Download H2 Optimal Cooperation of Homogeneous Agents Subject to Delyed

Proceedings of the 13th IFAC Workshop on Time Delay
Systems, Istanbul, Turkey, June 22-24, 2016
TA11.1
H2 Optimal Cooperation of Homogeneous Agents
Subject to Delyed Information Exchange ?
Daria Madjidian Leonid Mirkin Laboratory of Information and Decision Systems, Massachusetts Institute of
Technology, Cambridge, MA, 02139. E-mail: [email protected].
Faculty of Mechanical Engineering, Technion—IIT, Haifa 32000, Israel.
E-mail: [email protected].
Abstract: We consider a class of large-scale H2 coordination problems where subsystems are coupled
through a penalty on their average behavior and the information exchange between agents is subject
to a time delay. We derive explicit analytic expressions for the optimal control law and show that it is
scalable both in terms of implementation and the computational effort required to obtain it. Specifically,
the only centralized information processing required to implement the controller is averaging of local
agent information, and the computational effort required to obtain it scales linearly with the number of
agents. These results are a highly non-trivial extension to two previously studied special cases, where
the information exchange is either delay-free or the coordination constraints are hard.
Keywords: Distributed cntrol, low-rank coordination, time-delay systems, H2 optimization.
1. INTRODUCTION
same control signal (Brockett, 2010; Wood et al., 2008; Azuma
et al., 2013).
Control of complex systems, or distributed control, has received
significant attention over the last decade due to new technologies, networking and integration trends, efficiency demands,
etc. The research within this field is driven by a number of
multi-agent applications, where full, unstructured information
processing is not possible due to communication and scalability
restrictions.
Recently, we showed that the diagonal-plus-rank-one control
structure also appears as the optimal solution to a class of large
scale coordination problems, which arise in the control of wind
farms (Madjidian and Mirkin, 2014). More specifically, we
considered a group of homogeneous agents, where the objective
is to strike a trade-off between local performances of the agents
and a coordination requirement, expressed in terms of a desired
behavior of an average agent. The coordination requirement
may be formulated as both hard and soft constraint in the optimization problem. The former approach requires the constraint
to be satisfied at each time instance. The latter approach adds
coordination requirements as an additional weighted penalty
in the cost function. In both cases we showed that the only
global information needed to attain the optimal performance is
the knowledge of the state of the average agent. Moreover, the
computational effort required to obtain the optimal control law
is independent of the number of agents.
In a distributed control law, agents only have access to partial
information about the overall system. Most work within this
field assumes a sparse information structure, where agents only
exchange information with a limited subset of other agents. See
(Šiljak, 1978; Mesbahi and Egerstedt, 2010; Yüksel and Başar,
2013) and references therein. This type of control structure is
often well suited if it reflects the physical interaction pattern
of the agents (Bamieh et al., 2002). However, sparsity is not
the only way to obtain a scalable control law. One example of
a non-sparse, yet scalable, controller is a diagonal-plus-rankone configuration. As opposed to the sparse control structures,
which restrict the communication capability of each system
to a limited set of neighboring systems, this type of control
law reduces information processing by aggregating information
from all agents into a single quantity, e.g an average, which is
then made available to each of the agents.
To the best of our knowledge, the diagonal-plus-rank-one structure first explicitly appeared in (Hovd and Skogestad, 1994), as
the solution to a class of optimal control problems for symmetrically interconnected systems. A method to design this type
of control laws was proposed by Zečević and Šiljak (2005).
There are also number of works on rank-one control. This is
sometimes called broadcast or ensemble control, and refers to
groups of identical decoupled systems that are controlled by the
? This research was supported by the Bernard M. Gordon Center for Systems
Engineering at the Technion.
Copyright © 2016 IFAC
A potential limitation of the problem formulation of Madjidian
and Mirkin (2014) is that it assumes instantaneous information
exchange between agents. Initial steps to overcome this issue
were taken in (Madjidian et al., 2014), where the effect of
delays in the communication between agents was studied for
the special case of hard coordination constraints. A key feature
of the formulation in (Madjidian et al., 2014) is that, due to the
hard coordination constraint, the problem could be reduced to
a conventional H2 problem with a uniform delay. It was shown
that the optimal controller is then again of the diagonal-plusrank-one form.
