European Journal of Control (2010)1:54–69
# 2010 EUCADOI:10.3166/EJC.16.54–69
H2=H1 Control of Singularly Perturbed Systems: The State
Feedback Case
Kanti B. Datta1,�, Aparajita RaiChaudhuri2,��
1Department of Electrical Engineering, IIT Kharagpur, Kharagpur-721302, India;2Department of Electrical Engineering, B.E.S.U, Howrah-711103, India
The design of a mixed H2=H1 linear state variable
feedback (LSVF) suboptimal controller for a continuous-
time singularly perturbed system using reduced order
slow and fast subsystems is described. It is shown that
the designed controller based on reduced order models
and the corresponding performance index (PI) both
are Oð�Þ close to those synthesized using the full order
system. Each of the three algebraic matrix equations to
be sequentially solved in our study, contains three
matrix variables coupled in them. Two of these
equations can be made independent of the matrix gain
variable under mild restrictions.
Keywords: Singularly perturbed systems, H2=H1
control, optimal control, robust control
1. Introduction
In robust control problemwith parameter uncertainties
to have a reasonable performance level guaranteed, we
define performance measures as H2 or H1 norm cri-
teria. For parameter uncertainties one can at best
minimize one of themkeeping a constraint on the other.
Such approaches are referred to as constrained optimal
control problem (COCP). In H1-COCP, an H2 per-
formance is minimized simultaneously imposing a
constraint on the H1 norm. These problems go by the
name of mixed H2=H1 control. There exists a number
of different formulations of these latter problems. To
understand this, letTziwi, i ¼ 0; 1 denote the closed-loop
transfer functions from the exogenous input wi to the
controlled output zi. The H1-COCP is to find an
internally stabilizing controller which minimizes
kTz0w0k2 while maintaining kTz1w1
k1 < �: A controller
that solves this problem will ensure that the closed-loop
system is robustly stable to all finite gain stable per-
turbations �, interconnected to the system by
w1 ¼ �z1, such that k�k � 1=� where the k�k is the
induced operator norm obtained by taking L2½0;1Þnorms on w1 and z1. Currently no analytical solution to
this problem is available. However, the convex pro-
gramming approach of Boyd et al. [3] offers a feasible
numerical alternative for the solvability of H1-COCP.
A closely related problem was studied by Bernstein and
Haddad, in which they have considered w ¼ w0 ¼ w1
and established necessary conditions to minimize an
upper bound of kTz0wk2 subject to the constraint
kTz1wk1 < �: These conditions are later found to be
sufficient also by Yeh et al. [22], [2]. In the equalized
case, z0 ¼ z1, w0 ¼ w1 the problem of Bernstein and
Haddad is equivalent to themaximum entropy problem
investigated byMustafa andGlover [16]. In this case the
optimal controller is the so-called central controller.
In [7], another version of mixed H2=H1 problem is
studied for a system in which the signal w0 is assumed
to be white noise while w1 is a signal with bounded
power. This problem is equivalent to the dual of the
problem in [2].
*Correspondence to: K.B. Datta, E-mail: [email protected]**E-mail: [email protected]
Received 9 July 2008; Accepted 26 August 2009Recommended by J.C. Geromel, E.F. Camacho
Rotea and Khargonekar [20] parameterized the
set of all internally stabilizing LSVF controllers K
(in such cases we call K is admissible) for the
unconstrainedH2-optimal control problem: inf kTz0w0f
ðKÞk2 : K is admissibleg, and derived necessary and
sufficient conditions such that a controller in the above
set also satisfies kTz1w1k1 < 1. These conditions
involve two algebraic Riccati equations and a coupling
condition. On the other hand, Khargonekar and
Rotea [11] focused on the mixed H2=H1 control
problem as formulated by Bernstein and Haddad [2]
and developed a design procedure for a static state-
feedback gain controller by solving a finite-dimen-
sional convex programming problem over a bounded
set of real matrices.
In [14], a state-feedback mixed H2=H1 control
problem is solved via the solution of an associated
Nash game involving the two cost functions with w0 ¼w1 ¼ w and leads to a central state-feedback controller
as inminimumentropy problem ofMustafa andGlover
[16]. Moreover, in [14], it is assumed that the worst case
disturbance w�ðt; xÞ can be generated by a linear,
memoryless feedback strategy which is slightly restrict-
ive but necessary for a unique Nash equilibrium.
On the other hand, the H2-COCP for continuous
systems is excellently addressed in [8, 9, 19].
Based on a Riccati equation approach, Petersen and
McFarlane [19] developed a construction procedure
for an optimal state feedback quadratic guaranteed
cost control. Using the quadratic stabilizability
framework, Garcia et al. [8] studied theH2 guaranteed
cost control for singularly perturbed uncertain sys-
tems and showed how to construct a quadratic sta-
bilizing composite control minimizing an upper bound
on the H2 norm of a certain transfer matrix.
The high frequency phenomena and parameter
variations in a dynamical system can be handled by
considering a singularly perturbed uncertain model of
Kokotovic et al. [12]. In this case, the slow subsystem
takes into account low frequency uncertainties and the
fast dynamics describes the neglected high-frequency
part of the system. Important investigations into un-
certain singularly perturbed systems are made in [4, 5,
15]. In [5], the control of uncertain systems which
exhibit time-scale separation is studied by translating
the problem into a two-frequency scale decomposition
for H1 disk problems. A robust stabilizing composite
control for singularly perturbed systems with time-
varying uncertainties is developed in [4]. In [5], a
composite controller is designed with the objective to
guarantee stability of the uncertain system by the
construction of an appropriate Lyapunov function. In
these papers, no optimality properties have been
associated with these controllers. In [17, 18], optim-
ality through an H1 criterion is proposed under per-
fect state measurements for a well-knownmodel. In [1]
it is established that the mixed H2=H1 control prob-
lem is completely solved if a condition on the rank of
the input matrix is assumed. In [6], H1-COCP is
studied for a discrete singularly perturbed system and
a mixed H2=H1 suboptimal linear state-feedback
controller is designed based on reduced order models.
From above it appears thatH2=H1 optimization for a
continuous-time singularly perturbed system still
remains an unsolved problem.
The objective of the present paper is, therefore, to
design a mixed H2=H1 linear state variable feedback
(LSVF) suboptimal controller for a continuous-time
singularly perturbed system based on reduced order
slow and fast subsystems which are defined inde-
pendent of the small singular perturbation parameter
�. For the full-order case, our problem is to minimize
an upper bound of kTz0wk2 subject to the constraint
kTz1wk1 < � as in [2]. In the discrete-time case, see [6],
a weaker version of this problem is solved by assuming
that z0 ¼ z1, giving rise to a LSVF central controller.
The slow and fast subsystems introduced for our pres-
ent investigation are considerably different from those
used in quadratic regulator theory of Kokotovic et al.
[12]. The designed slow and fast LSVF gains are com-
bined to produce a composite feedback for the full-
order system with an approach quite different from
Kokotovic et al. [12]. The composite feedback gain and
also the corresponding quadratic performance index
(PI) are Oð�Þ close to the gain designed with full-order
system and the corresponding value of PI.
