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International Journal of Innovative
c
Computing, Information and Control ICIC International
2013 ISSN 1349-4198
Volume 9, Number 7, July 2013 pp. 2771–2788
SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR
SYSTEM BASED ON MATRIX SIGN FUNCTION
Chih-Cheng Huang1, Jason Sheng-Hong Tsai1,∗, Shu-Mei Guo2,∗
Yeong-Jeu Sun3 and Leang-San Shieh4
1Department of Electrical Engineering
2Department of Computer Science and Information Engineering
National Cheng Kung University
No. 1, University Road, Tainan City 701, Taiwan
n28981141@mail.ncku.edu.tw; ∗Corresponding authors: {shtsai; guosm}@mail.ncku.edu.tw
3Department of Electronic Engineering
I-Shou University
No.1, Sec. 1, Syuecheng Rd., Dashu District, Kaohsiung City 84001, Taiwan
yjsun@isu.edu.tw
4Department of Electrical and Computer Engineering
University of Houston
N308 Engineering Building 1 Houston, Texas 77204-4005, USA
lshieh@uh.edu
Received April 2012; revised August 2012
Abstract. The objective of this paper is to propose a constructive methodology for de-
termining the appropriate weighting matrices {Q,R}, which guarantees the solvability of
the generalized algebraic Riccati equation and for solving the generalized Riccati equa-
tion via the matrix sign function for the stabilizable singular system. A decomposition
technique is developed to decompose the singular system into a controllable reduced-order
regular subsystem and a non-dynamic subsystem. As a result, the well-developed analysis
and synthesis methodologies developed for a regular system can be applied to the reduced-
order regular subsystem. Finally, we transform the results obtained for the reduced-order
regular subsystem back to those for the original singular system. Illustrative examples
are presented to show the effectiveness and accuracy of the proposed methodology.
Keywords: Riccati equation, Singular system, Matrix sign function
1. Introduction. Singular systems are often encountered in many fields of science and
engineering systems, including circuits, economic systems, boundary control systems and
chemical processes [1]. Over the past decades, much effort has been invested in the anal-
ysis, synthesis and applications of singular systems due to the fact that singular systems
appear more nature to represent the real systems than the regular systems (state-space
systems) [1-5]. The real singular systems usually consist of the non-dynamic subsystems
and the dynamic subsystems, which are mathematically governed by the mixed represen-
tation of algebraic and differential equations. The complex nature of the singular systems
often encounters many difficulties in finding the analytical and numerical solutions to such
systems, particularly when there is a need for their control.
Over the past decades, the theory and design of linear quadratic regulator (LQR) for
optimal control of the regular systems have been well-developed and successfully applied
to many practical design problems [6-10]. Instead of tuning the controllers to satisfy the
desirable classical control specifications for regular systems, the optimal controller can be
easily designed by tuning the weighting matrices {Q,R} in the algebraic Riccati equation,
2771
2772 C.-C. HUANG, J. S.-H. TSAI, S.-M. GUO, Y.-J. SUN AND L.-S. SHIEH
for which many analytical and numerical solutions are available. The methodologies to
find specific weighting matrices {Q,R} for optimal control of regular systems have been
well-developed in the literature but not for singular systems, which is an open problem
to be solved.
The motivation of this paper is to propose a constructive methodology for determining
the appropriate weighting matrices {Q,R}, which guarantees the solvability of the gener-
alized algebraic Riccati equation and for solving the Riccati equation via the matrix sign
function method for the singular systems. A decomposition technique is developed to de-
compose the singular system into a reduced-order regular subsystem and a non-dynamic
subsystem. As a result, the well-known analysis and synthesis methodologies developed
for a regular system can be applied to the reduced-order regular subsystem. Finally, we
transform the results obtained for the reduced-order regular subsystem back to those for
the original singular system. The computationally fast and numerically stable matrix
sign function method is used to obtain the solution of the generalized algebraic Riccati
equation for optimal control of the linear continuous-time singular system.
Consider the stabilizable [1] n-th order generalized linear, time-invariant system char-
acterized by
Ex˙(t) = Ax(t) +Bu(t), (1)
where x(t) ∈ ℜn is the states, u ∈ ℜm is the control, E ∈ ℜn×n, A ∈ ℜn×n and B ∈ ℜn×m
are real constant matrices, and E is possibly singular. In recent studies, the algebraic
Riccati equation (ARE) for the regular system [11-19] has been generalized to the ARE
[18,19] with the nonsingular matrix E in (1). The generalized Riccati equation [19] is
given by
ATPE+ETPA−ETPBR−1BTPE+Q=On×n, (2)
where Q ∈ ℜn×n, R ∈ ℜm×m and P ∈ ℜn×n are real constant matrices. It should remark
that the generalized Riccati Equation (2) might have no solution, even if the selected Q
and R are positive-definite matrices, and E is a singular matrix.