In this paper we solve the general soft coordination problem
with delayed information exchange between agents. This formulation yields a highly non-trivial extension of previously
available results. Unlike the hard constraint case in (Madjidian
et al., 2014), the unorthodox, off-diagonal, delay pattern of the
controller does not reduce to a uniform delay. As a result, the
147
solution is substantially more involved, which is manifested
both in required technical tools and in the form of the optimal
controller. Our solution procedure uses the parametrization of
all delay-free solutions to reduce the problem to a one-block
finite-horizon H2 model-matching problem with a diagonal
controller. This produces closed-form controller formulae and
scalable computations. Specifically, the computational effort required to obtain the optimal control law scales linearly with the
number of agents. Moreover, just as in the delay-free case, the
only global information required to attain optimal performance
is an average of local agent information.
Another contribution of this paper is the extension of statefeedback results of (Madjidian and Mirkin, 2014; Madjidian
et al., 2014) to a more general setup. Namely, instead of perfect
state measurements we consider partial state measurements corrupred by noise. Moreover, measurement equations are allowed
to be heterogeneous in this paper, which extends the scope of
applicability of the result of Madjidian and Mirkin (2014).
Notation The transpose of a matrix M is denoted by M 0 and
its spectral radius by .M /. The notation diag.Mi / stands for
a block-diagonal matrix with matrices Mi , i D 1; : : : ; on its
diagonal. By ei we refer to the i th standard basis of a Euclidean
space and by In to the n n identity matrix (we drop the
dimension subscript when the context is clear). The notation
˝ is used for the Kronecker product of matrices:
3
2
a11 B a1m B
6
:: 7 ;
A ˝ B ´ 4 ::: : : :
: 5
ap1 B apm B
where aij stands for the .i; j / entry of A. Finally, the truncation
operator
A B
A B
A B
h
e sh
´
C 0
C 0
C eAh 0
and the completion operator
A B
A B
h
e sh ´
C 0
C e Ah 0
A B
C 0
e
sh
produce entire transfer functions having the (finite) impulse
responses C eAt B 1Œ0;h .t/ and C eA.t h/ B 1Œ0;h .t/, respectively,
where 1Œ0;h .t/ D 1 if t 2 Œ0; h and 0 otherwise.
2. PROBLEM FORMULATION
2.1 The setup
Consider the problem of coordinating agents subject to timedelay in the information exchange between them. The agents
are assumed to have linear time-invariant (LTI) dynamics, described by the following state-space dynamics:
xP i .t/ D Axi .t/ C Bwi wi .t/ C Bu ui .t/
(1)
yi .t/ D Cyi xi .t/ C Dyi wi wi .t/
where xi .t/ 2 Rn is a state vector, ui .t/ 2 Rm is a control
input, wi .t/ 2 Rqi is an exogenous input (disturbance), and
yi .t/ 2 Rpi is a measured output. Note that the local dynamics
and the effect of the control inputs on them are equal for each
agent, whereas the effect of the disturbances and measurement
equations may differ. The agents have identical local objectives,
´
w
G´w G´u
0 0 ˝ D
Gyw Gyu
´
y
K
u
Fig. 1. Aggregate standard H2 problem with coordination
penalty
which are quantified in terms of the H2 problems associated
with the regulated outputs
´i .t/ ´ C´ xi .t/ C D´u ui .t/:
(2)
These local objectives can be cast as the standard H2 problem
for the aggregate generalized plant with block-diagonal components,
2
3
I ˝ A Bw I ˝ Bu
G´w .s/ G´u .s/
D 4 I ˝ C´ 0 I ˝ D´u 5 ;
(3)
Gyw .s/ Gyu .s/
Cy
Dyw
0
which connects the aggregate inputs
P w and u with the aggregate
outputs ´ and y (e.g., w ´
i D1 ei ˝ wi ), where Bw ´
diag.Bwi /, Cy ´ diag.Cyi /, and Dyw ´ diag.Dyi wi /.
Coordination among the agents is imposed via another regulated variable,
X
i ui .t/
(4)
´ .t/ D D u.t/
N
´ D
i D1
for some weights i .
