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International Journal of Control
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Optimal control of discrete-time singularly perturbed systems
M. Bidani , N. E. Radhy & B. Bensassi Published online: 08 Nov 2010.
To cite this article: M. Bidani , N. E. Radhy & B. Bensassi (2002) Optimal control of discrete-time singularly perturbed systems, International Journal of Control, 75:13, 955-966
To link to this article: http://dx.doi.org/10.1080/00207170210156152
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Optimal control of discrete-time singularly perturbed systems
M. BIDANI{*, N. E. RADHY{ and B. BENSASSI{
In this paper, we present a new algorithm for solving the optimal control of discrete-time singularly perturbed systems.
The main idea of this algorithm is based on two steps. First, the Hamiltonian di erence equation is reduced to the backward recursive form rather than the forward recursive form. Second, the bilinear transformation is applied to transform the derived non-symmetric discrete-time Riccati equations into continuous-time equations. In order to improve the e ciency of this scheme, two matrix permutations are introduced into this algorithm by taking into account the previous work of Gajic and Shen (1991). Therefore, substantial numerical advantages are gained; namely, computa- tion and memory requirements. The F-8 aircraft model is used to illustrate the e ciency of the proposed method.
1. Introduction
Singular perturbations applied to the analysis and feedback control of continuous-time systems have matured over the last decades (Kokotovic et al. 1986).
The extension to discrete-time and sampled-data sys- tems has also been considered in di erent set-ups by many researchers (Litkouhi and Khalil 1984). Among the various forms, two main structures of singularly perturbed linear sampled-data systems were given: the fast time scale structure (see, for example, Butuzov and Vasileva 1971, Hoppensteadt and Miranker 1977, Blankenship 1981, Litkouhi 1983, Litkouhi and Khalil 1984, 1985, Mahmoud 1986, Khorasani and Azimi- Sadjadi 1987, Oloomi and Sawan 1987), and the slow time scale structure (see, for example, Phillips 1980, Naidu and Rao 1985, Mahmoud et al. 1986). Since the slow time scale version presupposes the asymptotic sta- bility of fast models, it seems that in the design proce- dure of stabilizing feedback controllers, the fast time scale version is much more appropriate (Litkouhi and Khalil 1985). We will consider in this work then the fast time scale structure of singularly perturbed sampled-data systems de®ned by Litkouhi and Khalil (1984).
The algebraic equations stated in Su et al. (1992) have the structure of general non-symmetri c continu- ous-time Riccati equations (NCRE), which for su - ciently small values of the singular perturbation parameter can be solved by performing iterations using the ®xed point (Kokotovic et al. 1980) and Newton methods (Su et al. 1992). However, if the sin- gular perturbation parameter is not small enough, the above methods will not produce the desired solutions,
and the decomposition method will not be accurate.
Some researchers have concentrated on this problem considering the case where the value of the singular per- turbation parameter is not very small; and recently two methods based on the asymptotic expansion and Taylor series (Derbel et al. 1994) and the eigenvector approach (Kecman et al. 1999) were proposed to overcome the previous problem.
The idea for realizing the exact decomposition on the discrete-time optimal control is based, for the ®rst time, on the use of the Hamiltonian di erence matrix as a forward recursion form (Gajic et al. 1995, Lim et al.
1995). Our idea here consists in using the backward recursion form of the Hamiltonian di erence matrix.
This scheme conducts to the order-reduction pure-slow and pure-fast non-symmetric discrete-time Riccati equa- tions (NDRE). It is known, however, that the partition of a discrete-time Riccati equation of a singularly per- turbed system produces a number of terms that make corresponding problems numerically ine cient, even though problem order reduction is achieved (see Gajic and Shen 1991). Following Gajic and Shen (1991), we take advantage of the bilinear transformation to trans- form the resulting NDRE into NCRE. The use of mild methods results in reduced memory requirements and gain of computations. The F-8 aircraft model demon- strates the e ciency of these improvements.
Throughout this paper, we will use the following natation. Given a matrix L, L T denotes the transpose of L, L
¡1the inverse of L. I
nis the n £ n identity matrix.
1.1. Preliminaries
The in®nite-time optimal linear regulator problem for the linear discrete-time singularly perturbed system presented in Litkouhi and Khalil (1984) under the fast time version
x 1 … n ‡ 1 † ˆ … I
n1‡ "A 11 † x 1 … n † ‡ "A 12 x 2 … n † ‡ "B 1 u … n † … 1 † x 2 …n ‡ 1† ˆ A 21 x 1 …n† ‡ A 22 x 2 …n† ‡ B 2 u…n† …2†
International Journal of ControlISSN 0020±7179 print/ISSN 1366±5820 online#2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals
DOI: 10.1080/00207170210156152
Finally accepted April 2002.