2. The Full-Order Problem
The singularly perturbed systems under consideration
is described by,
_xðtÞ ¼ AxðtÞ þ B1wðtÞ þ B2uðtÞ; ð1Þ
z1ðtÞ ¼ C1xðtÞ þD1uðtÞ ð2Þ
z0ðtÞ ¼ S1xðtÞ þ S2uðtÞ ð3Þ
xðtÞ ¼x1ðtÞ
x2ðtÞ
� �; A ¼
A11 A12
A21
�
A22
�
24
35;
B1 ¼B11
B21
�
24
35;B2 ¼
B12
B22
�
24
35; C1 ¼ ½C11 C12�
S1 ¼ ½S11 S12�;
and � > 0 is a small parameter, x1 2 Rn1 , x2 2 Rn2 ,
w 2 Rm1 is the disturbance input withE½wðtÞw0ðtÞ� ¼ I,
H2=H1 Control of SPS with LSVF 55
u 2 Rm2 is the controlled input, ziðtÞ 2 Rriði ¼ 0; 1Þare the controlled outputs. Other matrices defined in
(1)–(3) are of compatible dimensions with ðA;B2Þstabilizable, and ðC1;AÞ detectable. With a SVF
control law
u ¼ Kx ¼ K1 K2½ �x1x2
� �ð4Þ
the full-order closed-loop system (1)–(3) becomes
_xðtÞ ¼ ~AxðtÞ þ ~BwðtÞ ð5Þ
z1ðtÞ ¼ ~CxðtÞ ð6Þ
z0ðtÞ ¼ ~SxðtÞ ð7Þ
where
~A ¼
~A11~A12
~A21
�
~A22
�
24
35 ¼ Aþ B2K; ~B ¼
B11B21
�
" #; ð8Þ
~C ¼ ½ ~C11~C12� ¼ C1 þD1K, ~S ¼ S1 þ S2K, and for
i; j ¼ 1; 2 ~Aij ¼ Aij þ Bi2Kj; ~C1i ¼ C1i þD1Ki: The
transfer-function from wðtÞ to z1ðtÞ is Tz1wðsÞ ¼~CðsI� ~AÞ�1 ~B: Here, symbol tilde denotes all full-
order closed loop quantities containing K. The H2
performance functional to be minimized for the
system (5)–(7) is chosen as
JðKÞ ¼ limt!1
E
nx0ðtÞR1xðtÞ þ 2x0ðtÞR12uðtÞ þ u0ðtÞR2uðtÞ
o¼ lim
t!1E x0ðS1 þ S2KÞ
0ðS1 þ S2KÞx� �
¼ tr ~Q ~R ð9Þ
where Ef�g stands for expected value of f�g. Above we
defined,
R1 :¼ S01S1 R2 :¼ S0
2S2 R12 :¼ S01S2; S1 ¼ ½S11 S12�;
~R :¼ ðS1 þ S2KÞ0ðS1 þ S2KÞ and ~Q :¼ limt!1 E xðtÞf
xðtÞ0g is the closed-loop state covariance matrix,
which satisfies the algebraic Lyapunov equation, [13],~A ~Qþ ~Q ~A0 þ ~B ~B0 ¼ 0: It can be easily shown that (9)
is equal to the square of the H2-norm of a transfer
function given by Tz0wðsÞ ¼~SðsI� ~AÞ�1 ~B: So an
H2-optimization problem is to minimize (9). The aim
of mixedH2=H1 optimization problem is to minimize
H2 PI (9) subject to the constraint
kTz1wk1 � �; � > 0: ð10Þ
This problem is not analytically solvable. Since the
full-order algebraic Riccati equation (FOARE)
0 ¼ ~AQþQ ~A0 þ ��2Q ~C0 ~CQþ ~B ~B0 ð11Þ
implies (10), and Q � ~Q, see Theorem 1, an upper
bound J ðK;QÞ :¼ tr ðQ ~RÞ of H2 cost JðKÞ ¼ tr ~Q ~R is
minimized simultaneously satisfying (11). This is
known as auxiliary minimization problem [2]. So our
aim is now transformed to finding the control law (4)
in such a way that (a) the closed-loop system (1)–(2) is
asymptotically stable, (b) the closed-loop transfer
function Tz1w satisfies the bound (10), and (c) the PI
J ðK;QÞ is minimized.
With the structure of ~A as shown in (8), it is
established by Kokotovic et al. [12] that the structure
of Q is of the form given by (39),Q being the solution
of the full-order algebraic Riccati equation (11).
Theorem 1 [2]: Let K ¼ K1 K2½ � be given and assume
that 9Q 2 Rn�n, a nonnegative-definite matrix satisfy-
ing (11). Then (a) ð ~A; ~BÞ is stabilizable if and only if ~A
is asymptotically stable. In this case, (b) kTz1wk1 � �and (c) ~Q � Q: Consequently (d) JðKÞ � J ðK;QÞwhere, J ðK;QÞ ¼ tr½Q ~R�.
We now define S :¼ fðK;QÞ : Q > 0 and ~Aþ��2Q~C0 ~C is asymptotically stable}. The task is to
optimize J ðK;QÞ subject to the constraint (11) over
the open set S. The Lagrangian to be minimized is
LðK;Q;PÞ ¼ tr Q ~Rþ ð ~AQþQ ~A0 þQ ~R1Qþ ~VÞP� �
where ~R1 ¼ ��2 ~C0 ~C; ~V ¼ ~B ~B0; and P 2 Rn�n is the
symmetric Lagrange multiplier. The stationary condi-
tions@L
@Q¼ 0 and
@L
@K¼ 0 respectively result in
0 ¼ ð ~AþQ ~R1Þ0P þ Pð ~AþQ ~R1Þ þ ~R ð12Þ
0 ¼ S02ðS1 þ S2KÞ þ fB0
2 þ ��2D01ðC1 þD1KÞQgP
� �Q:
ð13Þ
We call (12), the Lagrange multiplier equation, and
(13), the gain equation for the full-order system. It
follows from (12) that P is nonnegative definite. As
the solution is sought in the asymptotic stability set S,
a unique solution of P in (12) exists. Under the usual
assumptions that D01½C1 D1� ¼ ½0 I� we have:
S02S2Kþ ��2KQP ¼ �B0
2P � S02S1;
which is the Sylvester equation. A full-order controller
K can be found by solving coupled equations (11), (12)
and ({13), see Section 4.
3. Design with Reduced Order Models
Preliminary to any separation of slow and fast
designs, the system (1)–(3) is decomposed into a slow
system model with n1 small eigenvalues and a fast
system model with n2 large eigenvalues. If xij and uj,
56 K.B. Datta and A RaiChaudhuri
i ¼ 1; 2 and j ¼ s; f are the slow and fast parts of
the corresponding variables related by xi ¼ xis þ xif,
i ¼ 1; 2 and u ¼ us þ uf, then the closed-loop slow and
fast subsystems can be described by (14), (15), (17),
and (18) given below where we have used � ¼ 0 and
us ¼ Ksxs and uf ¼ Kfxf in (1)–(3) to generate the
closed-loop subsystems. As (71) implies k ~C0ðsI�~A0Þ
�1 ~B0 þ ~D0k � �; the slow subsystem can be
defined more logically with the help of (71). Similarly,
the fast subsystem can be defined also with the help
of (44). However, definition of (16) and (19) given
below emerges from the study of the decomposition
of full-order Lagrange multiplier equation into slow
and fast components corresponding to the terms ~R0
in (87) and ~R22 in (80), respectively. For each of
these subsystems suboptimal H2=H1-controller
gains are designed separately. It is shown that the
composite controller obtained from a combination
of the suboptimal slow and fast gains generates the
full-order suboptimal H2=H1 controller’s gain to
Oð�Þ approximation. Symbol bar is used to denote
all closed loop quantities for slow and fast sub-
systems containing slow system gain Ks and fast
system gain Kf. This difference in notation is used
to emphasize the fact that the controller for a full-
order system and that of the reduced order models
are designed by separate techniques, but finally it
will be shown that Ks and Kf which form the
composite controller are Oð�Þ close to components
K1 and K2, respectively, of the full-order controller.
However, when we consider composite controller
gain Kc :¼ ½Ks Kf� in our discussion, we use the
subscript c to all matrices.
3.1. Reduced Order Models
We define the closed-loop slow subsystem by
_xs ¼ �Asxs þ �Bsw ð14Þ
zs1 ¼ �Csxs þ �Dsw ð15Þ
zs0 ¼ �Ssxs ð16Þ
and the closed-loop fast subsystem in fast time scale
by
� _xf ¼ �Afxf þ �Bfw ð17Þ
zf1 ¼ �Cfxf ð18Þ
zf0 ¼ �Sfxf ð19Þ
where
�As :¼ �A11 � �A12�A�122
�A21; �Bs :¼ B11 � �A12�A�122 B21
�Cs :¼ �C11 � �C12�A�122
�A21; �Ds :¼ � �C12�A�122 B21
�Af :¼ A22 þ B22Kf; �Bf :¼ B21; �Cf :¼ C12 þD1Kf
�Sf ¼ S12 þ S2Kf; �Ss :¼ ½ðS11 þ S2KsÞ � ðS12 þ S2KfÞ �E01�
�E01 ¼ ½ �A22 þ ��2Qf
�C012�C12�
�1½ �A21 þ ��2Qf�C012�C11�
�Ai1 :¼ Ai1 þ Bi2Ks; �C11 :¼ C11 þD1Ks;
�Ai2 :¼ Ai2 þ Bi2Kf; �C12 :¼ C12 þD1Kf;
for i ¼ 1; 2, where Ks and Kf are the slow and fast
controller gains respectively, and �A22 is invertible as
the fast subsystem is stable in closed-loop.
The design of the fast and slow controllers will be
described separately in the following subsections.