Combining local and coordination regulated variables, the
problem studied in this paper can be cast as
0 H2
the standard
problem for the setup in Fig. 1, where ´ 1 . The
goal is to internally stabilize the closed loop system
T´w .s/
G´w .s/
G´u .s/
D
C
0
T .s/
0
˝ D
K.s/ I Gyu .s/K.s/ 1 Gyw .s/: (5)
and minimize its H2 -norm. Here T´w and T reflect local and
coordination objectives,
respectively. By increasing D , e.g.,
p
by taking D D =.1 /I with " 1, we may enforce the
hard constraint uN D 0.
We assume that the the information exchange among agents is
subject to a delay of h time units. This restriction results in the
following constraint on K :
3
2
K11 .s/ e sh K12 .s/ e sh K1 .s/
6 e sh K21 .s/ K22 .s/ e sh K2 .s/ 7
7
6
(6)
K.s/ D 6
7
::
::
::
::
5
4
:
:
:
:
e
sh
K1 .s/ e
sh
K2 .s/ K .s/
for causal Kij and h > 0.
Remark 1. In principle, a more general form of coordination
than (4) can be considered. For example, following the formulation in (Madjidian and Mirkin, 2014), we P
may think of ´ based
on uN GN yN instead of uN , where yN ´ i i yi and GN .s/ is
some system. This modification, however, reduces to (4) via the
following shift of the control variables: ui D vi C GN yi . Thus,
we can consider the simpler form (4) without loss of generality.
O
148
Remark 2. If h D 0, the H2 problem in Fig. 1 is an outputfeedback extension of the state-feedback problem studied in
(Madjidian and Mirkin, 2014, ÷IV-A). It is relatively straightforward to use similar arguments as in (Madjidian and Mirkin,
2014) to show that the optimal control law has a diagonal-plusrank-one form, and that it can be obtained by solving three
algebraic Riccati equations (ARE) of size n, see Section 3. O
Remark 3. If h > 0 and we impose the constraint uN D 0
(corresponds to D D 1), the problem reduces to an outputfeedback version of the problem studied in (Madjidian et al.,
2014). As shown there, the combination of the hard constraint
with (6) implies that the diagonal elements of K must be
delayed as well. As a result, the problem reduces to a singledelay H2 problem, which is well understood.
O
2.2 The formal problem statement
The formal statement of the optimization problem considered
in this paper is then as follows:
T´w 2
minimize J ´ (7a)
T 2
subject to K of the form (6)
(7b)
We address (7) under the following assumptions:
I
F˛ 0
L
Fig. 2. Centralized part of the controller in the case of h D 0
whose solution is stabilizing if the matrix A ´ A C Bu F ,
0
where F ´ .I C D0 D / 1 .Bu0 X C D´u
C´ /, is Hurwitz,
and, for every i D 1; : : : ; ,
AYi C Yi A0 C Bwi Bw0 i
.Yi Cy0 i C Bwi Dy0 i wi /.Cyi Yi C Dyi wi Bw0 i / D 0;
(8c)
whose solution is stabilizing if the matrix ALi ´ A C Li Cyi ,
where Li ´ .Yi Cy0i C Bwi Dy0 i wi /, is Hurwitz. It is known
(Zhou et al., 1995, Ch. 13) that if A 1–6 hold, the stabilizing
solutions to (8) exist, are unique, and such that X˛ 0, X 0,
and Yi 0 (in fact, it is readily seen that X˛ X ).
Theorem 4. Let h D 0. The optimal attainable performance in
(7) is then
X
2
opt
D
tr L0i ..1 2i //X˛ C 2i X /Li C tr.C´ Yi C´0 / :
K.s/ D Fl .J.s/; Ru 1 Q.s//;
kQk22 2
2
opt
(9)
where Ru ´ .I 0 / ˝ I C .0/ ˝ .I C D0 D /1=2 and
2
3
I ˝ A C .I ˝ Bu /F C LCy L I ˝ Bu
5;
J.s/ D 4
F
0
I
Cy
I
0
A 4 : .Cyi ; A/ is detectable,
A j!I Bwi
has full row rank 8! 2 R,
A 5:
Cyi
D´i ui
A 6 : D´i ui D´0 i ui D I ,
where F ´ .I
and also
A 7 : 0 D 1.
