Solution of a Singular H
∞
Control Problem: A Regularization Approach
Valery Y. Glizer
1
and Oleg Kelis
1,2
1
Department of Applied Mathematics, ORT Braude College of Engineering,
51 Snunit Str., P.O.Box 78, Karmiel 2161002, Israel
2
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel
Keywords:
H
∞
Control Problem, Singular Problem, Regularization, H
∞
Partial Cheap Control Problem, Riccati Matrix
Algebraic Equation, Asymptotic Design of Controller.
Abstract:
We consider an infinite horizon H
∞
control problem for linear systems with additive uncertainties (distur-
bances). The case of a singular weight matrix for the control cost in the cost functional is treated. In such a
case, a part of the control coordinates is singular, meaning that the H
∞
control problem itself is singular. We
solve this problem by a regularization. Namely, we associate the original singular problem with a new H
∞
control problem for the same equation of dynamics. The cost functional in the new problem is the sum of the
original cost functional and an infinite horizon integral of the squares of the singular control coordinates with
a small positive weight. This new H
∞
control problem is regular, and it is a partial cheap control problem.
Based on an asymptotic analysis of this H
∞
partial cheap control problem, a controller solving the original
singular H
∞
control problem is designed. Illustrative example is presented.
1 INTRODUCTION
Controlled systems with uncertain dynamics are ex-
tensively studied in the literature (see e.g. ((Basar
and Bernard, 1991); (Chang, 2014); (Doyle et al.,
1990); (Fridman et al., 2014); (Glizer and Turetsky,
2012); (Petersen and Tempo, 2014); (Petersen et al.,
2000)) and references therein). Two classes of uncer-
tainties (disturbances) are usually distinguished: (1)
disturbances belonging to a known bounded set of
Euclidean space; (2) quadratically integrable distur-
bances. For controlled systems with quadratically in-
tegrable disturbances, the H
∞
control problem is fre-
quently considered (see e.g. ((Basar and Bernard,
1991); (Chang, 2014); (Doyle et al., 1989); (Petersen
et al., 2000)).
If the rank of the matrix of coefficients for the con-
trol variable in the output equation equals to the di-
mension of the control, then the solution of the H
∞
control problem can be reduced to a solution of a
game-theoretic Riccati matrix algebraic equation. If
the rank of the matrix of coefficients for the control in
the output is smaller than the dimension of the con-
trol, then the weight matrix for the control cost in the
cost functional of the H
∞
problem is singular mean-
ing that the mentioned above Riccati equation does
not exists. Such H
∞
control problems are called sin-
gular or nonstandard. Some cases of linear dynamics
singular H
∞
control problems were studied in the liter-
ature, using different approaches. Thus, in (Petersen,
1987), the H
∞
problem with no control in the out-
put was considered. For this problem, an ”extended”
game-theoretic Riccati matrix algebraic equation was
constructed. Based on the assumption of the exis-
tence of a proper solution to this equation, the solu-
tion of the considered H
∞
problem was derived. In
(Djouadi, 1998), an explicit expression for the opti-
mal controller was derived using an operator theory
approach and the Banach space duality. In (Stoorvo-
gel, 2000), a Riccati matrix inequality approach was
used to solve the problem. In (Chuang et al., 2011),
the controller design was based on a physical model
of the Atomic Force Microscope, studied in the paper.
In the present paper, we consider an infinite hori-
zon singular H
∞
control problem. A regularization of
this problem is proposed leading to a new H
∞
problem
with a partial cheap control. The latter is solved by
adapting a perturbation technique. Then, it is shown
on how accurately the controller, solving this H
∞
par-
tial cheap control problem, solves the original singu-
lar H
∞
control problem.
It should be noted that the regularization approach
has been widely applied in the literature for analy-
sis and solution of various control problem. Thus, in
((Bell and Jacobson, 1975); (Glizer, 2012c); (Glizer,
2012b); (Glizer, 2014); (Kurina, 1977)), different sin-
Glizer, V. and Kelis, O.
Solution of a Singular H
∞
Control Problem: A Regularization Approach.
DOI: 10.5220/0006397600250036
In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 25-36
ISBN: 978-989-758-263-9
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
25

gular optimal control problems were solved using this
approach. In (Turetsky et al., 2014), this approach
was applied for the design of a robust state-feedback
control in some trajectory tracking problem for uncer-
tain systems. In ((Shinar et al., 2014); (Glizer and Ke-
lis, 2015a); (Glizer and Kelis, 2015b); (Glizer, 2016),
(Glizer and Kelis, 2017)), some singular zero-sum
and non zero-sum differential games were solved by
application of the regularization approach. In a short
conference paper ((Glizer, 2013)), a singular H
∞
con-
trol problem for linear time delay systems was studied
in the case where the output equation of this problem
is independent of the control. In the present paper,
we consider the case where the output equation of a
singular H
∞
control problem depends on the control.
To the best of our knowledge, the rigorous analysis of
such a case of singular H
∞
control problems by appli-
cation of the regularization approach is carried out for
the first time in the literature in this paper.
2 PROBLEM STATEMENT
We consider the following controlled system:
dZ(t)
dt
= AZ(t) + Bu(t) + F w(t), Z(0) = 0, (1)
V (t) = col{C Z(t), M u(t)}, (2)
where t ≥ 0, Z(t) ∈ E
n
, u(t) ∈ E
r
, (n ≥ r), (u(t) is a
control); w(t) ∈ E
m
, (w(t) is a disturbance); V (t) ∈
E
p
, (V (t) is an output); A, B, F , C, M are given
constant matrices of dimensions n ×n, n ×r, n ×s,
p
1
×n, p
2
×r, (p
1
+ p
2
= p, r ≤ p
2
), respectively.
Assuming that w(t) ∈ L
2
[0,+∞;E
m
], let us con-
sider the following cost functional:
J (u,w) =

kV (t)k
L
2

2
−γ
2

kw(t)k
L
2

2
, (3)
where γ > 0 is a given constant.
The H
∞
control problem with a performance level
γ for the system (1)-(2) is to find a controller u
0
[Z(t)]
that ensures the inequality
J (u
0
,w) ≤ 0 (4)
along trajectories of (1) for all w(t) ∈ L
2
[0,+∞;E
m
].
Now, we consider the following two matrices:
D = C
T
C , N = M
T
M . (5)
The matrices D and N are symmetric matrices of
the dimensions n ×n and r ×r, respectively. Further-
more, they are at least positive semi-definite. More-
over, if rankM = r, then the matrix N is positive def-
inite. In the latter case, the matrix N is invertible.
Thus, we can write down the following Riccati alge-
braic equation for the n ×n-matrix P :
PA + A
T
P + P SP + D = 0, (6)
where S = γ
−2
F F
T
−BN
−1
B
T
.
Along with the equation (6), we consider the dif-
ferential equation
dZ(t)
dt
=