3.2. Fast Subsystem Design
For the system (17)–(19) the fast mixed H2=H1-
control problem is to find Kf such that
(i) �Af is asymptotically stable,
(ii) the fast transfer-function from wðtÞ to zf1ðtÞ,
Tzf1wðpÞ ¼�CfðpI� �AfÞ
�1 �Bf; p :¼ s�
satisfies the constraint
kTzf1wk1 � �; ð20Þ
and
(iii) the performance functional,
JfðKfÞ ¼ limt!1
E xf0ðS12 þ S2KfÞ
0ðS12 þ S2KfÞxf� �
¼ tr �Qf�Rf;
ð21Þ
is minimized, where
�Qf :¼ limt!1
E xfðtÞxfðtÞ0� �
is the closed-loop fast state covariance matrix, which
satisfies the fast algebraic Lyapunov equation,
�Af�Qf þ �Qf
�A0f þ
�Bf�B0f ¼ 0; ð22Þ
and
�Rf :¼ ðS12 þ S2KfÞ0ðS12 þ S2KfÞ: ð23Þ
It can be easily shown that (21) is equal to the square
of theH2-norm of the fast subsystem transfer function
given by Tzf0wðpÞ ¼�SfðpI� �AfÞ
�1 �Bf. To include the
H2=H1 Control of SPS with LSVF 57
disturbance attenuation constraint (20), as in the case
of the full-order problem, the Lyapunov’s equation
(22) is replaced with an algebraic Riccati equation, the
solution of which overbounds the closed-loop steady-
state covariance in view of the following theorem.
Theorem 2:Let Kf be given and assume that9Qf 2 Rn�n,
a nonnegative-definite matrix satisfying
0 ¼ �AfQf þQf�A0f þ ��2Qf
�C0f�CfQf þ �Bf
�B0f ð24Þ
Then
ðaÞ ð �Af; �BfÞ is stabilizable if and only if �Af is
asymptotically stable.
In this case,
ðbÞ kTzf1wk1 � �
and
ðcÞ �Qf � Qf:
Consequently,
ðdÞ JfðKfÞ � J fðKf;QfÞ;
where J fðKf;QfÞ ¼ tr½Qf�Rf�.
The proof of this theorem follows the line of the
proof of Theorem 1.
Define the open set Sfast :¼ fðKf;QfÞ : Qf > 0 and�Af þ ��2Qf
�C0f�Cf is asymptotically stable}. As dis-
cussed for the full-order problem the fast auxiliary
minimization problem is to minimize
J fðKf;QfÞ ¼ tr½Qf�Rf�; ð25Þ
an overbound of (21), where Qf is a solution to (24),
subject to the constraint (24) over the open set Sfast.
This is equivalent to minimizing the Lagrangian Lf
defined by
Lf ¼ tr Qf�Rf þ �AfQf þQf
�A0f þ ��2Qf
�C0f�CfQf þ �Bf
�B0f
n oPf
h i:
The conditions@Lf
@Qf
¼ 0 and@Lf
@Kf
¼ 0 give rise to
0 ¼ �Rf þ Pfð �Af þ ��2Qf�C0f�CfÞ þ ð �Af þ ��2Qf
�C0f�CfÞ
0Pf
ð26Þ
0 ¼ S02ðS12 þ S2KfÞ þ fB0
22 þ ��2D01ðC12 þD1KfÞQfgPf
� �Qf
ð27Þ
respectively. We call (26), the Lagrange multiplier
equation, and (27), the gain equation, for the fast
subsystem. A solution of Qf, Pf and the desired fast
controller gain Kf can be found from the coupled
equations (24), (26) and (27), see Section 4. As
the solution of (24) is sought over the open set Sfast,
(26) has a unique solution as ð �Af þ ��2Qf�C0f�CfÞ
is stable. Under the usual mild restriction:
D01½C1 D1� ¼ ½0 I�,Kf is the solution of the Sylvester’s
equation
S02S2Kf þ Kf�
�2QfPf ¼ �S02S12 � B0
22Pf:
3.3. Slow Subsystem Design
The solution of the slow suboptimal control problem
for the system (14)–(16) differs from the fast one. In
this case the structure of the ARE is more complex
and similar to the one in [10]. For clarity the solution
is described below in full.
Consider the system (14)–(16). The slow mixed
H2=H1-control problem is to find Ks such that
(i) the �As is asymptotically stable,
(ii) the slow transfer-function from wðtÞ to zs1ðtÞ,
Tzs1wðsÞ ¼�CsðsI� �AsÞ
�1 �Bs þ �Ds;
satisfies the constraint
kTzs1wk1 � �; ð28Þ
and
(iii) the performance functional,
JsðKsÞ ¼ limt!1
E xs0 �Rsxsf g
¼ tr �Qs�Rs;
ð29Þ
is minimized, where
�Qs :¼ limt!1
E xsðtÞxsðtÞ0� �
is the closed-loop slow state covariance matrix, which
satisfies the slow algebraic Lyapunov equation,
�As�Qs þ �Qs
�A0s þ
�Bs�B0s ¼ 0; ð30Þ
and
�Rs : ¼ �S0s�Ss
�Ss : ¼ ½ðS11 þ S2KsÞ � ðS12 þ S2KfÞ �E01�
�E01 ¼ ½ �A22 þ ��2Qf
�C012�C12�
�1½ �A21 þ ��2Qf�C012�C11�
ð31Þ
It can be easily shown that (29) is equal to the square
of the H2-norm of the slow subsystem transfer func-
tion given by Tzs0wðsÞ ¼�SsðsI� �AsÞ
�1 �Bs. To include
the disturbance attenuation constraint (28), as in the
58 K.B. Datta and A RaiChaudhuri
case of the full-order problem, the Lyapunov’s equa-
tion (30) is replaced with an algebraic Riccati equa-
tion, the solution of which overbounds the closed-
loop steady-state covariance in view of the following
theorem.
Theorem 3: Let Ks be given and assume that
9Qs 2 Rn�n, a nonnegative-definite matrix satisfying
0 ¼ �AsQs þQs�A0s þ ð �Ds
�B0s þ
�CsQsÞ0
ð�2I� �Ds�D0sÞ
�1ð �Ds�B0s þ
�CsQsÞ þ �Bs�B0s
ð32Þ
Then
ðaÞ ð �As; �BsÞ is stabilizable if and only if �As is
asymptotically stable.
In this case,
ðbÞ kTzs1wk1 � �
and
ðcÞ �Qs � Qs:
Consequently,
ðdÞ JsðKsÞ � J sðKs;QsÞ;
where J sðKs;QsÞ ¼ tr½Qs�Rs�.
The proof of this theorem follows the line of the
proof of Theorem 1.
Define the open set Sslow :¼ fðKs;QsÞ : Qs > 0 and
½ �As þ ð �Bs�D0s þQs
�C0sÞð�
2I� �Ds�D0sÞ
�1 �Cs� is asymptotic-
ally stable}. As discussed for the full-order problem
the slow auxiliary minimization problem is to minimize
J sðKs;QsÞ ¼ tr Qs�Rs½ �; ð33Þ
an overbound of (29), where Qs is a solution to (32),
subject to the constraint (32) over the open set Sslow.
This is equivalent to minimizing the Lagrangian Ls
defined by
LsðKs;Qs;PsÞ ¼ trnQs
�Rs þ��AsQs þQs
�A0s:ð
�Ds�B0s þ
�CsQsÞ0
ð�2I� �Ds�D0sÞ
�1ð �Ds�B0s þ
�CsQsÞ þ �Bs�B0s
�Ps
o:
Putting@Ls
@Qs
¼ 0 and@Ls
@Ks
¼ 0, we have
0 ¼ Ps�As þ ð �Bs
�D0s þQs
�C0sÞ
�M�1s
�Cs
� �þ �As þ ð �Bs
�D0s þQs
�C0sÞ
�M�1s
�Cs
� �0Ps þ �Rs;
ð34Þ
0 ¼ ½S2 � �S2�A�122�ðB22 þ ��2Qf
�C012D1Þ�
0 �SsQs
þ ½ �B2 þ ð �Bs�D0s þQs
�C0sÞ
�M�1s
�D1�0PsQs;
ð35Þ
respectively, where
�Ms :¼ ð�2I� �Ds�D0sÞ;
�B2 :¼ B12 � �A12�A�122 B22;
�D1 :¼ D1 � �C12�A�122 B22; �S2 :¼ S12 þ S2Kf;
Qs, Ps and Ks can be solved iteratively from (32), (34)
and (35) respectively, and Ms is assumed to be inver-
tible. We call (34), the Lagrange multiplier equation,
and (35), the gain equation, for the slow subsystem. As
the solution of (32) is sought over the open set Sslow,
(34) has a unique solution as �As þ ð �Bs�D0s þ
Qs�C0sÞ
�M�1s
�Cs is asymptotically stable. The slow
problem is solved as done earlier for the fast case.
However, the equations involved have more complex
forms. As is evident from the equations involved here,
the slow problem requires the knowledge of the fast
gain Kf and also of Qf, which are already available
from the solution of the fast problem.