* Author for correspondence. e-mail: Bidani@scientist.
com
{ Faculte des Sciences ain Chock, Laboratoire d’Auto- matique et Informatique Industrielle, B.P. 5366, Maarif, Route d’el Jadida Casablanca, Morocco.
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(where x 1 2 <
n1, x 2 2 <
n2and u 2 <
minclude, respec- tively, slow state, fast state and the input vector), is de®ned as the problem of ®nding a control u n … † 2 <
m, n 2 ‰ 0; 1 Š , so as to regulate the state
x 1 … † n x 2 … † n
" #
to the origin by way of minimising the cost functional
J ˆ "
2 X
1nˆ0
x 1 …n†
x 2 …n†
" #
T
Q x 1 …n†
x 2 …n†
" #
‡ u T … †Ru n n … † 0
@
1 A …3†
where the … n 1 ‡ n 2 † £ … n 1 ‡ n 2 † and m £ m weighting matrices Q and R penalize excessive values of state
x 1 … † n x 2 … † n
" #
and control u n … † respectively, and that R is a positive de®nite matrix and Q a positive semi-de®nite matrix.
The solution of the discrete linear quadratic regula- tor (DLQR) problem (1)±(3) is the optimal closed-loop control law (Litkouhi and Khalil 1984)
u…n† ˆ ¡ ¡ R ‡ B T PB
¡1B T PAx…n† …4†
where P is the solution of the discrete algebraic Riccati equation
P ˆ A T PA ‡ Q ¡ A T PB B ¡ T PB ‡ R
¡1B T PA …5†
and
A ˆ I
n1‡ "A 11 "A 12
A 21 A 22
" #
; B ˆ "B 1
B 2
" #
x n … † ˆ x 1 … † n x 2 … † n
" #
Note that the presence of a small positive parameter " in matrices A and B makes the resolution of DLQR ill conditioned and the matrix P ˆ P … † " may be partitioned as
P 1 =" P 2
P T 2 P 3
" #
(see Litkouhi and Khalil 1984). Then the expansion of the partitioned matrix products will produce a number of terms and make reduction of the computation limited even though equation (5) is exactly decomposed. Gajic and Shen (1991) had discussed this problem and they had used a bilinear interpolation to transform the discrete-time Riccati equation (5) into a continuous- time equation.
The corresponding continuous-time Riccati equation to (5), presented under fast time version, is
0 ˆ A T P ‡ PA ‡ Q ¡ PBR
¡1B T P …6†
with
A ˆ " A 1 " A 2
A 3 A 4
" #
; B ˆ " B 1
B 2
" #
The expressions of A, B, Q and R are deduced from Gajic and Shen (1991). We prefer to use the fast time version instead of the slow time version, since in our algorithm we will not use explicitly the perturbation parameter, " , and we suppose that " is not precisely determined.
Our goal here is to realize, in the ®rst step, the exact decomposition of DLQR (1)±(3) in order to overcome the sti ness problem. The second step consists in the use of a bilinear interpolation to substitute the resulting non-symmetric algebraic discrete Riccati equations with those of the non-symmetric continuous-time alge- braic Riccati equations. To achieve this goal, we have presented two schemes of the decomposition. The ®rst scheme is based on using the Hamiltonian di erence matrix as a forward recursion form. This scheme appears in Lim et al. (1995) and Gajic et al. (1995) and decomposes exactly the ill-de®ned discrete-time sin- gularly perturbed algebraic Riccati equation into two reduced-order pure-slow and pure-fast well-de®ned con- tinuous-time algebraic Riccati equations. The second scheme consists in using the Hamiltonian di erence matrix as the backward recursion. According to this idea, it appears that the optimal linear quadratic control problem of singularly perturbed discrete systems can be solved in terms of completely independent pure-slow and pure-fast discrete-time algebraic Riccati equations.
To improve the e ciency of the proposed exact decom- position method, two permutation matrices are intro- duced in the second scheme allowing signi®cantly reduced computational and memory requirements. The existence of some inverse matrices in the ®rst scheme makes it, if there exists some small eigenvalues of these matrices, less accurate and undesirably the computation needs a huge memory requirement to give satisfactory accuracy.
Due to the complete and exact decomposition, the proposed method is suitable for the parallel computing of optimal control in discrete-time singularly perturbed systems.