A −BN
−1
B
T
P

Z(t), t ≥ 0. (7)
As a particular case of the results of (Glizer,
2009b) (Lemma 2.1), we directly have the following
assertion.
Proposition 1. Let there exist a symmetric solution
P of the equation (6) such that the trivial solution to
the equation (7) is asymptotically stable. Then, the
controller
u
0
[Z(t)] = −N
−1
B
T
P Z(t) (8)
solves the H
∞
control problem (1)-(3).
Remark 1. Proposition 1 presents the solvability
conditions of the H
∞
control problem (1)-(3) and the
controller solving this problem. Due to the expres-
sions for the matrix S and the controller (8), one can
use this proposition only if the matrix N is invert-
ible, i.e., in the case where the rank of the matrix M
equals r (the dimension of the control vector). Other-
wise, Proposition 1 is not applicable to solution of the
H
∞
control problem (1)-(3).
The objective of this paper is to develop a method
of solution of the H
∞
control problem (1)-(3) in the
case where rankM = q < r. More precisely, we as-
sume that the matrix M has the block form
M =

M
1
,O
p
2
×(r−q)

, (9)
where the block M
1
is of dimension p
2
×q, O
n
1
×n
2
is
zero matrix of the dimension n
1
×n
2
, and
Λ
4
= M
T
1
M
1
= diag

λ
1
,...,λ
q

, λ
l
> 0, l = 1,...,q.
(10)
The H
∞
control problem (1)-(3) subject to (9) is a
singular (nonstandard) H
∞
control problem.
3 TRANSFORMATION OF THE
H
∞
CONTROL PROBLEM
(1)-(3),(9)-(10)
Let us partition the matrix B into the blocks as:
B =

B
1
,B
2

, (11)
where the matrices B
1
and B
2
are of dimensions n×q
and n ×(r −q), respectively.
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
26

In what follows, we assume:
(A1) the matrix B has full column rank r;
(A2) det B
T
2
DB
2
6= 0, where D is given in (5).
Let B
c
be a complement matrix to the matrix B,
i.e., the dimension of B
c
is n ×(n −r), and the block
matrix (B
c
,B) is nonsingular. Thus
e
B
c
= (B
c
,B
1
) (12)
is a complement to B
2
.
Consider the following matrices:
H = (B
T
2
DB
2
)
−1
B
T
2
D
e
B
c
, L =
e
B
c
−B
2
H . (13)
Using the matrix L, we transform the state in the
H
∞
problem (1)-(3),(9)-(10) as follows:
Z(t) = (L,B
2
)z(t), (14)
where z(t) ∈ E
n
is a new state.
Due to the results of (Glizer et al., 2007), the trans-
formation (14) is invertible.
Let us partition the matrix H into blocks as:
H =

H
1
,H
2

, (15)
where the blocks H
1
and H
2
are of the dimensions
(r −q) ×(n −r) and (r −q) ×q, respectively.
Based on the results of (Glizer and Kelis, 2015b)
(Lemma 1), we directly obtain the following two as-
sertions.
Proposition 2. Let the assumptions A1-A2 be valid.
Then, the transformation (14) converts the H
∞
control
problem (1)-(3),(9)-(10) to the new H
∞
control prob-
lem for the system
dz(t)
dt
= Az(t) + Bu(t) + Fw(t), z(0) = 0, t ≥ 0,
(16)
v(t) = col{Cz(t), Mu(t)}, (17)
and the cost functional
J(u,w) =

kv(t)k
L
2

2
−γ
2

kw(t)k
L
2

2
, (18)
where
A = (L, B
2
)
−1
A (L,B
2
), (19)
B = (L, B
2
)
−1
B =


O
(n−r)×q
O
(n−r)×(r−q)
I
q
O
q×(r−q)
H
2
I
r−q


,
(20)
F = (L, B
2
)
−1
F , (21)
C = C (L,B
2
), (22)
M = M . (23)
Let us partition the matrix C, given by (22), into
blocks as:
C =

C
1
,C
2

, (24)
where the blocks C
1
and C
2
are of the dimensions p
1
×
(n −r + q) and p
1
×(r −q), respectively.
Corollary 1. Let the assumptions A1-A2 be valid.
Then, the matrix D
4
= C
T
C has the block form
D =

C
T
1
C
1
C
T
1
C
2
C
T
2
C
1
C
T
2
C
2

=

D
1
O
(n−r+q)×(r−q)
O
(r−q)×(n−r+q)
D
2

, (25)
where
D
1
4
= C
T
1
C
1
= L
T
DL, D
2
4
= C
T
2
C
2
= B
T
2
DB
2
. (26)
The matrix D
1
is symmetric positive semi-definite,
while the matrix D
2
is symmetric positive definite.
Due to (9)-(10),(23), the new H
∞
control problem
(16)-(18) is singular. Similarly to the problem (1)-(3),
we say that the controller u
∗
[z(t)] solves the problem
(16)-(18) if it guarantees the fulfilment of the inequal-
ity
J

u
∗
,w

≤ 0 (27)
along trajectories of (16) for all w(t) ∈ L
2
[0,+∞;E
m
].
Lemma 1. Let the assumptions A1-A2 be valid. If the
controller u
0
[Z(t)] solves the H
∞
control problem (1)-
(3), then the controller u
0

L, B
2

z(t)

solves the H
∞
control problem (16)-(18). Vice versa, if the controller
u
∗
[z(t)] solves the H
∞
control problem (16)-(18), then
the controller u
∗
h

L, B
2

−1
Z(t)
i
solves the H
∞
con-
trol problem (1)-(3).
Proof. Let us start with the first statement of the
lemma. Since the controller u
0
[Z(t)] solves the prob-
lem (1)-(3), then the inequality (4) is satisfied along
trajectories of (1) for all w(t) ∈ L
2
[0,+∞;E
m
]. Now,
let us make the state transformation (14) in the prob-
lem (1)-(3). Due to this transformation and Propo-
sition 2, the problem (1)-(3) becomes the problem
(16)-(18). The inequality (4) becomes the inequal-
ity J

u
0

L, B
2

z(t)

,w(t)

≤0 along trajectories of
(16) for all w(t) ∈ L
2
[0,+∞;E
m
], meaning that the
controller u
0

L, B
2

z(t)

solves the problem (16)-
(18). This completes the proof of the first statement.
The second statement is proven similarly.
Remark 2. Due to Lemma 1, the initially formulated
problem (1)-(3) is equivalent to the new problem (16)-
(18). From the other hand, due to Proposition 2 (see
(20)) and Corollary 1, the latter is simpler than the
former. Therefore, in the sequel of this paper, we deal
with the H
∞
control problem (16)-(18). We consider
this problem as an original one and call it the Singu-
lar H
∞
Control Problem (SHICP).
Solution of a Singular H
∞
Control Problem: A Regularization Approach
27