3.4. Optimal Properties of H2=H1 Composite
Control
A composite controller Kc is constructed from the fast
controller Kf and the slow controller Ks as
Kc :¼ ½Ks j Kf�. Let J c denote the value of H2-PI
achieved by this controller Kc. The proposition below
establishes near-optimal properties of the composite
controllerKc and J c in comparison with the full-order
controller K and value of full-order H2-PI J ðK;QÞ.
Proposition 1: If Kf and Ks are the solutions of matrix
equations (27) and (35) respectively, then the com-
posite controller defined by
Kc :¼ ½Ks Kf�; ð36Þ
and J c :¼ J cðKc;QcÞ, the value of H2-PI achieved by
using this controller Kc are related to the full-order
controller K in (4), respectively, by
K ¼ Kc þOð�Þ;
J ðK;QÞ ¼ J cðKc;QcÞ þOð�Þ;
where Qc satisfies the CARE given by
0 ¼ ~AcQc þQc~A0c þ ��2Qc
~C0c~CcQc þ ~Bc
~B0c; ð37Þ
a subscript c denotes the quantities with a composite
control and
~Bc ¼ B1 ¼ ~B; ~Ac ¼ Aþ B2Kc; ~Cc ¼ C1 þD1Kc;
J cðKc;QcÞ ¼ tr Qc~Rc; ~Rc :¼ ðS1 þ S2KcÞ
0ðS1 þ S2KcÞ;
Qc1 ¼ Qs þOð�Þ; Qc3 ¼ Qf þOð�Þ and
Qc2 ¼ Q2 þOð�Þ:
ð38Þ
H2=H1 Control of SPS with LSVF 59
The proof is provided in Section 6.4. For this we need
to decompose the full-order Riccati equation, full-
order Lagrange Multiplier equation and full-order
gain equation into their slow and fast counterparts.
They are included in Sections 6.1, 6.2 and 6.3. The
controllers Ks and Kf are optimal only for the slow
and the fast subsystems respectively. The above Pro-
position shows that the composite controller Kc even
though does not contain � explicitly and defined in
terms ofKs and Kf, guarantees anOð�Þ approximation
of the optimal performance to the full order H2=H1
control problem. If we apply the control uðtÞ ¼KsxsðtÞ þ KfxfðtÞ, then there exists a solution Qc to
(37). Indeed, following the procedure given in Section
6.1, the Riccati equation (37) can be decomposed to
give us
Qc ¼Qs þOð�Þ Qc2
Q0c2 ��1ðQf þOð�ÞÞ
� �
where Qc2 can be determined with Oð�Þ approxima-
tion in terms of Qs and Qf, see (45). Since, ðKs;QsÞ 2Sslow and ðKf;QfÞ 2 Sfast, as � 7!0; Qc > 0, and by
virtue of Theorem 1, when (37) is satisfied,~Ac ¼ Aþ B2Kc, is asymptotically stable, where
Kc ¼ ½Ks Kf�. The following Proposition 2 is a direct
consequence of Proposition 1.
Proposition 2: Under the conditions of Proposition 1,
kTcz1w
k1 ¼ kTz1wk1 þOð�Þ;
where
Tcz1w
ðsÞ ¼ ~CcðsI� ~AcÞ ~Bc; and Tz1wðsÞ ¼~CðsI� ~AÞ ~B:
4. Numerical Experimentation
In this section a numerical result is presented to
illustrate the effectiveness of the design procedure
outlined so far.
Consider the system
_x1ðtÞ
� _x2ðtÞ
� �¼
�2 1 2
3 �1 2
2 �3 �2
264
375 x1ðtÞ
x2ðtÞ
� �
þ
1
1
0
264
375wðtÞ þ
1 1
1 0
0 1
264
375uðtÞ
zðtÞ ¼0:2000 0:1000 0:4000
0:4445 0 1:4845
� �xðtÞ
þ0:10 0:40
0:50 0:11
� �uðtÞ
For this system theH2=H1 control problem described
in Section 2 was solved using the Robust Control
Toolbox in MATLAB. The full-order controllerK for
� ¼ 0:1 and � ¼ 1 was found to be
K ¼�1:3401 �1:6377 �2:81850:0906 0:9996 �0:3402
� �
The composite controller found using the procedure
described in Section 3 was
Kc ¼ ½Ks Kf�
¼�1:6149 �1:6518 �2:9090
0:2513 0:9146 �0:2728
� �
It is evident that K and Kc are Oð�Þ close.
5. Conclusion
In this paper it is shown how to design a mixed
H2=H1 LSVF controller for a singularly perturbed
system based on reduced order models of slow and
fast subsystems. The design of LSVF controller of the
slow subsystem depends on Kf and Qf of the fast
subsystem because the H1 design constraint is placed
on the closed-loop system. An independent design of
the slow and fast reduced order subsystems so as to
satisfy the design requirements of the full-order
system still remains an unsolved problem. The three
design equations for the variables Q, P and K are
coupled equations containing these variables which
are to be solved recursively. With the assumption that
z0 ¼ z1 and under some mild restrictions, it can be
shown that the equations for Q, and P can be made
independent of K for the full-order and also for the
slow and fast subsystems. In this case we get a con-
troller known as a central controller. The H2=H1
theory of singularly perturbed system can be studied
by representing it as a descriptor variable system [21],
which remains a topic of future investigation.
6. Appendix
6.1. Decomposition of the Full-Order
Riccati Equation
The decomposition of full-order ARE (11) into slow
and fast equations is done assuming, see [12],
Q ¼Q1 Q2
Q02 ��1Q3
� �: ð39Þ
60 K.B. Datta and A RaiChaudhuri
Substituting from (39), the terms of (11) are now
simplified separately as shown below:
~AQ ¼~A11
~A12
��1 ~A21 ��1 ~A22
" #Q1 Q2
Q02 ��1Q3
� �
¼~A11Q1 þ ~A12Q
02
~A11Q2 þ ��1 ~A12Q3
��1ð ~A21Q1 þ ~A22Q02Þ ��2ð� ~A21Q2 þ ~A22Q3Þ
" #
ð40Þ
~B ~B0 ¼B11
��1B21
� �B011 ��1B0
21
� �
¼B11B
011 ��1B11B
021
��1B21B011 ��2B21B
021
" # ð41Þ
��2Q ~C0 ~CQ ¼ ��2Q1 Q2
Q02 ��1Q3
� �~C011
~C012
" #
~C11~C12
� � Q1 Q2
Q02 ��1Q3
� �
¼ ��2 Q1~C011 þQ2
~C012
Q02~C011 þ ��1Q3
~C012
" #
½ ~C11Q1 þ ~C12Q02
~C11Q2 þ ��1 ~C12Q3�
Writing the elements of (11) in the form of separate
equations with Oð�Þ approximation we have
ðiÞ 0 ¼ ð ~A11 þ ��2Q2~C012~C11ÞQ1
þQ1ð ~A11 þ ��2Q2~C012~C11Þ
0
þ ~A12Q02 þQ2
~A012 þ ��2½Q1
~C011~C11Q1
þQ2~C012~C12Q
02� þ B11B
011
ð42Þ
ðiiÞ 0 ¼ ~A12Q3 þQ1~A021 þQ2
~A022 þ B11B
021
þ ��2ðQ1~C011 þQ2
~C012Þ
~C12Q3 þOð�Þð43Þ
ðiiiÞ 0 ¼ ~A22Q3 þQ3~A022 þ B21B
021
þ ��2Q3~C012~C12Q3 þOð�Þ
ð44Þ
Afterwards (44) will turn out to be the algebraic
Riccati equation for designing suboptimal H2=H1
controller for the fast subsystem to Oð�Þ approxima-
tion. However, in the process of decomposition of
Lagrange’s multiplier equation and the gain equation
into slow-fast components, we will need some of the
intermediate steps in the process of derivation of (71).
With this hindsight, we take pain to give a sketch of
the derivation of (71) for the benefit of the readers.