2. Exact decomposition of the system
2.1. First scheme (Gajic et al. 1995, Lim et al. 1995) Assumption 1: The matrix A 22 is non-singular .
Under Assumption 1, this scheme is based on the classical application of the forward recursion form of the Hamiltonian di erence matrix
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x n … ‡ 1 † p n … ‡ 1 †
" #
ˆ H x n … † p n … †
" #
…7†
with
Hˆ A ‡ BR
¡1B T A
¡TQ ¡BR
¡1B T A
¡T¡A
¡TQ A
¡T" #
to resolve the DLQR sti ness problem of the system (1)±(3). Hence Lim et al. (1995) and Gajic et al. (1995) introduced a permutation matrix E 1 ,
E 1 ˆ
I
n10 0 0
0 0 "I
n10
0 I
n20 0 0 0 0 I
n22 6 6 6 6 6 4
3 7 7 7 7 7 5
and put
p ˆ p 1 ="
p 2
" #
with p 1 2 <
n1and p 2 2 <
n2, to change the forward recur- sion form (7) to
x 1 …n ‡ 1†
p 1 …n ‡ 1†
x 2 …n ‡ 1†
p 2 …n ‡ 1†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
ˆ I 2n
1‡ "T 1 "T 2
T 3 T 4
" # x 1 …n†
p 1 …n†
x 2 …n†
p 2 …n†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
… 8 †
where
T 1 ˆ A 1 S 1
Q 1 A T 11 2
4
3
5 ; T 2 ˆ A 2 S 2
Q 2 A T 21 2
4
3 5
T 3 ˆ A 3 S 3
Q 3 A T 12 2
4
3
5 ; T 4 ˆ A 4 S 4
Q 4 A T 22 2
4
3 5
Note that we only have an implicit expression of matrices, with `bar’, formed by the computer in the pro- cess of calculations.
To pursue the ®rst decompose scheme, the Chang’s transformation K (Chang 1972)
K ˆ I 2n
1¡ "H L ¡ "H
L I 2n
22 4
3 5
² 1 …n†
² 2 …n†
& 1 …n†
& 2 …n†
2 6 6 6 6 6 6 4
3 7 7 7 7 7 7 5
ˆ K x 1 …n†
p 1 …n†
x 2 …n†
p 2 …n†
2 6 6 6 6 6 6 4
3 7 7 7 7 7 7 5
is applied to the system (8) entailing two completely decoupled subsystems, a pure-slow subsystem
² 1 …n ‡ 1†
² 2 …n ‡ 1 †
" #
ˆ ¡ I 2n
1‡ "T 1 ¡ T 2 L ² 1 …n†
² 2 …n†
" #
ˆ a 1 a 2
a 3 a 4
" #
² 1 …n†
² 2 …n†
" #
…9†
and a pure-fast subsystem
± 1 …n ‡ 1†
± 2 …n ‡ 1†
" #
ˆ T 4 ‡ "L T 2
¡ ± 1 …n†
± 2 …n†
" #
ˆ b 1 b 2
b 3 b 4
" #
± 1 …n†
± 2 …n†
" #
…10†
provided that matrices L and H satisfy 0 ˆ I 2n
2¡ T 4
¡ L ‡ T 3 ‡ " L ¡ T 1 ¡ T 2 L
0 ˆ H I 2n
2¡ T 4 ¡ "L T 2
¡ ‡ T 2 ‡ " ¡ T 1 ¡ T 2 L
H By assuming that the pure-slow subsystem (9) and the pure-fast subsystem (10) are stabilizable-detectable , they therefore de®ne a pure-slow pencil, P
rs, and a pure- fast pencil, P
rf, to achieve the exact decomposition for the optimal closed-loop linear discrete system (1)±(3) as
² 2 … † n
± 2 … † n
" #
ˆ P
rs0 0 P
rf" #
² 1 … † n
± 1 … † n
" #
… 11 †
which derive two completely decoupled subsystems
² 1 …n ‡ 1† ˆ … a 1 ‡ a 2 P
rs† ² 1 …n†
± 1 …n ‡ 1† ˆ b 1 ‡ b 2 P
rf¡ ± 1 …n†
and two (the pure-slow and the pure-fast) non-sym- metric continuous-time algebraic Riccati equations
P
rsa 1 ¡ a 4 P
rs¡ a 3 ‡ P
rsa 2 P
rsˆ 0 …12†
P
rfb 1 ¡ b 4 P
rf¡ b 3 ‡ P
rfb 2 P
rfˆ 0 …13†
Finally, a matrix E 2 ,
E 2 ˆ
I
n10 0 0 0 0 I
n20 0 I
n10 0 0 0 0 I
n22 6 6 6 6 6 4
3 7 7 7 7 7 5
is introduced to realize the permutation
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² 1 …n†
± 1 …n†
² 2 …n†
± 2 …n†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
ˆ E 2
² 1 …n†
² 2 …n†
± 1 …n†
± 2 …n†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
so that such transformation matrices P and O de®ned as
P ˆ P 1 P 2
P 3 P 4
" #
ˆ E 2 KE 1
O ˆ O 1 O 2
O 3 O 4
" #
ˆ E 1
¡1K
¡1E 2
¡1yield the following exchange between variables of the original system and the decoupled system:
² 1 … † n
± 1 … † n
" #
ˆ … P 1 ‡ P 2 P †x n … †
² 2 … † n
± 2 … † n
" #
ˆ … P 3 ‡ P 4 P †x n … †
P
rs0 0 P
rf" #
ˆ … P 3 ‡ P 4 P † … P 1 ‡ P 2 P †
¡1and
P ˆ O 3 ‡ O 4
P
rs0 0 P
rf" #
³ !