Assumptions A 1,2,4,5 are standard assumptions necessary for
the well-posedness of the uncoordinated local problems. The
normalization assumptions in A 3,6,7 are introduced to simplify
the exposition and can be relaxed.
3. DELAY-FREE SOLUTION
We start with extending the results of (Madjidian and Mirkin,
2014) to the output-feedback setting. Although not a technical
challenge, this extension is the starting point of our treatment
of the delayed version of the problem.
To formulate the solution, we need the following algebraic
Riccati equations (AREs):
(8a)
whose solution is stabilizing if the matrix A˛ ´ A C Bu F˛ ,
0
where F˛ ´ .Bu0 X˛ C D´u
C´ /, is Hurwitz,
.X Bu C C´0 D´u /
0
.I C D0 D / 1 .Bu0 X C D´u
C´ / D 0;
A
F
1
and, for every opt , the set of all -suboptimal controllers
is parametrized as
for every i D 1; : : : ; ,
A0 X C X A C C´0 C´
N
1 L1
i D1
A 1 : .A; Bu / is stabilizable,
A j!I Bu
has full column rank 8! 2 R,
A 2:
C´
D´u
0
A 3 : D´u
D´u D I ,
A0 X˛ C X˛ A C C´0 C´
0
.X˛ Bu C C´0 D´u /.Bu0 X˛ C D´u
C´ / D 0;
(8b)
0 / ˝ F˛ C .0 / ˝ F and L ´ diag.Li /.
Proof. Key steps are to show that the solution to the filtering
ARE associated with the generalized plant in Fig. 1 is diag.Yi /,
which is straightforward, and the solution to the corresponding
control ARE is .I 0/ ˝ X˛ C .0/ ˝ X , which follows by
coordinate transformations similar to those used in (Madjidian
and Mirkin, 2014). The rest follows by (Zhou et al., 1995,
Thm. 14.8).
The optimal controller of Theorem 4, the one corresponding to
Q D 0, can be implemented in the following observer-based
form:
P i .t/ D Ai .t/ C Bu ui .t/ Li i .t/;
(10)
ui .t/ D F˛ i .t/ C i .F F˛ /.t/;
N
where i .t/
P ´ yi .t/ Cyi i .t/ is the i th “innovation” and
.t/
N
´
i D1 i i .t/ is the “averaged” observer state. This
controller inherits the main feature of the state-feedback controller derived in (Madjidian and Mirkin, 2014). Namely, the
information exchange between agents is done via an averaging
operation. The only difference is that the actual states are now
replaced with their observations.
P Note also that N verified the
PN
equation .t/
D A .t/
N
i D1 i Li i .t/. Hence, the output
equation in (10) can be rewritten as
ui .t/ D F˛ i .t/
i .t/;
N
where N is generated by the system presented in Fig. 2. This
signal will play an important role in the case of h > 0.
149
The following result is required to characterize the local performance under the controller of Theorem 4.
Lemma 5. The H2 norm of the closed-loop system from the
aggregate exogenous signal w to the regulated variable ´i
attainable with the controller of Theorem 4 is
ui
2
i
kT´i w k22 D i;2opt C k.ei ˝ I /.J1 C J2 Ru 1 Q/k22 ;
i;2opt
tr.L0i X˛ Li /
tr.C´ Yi C´0 /
where
D
C
performance of the i th agent and
A
J1 .s/ J2 .s/ D
˝ .F F˛ /
is the optimal local
It follows from Lemma 5 that under the optimal controller
(Q D 0) we have that
tr.j2 Lj0 Xv Lj /;
j D1
where Xv 0 is the solution to the Lyapunov equation
F˛ /0 .F
F˛ / D 0:
(11)
It is then readily seen that
X
X
kT´i w k22 D
tr 2i L0i X´ Li ;
2
kT´ w k22 D opt
i D1
where X´ ´ X
X˛
A0 X´
i D1
Xv . Because X´ verifies
C X´ A C F0 D0 D F D 0
(this can be verified by some algebra), we have X´ 0.
