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 [email protected]
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
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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].
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