From (43), solving for Q2 we get
�Q2 ¼ Q1E1 þ E2 þOð�Þ; ð45Þ
where
E1 :¼ ð ~A021 þ ��2 ~C0
11~C12Q3Þ ~A
0�1
22� ð46Þ
E2 :¼ ð ~A12Q3 þ B11B021Þ
~A0�1
22� ð47Þ
~A22� :¼ ~A22 þ ��2Q3~C012~C12
¼ ðIþ ��2Q3~C012~C12
~A�122 Þ
~A22:ð48Þ
Substituting (45) forQ2 in (42) and simplifying, we are
led to an equation with Q2 removed as
0 ¼ AQ1 þQ1A0 þQ1UQ1 þ B0BþOð�Þ ð49Þ
where
A :¼ ~A11 � ~A12E01 � ��2E2
~C012~C11
þ ��2E2~C012~C12E
01
ð50Þ
U ¼ ��2C0C ð51Þ
B0B :¼ � ~A12E02 � E2
~A012 þ B11B
011
þ ��2E2~C012~C12E
02
ð52Þ
C :¼ ~C11 � ~C12E01 ð53Þ
Making use of (46), we have
C0 ¼ ~C011½I� ��2 ~C12Q3
~A0�122�
~C012� �
~A021~A0�1
22�~C012:
Defining the bracketed term by H and simplifying
with the help of (48) we have
H :¼ I� ��2 ~C12Q3~A0�122�
~C012
¼ I� ��2 ~C12Q3~A0�122 ðIþ ��2 ~C0
12~C12Q3
~A0�122 Þ�1Þ�1 ~C0
12
¼ ðIþ ��2 ~C12Q3~A0�122
~C012Þ
�1
ð54Þ
It is not difficult to show that
~A�122� :¼ ð ~A22 þ ��2Q3
~C012~C12Þ
�1
¼ ~A�122 ½I� ��2Q3
~C012~C12ð ~A22 þ ��2Q3
~C012~C12Þ
�1�
¼ ~A�122 ½I� ��2Q3
~C012H
0 ~C12~A�122 �
ð55Þ
H2=H1 Control of SPS with LSVF 61
~C11Q1 þ ~C12Q02 ¼ CQ1 � ~C12E
02 ð56Þ
In view of this simplification in (54), C0 becomes
C0 ¼ ~C011½Iþ ��2 ~C12Q3
~A0�122
~C012�
�1
� ~A021~A0�1
22 ½Iþ ��2 ~C012~C12Q3
~A0�122 ��1 ~C 0
12
¼ ~C00H;
where
~C0 :¼ ~C11 � ~C12~A�122
~A21
With this simplification, U takes the form
U ¼ ��2 ~C00HH0 ~C0:
Now, with the help of (44) we can write
ðHH0Þ�1 ¼ ½Iþ ��2 ~C12~A�122 Q3
~C012�½Iþ ��2 ~C12Q3
~A0�122
~C012�
¼ Iþ ��2 ~C12~A�122 ½
~A22Q3 þQ3~A022
þ ��2Q3~C012~C12Q3� ~A
0�122
~C012
¼ I� ��2 ~C12~A�122 B21B
021~A0�122
~C012;
¼ I� ��2 ~D0~D00
¼: ��2 ~M0
ð57Þ
where
~D0 :¼ � ~C12~A�122 B21; ~M0 :¼ �2I� ~D0
~D00: ð58Þ
This gives us
U ¼ ��2C0C ¼ ~C00~M�10
~C0 ð59Þ
To simplify the expression for A we will use the fol-
lowing relations which are direct outcomes of (46),
(48) and (54):
E1 ¼ ð ~A021 þ ��2 ~C0
11~C12Q3 � ��2E1
~C012~C12Q3Þ ~A
0�1
22
ð60Þ
~C12~A�1
22� ¼ H0 ~C12~A�1
22 ð61Þ
~C12Q3~A0�1
22� ¼ H ~C12Q3~A0�1
22 ð62Þ
Substituting the expressions from (47)–(48), (60)
in (50) and simplifying with (44), (53) and (61), we
obtain
A ¼ ~A11 � ~A12~A�122 ð
~A21 þ ��2Q3~C012~C11
� ��2Q3~C012~C12E
01Þ � ��2E2
~C012ð
~C11 � ~C12E01Þ
¼ ~A0 � ��2ð ~A12~A�122 Q3 þ E2g ~C
012C
¼ ~A0 � ��2½ ~A12~A�122 Q3 þ ð ~A12Q3 þ B11B
021Þ
ð ~A022 þ ��2 ~C0
12~C12Q3Þ
�1� ~C012C
¼ ~A0 � ��2½ ~A12~A�122 fQ3ð ~A
022 þ ��2 ~C0
12~C12Q3Þ
þ ~A22Q3g þ B11B021�ð
~A022 þ ��2 ~C0
12~C12Q3Þ
�1 ~C012C
¼ ~A0 þ ��2 ~B0~D00HC
¼ ~A0 þ ~B0~D00~M�10
~C0
ð63Þ
where
~A0 :¼ ~A11 � ~A12~A�122
~A21;
~B0 :¼ ~B11 � ~A12~A�122
~B21; ~C0 :¼ ~C11 � ~C12~A�122
~A21
ð64Þ
To simplify B0B, we write E2 with the help of (47) and
(48) in the form
E2 ¼ ð ~A12Q3 þ B11B021 � ��2E2
~C012~C12Q3Þ ~A
0�122
ð65Þ
Inserting this expression for E2 in (52)
B0B ¼ B11B011 �
~A12~A�122 ð
~A12Q3 þ B11B021
� ��2E2~C012~C12Q3Þ
0 � ð ~A12Q3 þ B11B021
� ��2E2~C012~C12Q3Þ ~A
0�122
~A012 þ ��2E2
~C012~C12E
02
ð66Þ
With the help of (44), the second and fifth terms in
(66) can be written as
� ~A12~A�122 ðQ3
~A022 þ
~A22Q3Þ ~A0�122
~A012
¼ ~A12~A�122 ðB21B
021 þ ��2Q3
~C012~C12Q3Þ ~A
0�122
~A012
Hence,
B0B ¼ B11B011 �
~A12~A�122 ðB11B
021Þ
0 � ðB11B021Þ
~A0�122
~A012
þ ð ~A12~A�122 B21Þð ~A12
~A�122 B21Þ
0
þ ��2 ~A12~A�122 ðE2
~C012~C12Q3Þ
0
þ ��2ðE2~C012~C12Q3Þ ~A
0�122
~A012
þ ��2E2~C012~C12E
02~A12
~A�122 Q3
~C012~C12Q3
~A0�122
~A012
¼ ðB11 � ~A12~A�122 B21ÞðB11 � ~A12
~A�122 B21Þ
0
þ ��2ðE2 þ ~A12~A�122 Q3Þ ~C
012~C12ðE2 þ ~A12
~A�122 Q3Þ
ð67Þ
62 K.B. Datta and A RaiChaudhuri
Now,
E2 þ ~A12~A�122 Q3
¼ ð ~A12Q3 þ B11B021Þð
~A22 þ ��2 ~C012~C12Q3Þ
�1 ~A12~A�122 Q3
¼ ½ ~A12~A�122 f
~A22Q3 þQ3ð ~A0�122 þ ��2 ~C0
12~C12Q3Þg
þ B11B021�ð
~A22 þ ��2 ~C012~C12Q3Þ
�1
¼ ~A12~A�122 B11B
021 þ B11B
021Þð
~A22 þ ��2 ~C012~C12Q3Þ
�1
¼ ~B0B021ð
~A22 þ ��2 ~C012~C12Q3Þ
�1
ð68Þ
So,
ðE2 þ ~A12~A�122 Q3Þ ~C
012
¼ ~B0B021~A0�122
~C012ðIþ ��2 ~C12Q3
~A0�122
~C012Þ
�1
¼ � ~B0~D00H
ð69Þ
Consequently, we have,
B0B ¼ ~B0~B00 þ ��2 ~B0
~D00HH0 ~D0
~B00
¼ ~B0½Iþ ~D00~M�10
~D0� ~B00
ð70Þ
We shall now replace (63)–(70) in (49) to obtain the
following Riccati equation in Q1 which has now been
made free from Q3. However, it contains K1 and K2.