O 1 ‡ O 2
P
rs0 0 P
rf" #
³ !
¡12.2. Second scheme
This scheme is more attractive since it allows the computation of the DLQR problem gain by resorting to the non-symmetric pure-slow and pure-fast discrete Riccati equations instead of computing them by the non-symmetri c continuous-time equations (12) and (13). Furthermore matrices like T
i; i ˆ 1; 2; 3; 4 are explicitly determined. To illustrate this scheme, let us rearrange the Hamiltonian di erence matrix to get the backward recursion form
x…n ‡ 1†
p…n†
" #
ˆ A ¡BR
¡1B T
Q A T
" #
x…n†
p…n ‡ 1†
" #
…14†
Then we obtain x 1 …n ‡ 1†
p 1 …n†
x 2 …n ‡ 1†
p 2 …n†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
ˆ I 2n
1‡ "T 1 "T 2
T 3 T 4
" # x 1 …n†
p 1 …n ‡ 1†
x 2 …n†
p 2 …n ‡ 1†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
…15†
where
T 1 ˆ A 11 ¡B 1 R
¡1B T 1
Q 1 A T 11
2 4
3
5 ; T 2 ˆ A 12 ¡B 1 R
¡1B T 2
Q 2 A T 21
2 4
3 5
T 3 ˆ A 21 ¡B 2 R
¡1B T 1
Q T 2 A T 12
2 4
3
5 ; T 4 ˆ A 22 ¡B 2 R
¡1B T 2
Q 3 A T 22
2 4
3 5
Q ˆ Q 1 Q 2
Q T 2 Q 3
" #
by choosing
p ˆ p 1 ="
p 2
" #
with p 1 2 <
n1and p 2 2 <
n2and interchanging the second and third rows in (14).
To perform the decomposition of the transformed system (15), consider the Chang matrix (Chang 1972) de®ned by
K ˆ I 2n
1¡ " HL ¡ " H
L I 2n
2" #
K
¡1ˆ I 2n
1"H
¡L I 2n
2¡ "LH
" #
9 >
> >
> >
> =
> >
> >
> >
;
…16†
such that the dichotomy of (15) is derived in two steps.
First
± 1 …n†
± 2 …n ‡ 1†
" #
ˆ x 2 …n†
p 2 …n ‡ 1†
" #
‡ L x 1 …n†
p 1 …n ‡ 1†
" #
is substituted, where L is chosen to satisfy I 2n
2¡ T 4
¡ L ‡ T 3 ‡ "L … T 1 ¡ T 2 L † ˆ 0 …17†
so that the lower left corner block is eliminated. Second
² 1 …n†
² 2 …n ‡ 1†
" #
ˆ x 1 …n†
p 1 …n ‡ 1†
" #
¡ "H ± 1 …n†
± 2 …n ‡ 1†
" #
is substituted, where H is chosen to satisfy
H I 2n
2¡ T 4 ¡ "LT 2
¡ ‡ T 2 ‡ " … T 1 ¡ T 2 L †H ˆ 0 …18†
so that the upper right corner block is eliminated.
Assumption 2: …I
n2¡ A 22 † is non-singular throughout this paper.
The known techniques for solving equations (17) and (18), under Assumption 2, that is the matrix I 2n
2¡ T 4
¡
is non-singular; are: the ®xed point method (Kokotovic et al. 1980), the Newton method (Su et al. 1992, Gajic and Shen 1993), the asymptotic expansion and Taylor series methods (Derbel et al. 1994), and
®nally the eigenvector approach (Kecman et al. 1999).