4 REGULARIZATION OF THE
SHICP
4.1 Partial Cheap Control H
∞
Problem
To study the SHICP, we replace it with a regular H
∞
control problem, which is close in a proper sense to
the SHICP. This new H
∞
control problem has the
same equation of dynamics (16). However, the out-
put equation in the new problem differs from the one
in the SHICP. This output equation has the ”regular”
form, i.e., the rank of the matrix of coefficients for the
control in this equation equals r (the dimension of the
control vector), and it is close to the one in the SHICP.
Since r ≤ p
2
, then q < p
2
and r −q ≤ p
2
. There-
fore, there exists a

p
2
×(r −q)

-matrix M
2
such that
M
T
2
M
1
= O
(r−q)×q
, M
T
2
M
2
= I
r−q
. (28)
Based on this observation, we choose the regular out-
put equation as:
v
ε
(t) = col{Cz(t), M
ε
u(t)}, (29)
where
M
ε
=

M
1
,εM
2

, (30)
ε > 0 is a small parameter.
Using (10), (28) and (30), we obtain
N
ε
4
= M
T
ε
M
ε
=


Λ O
q×(r−q)
O
(r−q)×q
ε
2
I
r−q


, (31)
rankN
ε
= r, and N
ε
is positive definite for all ε > 0.
The cost functional, corresponding to the output
equation (29), is
J
ε
(u,w) =

kv
ε
(t)k
L
2

2
−γ
2

kw(t)k
L
2

2
=
Z
+∞
0

z
T
(t)Dz(t) + u
T
(t)N
ε
u(t)
−γ
2
w
T
(t)w(t)

dt, (32)
where the matrix D is given by (25)-(26).
Remark 3. Due to the smallness of the parameter ε
and the form of the matrix N
ε
, the H
∞
control problem
(16),(29),(32) is a partial cheap control problem, i.e.,
the problem where the cost only of some (but not all)
control coordinates is small. A total cheap control
problem, i.e., the problem where the cost of all con-
trol coordinatesis small, has been extensively inves-
tigated in the literature in different settings. Thus, in
((Bikdash et al., 1993); (Glizer, 1999); (Glizer, 2005);
(Glizer, 2009a); (Glizer, 2012c); (Glizer, 2012b);
(Glizer, 2014); (Kurina and Hoai, 2014); (Mahade-
van and Muthukumar, 2011); (O’Malley and Jame-
son, 1977); (Popescu and Gajic, 1999); (Saberi and
Sannuti, 1987); (Seron et al., 1999)) (see also refer-
ences therein) various optimal control problems for
both, finite and infinite horizon total cheap control
cost functionals, were studied. In (Turetsky et al.,
2014), a robust trajectory tracking problem with a to-
tal cheap control was considered. In ((Glizer, 2000);
(Glizer, 2016); (Glizer and Kelis, 2015a); (Glizer and
Kelis, 2017); (Petersen, 1986); (Shinar et al., 2014);
(Starr and Ho, 1969); (Turetsky and Glizer, 2007))
different differential games with total cheap control of
at least one player were analyzed. In ((Glizer, 2009b);
(Glizer, 2012a); (Glizer, 2013)), some H
∞
total cheap
control problems were solved. However, partial cheap
control problems were considered only in few works
in the literature. Thus, in (O’Reilly, 1983) and (Glizer
and Kelis, 2016), an infinite horizon linear-quadratic
optimal control problem with partial cheap control
for homogeneous and nonhomogeneous systems, re-
spectively, was studied. In (Glizer and Kelis, 2015b),
a zero-sum linear-quadratic differential game with
partial cheap control for the minimizing player was
analyzed. To the best of our knowledge, an H
∞
con-
trol problem with partial cheap control has not yet
been considered in the literature. In what follows, we
call the problem (16), (29), (32) the H
∞
Partial Cheap
Control Problem (HIPCCP).
Remark 4. We say that the controller u
∗
ε
[z(t)] solves
the HIPCCP if it guarantees the fulfilment of the in-
equality
J
ε

u
∗
ε
,w

≤ 0 (33)
along trajectories of (16) for all w(t) ∈L
2
[0,+∞;E
m
].
4.2 Solvability Conditions of the
HIPCCP
Since the matrix N
ε
is positive definite for all ε > 0,
then we can apply Proposition 1 to solve the HIPCCP.
For this purpose, we write down the Riccati matrix
algebraic equation
PA + A
T
P + P(S
w
−S
u
(ε))P + D = 0, (34)
and the linear system
dz(t)
dt
=

A −S
u
(ε)P

z(t), t ≥ 0 (35)
where
S
w
= γ
−2
FF
T
, S
u
(ε) = BN
−1
ε
B
T
, (36)
the matrix D is defined in Corollary 1.
By virtue of Proposition 1, we have the following
assertion.
Proposition 3. Let, for a given ε > 0, the equation
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
28

(34) have a symmetric solution P = P(ε) such that
the trivial solution of the system (35) for P = P(ε) is
asymptotically stable. Then, the controller
u
∗
ε
[z(t)] = −N
−1
ε
B
T
P(ε)z(t) (37)
solves the HIPCCP.
5 ASYMPTOTIC ANALYSIS OF
THE EQUATION (34)
5.1 Equivalent Transformation of (34)
Substitution of the block representations of the ma-
trices B and N
ε
(see the equations (20) and (31)) into
the expression for S
u
(ε) (see (36)), yields after a rou-
tine algebra the following block representation of this
matrix:
S
u
(ε) =


S
u
1
S
u
2
S
T
u
2
(1/ε
2
)S
u
3
(ε)


, (38)
where
S
u
1
=


O
(n−r)×(n−r)
O
(n−r)×q
O
q×(n−r)
Λ
−1


,
S
u
2
=


O
(n−r)×(r−q)
Λ
−1
H
T
2


,
S
u
3
(ε) = ε
2
H
2
Λ
−1
H
T
2
+ I
r−q
,
(39)
Λ is given by (10).
Due to (38)-(39), the left-hand side of the equation
(34) has a singularity at ε = 0. In order to remove this
singularity, we seek the solution P(ε) of this equation
in the block-form
P(ε) =


P
1
(ε) εP
2
(ε)
εP
T
2
(ε) εP
3
(ε)


, (40)
where the matrices P
1
(ε), P
2
(ε) and P
3
(ε) have the
dimensions (n −r + q) ×(n − r + q), (n −r + q) ×
(r −q) and (r −q) ×(r −q), respectively, and
P
T
1
(ε) = P
1
(ε), P
T
3
(ε) = P
3
(ε). (41)
We also partition the matrices A and S
w
into blocks
as:
A =