0 ¼ ð ~A0 þ ~B0~D00~M�10
~C0ÞQ1 þQ1ð ~A0 þ ~B0~D00~M�10
~C0Þ0
þQ1~C00~M�10
~C0Q1 þ ~B0½Iþ ~D00~M�10
~D0� ~B00 þOð�Þ;
which after minor simplification finally reduces to
0 ¼ ~A0Q1 þQ1~A00 þ ð ~D0
~B00 þ
~C0Q1Þ0 ~M�1
0
ð ~D0~B00 þ
~C0Q1Þ þ ~B0~B00 þOð�Þ:
ð71Þ
6.2. Decomposition of the Lagrange Multiplier
Equation
The Lagrange multiplier equation (12), which is in fact
a Lyapunov equation, is reproduced below
0 ¼ ð ~AþQ ~R1Þ0P þ Pð ~AþQ ~R1Þ þ ~R; ð72Þ
and has now to be decomposed into slow and fast
subsystem components as done for algebraic Riccati
equation (11) in the last section. It should be noted
that all through Oð�Þ approximation is used wherever
necessary. The structure of P is assumed to be of the
form, see [12],
P ¼P1 �P2
�P02 �P3
� �:
Proceeding with simplification of the terms in (72) we
have successively,
where we define,
A11 :¼ ~A11 þ ��2ðQ1~C011 þQ2
~C012Þ
~C11
A21 :¼ ~A21 þ ��2Q3~C012~C11;
A12 :¼ ~A12 þ ��2ðQ1~C011 þQ2
~C012Þ
~C12;
A22 :¼ ~A22 þ ��2Q3~C012~C12;
ð74Þ
so that,
Pð ~AþQ ~R1Þ ¼P1 �P2
�P02 �P3
� �A11 A12
��1A21 ��1A22
� �
¼P1A11 þ P2A21 P1A12 þ P2A22
P3A21 þOð�Þ P3A22 þOð�Þ
� �:
ð75Þ
Also,
~R ¼ ðS1 þ S2KÞ0ðS1 þ S2KÞ
¼ S11 þ S2K1 S12 þ S2K2½ �0
S11 þ S2K1 S12 þ S2K2½ �
¼~S01~S1
~S01~S2
~S02~S1
~S02~S2
" #
¼:~R11
~R12
~R0
12~R22
" #ð76Þ
where
~S1 :¼ ðS11 þ S2K1Þ; ~S2 :¼ ðS12 þ S2K2Þ;
~R11 :¼ ~S01~S1; ~R12 :¼ ~S0
1~S2;
~R0
12 :¼~S02~S1; ~R22 :¼ ~S0
2~S2:
ð77Þ
ð ~AþQ ~R1Þ ¼~A11 þ ��2ðQ1
~C011 þQ2
~C012Þ
~C11~A12 þ ��2ðQ1
~C011 þQ2
~C012Þ
~C12
��1ð ~A21 þ ��2Q3~C012~C11Þ þOð1Þ ��1ð ~A22 þ ��2Q3
~C012~C12Þ þOð1Þ
� �: ð73Þ
H2=H1 Control of SPS with LSVF 63
Using (75) and (76) in (72), the elements in the
resultant matrix are respectively:
0 ¼ ~R11 þ P1A11 þ P2A21 þA011P1 þA
021P
02 þOð�Þ
ð78Þ
0 ¼ ~R12 þ P1A12 þ P2A22 þA021P3 þOð�Þ
ð79Þ
0 ¼ ~R22 þ P3A22 þA022P3 þOð�Þ ð80Þ
It will be shown later that equation (80) is similar to
the Lagrange multiplier equation for a properly
defined fast subsystem of (1)–(3). We will now use
(79) and (80) to eliminate P2 and P3 from (78).
Solving (79) for P2 we get,
P2 ¼ P1X1 þ X2 þOð�Þ; ð81Þ
where
X1 ¼ �A12A�122 ; X2 ¼ �ð ~R12 þA
021P3ÞA
�122 ð82Þ
Substituting P2 from (81) into (78) gives us
0 ¼ ~R11 þ P1A11 þA011P1 þ ðP1X1 þ X2ÞA21
þA021ðP1X1 þ X2Þ
0 þOð�Þ
¼ ~S01~S1 þ P1A0 þA
00P1 þ X2A21 þ ðX2A21Þ
0
ð83Þ
where
A0 :¼ A11 þ X1A21:
Equation (83) contains P3 in X2 but P2 is eliminated.
The terms containing P3 may be simplified as shown
below. We have using (80),
~S01~S1 þ X2A21 þA
021X
02
¼ ~S01~S1 � ð~S0
1~S2 þA
021P3ÞA
�122 A21
�A021A
0�122 ð~S0
2~S1 þ P3A21Þ
¼ ~S01~S1 � ~S0
1~S2A
�122 A21 �A
021A
0�122
~S02~S1
þA021A
0�122
~S02~S2A
�122 A21
¼ ð~S1 � ~S2A�122 A21Þ
0ð~S1 � ~S2A�122 A21Þ:
With the above simplification (83) transforms to
0 ¼ P1A0 þA0P1 þ ð~S1 � ~S2A�122 A21Þ
0
ð~S1 � ~S2A�122 A21Þ þOð�Þ;
ð84Þ
in which P2 and P3 have been eliminated. We
will now remove Q2 and Q3 from A0 in (84)
using (43)–(44). Comparing (46) and (74), it is obvious
that
E01 ¼ A
�122 A21
So, making substitutions from (45) and (50) we
have
A0 ¼ A11 �A12A�122 A21
¼ ~A11 þ ��2ðQ1~C011 þQ2
~C012Þ
~C11
� f ~A12 þ ��2ðQ1~C011 þQ2
~C012Þ
~C12gE01
¼ ~A11 � ~A12E01 þ ��2ðQ1
~C011 þQ2
~C012Þð
~C11 � ~C12E01Þ
¼ ~A11 � ~A12E01 � ��2E2
~C012ð
~C11 � ~C12E01Þ
þ ��2Q1ð ~C11 � ~C12E01Þ
0ð ~C11 � ~C12E01Þ
¼ Aþ ��2Q1C0C;
ð85Þ
in which A and C are given in (63) and (53), see also
(59).
Putting (85) in (84), the equation for P1 is
0 ¼ P1ðAþ ��2Q1C0CÞ þ ðAþ ��2Q1C
0CÞ0P1
þ ð~S1 � ~S2A�122 A21Þ
0ð~S1 � ~S2A�122 A21Þ þOð�Þ;
ð86Þ
which after substitution from (63) for A and from (59)
for C0C finally reduces to
0 ¼ P1½ ~A0 þ ð ~B0~D00 þQ1
~C00Þ
~M�10
~C0�
þ ½ ~A0 þ ð ~B0~D00 þQ1
~C00Þ
~M�10
~C0�0P1
þ ~R0 þOð�Þ;
ð87Þ
where ~R0 ¼ ð~S1 � ~S2A�122 A21Þ
0ð~S1 � ~S2A�122 A21Þ,which
is similar to �Rs in which K1;K2 and Q3 are now
replaced with Ks;Kf and Qf respectively. The above
equation will be shown in a subsequent section to
represent the Lagrange multiplier equation of a
properly defined slow subsystem of (1)–(3).
6.3. Decomposition of the Gain Equation
The full-order controller gain K satisfies
0 ¼ S02ðS1 þ S2KÞQþ ½B0
2 þ ��2D01ðC1 þD1KÞQ�PQ;
ð88Þ
which will be decomposed into two parts, one
approximating K1, the controller’s gain for slow
64 K.B. Datta and A RaiChaudhuri
subsystem and the other K2, the controller’s gain for
fast subsystem. Simplifying the individual terms of
(88) we have,
ðIÞ S02S1Q ¼ S0
2 S11 S12½ �Q1 Q2
Q02 ��1Q3
� �
¼ S02 S11Q1 þ S12Q
02 S11Q2 þ
1
�S12Q3�
�
ðIIÞ S02S2KQ ¼ S0
2S2 K1Q1 þ K2Q02
�K1Q2 þ ��1K2Q3
�
ðIIIÞ B02PQ ¼ B0
12
1
�B022
� �P1 �P2
�P02 �P3
� �Q1 Q2
Q02 ��1Q3
� �¼ B0
12P1Q1 þ B022ðP
02Q1 þ P3Q
02Þ
�þOð�Þ ��1B0
22P3Q3 þOð1Þ�
ðIVÞ ��2D01ðC1 þD1KÞQPQ
¼ ��2 D01f
~C11Q1P1Q1 þ ~C12ðQ02P1Q1
�þQ3P
02Q1 þQ3P3Q
02Þg;
��1D01~C12Q3P3Q3 þOð1Þ�
From the terms (I)–(IV) simplified above, assembling
together ð1; 2Þ elements, we have
0 ¼ ½S02ðS12 þ S2K2Þ þ fB0
22 þ ��2D01ðC12
þD1K2ÞQ3gP3�Q3 þOð�Þ;ð89Þ
which turns out to be the controller’s gain for the fast
subsystem described by (27) to Oð�Þ approximation.