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These methods are preferred in turn according to the value of ":
Assuming the existence of such matrices L and H satisfying (17) and (18), we get two completely decoupled subsystems while applying the Chang trans- formation (16)
² 1 …n†
² 2 …n ‡ 1†
± 1 …n†
± 2 …n ‡ 1†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
ˆ K
x 1 …n†
p 1 …n ‡ 1†
x 2 …n†
p 2 …n ‡ 1†
2 6 6 6 6 6 4
3 7 7 7 7 7 5
…19†
to the system (15). The pure-slow subsystem
² 1 …n ‡ 1†
² 2 …n†
" #
ˆ a 1 a 2
a 3 a 4
" #
² 1 …n†
² 2 …n ‡ 1†
" #
…20†
and the pure-fast subsystem
± 1 …n ‡ 1†
± 2 …n†
" #
ˆ b 1 b 2
b 3 b 4
" #
± 1 …n†
± 2 …n ‡ 1†
" #
… 21 †
where
a 1 a 2
a 3 a 4
" #
ˆ I 2n
1‡ " … T 1 ¡ T 2 L †
b 1 b 2
b 3 b 4
" #
ˆ T 4 ‡ " LT 2
We derive then the closed-loop form of two completely decoupled slow and fast subsystems (20) and (21)
² 1 … n ‡ 1 † ˆ a 1 ‡ a 2 P
rsI
n1¡ a 2 P
rs¡
¡1a 1
³ ´
² 1 … n † … 22 †
± 1 …n ‡ 1† ˆ b 1 ‡ b 2 P
rfI
n2¡ b 2 P
rf¡
¡1b 1
³ ´
± 1 …n† …23†
with
² 2 …n ‡ 1†
± 2 …n ‡ 1†
" #
ˆ P
rsI
n1¡ a 2 P
rs¡
¡1a 1 0
0 P
rf¡ I
n2¡ b 2 P
rf ¡1b 1 2
4
3 5
£ ² 1 …n†
± 1 …n†
" #
…24†
and two reduced-order non-symmetric algebraic dis- crete-time Riccati equations
P
rsˆ a 4 P
rsa 1 ‡ a 3 ‡ a 4 P
rsI
n1¡ a 2 P
rs¡
¡1a 2 P
rsa 1 …25†
P
rfˆ b 4 P
rfb 1 ‡ b 3 ‡ b 4 P
rfI
n2¡ b 2 P
rf¡
¡1b 2 P
rfb 1 …26†
Thereafter, the existence of unique solution P
rs(resp.
P
rf) corresponding to equation (25) (resp. (26)) is guar- anteed by the following lemmas.
Assumption 3: The fast subsystem A 22 ; B 2 ; Q 3
… p † … 21 † is stabilizable-detectable .
Lemma 1: Under Assumption 3 there exists " 1 > 0 such that for any " µ " 1 a unique solution of … 26 † exists.
Proof: First, the computation of b
i… i ˆ 1; 2; 3; 4 † yields
b 1 b 2
b 3 b 4
" # ˆ
"
A 22 ‡ " … L 1 A 12 ‡ L 2 Q 2 †
Q 3 ‡ " … L 3 A 12 ‡ L 4 Q 2 †
¡B 2 R
¡1B T 2 ‡ " L 2 A T 21 ¡ L 1 B 1 R
¡1B T 2
¡
A T 22 ‡ " L 4 A T 21 ¡ L 3 B 1 R
¡1B T 2
¡
#
with
L ˆ L 1 L 2
L 3 L 4
" #
Hence, the O… " † approximation of b
i… i ˆ 1; 2; 3; 4 † b 1 b 2
b 3 b 4
" #
ˆ A 22 ¡B 2 R
¡1B T 2
Q 3 A T 22
" #
yields the symmetric discrete-time Riccati equation represented by
P
rfˆ A T 22 P
rfA 22 ‡ Q 3
¡ A T 22 P
rfB 2 ¡ B T 2 P
rfB 2 ‡ R
¡1B T 2 P
rfA 22 … 27 † Therefore the unique solution P
rfof the equation (27) exists if the system … A 22 ; B 2 ;
Q 3
p † is stabilizable and detectable. Using the implicit function theorem (Ortega and Rheinboldt 1970), the existence and the uniqueness of the solution of equation (26) is then guaranteed. Thereby the proof of this lemma is accom-
plished. &
Assumption 4: The slow subsystem A
o; B
oR
¡1oB T
op
¡ ; C
o† is stabilizable-detectabl e with
A
oˆ I
n1‡ " A 11 ‡ A 12 I
n2¡ A 22
¡
¡1A 21
³ ´
B
oˆ " B 1 ‡ A 12 I
n2¡ A 22
¡
¡1B 2
³ ´
C
oˆ C 1 ‡ C 2 I
n2¡ A 22
¡
¡1A 21
R
oˆ R ‡ D T
oD
oand
D
oˆ C 2 I
n2¡ A 22
¡
¡1B 2 :
Lemma 2: Under Assumption 4 there exists " 2 > 0 such that for any " µ " 2 a unique solution of …25† exists.