4. ADDING DELAYS
We need another set of AREs:
A0 Xi C Xi A C C´0 C´
.Xi Bu C C´0 D´u /
0
.I C 2i D0 D / 1 .Bu0 Xi C D´u
C´ / D 0;
(12)
whose solution is stabilizing if the matrix Ai ´ A C Bu Fi ,
0
C´ /, is Hurwitz.
where Fi ´ .I C 2i D0 D / 1 .Bu0 Xi C D´u
It is readily seen that X˛ Xi X . It can also be verified that
Zi ´ Xi
.1
2i /X˛
N
A
F
I
F˛ 0
e
1 LQ 1
1
LQ sh
Fig. 4. Centralized part of the controller in the case of h > 0
is the generator of all suboptimal controllers for the uncoordinated version of the problem (the one with D D 0). The result
then follows by applying the arguments of (Zhou et al., 1995,
Sec. 14.6).
A0 Xv C Xv A C .F
LQ i Bu
A˛
Fi F˛ 0 I
Fig. 3. The optimal controller for the i th agent
I ˝ A˛ C LCy L I ˝ Bu
5
I ˝ F˛
0
I
Cy
I
0
X
i N
.0 ˝ I /L 0 ˝ Bu
:
0
I
3
kT´i w k22 D i;2opt C 2i
sh
e
where
J˛ .s/ D 4
i
˘i .s/
Proof. Based of the readily verifiable fact that the controller
parametrization of Theorem 4 can be equivalently expressed as
K.s/ D Fl J˛ .s/; J1 .s/ C J2 .s/Ru 1 Q.s/ ;
2
yi
3
Li Bu
0 I 5
I 0
Ai C Li Cyi
4
Fi
Cyi
2i X 0
(Zi ! 0 as both i # 0 and i " 1). We also need the matrix
functions
Z t
0
eAi Bu .I C 2i D0 D / 1 Bu0 eAi d:
Wi .t/ ´
0
The main result of this paper, whose proof is postponed to
Section 5, is then formulated as follows:
Theorem 6. For every h 0, .Wi .h/Zi / < 1 for all i D
1; : : : ; , the optimal attainable performance in (7) is
2
2
h;
opt D opt C
X
tr L0i Zi
0
eAi h Zi .I
Wi .h/Zi /
1 Ai h
e
i D1
Li
and the optimal controller is as depicted in Fig. 3, where
Qi
Zi L
A0i
sh
e
;
˘i .s/ ´ h
.I C 2i D0 D / 1 Bu0 0
Q i ´ .I
L
Wi .h/Zi / 1 eAi h Li , and N is generated by the
system presented
P in Fig. 4 from the weighted average of local
“innovations” iD1 i LQ i i .
The local part of the controller of Theorem 6 comprises two
finite-dimensional systems, a delay, and a distributed-delay
element ˘i . The latter is an infinite-dimensional system, which
can nevertheless be efficiently implemented, see (Troeng and
Mirkin, 2013) and the references therein. The centralized part
presented in Fig. 4 is based on the weighted average of the local
innovations i . This averaging is an extension of those required
in the state-feedback cases studied in (Madjidian and Mirkin,
2014; Madjidian et al., 2014). The presence of the delay in
Fig. 4 renders the resulting controller compatible with (6), as
expected.
Remark 7. Because the matrices Ai are Hurwitz, the increase
of the delay h reduces the gains LQ i . As a result, both i and N
vanish as h ! 1 and we end up with the diagonal controllers
Ai C Li Cyi L1
:
Fi
0
It can be shown that these are the optimal controllers for (7a)
under the constraint that K is block-diagonal, i.e., when no
coordination between the agents is permitted.
O
Remark
8.
As
mentioned
in
Section
2,
the
choice
D
D
p
=.1 /I for " 1 enforces the hard constraint uN D 0.
This formulation, which was studied in (Madjidian et al., 2014)
and is well posed only if A is Hurwitz, results in Xi D X ,
Fi D F D 0, and ˘i D 0. The latter simplifies the implementation of the optimal controller.