On the other hand, the (1,1)-elements lead to
0 ¼ S02ðS11Q1 þ S12Q
02Þ þ S0
2S2ðK1Q1 þ K2Q02Þ
þ ðB022P3 þ ��2D0
1~C12Q3P3ÞQ
02
þ ½B012 þ ��2D0
1ð~C11Q1 þ ~C12Q
02Þ�P1Q1
þ ðB022 þ ��2D0
1~C12Q3ÞP
02Q1 þOð�Þ;
ð90Þ
which will be shown to be the controller’s gain for the
slow subsystem described by (35) to Oð�Þ approx-
imation. Now, the various terms of (90) is rewritten
by setting
�Q2 ¼ Q1E1 þ E2 þOð�Þ;
P2 ¼ P1X1 þ X2 þOð�Þ
as defined in (45) and (81) in the following way toOð�Þapproximation:
ðAÞ S02ðS11Q1 þ S12Q
02Þ þ S0
2S2ðK1Q1 þ K2Q02Þ
ðBÞ ðB022P3 þ ��2D0
1~C12Q3P3ÞQ
02
¼ �ðB022P3 þ ��2D0
1~C12Q3P3ÞE
01Q1
� ðB022P3 þ ��2D0
1~C12Q3P3ÞE
02
ðCÞ ½B012 þ ��2D0
1ð~C11Q1 þ ~C12Q
02Þ�P1Q1
¼ B012P1Q1 þ ��2D0
1CQ1P1Q1
� ��2D01~C12E
02P1Q1
ðDÞ ðB022 þ ��2D0
1~C12Q3ÞP
02Q1
¼ ðB022 þ ��2D0
1~C12Q3ÞX
01P1Q1
þ ðB022 þ ��2D0
1~C12Q3ÞX
02Q1;
where in (C) we have used C ¼ ~C11 � ~C12E01 from (53).
IfQ is struck out from the right hand side of (88), then
the coefficient of Q02 in (A) and all terms contained in
(B) will not appear in the (1,1) term. These terms are,
in fact, eliminated if (89) is valid after Q3 is removed.
However, in the coefficient of Q02 in (A) and terms in
(B) will cause a simplification while simplifying coef-
ficients of Q1,P1Q1, and Q1P1Q1. The various terms
in (A), (B), (C), and (D) are now simplified consider-
ing the coefficients of Q1,P1Q1, and Q1P1Q1 suc-
cessively.
6.3.1. (I) Coefficients of Q1
Stacking together the terms in (A), (B), and the coef-
ficient of Q1 in (D), we have
S02~S1Q1 � S0
2~S2ðE
01Q1 þ E0
2Þ
� ðB022P3 þ ��2D0
1~C12Q3P3ÞE
01Q1
� ðB022P3 þ ��2D0
1~C12Q3P3ÞE
02
� ðB022 þ ��2D0
1~C12Q3Þ ~A
0�122�½
~S02~S1 þ P3A21�Q1
Taking into account (80) : 0 ¼ ~R22 þ P3A22 þA
022P3 þOð�Þ we see that the coefficient of E0
2 is Oð�Þ,and hence the above expression becomes
H2=H1 Control of SPS with LSVF 65
S02ð~S1 � ~S2E
01ÞQ1 � ðB 0
22 þ ��2D01~C12Q3Þ
½P3A�122 A21 þA
0�122 ð~S0
2~S1 þ P3A21Þ�Q1
¼ S02ð~S1 � ~S2E
01ÞQ1 � ðB 0
22 þ ��2D01~C12Q3Þ
½A0�122 ðA0
22P3 þ P3A22ÞA�122 A21 þA
0�122
~S02~S1�Q1
¼ ½S 02 � ðB 0
22 þ ��2D01~C12Q3ÞA
0�122
~S02�ð
~S1 � ~S2E01ÞQ1
ð91Þ
6.3.2. (II) Coefficients of P1Q1
The terms in the coefficient of P1Q1 are
B 012 � ��2D0
1~C12E
02 þ ðB0
22 þ ��2D01~C12Q3ÞX
01:
Now, observing that ~A11� ¼ A22, substituting for X1
from (82) and for Q2 from (45), simplifying with (53),
(61), (62) and (55) we have,
B 012 þ B 0
22X01
¼ B012 � B 0
22~A0�122�½
~A012�
�2 ~C 012f
~C11Q1 � ~C12ðE01Q1 þ E 0
2Þg�
¼ B012 � B 0
22ðI� ��2 ~A0�122
~C 012H
~C12Q3Þ ~A0�122
~A012
� ��2B 022~A0�122�
~C 012ðH
0 ~C 00Q1 � ~C12E
02Þ
The coefficient of Q1 above is, in fact, a coefficient of
Q1P1Q1 , and so transferring it to be included in (III),
we have
B 012 þ B 0
22X01 � coefficient of Q1
¼ ~B02 þ ��2B0
22~A0�122
~C012H
~C12Q3~A0�122
~A012
þ ��2B022~A0�122
~C012HH0 ~C12
~A�122 ðQ3
~A012 þ B21B
011Þ
¼ ~B02 þ ��2B0
22~A0�122
~C012M
�10
~C12~A�122
½ ~A22Q3 þ ��2Q3~C012~C12Q3 þQ3
~A022�
~A0�122
~A012
þ ��2B022~A0�122
~C012HH0 ~C12
~A�122 B21B
011
¼ ~B02 þ B0
22~A0�122
~C012M
�10
~C12~A�122 B21ðB
011 � B0
11~A0�122
~A012Þ
¼ ~B02 � B0
22~A0�122
~C012M
�10
~D0~B00;
where ~B2 ¼ B12 � ~A12~A�122 B22. The remaining terms
are
��2D01~C12ðQ3X
01 � E 0
2Þ
¼ ��2D 01~C12½Q3
~A0�122�f
~A012 þ ��2 ~C0
12ð~C11Q1
þ ~C12Q02Þg þ E 0
2�
¼ ��2D01~C12½Q3
~A0�122�f
~A012 þ ��2 ~C0
12ð~C11 � ~C12E
01ÞQ1
� ��2 ~C012~C12E
02g þ E 0
2�
So
��2D01~C12ðQ3X
01 � E 0
2Þ � coefficient of Q1
¼ ��2D01~C12½Q3
~A0�122�
~A012 � f��2Q3
~A0�122�
~C 012~C12 � IgE 0
2�
¼ ��2D01½H
~C12Q3~A0�122
~A012 � f��2H ~C12Q3
~A0�122
~C 012~C12
� ~C12g ~A0�122�ðQ3
~A012 þ B21B
011Þ�
¼ ��2D01½H
~C12Q3~A0�122
~A012 þHH0 ~A�1
22 ðQ3~A012 þ B21B
011Þ�
¼ ��2D01HH0½ðIþ ��2 ~C12
~A�122 Q3
~C12~C012Þ
~C12Q3~A0�122
~A012 þ
~C12~A�122 Q3
~A012 þ
~C12~A�122 B21B
011�
¼ ��2D01HH0½ ~C12
~A�122 ð
~A22Q3 þ ��2Q3~C 012~C12Q3
þQ3~A022Þ
~A0�122
~A012 þ
~C12~A�122 B21B
011�
¼ ���2D01HH0 ~C12
~A�122 B21ðB
011 � B0
21~A0�122
~A 012Þ
¼ D01M
�10
~D0~B00
ð92Þ
6.3.3. (III) Coefficients of Q1P1Q1
Simplification of coefficients of Q1P1Q1 appending
those obtained from (II) gives us
��2½D01ðI�
~C12Q3~A0�122�
~C012Þ � B0
22~A0�122�
~C012�C
¼ ��2½D01ðI�H ~C12Q3
~A0�122
~C012Þ � B0
22~A0�122
~C012H�C
¼ ��2½D01H� B0
22~A0�122
~C012H�C
¼ ~D01M
�10
~C0
ð93Þ
Hence, combining the terms in (91)–(93), the (1,1)-
element in the full-order gain equation, with~S0 ¼ ð~S1 � ~S2E
01Þ, becomes
0 ¼ ½S2 � ~S2A�122 ðB22 þ ��2Q3
~C012D1Þ�
0 ~S0Q1
þ ½ ~B2 þ ð ~B0~D00 þQ1
~C00ÞM
�10
~D1�0P1Q1 þOð�Þ;
ð94Þ
which turns out to be the controller’s gain for the slow
subsystem described by (35) to Oð�Þ approximation.
