−2B′

1dX(kτ)B1d]w∗

d(k)

= γ−2B′

1dX(kτ){Adx(kτ) + B2du∗

d(k)

+B3drd(k)} + γ−2B′

1dθ(kτ+).

Suppose

T1(k) = γ2I − B′

1dX(kτ)B1d > aXI (25)

for some ax > 0. Then we obtain the impulsive part (9)

of the worst case disturbance. From the condition (16) of

optimality,

B′

2d{X(kτ)x(kτ+) + θ(kτ+)}= −D′

12dD12du∗

d(k) − D′

12dD13drd(k).

Using the impulsive part (18) of the canonical equation,

B′

2dX(kτ){Adx(kτ) + B1dw∗

d(k)

+B2du∗

d(k) + B3drd(k)} + B′

2dθ(kτ+)

= −D′

12dD12du∗

d(k) − D′

12dD13drd(k).

By substituting the worst case disturbance w∗

d(k), which we

have already obtained, into the above equality, we obtain

the impulsive part (10) of the control strategy. Using the

canonical equation (20),

X(kτ−)x(kτ) + θ(kτ) = λ(kτ)

= A′

d{X(kτ)x(kτ+) + θ(kτ+)}+C ′

1d{C1dx(kτ) + D12du∗

d(k) + D13drd(k)}= A′

dX(kτ){Adx(kτ) + B1dw∗

d(k) + B2du∗

d(k)

+B3drd(k)} + A′

dθ(kτ+)

+C ′

1d{C1dx(kτ) + D13drd(k)}.By the arbitrariness of x(k), with regard to the first order

terms of x(k), we obtain the impulsive part of the Riccati

equation

X(kτ−) = A′

dX(kτ)Ad + C′

1dC1d

+R′

1T−11 R1(k) − F ′

2V2F2(k). (26)

With regard to the terms without depending on x(k), we

also obtain the impulsive part of the noncausal compensator

dynamics (11)

θ(kτ) = A′

d(k)θ(kτ+) + Bd(k)rd(k). (27)

The combination of (23), (25) and (26) yields the im-

pulsive Riccati equation (3)-(5), which gives the necessary

conditions for the solvability of the H∞ noncausal tracking

problem. The combination of (24) and (27) yields the non-

causal compensator dynamics (11) in the case of QT = O.

Remark 4.1: As described in the Remark 3.1, The pre-

view compensator dynamics (12) in the fixed preview case

has the same form as the noncausal compensator dynamics

(11) in the noncausal case. We can obtain the preview

compensator dynamics (12) with the receding terminal con-

dition θc(tf ) = 0 by restricting the final time T and N

(Nτ < T < (N + 1)τ ) in the Lagrangian JT to t + h and

N1 (N1τ < t + h < (N1 + 1)τ ).

Remark 4.2: Notice that, in the case with Ad = I ,

B1d = O, B2d = O and B3d = O, by adopting the

derivation method by variational calculus presented in this

section, we obtain the same Riccati differential equation and

noncausal compensator dynamics for the linear continuous-

time systems as the ones given by U. Shaked et al.([14]) Also

notice that, in the case with A = O, B1 = 0, B2 = O and

B3 = O, by adopting the derivation method presented in this

2354

section, we obtain the same Riccati difference equation and

noncausal compensator dynamics for the linear discrete-time

systems as the ones given by A. Cohen et al.([2])

Remark 4.3: Also notice that this derivation method of

noncausal compensator dynamics for tracking control sys-

tems is equaivalent to the one of smoothers (noncausal

estimators) for systems with observation and noises by

maximum likelihood (ML) and variational calculus approach.

Consider the following linear system with impulsive effects

and sampled-observation.

dx(t) = Ax(t)dt + Gcw(t)dt t �= kτ,

x(0) = x0, 0 < t ≤ Nτ

x(kτ+) = Ad(k)x(kτ) + Gdwd(k)

y(k) = Hd(k)x(kτ) + v(k)

where x ∈ Rn is the state, w ∈ R

p and wd ∈ Rpd are

the exogenous noises, v ∈ Rk is the measurement noise,

y ∈ Rk is the measured output, x0 is an unknown initial state

and x0 ∼ N (x0, Σ0). w(t), wd(k) and v(k) are mutually

uncorrelated zero mean Gaussian white noises with constant

covariance matrices Qc ≥ O, Qd ≥ O and Rd ≥ O. It is

well known that, in the case of the smoothing problem, we

set

xs(t|N) = xf (t|k) + Pf (t|k)λf (t)

where xs(·|T ) is the state of smoother dynamics, xf (·|t) is

the state of forward filter dynamics, Pf (·|·) is the covariance

matrix of the forward daynamics and λf (·) is the Lagrange

multiplier for plant dynamics constraint. Using this rela-

tionship, we can obtain the following (backward) smoother

dynamics.

dxs(t|N) = Axs(t|N)dt

+GcQcG′

cP−1f (t|k)[xs(t|N) − xf (t|k)]dt, t �= kτ

xs(kτ+|N) = Adxs(kτ |N)

+GdQdG′

dP−1f (kτ+|k)[xs(kτ+|N) − xf (kτ+|k)]

In detail, refer to [12]. This derivation method of the optimal

smoother is equivalent to the one of noncausal LQ preview

compensator. In the case of H∞ setting, we can easily the

similar equivalence.