O
Remark 9. The i th controller of Theorem 6 can be alternatively
presented in the form depicted in Fig. 5, where
150
ui
2
i
A˛ C Li Cyi
4
F˛
Cyi
˘0i .s/
3
Li Bu
0 I 5
I 0
3
I ˝ A C .I ˝ Bu /FQ C LCy L I ˝ Bu
5
JQ .s/ ´ 4
FQ
0
I
Cy
I
0
2
yi
i
for some block-diagonal FQ D diag.FQi / such that all A C Bu FQi
are Hurwitz. The choice of the same observer gain L as in J.s/
is motivated by the fact that in this case
Q
Q.s/ D T1 .s/ C T2 .s/Q.s/;
i N
Fig. 5. The optimal controller for the i th agent (alternative
form)
2
3
Qi
L
Ai
Bu Ri 1 Bu0
Q i 5 e sh ;
˘0i .s/ ´ h 4
Zi L
0
A0i
1 0
F˛ Fi Ri Bu
0
˚
Ri ´ I C 2i D0 D , and the centralized term N is generated by
the system presented in Fig. 4 (note the gain F˛ instead of Fi
in the main generator). This can be seen using the arguments in
the beginning of the proof of Lemma 5 and for the optimal Q
given by Lemma 11, from which
J1 .s/ C J2 .s/Ru 1 Q.s/ D diag ˘0i .s/
Q
A
.0 ˝ I /L
. ˝ I /
e sh :
F F˛
0
The latter expression can also be used to calculate the local
costs via Lemma 5.
O
5. PROOF OF THEOREM 6 (OUTLINE)
Our starting point in the proof of the main result is the observation that the set of controllers described by (6) is a subset of
unconstrained causal controllers addressed in Theorem 4. As
a result, for any h > 0, the optimal solution of (7) is covered
by parametrization (9). We can then solve (7) by finding the
minimal-norm Q for which controller (9) is of the form (6).
To this end, note that the mapping Q 7! K defined by (9) is invertible and the inverse mapping is described by the LFT (Zhou
Q K/,
et al., 1995, Lemmas 10.4 (c) and 10.3 (i)) Q D Ru Fl .G;
where
2
3
I ˝ A L I ˝ Bu
1
Q
5;
G.s/
´ P12 J .s/P12 D 4 F
0
I
Cy
I
0
and P12 ´ I0 I0 . Therefore, (7) is equivalent to the problem of
minimizing the H2 norm of Q by a controller of the form (6).
This is a constrained standard H2 problem for the generalized
plant GQ , similar to (7). A big advantage of this transformation is
that the problem associated with GQ is a one-block problem (this
fact was first explicitly exploited in (Meinsma et al., 2002)). As
will be demonstrated below, this property plays an important
role in handling the off-diagonal delays in K transparently.
Now, the .2; 2/ part of GQ , which actually equals Gyu , is blockdiagonal. It is then readily seen that constraint (6) is quadratically invariant (Rotkowitz and Lall, 2006) with respect to this
plant. This, in turn, implies that the Youla parametrization of
all stabilizing controllers preserves structure (6) in its free parameter, provided the parametrization is also based on blockdiagonal elements. We consider the set of all stabilizing K of
form (6):
Q
K.s/ D Fl JQ .s/; Q.s/
;
Q
where Q.s/
is constrained as in (6) and
where
I ˝ A C .I ˝ Bu /FQ L I ˝ Bu
T1 .s/ T2 .s/ ´
Ru .FQ F /
0
Ru
and Ru and F are defined in Theorem 4. Our problem is now to
find QQ that minimizes kQk2 .
Let us now split
Q
Q.s/
D QQ d .s/ C e
sh
QQ h .s/
for some block-diagonal FIR, with support in Œ0; h, QQ d D
diag.QQ d;i / and a causal and stable, but otherwise unconstrained
QQ h . The following result can then be formulated:
Lemma 10. Denote by qQd;i .t/ the impulse response of QQ d;i and
Q D diag.L
Q i /, where
L
Z h
Q
Q i ´ e.ACBu FQi /h Li
e.ACBu Fi /.h / Bu qQd;i . /d:
L
0
Then
˚
Q.s/ D h T1 .s/ C T2 .s/QQ d .s/
C e sh TQ1 .s/ C T2 .s/QQ h .s/ ;
where
TQ1 .s/ ´
I ˝ A C .I ˝ Bu /FQ
Ru .FQ F /
(13)
Q
L
:
0
Proof. Omitted because of space limitations.