6.4. Proof of Proposition 1
The set of equations (44), (87) and (89) respectively
closely approximate the fast system equations (24),
(26) and (27). Hence it follows that
K2 ¼ Kf þOð�Þ:
Similarly the set of equations (71), (80) and (94)
respectively are close approximations of (32), (34) and
(35). So,
K1 ¼ Ks þOð�Þ:
66 K.B. Datta and A RaiChaudhuri
Hence, K ¼ Kc þOð�Þ. With the composite controller
Kc, the closed-loop system becomes
_xðtÞ ¼ ~AcxðtÞ þ ~BcwðtÞ;
z1ðtÞ ¼ ~CcxðtÞ;
where
~Ac ¼~Ac11
~Ac12
��1 ~Ac21 ��1 ~Ac22
" #
~Acij ¼ Aij þ Bi2Kcj; ~Bc ¼ B1;
~Cc ¼ ½ ~Cc11~Cc12� ¼ C1 þD1Kc; ~Cc1i ¼ C1i þD1Kci
Kc1 ¼ Ks; Kc2 ¼ Kf; ~Ac22 ¼ �Af; B21 ¼ �Bf; ~Cc12 ¼ �Cf:
Since Qc satisfies (37), proceeding as in Section 6.1
for the decomposition of ARE we have
0 ¼ ~AcsQc1 þQc1~A0cs þ ð ~Dcs
~B0cs þ
~CcsQc1Þ0
~M�1cs ð
~Dcs~B0cs þ
~CcsQc1Þ þ ~Bcs~B0cs þOð�Þ;
ð95Þ
0 ¼ ~Ac22Qc3 þQc3~A0c22 þ ��2Qc3
~C0c12
~Cc12Qc3
þ B21B021 þOð�Þ;
ð96Þ
where
~Acs :¼ ~Ac11 � ~Ac12~A�1c22
~Ac21;
~Bcs :¼ ~B11 � ~Ac12~A�1c22
~B21;
~Ccs :¼ ~Cc11 � ~Cc12~A�1c22
~Ac21
~Dcs :¼� ~Cc12~A�1c22B21;
~Mcs :¼ �2 � ~Dcs~D0cs
As ~Ac22 ¼ �Af; B21 ¼ �Bf; ~Cc12 ¼ �Cf, it follows from
(24) and (96) that Qc3 ¼ Qf þOð�Þ. Similarly, com-
paring (32) and (95) we haveQc1 ¼ Qs þOð�Þ. Also as
in Section 6.1, see (45), we can show
Qc2 ¼ Qc1E1ðKf; Qc3Þ þ E2ðKs;Kf;Qc3Þ þOð�Þ;
where E1ðKf;Qc3Þ is obtained replacing K2 and Q3 in
E1 with Kf and Qc3 and E2ðKs, Kf;Qc3Þ is obtained
replacing K1;K2;Q3 in E2 with Ks;Kf, and Qc3 using
(46) and (47) respectively. This argument shows
that Qc2 ¼ Q2 þOð�Þ. To show that J ðK;QÞ ¼J cðKc;QcÞþ Oð�Þ, we note that, since K ¼ Kc þOð�Þ,we have ~C ¼ ~Cc þOð�Þ, and ~A ¼ ~Ac þOð�Þ in view of
(5), (6) and (38). Now, (37) can be written as,
0 ¼ ð ~Ac þ ��2Qc~C0c~CcÞQc þQcð ~Ac þ ��2Qc
~C0c~CcÞ
0
� ��2QcðQc~C0c~CcÞ
0 þ ~Bc~B0c:
ð97Þ
Writing (11) in the form
0 ¼ ð ~Ac þ ��2Qc~C0c~CcÞQþQð ~Ac þ ��2Qc
~C0c~CcÞ
0
þ ��2Q ~C0c~CcQþ ~Bc
~B0c � ��2Qc
~C0c~CcQ
� ��2QðQc~C0c~CcÞ
0 þOð�Þ;
ð98Þ
we have, subtracting (98) from (97)
0 ¼ ð ~Ac þ ��2Qc~C0c~CcÞðQc �QÞ
þ ðQc �QÞð ~Ac þ ��2Qc~C0c~CcÞ
0
� ��2ðQc �QÞ ~C0c~CcðQc �QÞ þOð�Þ:
ð99Þ
By implicit function theorem, it can be shown that
Qc in (37) possesses a power series at � ¼ 0 represented
by
Qc ¼Qc1 Qc2
Q0c2 ��1Qc3
� �þX1i¼1
�i
i!
QðiÞc1 Q
ðiÞc2
QðiÞ0
c2 ��1QðiÞc3
" #
ð100Þ
and so also Q. The representation of Q is the same
as Qc but replacing Qc with Q in (100). Thus V :¼Qc �Q can be expanded as
V ¼V1 V2
V02 ��1V3
� �þX1i¼1
�i
i!
VðiÞ1 V
ðiÞ2
VðiÞ2
0��1V
ðiÞ3
" #;
ð101Þ
which satisfies in view of (99),
0 ¼ ð ~Ac þ ��2Qc~C0c~CcÞVþ Vð ~Ac þ ��2Qc
~C0c~CcÞ
0
� ��2V ~C0c~CcVþOð�Þ:
ð102Þ
Putting (101) into (102) and simplifying, the zero
order term for the (2,2) block element in the resulting
matrix equation obtained by putting � ¼ 0 becomes
0 ¼ ð ~Ac22 þ ��2Qc3~C0c12
~Cc12ÞV3
þ V3ð ~Ac22 þ ��2Qc3~C0c12
~Cc12Þ0
� ��2V3~C0c12
~Cc12V3:
ð103Þ
Assuming that ð �Af; �BfÞ is stabilizable and ð �Af; �CfÞ is
detectable, it follows from (24) that �Af þ ��2Qf�C0f�Cf is
stable. Since, �Af ¼ ~Ac22; �Cf ¼ ~Cc12 and Qc3 ¼ Qf, so~Ac22 þ ��2Qc3
~C0c12
~Cc12 is stable and ð ~Ac22; ~Cc12Þ is
detectable, which implies that the solution of the
above equation (103) gives rise to V3 ¼ 0. In view of
H2=H1 Control of SPS with LSVF 67
this, the (1,2) and (1,1) elements in (102) at � ¼ 0 are
respectively given by
0 ¼V2ð ~Ac22 þ ��2Qc3~C0c12
~Cc12Þ0
þ V1ð ~Ac21 þ ��2Qc3~C 0c12
~Cc11Þ0
0 ¼ ½ ~Ac11 þ ��2ðQc1~C 0c11 þQc2
~C 0c12Þ
~Cc11�V1
þ ½ ~Ac12 þ ��2ðQc1~C 0c11 þQc2
~C 0c12Þ
~Cc12�V02
þ V1½ ~Ac11 þ ��2ðQc1~C 0c11 þQc2
~C 0c12Þ
~Cc11�0
þ V2½ ~Ac12 þ ��2ðQc1~C 0c11 þQc2
~C 0c12Þ
~Cc12�0
� ��2½ðV1~C 0c11 þ V2
~C 0c12Þ
~Cc11V1
þ ðV1~C 0c11 þ V2
~C 0c12Þ
~Cc12V02�:
With the help of first equation eliminatingV2 from the
second equation, left coefficient of V1 becomes,
Cc ¼ ~Ac11 � ~Ac12E0c1 � ��2Ec2
~C 0c12ð
~Cc11 � ~Cc12E0c1Þ
0�
þ ��2Qc1ð ~Cc11 � ~Cc12E0c1Þ
0ð ~Cc11 � ~Cc12E0c1Þ
¼ Ac þ ��2Qc1CcC0c
¼ ~Acs þ ð ~Bcs~D0cs þQc1
~C 0csÞ
~M�1cs
~Ccs
where we have used the notations,
Ac :¼ ~Ac11 � ~Ac12E0c1 � ��2Ec2
~C0c12ð
~Cc11 � ~Cc12E0c1Þ
0
Cc :¼ ~Cc11 � ~Cc12E0c1; �Qc2 ¼ Qc1Ec1 þ Ec2 þOð�Þ;
Ec1 :¼ ð ~A0c21 þ ��2 ~C0
c11~Cc12Qc3Þð ~Ac22 þ ��2Qc3
~C0c12
~Cc12Þ0�1;
Ec2 :¼ ð ~Ac12Qc3 þ B11B021Þð
~Ac22 þ ��2Qc3~C0c12
~Cc12Þ0�1;
which follows from the observation that the decom-
position of (37) will generate similar equations as
given in Section 6.1 inserting only an additional sub-
script c in the variables. Simplifying other terms, the
(1,1) element finally becomes
0 ¼ CcV1 þ V1C0c � V1
~C0cs~M�1cs
~CcsV1:
Since Cc is stable, and ð ~Acs; ~CcsÞ is detectable, which
follow from the property of the slow-subsystem, the
solution of the above equation gives rise to V1 ¼ 0,
and consequently, V2 ¼ 0. Now, the first order term
for the (2,2) block element obtained from (101) and
(102) with V1 ¼ 0 ;V2 ¼ 0 and V3 ¼ 0 is
0 ¼ ð ~Ac22 þ ��2Qc3~C0c12
~Cc12ÞVð1Þ3
þ Vð1Þ3 ð ~Ac22 þ ��2Qc3
~C0c12
~Cc12Þ0
� ��2Vð1Þ3
~C0c12
~Cc12Vð1Þ3 þOð�Þ;
which is of the same form as (103) except for the Oð�Þ
term. Arguing similarly we have, Vð1Þ3 ¼ Oð�Þ: Thus
it is evident that V ¼ Qc �Q ¼ Oð�Þ: Hence,
J ðK;QÞ � J cðKc;QcÞ ¼ tr½ðQ�QcÞ ~B ~B0� ¼ Oð�Þ:
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