A
1
A
2
A
3
A
4


, S
w
=


S
w
1
S
w
2
S
T
w
2
S
w
3


,
(42)
where the blocks A
1
, A
2
, A
3
and A
4
have the dimen-
sions (n −r + q) ×(n −r + q), (n −r + q) ×(r −q),
(r −q) × (n −r + q) and (r −q) × (r −q), respec-
tively; the blocks S
w
1
, S
w
2
and S
w
3
have the form
S
w
1
= γ
−2
F
1
F
T
1
, S
w
2
= γ
−2
F
1
F
T
2
, S
w
3
= γ
−2
F
2
F
T
2
,
(43)
F
1
and F
2
are the upper and lower blocks of
the matrix F of the dimensions (n −r + q) ×s and
(r −q) ×s, respectively, i.e.,
F =

F
1
F
2

. (44)
Substitution of the equations (25), (38), (40) and
(42) into the equation (34) converts this equation after
a routine algebra into the following equivalent set:
P
1
(ε)A
1
+ εP
2
(ε)A
3
+ A
T
1
P
1
(ε) + εA
T
3
P
T
2
(ε)
+P
1
(ε)(S
w
1
−S
u
1
)P
1
(ε)
+εP
2
(ε)

S
T
w
2
−S
T
u
2

P
1
(ε)
+εP
1
(ε)(S
w
2
−S
u
2
)P
T
2
(ε)
+P
2
(ε)

ε
2
S
w
3
−S
u
3
(ε)

P
T
2
(ε) + D
1
= 0,
P
1
(ε)A
2
+ εP
2
(ε)A
4
+ εA
T
1
P
2
(ε) + εA
T
3
P
3
(ε)
+εP
1
(ε)(S
w
1
−S
u
1
)P
2
(ε)
+ε
2
P
2
(ε)

S
T
w
2
−S
T
u
2

P
2
(ε)
+εP
1
(ε)(S
w
2
−S
u
2
)P
3
(ε)
+P
2
(ε)

ε
2
S
w
3
−S
u
3
(ε)

P
3
(ε) = 0,
εP
T
2
(ε)A
2
+ εP
3
(ε)A
4
+ εA
T
2
P
2
(ε) + εA
T
4
P
3
(ε)
+ε
2
P
T
2
(ε)(S
w
1
−S
u
1
)P
2
(ε)
+ε
2
P
3
(ε)

S
T
w
2
−S
T
u
2

P
2
(ε)
+ε
2
P
T
2
(ε)(S
w
2
−S
u
2
)P
3
(ε)
+P
3
(ε)

ε
2
S
w
3
−S
u
3
(ε)

P
3
(ε) + D
2
= 0. (45)
5.2 Zero-order Asymptotic Solution of
the Set (45)
We look for the zero-order asymptotic solution

P
10
,P
20
,P
30

of the system (45). Equations for the
terms of this asymptotic solution are obtained by set-
ting formally ε = 0 in (45), which yields the set of the
equations
P
10
A
1
+A
T
1
P
10
+P
10
(S
w
1
−S
u
1
)P
10
−P
20
P
T
20
+D
1
= 0,
(46)
P
10
A
2
−P
20
P
30
= 0, (47)
(P
30
)
2
−D
2
= 0. (48)
The equation (48) has the solution
P
30
= P
∗
30
4
=

D
2

1/2
, (49)
where the superscript ”1/2” denotes the unique sym-
Solution of a Singular H
∞
Control Problem: A Regularization Approach
29

metric positive definite square root of the correspond-
ing symmetric positive definite matrix.
Due to (49), the equation (47) yields the expres-
sion for P
20
P
20
= P
10
A
2

D
2

−1/2
, (50)
where the superscript ” − 1/2” denotes the inverse
matrix for the unique symmetric positive definite
square root of the corresponding symmetric positive
definite matrix.
Now, substituting (50) into (46), we obtain after
some rearrangement the equation with respect to P
10
P
10
A
1
+ A
T
1
P
10
+ P
10
S
1
P
10
+ D
1
= 0, (51)
where
S
1
= S
w
1
−S
u
1
−A
2
D
−1
2
A
T
2
. (52)
Consider the matrix
¯
M =

M
1
O
p
2
×(r−q)
O
p
1
×q
C
2

. (53)
Using the equation (10) and Corollary 1 yields
¯
N
4
=
¯
M
T
¯
M =

Λ O
q×(r−q)
O
(r−q)×q
D
2

. (54)
By virtue of the results of (Glizer and Kelis,
2015b) (Lemma 5), the matrix S
1
can be represented
in the form
S
1
= S
w
1
−
¯
B
¯
N
−1
¯
B
T
, (55)
where
¯
B =

e
B , A
2

,
e
B =

O
(n−r)×q
I
q

. (56)
In what follows, we assume:
(A3) Riccati matrix algebraic equation (51) has a
symmetric solution P
10
= P
∗
10
such that the trivial so-
lution of the system
dx(t)
dt
=

A
1
−
¯
B
¯
N
−1
¯
B
T
P
∗
10

x(t), t ≥ 0 (57)
is asymptotically stable.
5.3 H
∞
-Control Interpretation of the
Equation (51)
Consider the H
∞
control problem for the system
d ¯x(t)
dt
= A
1
¯x(t) +
¯
B ¯u(t) + F
1
¯w(t), t ≥ 0, ¯x(t) = 0,
(58)
¯v(t) = col{C
1
¯x(t),
¯
M ¯u}, (59)
where ¯x(t) ∈ E
n−r+q
is a state vector; ¯u(t) ∈ E
r
is a
control, ¯w(t) ∈ E
m
is a disturbance.
The cost functional of this problem is
¯
J( ¯u, ¯w) =

k¯v(t)k
L
2

2
−γ
2

k ¯w(t)k
L
2

2
=
Z
+∞
0

¯x
T
(t)D
1
¯x(t) + ¯u
T
(t)
¯
N ¯u(t)
−γ
2
¯w
T
(t) ¯w(t)

dt. (60)
We call the H
∞
control problem (58)-(60) the Re-
duced H
∞
Control Problem (RHICP).
We say that the controller ¯u
∗
[ ¯x(t)] solves the
RHICP if it guarantees the fulfilment of the inequality
¯
J

¯u
∗
, ¯w

≤ 0 (61)
along trajectories of (58) for all ¯w(t) ∈ L
2
[0,+∞;E
m
].
Subject to the assumption (A3) and by virtue of
Proposition 1, the RHICP is solvable. The controller
¯u
∗
[ ¯x(t)] = −
¯
N
−1
¯
B
T
P
∗
10
¯x(t) (62)
solves this H
∞
control problem.
Thus, the equation (51) is connected with the
RHICP by the solvability conditions of the latter.
Note, that the RHICP can be derived in another
way, directly from the HIPCCP. Namely, let us parti-
tion the state vector z(t) and the control vector u(t) of
the latter problem into blocks as:
z(t) = col

x(t),y(t)