V. CONCLUDING REMARKS

In this paper we have studied the H∞ tracking control

theory considering the preview information by state feedback

for the linear continuous-time systems with impulsive parts,

which are of high practice. The compensators introducing

the preview information of the reference signal have been

also described by the linear impulsive systems. The theory

presented in this paper can be also applied to the systems

with the input realized through a zero-order hold.

The author had presented the solution of the H∞ preview

tracking control theory by state feedback and output feedback

for the linear continuous-time systems with impulsive effects

([10], [11]). However it had not been yet fully investigated

why the preview compensator dynamics has such form.

Therefore in this paper he has presented the direct derivation

method of the form of the preview compensator dynamics by

using the variational calculus with the dynamics constraint.

As described in the Remark 4.2, the theory presented in

this paper covers both of the theories for the continuous-

and discrete-time systems, and so the form of the preview

compensator dynamics by the derivation method presented

in this paper corresponds to the ones by another derivation

method by U. Shaked et al. ([2], [14]) by restricting the form

of the system matrices in (1).

The derivation method of preview compensator dynamics

presented in this paper can be applied to more various types

of systems and problem settings. They will be reported

elsewhere. Also refer to [13].

REFERENCES

[1] D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect:

Stability, Theory and Applications, Ellis Horwood Limited, New York;1989.

[2] A. Cohen and U. Shaked, Linear Discrete-Time H∞-Optimal Trackingwith Preview, IEEE Trans. Automat. Contr., vol. 42, no. 2, 1997, pp.270-276.

[3] E. Gershon, D. J. N. Limebeer, U. Shaked and I. Yaesh, StochasticH∞ Tracking with Preview for State-Multiplicative Systems, IEEE

Trans. Automat. Contr., vol. 49, no. 11, 2004, pp. 2061-2068.[4] E. Gershon, U. Shaked and I. Yaesh, H∞ tracking of linear continuous-

time systems with stochastic uncertainties and preview, Int. J. Robust

and Nonlinear Control, vol. 14, no. 7, 2004, pp. 607-626.[5] A. Ichikawa and H. Katayama, Linear Time Varying Systems and

Sampled-data Systems, LNCIS, Vol. 265, Springer, London; 2001.[6] L. A. Montestruque and P. J. Antsaklis, On The Model-Based Control

of Networked Systems, Automatica, vol. 39, no. 10, 2003, pp. 1837-1843.

[7] L. A. Montestruque and P. J. Antsaklis, Stability of Model-BasedNetworked Control Systems with Time-Varying Transmission Times,IEEE Trans., Automat., Contr., vol. 49, no. 9, 2004, pp. 1562-1572.

[8] G. Nakura, ”H∞ Tracking with Preview for Linear Continuous-TimeMarkovian Jump Systems”, Proceedings of the SICE 8th Annual

Conference on Control Systems, Kyoto, Japan, 2008, 073-2-1 (CD-ROM).

[9] G. Nakura, ”Noncausal Optimal Tracking for Linear Switched Sys-tems”, Proceedings of The 11th International Workshop of Hybrid

Systems : Computation and Control (HSCC2008), St. Louis, MO, USA

(2008, 4, 22-24), LNCS, vol.4981, Springer, Berlin, Heidelberg; 2008,pp. 372-385.

[10] G. Nakura, ”H∞ Tracking with Preview for Linear Systems withImpulsive Effects -State Feedback and Full Information Cases-”,Proceedings of the 17th IFAC World Congress, Seoul, Korea, 2008,TuA08.4 (CD-ROM).

[11] G. Nakura, ”H∞ Tracking with Preview by Output Feedback forLinear Systems with Impulsive Effects”, Proceedings of the 17th IFAC

World Congress, Seoul, Korea, 2008, TuA08.5 (CD-ROM).[12] G. Nakura, ”Fixed-Interval Optimal Smoothing for Linear Impulsive

Systems”, Proceedings of the 41st ISCIE International Symposium on

Stochastic Systems Theory and Its Applications (SSS09), Kobe, Japan,2009, pp. 140-147.

[13] G. Nakura, ”On Noncausal H∞ Tracking Control for LinearContinuous-Time Markovian Jump Systems”, Proceedings of the 41st

ISCIE International Symposium on Stochastic Systems Theory and Its

Applications (SSS09), Kobe, Japan, 2009, pp. 172-177.[14] U. Shaked and C. E. de Souza, Continuous-Time Tracking Problems

in an H∞ Setting: A Game Theory Approach, IEEE Trans. Automat.

Contr., vol. 40, no. 5, 1995, pp.841-852.[15] Y. Xiao, D. Cheng and H. Qin, Optimal impulsive control in periodic

ecosystem, Systems and Control Letters, vol. 55, no. 7, 2006, pp. 558-565.

[16] T. Yang, Impulsive Control Theory, LNCIS, Vol. 272, Springer-Verlag,Berlin; 2001.

2355