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Proof: With L ˆ ¡ … I 2n
2¡ T 4 †
¡1T 3 , the O… " † approxi- mation of a
i0s i … ˆ 1; 2; 3; 4 † pure-slow subsystem pro- duces
a 1 a 2
a 3 a 4
" #
ˆ I 2n
1‡ "T 1
¡ ‡ "T 2 I 2n
2¡ T 4
¡
¡1T 3
which yields
a 1 ˆ I
n1‡ " …A 11 ‡ A 12 I
n2¡ A 22
¡
¡1A 21 † a 2 ˆ ¡B
oR
¡1oB T
oB
oˆ " …B 1 ‡ A 12 I
n2¡ A 22
¡
¡1B 2 † C
oˆ C 1 ‡ C 2 …I
n2¡ A 22 †
¡1A 21
D
oˆ C 2 …I
n2¡ A 22 †
¡1B 2
R
oˆ R ‡ D T
oD
o; a 4 ˆ a T 1 ; a 3 ˆ C T
oC
oand that C 1 is a full-rank factorization of Q 1 (i.e.
Q 1 ˆ C T 1 C 1 and rank (Q 1 )ˆ rank (C 1 )) and C 2 is a full-rank factorization of Q 3 (i.e. Q 3 ˆ C 2 T C 2 and rank (Q 3 )ˆ rank (C 2 )) then it results in the symmetric discrete-time Riccati equation
P
rsˆ a T 1 P
rsa 1 ‡ a 3 ‡ a T 1 P
rsI
n1¡ a 2 P
rs¡
¡1a 2 P
rsa 1 …28†
The solution P
rsof equation (28) exists if the system a 1 ; B
o; C
o… † is stabilizable and detectable (see, for instance, Laub 1979 and Pappas et al. 1980).
Using the implicit function theorem (Ortega and Rheinboldt 1970), the existence and the uniqueness of the solution of (25) is then guaranteed which ends the
proof of this lemma. &
2.3. Conversion of NDRE into NCRE
It seems that equations (27) and (28) are more di - cult to resolve than the corresponding continuous-time equations. Therefore, we adopt the bilinear interpola- tion (Gajic and Shen 1991) to transform equations (27) and (28) into corresponding continuous-time equa- tions.
Hence, as a
c1a
c2a
c3a
c4" #
ˆ I
n1¡ 2¯
a¡ 2¯
aa 2 I
n1‡ a 4
¡
¡1¡2³
aa 3 I
n1‡ a 1
¡
¡12³
a¡ I
n12 4
3 5
with
¯
aˆ I
n1‡ a 1 ¡ a 2 I
n1‡ a 4
¡
¡1a 3
³ ´
¡1³
aˆ I
n1‡ a 4 ¡ a 3 I
n1‡ a 1
¡
¡1a 2
³ ´
¡1we compute P
rsfrom
P
rsa
c1¡ a
c4P
rs¡ a
c3‡ P
rsa 2 P
rsˆ 0 …29†
and from b
c1b
c2b
c3b
c4" #
ˆ I
n2¡ 2¯
b2¯
bb 2 I
n2‡ b 4
¡
¡1¡2³
bb 3 I
n2‡ b 1
¡
¡12³
b¡ I
n22 4
3 5
with
¯
bˆ I
n2‡ b 1 ¡ b 2 I
n2‡ b 4
¡
¡1b 3
³ ´
¡1³
bˆ I
n2‡ b 4 ¡ b 3 I
n2‡ b 1
¡
¡1b 2
³ ´
¡1we compute P
rfas
P
rfb
c1¡ b
c4P
rf¡ b
c3‡ P
rfb
c2P
rfˆ 0 …30†
These transformation s hold if and only if the follow- ing assumption is satis®ed.
Assumption 5: The matrices I …
n1‡ a 1 † and I …
n1‡ a 4 † of pure-slow subsystem …20† and matrices I …
n2‡ b 4 † and
I
n2‡ b 1
… † of pure-fast subsystem …21† are invertible.
To accomplish this second scheme of exact decom- position we should introduce, with the same analogy as the ®rst scheme, two new transformations
² 1 … † n
± 1 … † n
² 2 … n ‡ 1 †
± 2 … n ‡ 1 †
2 6 6 6 6 6 6 4
3 7 7 7 7 7 7 5
ˆ G x n … † p n … ‡ 1 †
" #
ˆ E 2 KE 1
x n … † p n … ‡ 1 †
" #
and
x n … † p n … ‡ 1 †
" #
ˆ C
² 1 … † n
± 1 … † n
² 2 … n ‡ 1 †
± 2 … n ‡ 1 †
2 6 6 6 6 6 6 4
3 7 7 7 7 7 7 5
ˆ E 1
¡1K
¡1E
¡12
² 1 … † n
± 1 … † n
² 2 … n ‡ 1 †
± 2 … n ‡ 1 †
2 6 6 6 6 6 6 4
3 7 7 7 7 7 7 5
in such way that the original coordinates communicate with the pure-slow and pure-fast coordinates.