, x(t) ∈ E
n−r+q
, y(t) ∈ E
r−q
,
(63)
u(t) = col

u
1
(t),u
2
(t)

, u
1
(t) ∈ E
q
, u
2
(t) ∈ E
r−q
.
(64)
Now, using the block representations of the matri-
ces B, C, A, F, M
ε
, N
ε
,
e
B (see (20), (24), (42), (44),
(30), (31), (56)) and Corollary 1, we can rewrite the
HIPCCP (16), (29), (32) in the following equivalent
form:
dx(t)
dt
= A
1
x(t) + A
2
y(t)
+
e
Bu
1
(t) + F
1
w(t), t ≥ 0, x(0) = 0,
dy(t)
dt
= A
3
x(t) + A
4
y(t)
+H
2
u
1
(t) + u
2
(t) + F
2
w(t), t ≥ 0, y(0) = 0,
(65)
v
ε
(t) = col{C
1
x(t) +C
2
y(t),M
1
u
1
(t) + εM
2
u
2
(t)},
(66)
J
ε
(u,w) =

kv
ε
(t)k
L
2

2
−γ
2

kw(t)k
L
2

2
=
Z
+∞
0

x
T
(t)D
1
x(t) + y
T
(t)D
2
y(t)
+u
T
1
(t)Λu
1
(t) + ε
2
u
T
2
(t)u
2
(t)
−γ
2
w
T
(t)w(t)

dt. (67)
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
30

The transformation of the control
e
u
2
(t) = εu
2
(t)
converts the H
∞
control problem (65)-(67) to the
equivalent problem
dx(t)
dt
= A
1
x(t) + A
2
y(t)
+
e
Bu
1
(t) + F
1
w(t), t ≥ 0, x(0) = 0,
ε
dy(t)
dt
= ε

A
3
x(t) + A
4
y(t) + H
2
u
1
(t)

+
e
u
2
(t) + εF
2
w(t), t ≥ 0, y(0) = 0,
(68)
e
v(t) = col{C
1
x(t) +C
2
y(t),M
1
u
1
(t) + M
2
e
u
2
(t)},
(69)
e
J(u
1
,
e
u
2
,w) =

k
e
v(t)k
L
2

2
−γ
2

kw(t)k
L
2

2
=
Z
+∞
0

x
T
(t)D
1
x(t) + y
T
(t)D
2
y(t) (70)
+u
T
1
(t)Λu
1
(t) +
e
u
T
2
(t)
e
u
2
(t) −γ
2
w
T
(t)w(t)

dt.
The problem (68)-(70) is an H
∞
control problem
with a singularly perturbed dynamics. Namely, the
system (68) is singularly perturbed.
Remark 5. H
∞
control problems with singularly per-
turbed dynamics were studied extensively in the liter-
ature in various settings (see e.g. (Glizer and Frid-
man, 2000) and references therein).
Now, we are going to show that the slow sub-
problem, associated with the problem (68)-(70), co-
incides with the RHICP. This slow subproblem is ob-
tained similarly to ((Glizer, 2009b)) by setting for-
mally ε = 0 in (68)-(70). Note, that the setting ε = 0
in the second equation of (68) yields
e
u
2
(t) = 0, t ≥ 0.
Based on this observation and re-denoting x(t), y(t),
u
1
(t), w(t),
e
v(t),
e
J with x
s
(t), y
s
(t), u
1,s
(t), w
s
(t),
e
v
s
(t),
e
J
s
, respectively, we obtain
dx
s
(t)
dt
= A
1
x
s
(t) + A
2
y
s
(t)
+
e
Bu
1,s
(t) + F
1
w
s
(t), t ≥ 0, x
s
(0) = 0, (71)
e
v
s
(t) = col{C
1
x
s
(t) +C
2
y
s
(t),M
1
u
1,s
(t)}, (72)
e
J
s
=

k
e
v
s
(t)k
L
2

2
−γ
2

kw
s
(t)k
L
2

2
=
Z
+∞
0

x
T
s
(t)D
1
x
s
(t) + y
T
s
(t)D
2
y
s
(t)
+u
T
1,s
(t)Λu
1,s
(t) −γ
2
w
T
s
(t)w
s
(t)

dt.(73)
Remark 6. Since the variable y
s
(t) in the prob-
lem (71)-(73) does not satisfy any equation for t ∈
[0,+∞), we can choose this variable to satisfy a
desirable property of the system (71)-(72). This
means that the variable y
s
(t) can be considered
as an additional control variable in this system.
Thus, the functional (73), calculated along trajec-
tories of this system, depends on the control vari-
able ˆu
s
(t)
4
= col

u
1,s
(t),y
s
(t)

and the disturbance
w
s
(t) ∈ L
2
[0,+∞;E
q
], i.e.,
e
J
s
=
e
J
s
( ˆu
s
,w
s
). Thus, the
slow subproblem, associated with the HIPCCP, is to
find a controller ˆu
∗
s
[x
s
(t)] that ensures the inequal-
ity
e
J
s
( ˆu
∗
s
,w
s
) ≤ 0 along trajectories of (71) for all
w
s
∈L
2
[0,+∞;E
m
]. This H
∞
control problem is called
the Slow H
∞
Control Subproblem (SHICSP) associ-
ated with the HIPCCP.
Due to Remark 6, the RHICP equation of dynam-
ics (58) and the integral form of the cost functional
(60) for ¯u(t) = ˆu
s
(t) coincide with the equation of dy-
namics (71) and the integral form of the cost func-
tional (73), respectively, in the SHICSP associated
with the HIPCCP. This means that the RHICP and the
SHICSP are identical to each other. Thus, due to (62),
the controller
ˆu
∗
s
[x
s
(t)]
4
= col

u
∗
1,s
[x
s
(t)],y
∗
s
[x
s
(t)]

= −
¯
N
−1
¯
B
T
P
∗
10
x
s
(t) (74)
solves the SHICSP. Using (54) and (56), the blocks
u
∗
1,s
[x
s
(t)] and y
∗
s
[x
s
(t)] of this controller can be rep-
resented in the form u
∗
1,s
[x
s
(t)] = −Λ
−1
e
B
T
P
∗
10
x
s
(t),
y
∗
s
[x
s
(t)] = −D
−1
2
A
T
2
P
∗
10
x
s
(t).
5.4 Justification of the Asymptotic
Solution to the Equation (34)
We assume that:
(A4) The trivial solution of the system
dx(t)
dt
=