For instance, let[ ] [ ] [ ]
I O A O B
E= κ , A= s , B= s ,
O E O I B
f n×n n−κ n×n f n×m
[Q 0 ]
Q= s , R >O,
0 Qf m×m
[ ] n×n
and P = Ps 0 , where I denotes the κ×κ identity matrix and E is in the Jordan
0 P κ f
f n×n
canonical form. From (2), we have
[ T ] [P A O ] PBR−1BTP P B R−1BTP E
A P O s s s s s s s s f f f
s s + −
O P E O ETP ETP B R−1BTP ETP B R−1BTP E
f f f f f f f s s f f f s f f
[Q O] [O O ]
+ s = k ,
O Q O O
f n−k
which implies
T −1 T
A P +P A +P B R B P +Q =O , (3)
s s s s s s s s s κ
P B R−1BTP E =O , (4)
s s f f f κ×(n−κ)
ETP B R−1BTP =O , (5)
f f f s s (n−κ)×κ
SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2773
P E +ETP +ETP B R−1BTP E +Q =O . (6)
f f f f f f f s f f f (n−κ)
For P > 0 and any non-null matrices B and B , (4) yields P × E = O , which
s f s f f (n−κ)
induces, for example,
0 0 ∗×0 1 0=O , (7)
0 0 ∗ 0 0 1 3
0 0 ∗ 0 0 0
where “∗” denotes free variables. Similarly, (5) gives ET × P = O , which induces,
f f (n−κ)
for example,
0 0 0 0 1 0
1 0 0×0 0 1=O3. (8)
0 1 0 ∗ ∗ ∗
As a result, the pairs of (7) and (8) indicate that P is a null matrix, where the last-right-
f
bottom element denotes as a free variable. This also implies that P is not a positive-
definite matrix.
In general, the respective E and P can be given by
f f
E =block diagonal {E ,E ,··· ,E } (9a)
f f1 f2 fl
and
Pf = block diagonal {Pf ,Pf ,··· ,Pf }. (9b)
1 2 l
For example, let
0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0
E = , E =0 0 0 , and E = 0 0 0 1 0 ,
f f f
i 0 0 0 1 j k
0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0
where 1 ≤ i < j < k ≤ l, which gives
0 0 0 0 0 0 0 0 0
0 0 0 0 ∗ ∗ ∗ 0 ∗ 0 0 0
∗ ∗ ∗ 0 0 0 0 0
P = , P = , and P = ,
fi fj f
0 0 0 0 k
0 0 0 ∗ ∗ ∗ ∗ 0 0 0 0 0
0 0 0 0 ∗
where “∗” denotes free variables. The triple (4)-(6) also gives Qf = O. From the above
illustrative examples, we can conclude that P and Q are not positive-definite matrices.
Therefore, even if the selected Q and R are positive-definite matrices, and E is a singular
matrix, the generalized Riccati Equation (2) might have no solution.
By utilizing the neural network approaches [20-23] but without explicitly providing a
constructive way for determining the weighting matrices {Q,R}, various solution methods
for the generalized Riccati equation in (2) can be found in [20-23]. This paper proposes
a constructive method to determine the weighting matrices {Q,R} for the solution of the
generalized Riccati equation in (2) for singular systems via the computationally fast and
numerically stable matrix sign function method.
2. Problem Formulation and Main Result. Considerthecontrollable linear continu-
ous-time singular system
E x˙(t) = A x(t) + B u(t), (10)
r r r
wherex(t) ∈ ℜn is the states, u(t) ∈ ℜm is the control, Er ∈ ℜn×n is a singular matrix, and
A ∈ℜn×n and B ∈ℜn×m are real constant matrices. The singular system is assumed to
r r
2774 C.-C. HUANG, J. S.-H. TSAI, S.-M. GUO, Y.-J. SUN AND L.-S. SHIEH
be controllable at finite and impulsive modes. The singular system can be transformed
into the slow and fast subsystem models [24], such as (Appendix A)
ˆ˙ ˆ ˆ
Exˆ(t) = Axˆ(t) + Bu(t), (11)
where
[ ] [I O] [ ˆ ] ˆ
xˆ κ A O B
s ˆ ˆ s ˆ s
xˆ = xˆ , E= ˆ , A= , B= ˆ ,
f O E O I B
n×1 f n×n n−κ n×n f n×m
ˆ
the Os denote null matrices with appropriate sizes, E is in the Jordan canonical form
∑ f
d ˆ
with d blocks of sizes u ,u ,··· ,u , and u = column (row) number of E .
1 2 d i=1 i f
Lemma2.1. Given the linear controllable continuous-time singular system (10), the gen-
eralized algebraic Riccati equation for the steady-state linear quadratic regulator is
ATP E +ETP A −ETP B R−1P E +Q =O . (12)
r r r r r r r r r r r r r n
Proof: For the finite-time linear quadratic regulator (LQR) problem, let the quadratic
cost function for the singular system (10) be chosen as
∫ T
1 f [ ]
minJ = xT(t)Q x(t)+uT(t)R u(t) dt, (13)
c r r
u(t) 2 0
where Q ≥ O, R > O, and T is the final time. Here, the Pontryagin’s maximum
r r f
principle [9] is used to solve this optimization problem. Define a Hamiltonian as
1 ( T T ) T
H(t) = x (t)Q x(t)+u (t)R u(t) +λ (A x(t)+B u(t)),
2 r r r r
where λ(t) ∈ ℜn×1 is an un-determined multiplier function. The state and costate equa-
tions are respectively given as
∂H(t) = A x(t)+B u(t) = E x˙(t),
∂λ(t) r r r
∂H(t) T T ˙
=Qrx(t)+A λ(t)=−E λ(t),
∂x(t) r r
and the stationary condition is
∂H(t) = R u(t)+BTλ(t) = O.
∂u(t) r r
Solving the last equation yields the optimal control law in terms of the costate equation
as
u(t) = −R−1BTλ(t). (14)
r r
Substituting (14) into (10) yields
E x˙ = A x(t)−B R−1BTλ(t),
r r r r r
which can be combined with the costate equation to give the homogeneous Hamiltonian
system as [ ] [ ][ ]
Erx˙(t) A −B R−1BT x(t)
= r r r r . (15)
˙ T λ(t)
E λ(t) −Q −A
r r r
The coefficient matrix in (15) is called the Hamiltonian matrix. Let
λ(t) = Pr(t)Erx(t),
which implies ETλ(t) = ETP (t)E x(t) and
r r r r
u(t) = −R−1BTP E x(t), (16)
r r r r
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