Hence, with the partition of newly de®ned transfor- mations G as
Downloaded by [UZH Hauptbibliothek / Zentralbibliothek Zürich] at 19:39 19 August 2013
² 1 … † n
± 1 … † n
² 2 … n ‡ 1 †
± 2 … n ‡ 1 †
2 6 6 6 6 6 4
3 7 7 7 7 7 5
ˆ G 1 G 2
G 3 G 4
" #
x n … † p n … ‡ 1 †
" #
…31†
and C as
x n … † p n … ‡ 1 †
" #
ˆ C 1 C 2
C 3 C 4
" # ² 1 … † n
± 1 … † n
² 2 … n ‡ 1 †
± 2 … n ‡ 1 †
2 6 6 6 6 6 4
3 7 7 7 7 7 5
…32†
we succeed to the exchange of the variable between the pure-slow, ² 1 , P
rs, and pure-fast, ± 1 , P
rf, coordinates, on the one hand, and the coordinates of x and P, on the other hand
² 1 …n†
± 1 …n†
" #
ˆ … G 1 ‡ G 2 P I ¡
n1‡n2‡ BR
¡1B T P
¡1A†x…n† …33†
² 2 …n†
± 2 …n†
" #
ˆ … G 3 ‡ G 4 P I ¡
n1‡n2‡ BR
¡1B T P
¡1A†x…n† …34†
P
rsI
n1¡ a 2 P
rs¡
¡1a 1 0
0 P
rf…I
n2¡ b 2 P
rf†
¡1b 1
2 4
3 5
ˆ … G 3 ‡ G 4 P I ¡
n1‡n2‡ BR
¡1B T P
¡1A†
£…G 1 ‡ G 2 P I ¡
n1‡n2‡ BR
¡1B T P
¡1A†
¡1and reversely
x…n† ˆ C
1‡ C
2P
rsI
n1¡ a
2P
rs¡
¡1a
10
0 P
rfI
n2¡ b
2P
rf¡
¡1b
12 4
3 5 0
@
1 A ²
1…n†
±
1…n†
" #
P I ¡
n1‡n2‡ BR
¡1B
TP
¡1A ˆ C
3‡ C
4P
rsI
n1¡ a
2P
rs1¡
¡1a
10
0 P
rfI
n2¡ b
2P
rf¡
¡1b
12 4
3 5 0
@
1 A
£ C
1‡ C
2P
rsI
n1¡ a
2P
rs¡
¡1a
10
0 P
rfI
n2¡ b
2P
rf¡
¡1b
12 4
3 5 0
@
1 A
¡1
To improve the e ciency of this scheme in order to reduce the computation and to make it independent of the value of the singular perturbation parameter " , we have to de®ne two permutation matrices E 3
² 1 …n†
² 2 …n†
± 1 …n†
± 2 …n†
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
ˆ
I
n10 0 0 a 3 a 4 0 0 0 0 I
n20 0 0 b 3 b 4
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
² 1 …n†
² 2 …n ‡ 1†
± 1 …n†
± 2 …n ‡ 1†
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
ˆ E 3
² 1 …n†
² 2 …n ‡ 1†
± 1 …n†
± 2 …n ‡ 1†
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
and E 4
x 1 …n†
p 1 …n†
x 2 …n†
p 2 …n†
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
ˆ
I
n10 0 0
"Q 1 I
n1‡ "A T 11
¡ "Q 2 "A T 21
0 0 I
n20
Q T 2 A T 12 Q 3 A T 22
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
x 1 …n†
p 1 …n ‡ 1†
x 2 …n†
p 2 …n ‡ 1†
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
ˆ E 4
x 1 … n † p 1 …n ‡ 1†
x 2 …n†
p 2 …n ‡ 1†
2 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
Hence, the Chang transformation used by the ®rst scheme can be restituted by the equality
K ˆ E 3 KE 4
¡1leading therefore to the following exchange, instead of those de®ned by G (31) and C (32)
x…n† ˆ …P 1 ‡ P 2 P†
¡1² 1 …n†
± 1 …n†
" #
ˆ O 1 ‡ O 2
P
rs0 0 P
rf2 4
3 5 0
@
1 A ² 1 …n†
± 1 …n†
" #
P ˆ O 3 ‡ O 4
P
rs0 0 P
rf" #
³ !
O 1 ‡ O 2
P
rs0 0 P
rf" #
³ !
¡1Downloaded by [UZH Hauptbibliothek / Zentralbibliothek Zürich] at 19:39 19 August 2013
Hence the discrete feedback gain, G, corresponding to the system (1)±(3) is given by
G ˆ ¡ R ‡ B T PB
¡1B T PA
By considering matrices A, B, R and Q, the outline of the parallel linear quadratic regulator algorithm is designed as follows.