A
1
+ S
1
P
∗
10

x(t), x(t) ∈ E
n−r+q
, t ≥ 0
(75)
is asymptotically stable, where P
∗
10
is the solution of
the equation (51) mentioned in the assumption (A3).
Let us denote
P
∗
20
4
= P
∗
10
A
2

D
2

−1/2
. (76)
Lemma 2. Let the assumptions (A1)-(A2),(A4) be
valid. Then, there exists a positive number ε
0
, such
that for all ε ∈ (0, ε
0
] the equation (34) has the sym-
metric solution P(ε) of the block-form (40), and the
blocks P
i
(ε), (i = 1, 2, 3) of this solution satisfy the in-
equalities
kP
i
(ε) −P
∗
i0
k ≤ aε, i = 1,2,3, ε ∈ (0, ε
0
], (77)
where a > 0 is some constant independent of ε.
Solution of a Singular H
∞
Control Problem: A Regularization Approach
31

Proof. Based on the Implicit Function Theorem
(Schwartz, 1967) (Chapter III, paragraph 8), the
lemma is proven similarly to (Kokotovic et al., 1986)
(Theorem 4.2).
Lemma 3. Let the assumptions (A1)-(A4) be valid.
Then, there exists a positive number ε
1
≤ε
0
, such that
for any ε ∈ (0,ε
1
] the trivial solution of the system
(35) with P = P(ε) is asymptotically stable.
Proof. Substitution of the block representations of
the vector z(t) and the matrices S(ε), P(ε), A (see (63)
and (38), (40), (42)) into the system (35) yields after
a routine algebra the following equivalent system:
dx(t)
dt
=

A
1
−S
u
1
P
1
(ε) −εS
u
2
P
T
2
(ε)

x(t)
+

A
2
−εS
u
1
P
2
(ε) −εS
u
2
P
3
(ε)

y(t), t ≥ 0,
ε
dy(t)
dt
=

εA
3
−εS
T
u
2
P
1
(ε) −S
u
3
(ε)P
T
2
(ε)

x(t)
+

εA
4
−ε
2
S
T
u
2
P
2
(ε) −S
u
3
(ε)P
3
(ε)

y(t), t ≥ 0.
(78)
Remember that the parameter ε > 0 is small.
Therefore, the system (78) is singularly perturbed
(Kokotovic et al., 1986). To prove the asymptotic sta-
bility of the trivial solution to this system, we use the
results of (Kokotovic et al., 1986) (Corollary 3.1). By
virtue of these results, if the trivial solutions of the
slow and fast subsystems associated with the singu-
larly perturbed system (78) are asymptotically stable,
then for all sufficiently small ε > 0 the trivial solution
of the system (78) itself is asymptotically stable.
The slow subsystems associated with the system
(78) is obtained in the following two steps. First, set-
ting formally ε = 0 in (78), using the equation (39),
the inequalities (77), and re-denoting x(t) and y(t)
with x
s
(t) and y
s
(t), respectively, we obtain the sys-
tem
dx
s
(t)
dt
=

A
1
−S
u
1
P
∗
10

x
s
(t) + A
2
y
s
(t), t ≥ 0,
0 =

P
∗
20

T
x
s
(t) + P
∗
30
y
s
(t), t ≥ 0.
(79)
Then, eliminating y
s
(t) from (79), and using the
equations (49) and (50) yield the slow subsystem as-
sociated with (78)
dx
s
(t)
dt
=

A
1
−

S
u
1
+A
2
(D
2
)
−1
A
T
2

P
∗
10

x
s
(t), t ≥0.
(80)
Comparing the equations (52) and (55), we ob-
tain that S
u
1
+ A
2
(D
2
)
−1
A
T
2
=
¯
B
¯
N
−1
¯
B
T
. Therefore,
the differential equation (80) coincides with the dif-
ferential equation (57). Hence, due to the assumption
(A3), the trivial solution of (80) is asymptotically sta-
ble.
The fast subsystem associated with (78) is ob-
tained from the second equation of this system in the
following formal way. First, we remove from this
equation the term depending on x(t). Second, we
make in the obtained equation the transformation of
variables t = εξ, y
f
(ξ) = y(εξ), where ξ and y
f
(ξ)
are new independent variable and state variable. Fi-
nally, setting formally ε = 0 in the transformed equa-
tion yields the fast subsystem
dy
f
(ξ)
dξ
= −P
∗
30
y
f
(ξ), ξ ≥ 0. (81)
Since the matrix P
∗
30
= (D
2
)
1/2
is positive defi-
nite, the trivial solution of the differential equation
(81) is asymptotically stable. Therefore, by virtue
of the above mentioned results of (Kokotovic et al.,
1986), there exists a positive number ε
1
such that,
for all ε ∈ (0,ε
1
], the trivial solution of the system
(78) is asymptotically stable. Since the system (35)
with P = P(ε) is equivalent to (78), the trivial solu-
tion of the former also is asymptotically stable for all
ε ∈ (0,ε
1
]. Thus, the lemma is proven.
Corollary 2. Let the assumptions (A1)-(A4) be valid.
Then, for all ε ∈ (0,ε
1
], the controller (37) solves the
HIPCCP.
Proof. The corollary is a direct consequence of
Proposition 3, Lemma 2 and Lemma 3.
6 SOLUTION OF THE SHICP
6.1 Controller for the SHICP: Formal
Design
First of all, let us note the following. Due to the equa-
tions (17), (18), (29), (32),
J

u
∗
ε
[z(t)],w(t)

≤ J
ε

u
∗
ε
[z(t)],w(t)

(82)
along trajectories of the equation (16) for all ε ∈
(0,ε
1
] and all w(t) ∈ L
2
[0,+∞;E
m
]. Therefore, the
controller u
∗
ε
[z(t)], solving the HIPCCP (see Corol-
lary 2), also solves the SHICP. However, the design
of u
∗
ε
[z(t)] is a complicated task, because it requires
the solution of a high dimension system of nonlin-
ear algebraic equations depending on a parameter. To
overcome this difficulty, we propose in this subsection
another (simplified) controller for the SHICP.
Consider the matrix
P
∗
0
(ε) =

P
∗
10
εP
∗
20
ε

P
∗
20

T
εP
∗
30

, ε ∈ (0,ε
1
]. (83)
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
32

Based on the matrix P
∗
0
(ε), we consider the fol-
lowing auxiliary controller, obtained from the con-
troller u
∗
ε
[z(t)] (see (37)) by replacing P(ε) with
P
∗
0
(ε):
u
aux
[z(t)] = −N
−1
ε
B
T
P
∗
0
(ε)z(t). (84)
Substitution of the block representations for the
matrices B, N
ε
, P
∗
0
(ε) and the vector z(t) (see (20),
(31), (83) and (63)) into (84), and use of the block
form of the matrix
e
B (see (56)) yield after a routine al-
gebra the block representation for the vector u
aux
[z(t)]
u
aux
[z(t)] = −