Lemma 10 says that Q can always be split into two parts
that are orthogonal in H2 (they have non-overlapping impulse
responses). Moreover, the second term in the right-hand side of
(13) can be made zero for every QQ d (this is the very reason for
reducing (7) to the one-block problem at the beginning of this
section). Indeed, it is readily seen that T2 is square and stably
invertible, so the choice
Q
I ˝ A C .I ˝ Bu /F L
QQ h .s/ D T2 1 TQ1 .s/ D
(14)
FQ F
0
is admissible and optimal. Because the choice of QQ h does not
affect the first term in th right-hand side of (13), the problem of
minimizing kQk2 reduces to the problem of minimizing kQ0 k2
by a block-diagonal QQ d , where
˚
Q0 .s/ ´ h T1 .s/ C T2 .s/QQ d .s/
(as a matter of fact, the optimal Q is then FIR).
The latter problem can be split into independent problems by
noting that the i th block column of Q0 depends only on QQ d;i .
Indeed, because QQ d .ei ˝ I / D .ei ˝ I /QQ d;i ,
˚
Q0 .s/.ei ˝ I / D h T1i .s/ C T2i .s/QQ d;i .s/ ;
where T1i ´ T1 .ei ˝ I / and T2i ´ T2 .ei ˝ I / with
A C Bu FQi Li Bu
T1i .s/ T2i .s/ D
Ci C Di FQi 0 Di
(15)
151
with Ci ´ Ru F .ei ˝ I / and Di ´ Ru .ei ˝ I /. Thus, we end
up with optimization problems
˚
minimize h T1i .s/ C T2i .s/QQ d;i .s/ 2 ;
(16)
which are finite-horizon H2 model-matching problems. If we
then choose FQi D Fi , where Fi is the gain associated with
(12), T2iÏ .s/T2i .s/ D Di0 Di D I C 2i D0 D (Zhou et al., 1995,
Thm. 13.32) and the solution to (16) is simplified. This solution
is given by the following lemma:
Lemma 11. Let FQi D Fi , then the optimal QQ d;i D ˘i defined
in Theorem 6, the i th column of the resulting Q is
2
3
LQ i
Ai
Bu Ri 1 Bu0
Q i 5 e sh ;
Qi .s/ D Ru h 4 0
Zi L
A0i
1 0
Mi ei ˝ .Ri Bu /
0
˚
where Ri ´ I C 2i D0 D and Mi ´ F .ei ˝ I /
H2 norm of this optimal Qi is
0
kQi k22 D tr L0i Zi
eAi h Zi .I
Wi .h/Zi /
ei ˝ Fi , the
1 Ai h
e
Li ;
which is well-defined because .Wi .h/Zi / < 1, and LQ i defined
in Lemma 10 coincides with that defined in Theorem 6.
Proof. Omitted because of space limitations.
Thus, combining the optimal QQ d;i with QQ h from (14) and taking
into account that
I ˝ A C .I ˝ Bu /F D .I
0 / ˝ A˛ C .0 / ˝ A ;
the optimal
I ˝ A C .I ˝ Bu /F LQ
Q
e sh
Q.s/
D diag.˘i .s// C
FQ F
0
LQ i
A˛
sh
D diag ˘i .s/ C
e
Fi F˛ 0
Q
.I 0 / ˝ A˛ C .0 / ˝ A L
C
e sh
.0 / ˝ .F F˛ /
0
A˛
LQ i
e sh
D diag ˘i .s/ C
diag.i I /
Fi F˛ 0
3
2
A˛
Bu
6 F1 F˛ I 7 Q
A
.0 ˝ I /L
7
6
e sh
6
::
:: 7
0
4
:
: 5 F F˛
F
F˛
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I
which can be implemented as shown in Figs. 3 and 4.
To complete the proof of Theorem 6 we only need to sum up
the norms kQi k22 , which yields the increment of the optimal
cost dues to the presence of delays.
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