Step 1. Forming matrices T 1 , T 2 , T 3 and T 4 .
Step 2. If Assumptions 1±5 are satis®ed then computing matrices L, H , P
rsand P
rf, following the small-
ness perturbation parameter, by iterative methods or eigenvector and Schur approach methods.
Step 3. Computing the matrices E 1 , E 2 , E 3 , E 4 , O and K
¡1(see below). Note that, in this algorithm, "
is not used at all which con®rms the robustness of the proposed algorithm toward the perturba- tion parameter.
We designate by 0 1 : n
i; 1 : n
j¡ a n
i£ n
j¡ zeros
matrix and by F 1 : n
i; 1 : n
j¡ a n
i£ n
j¡ truncated
matrix of F .
E
4ˆ
I
n10…1 : n
1;1 : n
1† 0 1 … : n
1;1 : n
2† 0 1 … : n
1; 1 : n
2† Q … 1 : n
1; 1 : n
1† A … 1 : n
1; 1 : n
1†
TQ … 1 : n
1; n
1‡1 : n
1‡n
2† A n …
1‡1 : n
1‡n
2;1 : n
1†
T0 1 … : n
2; 1 : n
1† 0 1 … : n
2;1 : n
1† I
n20 1 … : n
2; 1 : n
1† Q … 1 ‡ n
1: n
1‡n
2; 1 : n
1† A … 1 : n
1; n
1‡1 : n
1‡n
2†
TQ n …
1‡1 : n
1‡n
2; n
1‡1 : n
1‡n
2† A n …
1‡1 : n
1‡n
2; n
1‡1 : n
1‡n
2†
T2
6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
E
3ˆ
I
n10 1 … : n
1; 1 : n
1† 0 1 … : n
1; 1 : n
2† 0 1 … : n
1;1 : n
2† a n …
1‡1 : 2n
1; 1 : n
1† a n …
1‡1 : 2n
1; n
1‡1 : 2n
1† 0 1 … : n
1; 1 : n
2† 0 1 … : n
1;1 : n
2† 0 1 … : n
2; 1 : n
1† 0 1 … : n
2; 1 : n
1† I
n20 1 … : n
2;1 : n
2† 0 1 … : n
2; 1 : n
1† 0 1 … : n
2; 1 : n
1† b n …
2‡ 1 : 2n
2; 1 : n
2† b n …
2‡ 1 : 2n
2;n
2‡ 1 : 2n
2† 2
6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
E
1ˆ
I
n10 1 … : n
1; 1 : n
2† 0 1 … : n
1;1 : n
1† 0 1 … : n
1; 1 : n
2† 0 1 … : n
1; 1 : n
1† 0 1 … : n
1; 1 : n
2† I
n10 1 … : n
1; 1 : n
2† 0 1 … : n
2; 1 : n
1† I
n20 1 … : n
2;1 : n
1† 0 1 … : n
2; 1 : n
2† 0 1 … : n
2; 1 : n
1† 0 1 … : n
2; 1 : n
2† 0 1 … : n
2;1 : n
1† I
n22 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
E
2ˆ E
1Tˆ
I
n10 1 … : n
1; 1 : n
1† 0 1 … : n
1; 1 : n
2† 0 1 … : n
1;1 : n
2† 0 1 … : n
2; 1 : n
1† 0 1 … : n
2; 1 : n
1† I
n20 1 … : n
2;1 : n
2† 0 1 … : n
1; 1 : n
1† I
n10 1 … : n
1; 1 : n
2† 0 1 … : n
1;1 : n
2† 0 1 … : n
2; 1 : n
1† 0 1 … : n
2; 1 : n
1† 0 1 … : n
2; 1 : n
2† I
n22 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 5
K
¡1ˆ I
2n1H
¡L I
2n2¡ LH 2
4
3
5 ; K
¡1ˆ E
4K
¡1E
¡13; O ˆ E
T1K
¡1E
1P ˆ (
O … n
1‡n
2‡1 : 2 … n
1‡ n
2†; 1 : n
1‡n
2† ‡ O … n
1‡ n
2‡ 1 : 2 … n
1‡ n
2†;
£ n
1‡ n
2‡ 1 : 2 … n
1‡ n
2†† P
rs0 1 … : n
1; 1 : n
2† 0 1 … : n
2; 1 : n
1† P
rf2 4
3 5 9 =
;
£ (
O … 1 : n
1‡ n
2; 1 : n
1‡ n
2† ‡ O … 1 : n
1‡ n
2; n
1‡ n
2‡ 1 : 2 … n
1‡ n
2† † P
rs0 1 … : n
1; 1 : n
2† 0 1 … : n
2; 1 : n
1† P
rf2 4
3 5 9 =
;
¡1