Λ
−1
h
K
1
(ε)x(t) + εK
2
(ε)y(t)
i
1
ε
h

P
∗
20

T
x(t) + P
∗
30
y(t)
i




,
(85)
where Λ and H
2
are defined in (10) and (15), re-
spectively, K
1
(ε)
4
=
e
B
T
P
∗
10
+ εH
T
2

P
∗
20

T
, K
2
(ε)
4
=
e
B
T
P
∗
20
+ εH
T
2
P
∗
30
.
Now, calculating the point-wise (with respect to
z(t) ∈E
n
) limit of the upper block in (85) for ε → 0
+
,
we obtain the simplified controller for the SHICP
u
∗
ε,0
[z(t)] =



−Λ
−1
e
B
T
P
∗
10
x(t)
−
1
ε
h

P
∗
20

T
x(t) + P
∗
30
y(t)
i



. (86)
Remark 7. Comparing the controller u
∗
ε,0
[z(t)] with
the controller ˆu
∗
s
[x
s
(t)], solving the SHICSP (see
(74)), one can conclude that the upper blocks of these
controllers coincide with each other.
6.2 Properties of the Controller (86)
Let for given ε > 0 and w(t) ∈ L
2
[0,+∞;E
m
],
z
∗
0

t,ε; w(·)

, t ≥ 0 be the solution of the initial-value
problem (16) with u(t) = u
∗
ε,0
[z(t)]. Let
K
∗
0
4
=


P
∗
20

T
,P
∗
30

. (87)
Theorem 1. Let the assumptions (A1)-(A4) be valid.
Then, there exists a positive number ε
∗
0
such that for
all ε ∈ (0,ε
∗
0
] and w(t) ∈ L
2
[0,+∞;E
m
] the following
inequality is satisfied
J

u
ε,0
,w

≤
−
Z
+∞
0

z
∗
0

t,ε; w(·)

T

K
∗
0

T
K
∗
0
z
∗
0

t,ε; w(·)

dt (88)
along trajectories of the system (16).
Proof. To save the space, we present here a sketch of
the proof.
First, we are going to show that the controller
u
∗
ε,0
[z(t)] solves the HIPCCP for all sufficiently small
ε > 0, i.e., for all such ε and all w(t) ∈ L
2
[0,+∞;E
m
],
the following inequality is satisfied:
J
ε

u
∗
ε,0
,w) ≤ 0. (89)
Substitution of u
∗
ε,0
[z(t)] into this problem and use
of the block representations for the matrices A, B, F,
D, N
ε
, and for the vectors z(t), u
∗
ε,0
[z(t)] (see (42),
(20), (44), (25), (31) and (63), (86)) transform the
equation of dynamics (16) and the functional (32) of
the HIPCCP as follows:
dz(t)
dt
=
ˆ
A(ε)z(t) + Fw(t), z(0) = 0, (90)
ˆ
J
ε
(w)
4
= J
ε
(u
∗
ε,0
,w)
=
Z
+∞
0

z
T
(t)
ˆ
Dz(t) −γ
2
w
T
(t)w(t)

dt, (91)
where
ˆ
A(ε) =

ˆ
A
1
ˆ
A
2
ˆ
A
3
(ε)
ˆ
A
4
(ε)

,
ˆ
D =

ˆ
D
1
ˆ
D
2
ˆ
D
T
2
ˆ
D
3

,
ˆ
A
1
= A
1
−
e
BΛ
−1
e
B
T
,
ˆ
A
2
= A
2
,
ˆ
A
3
(ε) = A
3
−
H
2
Λ
−1
e
B
T
P
∗
10
−(1/ε)

P
∗
20

T
,
ˆ
A
4
(ε) = A
4
−(1/ε)P
∗
30
,
ˆ
D
1
= D
1
+ P
∗
10
e
BΛ
−1
e
B
T
P
∗
10
,
ˆ
D
2
= P
∗
20
P
∗
30
,
ˆ
D
3
= D
2
+

P
∗
30

2
.
Due to (91), the inequality (89) is equivalent to the
following inequality for all sufficiently small ε > 0
and all w(t) ∈ L
2
[0,+∞;E
m
]:
ˆ
J
ε
(w) ≤ 0. (92)
Similarly to Corollary 3, it is shown that the trivial
solution to the differential equation in (90) is asymp-
totically stable for all sufficiently small ε > 0. This
observation yields the following limit equality for any
w(t) ∈L
2
[0,+∞;E
m
] and any sufficiently small ε > 0:
lim
t→+∞
z
0

t,ε; w(·)

= 0. (93)
Now, let us consider the Riccati matrix algebraic
equation with respect to the matrix
ˆ
P
ˆ
P
ˆ
A(ε) +
ˆ
A
T
(ε)
ˆ
P +
ˆ
PS
w
ˆ
P +
ˆ
D = 0. (94)
Similarly to Lemma 2, it is shown that for all suffi-
ciently small ε > 0 the equation (94) has a symmetric
solution
ˆ
P =
ˆ
P
∗
(ε). Using this observation, we con-
sider the Lyapunov-like function V (z,ε) = z
T
ˆ
P
∗
(ε)z,
z ∈ E
n
. Analyzing the behavior of V (z,ε) along tra-
jectories of the equation in (90) and using the equa-
tion (93), we prove the validity of the inequality (92)
and, therefore, the inequality (89). The latter, along
with the equations (18), (32), (87) and (91), yields the
statement of the theorem.
Solution of a Singular H
∞
Control Problem: A Regularization Approach
33

Theorem 2. Let the assumptions (A1)-(A4) be valid.
Then, there exists a positive number ε
∗
1
such that for
all ε ∈(0,ε
∗
1
] and w(t) ∈ L
2
[0,+∞;E
m
] the integral in
the right-hand part of (88), being nonnegative, satis-
fies the inequality
Z
+∞
0

z
∗
0

t,ε; w(·)

T

K
∗
0

T
K
∗
0
z
∗
0

t,ε; w(·)

dt
≤ aε

kw(t)k
L
2

2
, (95)
where a > 0 is some constant independent of ε and
w(·).
Proof. Here, we also present a sketch of the proof.
Let the n×n-matrix Φ(t,ε) be the fundamental matrix
solution of the equation dz(t)/dt =
ˆ
A(ε)z(t), i.e., this
matrix satisfies the following initial-value problem:
dΦ(t,ε)
dt
=
ˆ
A(ε)Φ(t,ε), t ≥ 0, Φ(0, ε) = I
n
. (96)
Then,
z
∗
0

t,ε; w(·)

=
Z
t
0
Φ(t −σ,ε)Fw(σ)dσ, t ≥0. (97)
Let us partition the vector-valued function
z
∗
0

t,ε; w(·)

into blocks as z
∗
0

t,ε; w(·)

=
col

x
∗
0

t,ε; w(·)

,y
∗
0

t,ε; w(·)


, x
∗
0

t,ε; w(·)

∈
E
n−r+q
, y
∗
0

t,ε; w(·)

∈ E
r−q
. Now, a proper asymp-
totic analysis of the problem (96), the use of the
equation (97) and the Cauchy-Bunyakovsky-Schwarz
integral inequality yield the existence of a posi-
tive number ε
∗
1
such that for all ε ∈ (0,ε
∗
1
] and all
w(t) ∈ L
2
[0,+∞;E
m
] the following inequalities are
satisfied:


x
∗
0

t,ε; w(·)

− ˆx

t;w(·)



≤a
1
ε
1/2
kw(t)k
L
2
, t ≥0,
(98)


y
∗
0

t,ε; w(·)

− ˆy

t;w(·)



≤a
1
ε
1/2
kw(t)k
L
2
, t ≥0,
(99)
where a
1
> 0 is some constant independent of ε and
w(·),
ˆx

t;w(·)

=
Z
t
0
ˆ
Φ
x
(t −σ)F
1
w(σ)dσ, t ≥ 0, (100)
ˆy

t;w(·)

=
Z
t
0
ˆ
Φ
y
(t −σ)F
1
w(σ)dσ, t ≥ 0, (101)
the (n −r + q) ×(n −r + q)-matrix-valued function
ˆ
Φ
x
(t) is the solution of the initial-value problem
d
ˆ
Φ
x
(t)
dt
=

ˆ
A
1
−A
2
(P
∗
30
)
−1
(P
∗
20
)
T

ˆ
Φ
x
(t), t ≥ 0,
ˆ
Φ
x
(0) = I
n−r+q
,
and the (r −q) ×(n −r + q)-matrix-valued function
ˆ
Φ
y
(t) has the form
ˆ
Φ
y
(t) = −(P
∗
30
)
−1
(P
∗
20
)
T
ˆ
Φ
x
(t).
The latter expression, along with the equations (100)-
(101), yields for all t ≥ 0, w(t) ∈ L
2
[0,+∞;E
m
]:
(P
∗
20
)
T
ˆx

t;w(·)

+ P
∗
30
ˆy

t;w(·)

= 0. (102)
Now, using the equations (87), (102) and the in-
equalities (98)-(99), we obtain the following inequal-
ity for all ε ∈ (0,ε
∗
1
] and all w(t) ∈ L
2
[0,+∞;E
m
]:


K
∗
0
z
0

t,ε; w(·)



≤ a
2
ε
1/2
kw(t)k
L
2
,
where a
2
> 0 is some constant independent of ε and
w(·). This inequality directly yields the inequality
(95).
The following corollary is a direct consequence of
Theorem 2.
Corollary 3. Let the assumptions (A1)-(A4) be valid.
Then, there exists a positive number ε
∗
2
and a function
g(ε), (0 ≤ g(ε) ≤ aε, ε ∈ (0,ε
∗
2
], the constant a > 0
is defined in Theorem 2), such that for all ε ∈ (0, ε
∗
2
]
the controller u
∗
ε,0
[z(t)] solves the singular H
∞
control
problem for the system (16) with the following func-
tional:
J
g
(u,w) =
Z
+∞
0
h
z
T
(t)Dz(t) −(γ
g
(ε))
2
w
T
(t)w(t)
i
dt,
where the performance level γ
g
(ε) has the form
γ
g
(ε) =
p
γ
2
−g(ε) > 0.
Remark 8. Due to Theorems 1, 2 and Corollary 3,
the controller u
∗
ε,0
[z(t)] solves not only the original
SHICP (16), (18), but also the singular H
∞
control
problem with the same dynamics (16) and the new
cost functional J
g
(u,w). The latter has a smaller per-
formance level γ
g
(ε) than SHICP. This performance
level satisfies the limit equality lim
ε→0
+
γ
g
(ε) = γ.
7 EXAMPLE
To illustrate the theoretical results of the paper, we
consider the following example of the problem (16)-
(18) with the data: n = r = m = 2, q = 1 and
A =

0 2
8 1

, B =

1 0
2 1

, D =

2 0
0 1

,
F =

−1 2
−4 3

, M =

1 0
0 0

, γ = 1. (103)
Using these data, we obtain:
u
∗
ε,0
[z(t)] =
−
√
2x(t)
−
1
ε

4
√
2x(t) + y(t)

!
, (104)
where z(t) = col

x(t),y(t)

.
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
34

In Table 1, the minimum H
∞
performance level γ
ε
(H
∞
-norm) of the system (16)-(17), subject to the data
(103) and the control u(t) = u
∗
ε,0
[z(t)], is presented for
various values of ε.
Table 1: Minimum H
∞
performance level.
ε 0.5 0.25 0.1 0.05 0.025
γ
ε
2.455 1.736 1.193 0.994 0.988
It is seen that γ
ε
decreases for the decreasing ε.
Moreover, for sufficiently small ε, the value of γ
ε
be-
comes smaller than the performance level γ = 1 in the
H
∞
control problem of this example.
8 CONCLUSIONS
An H
∞
control problem for a linear system was con-
sidered. The feature of the problem is that the ma-
trix of coefficients for the control in the quadratic cost
functional is singular but, in general, non-zero. The
control coordinates presenting in the cost functional
are regular, while the other ones are singular. Un-
der proper assumptions, the linear system was trans-
formed equivalently to the system consisting of three
modes. The first mode is not controlled directly, the
second mode is controlled by the regular control co-
ordinates, while the third mode is controlled by the
entire control. Due to this transformation, the ini-
tially formulated H
∞
control problem was converted
to a new singular H
∞
control problem. This new prob-
lem was solved by a regularization approach, i.e., by
its approximate transformation to an auxiliary regular
H
∞
control problem. The latter has the same equa-
tion of dynamics and a similar cost functional aug-
mented by an integral of the squares of the singular
control coordinates with a small positive weight ε
2
,
(ε > 0). Hence, the auxiliary problem is an H
∞
par-
tial cheap control problem. An asymptotic solution
of the ε-dependent Riccati matrix algebraic equation,
associated with this partial cheap control problem by
the solvability conditions, was constructed and justi-
fied. Based on this asymptotic solution, a simplified
controller for the H
∞
partial cheap control problem
was designed. It was shown that this controller also
solves the singular H
∞
control problem. Moreover,
it was shown that this controller also solves a singular
H
∞
control problem with a smaller performance level,
depending on ε. This smaller performance level tends
to the original one for ε → 0
+
.
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