Optimal control of multi delay systems via orthogonal functions

International Journal of Advanced Research in Engineering & Technology
Volume 1 • Issue 1 • May 2010 • pp. 1 – 24
http://iaeme.com/ijaret.html

I JARET
(c) I A E M E

Optimal Control of Multi-Delay
Systems via Orthogonal Functions
B.M.Mohan∗and Sanjeeb Kumar Kar
Department of Electrical Engineering
Indian Institute of Technology
Kharagpur-721302, India
e-mail: mohan@ee.iitkgp.ernet.in.
e-mail: skkar@ee.iitkgp.ernet.in.

Abstract: Based on using block-pulse functions (BPFs)/shifted Legen-
dre polynomials (SLPs) a unified approach for computing optimal control
law of linear time-invariant/time-varying time-delay free/time-delay dynamic
systems with quadratic performance index is discussed in this paper. The
governing delay-differential equations of dynamic systems are converted into
linear algebraic equations by using the operational matrices of differentiation,
integration, delay and product of orthogonal functions (BPFs and SLPs).
Thus, the problem of finding optimal control law is reduced to the problem
of solving algebraic equations obtained via the operational matrices. Nu-
merical examples are included to demonstrate the applicability of the unified
approach.
Keywords: Optimal control, linear systems, time-invariant systems,
time-varying systems, time-delay systems, orthogonal functions, block-pulse
functions, Legendre polynomials.

1 Introduction
Time-delay systems are those systems in which time delays exist between
the application of input or control to the system and their resulting effect on
it. They arise either as a result of inherent delays in the components of the
system or as a deliberate introduction of time delay into the system for con-
trol purposes. Examples of such systems are electronic systems, mechanical
∗
Corresponding author

1


systems, biological systems, environmental systems, metallurgical systems
and chemical systems. Few practical examples [8] are, controlling the speed
of a steam engine running an electric power generator under varying load
conditions, control of room temperature, cold rolling mill, spaceship control,
hydraulic system etc.
As it appears from the literature, extensive work was done on the problem
of optimal control of linear continuous-time dynamical systems containing
time delays. Palanisamy and Rao [2] appear to be the first to study the
optimal control problem via a class of piecewise constant basis functions -
Walsh functions. They considered time-invariant systems with one delay in
state and one delay in control. Hwang and Shih [4] considered time-varying
systems containing one delay in state and one delay in control, and studied
optimal control problem of such systems via BPFs. Solutions obtained in
[2, 4] are piecewise constant. In order to obtain smooth solution, Horng and
Chou [5] used Chebyshev polynomials of first kind and studied time-invariant
systems with one delay in state only.
Hwang and Chen [6] dealt with time-varying systems with multiple delays
in state and control by using SLPs. Perng [7] investigated the same problem
in [2] by applying SLPs. Tsay et al [9] considered the same problem in [4],
approximated the time-delay terms using Pade approximation, and obtained
the optimal control law using general orthogonal polynomials. Razzaghi et al
[10] replaced Chebyshev polynomials by SLPs and studied the same problem
in [5].
In the recent years, people came up with a new idea of defining hybrid
functions (with BPFs & SLPs) and utilizing the same for studying problems
in Systems and Control. The so called hybrid functions approach was first
introduced by Marzban and Razzaghi [12] to study the optimal control prob-
lem of time-varying systems with time-delay in state only. Subsequently,
this approach was extended to time-invariant systems in [5] by Wang [13].
Khellat [14] used linear Legendre multiwavelets to solve the timevarying sys-
tems having single delay in state only. Hybrid functions (BPFs and Taylor
polynomials) were employed [15] to study the same problem considered in
[12]. Wang [16] used general Legendre wavelets to solve the optimal control
problem of timevarying singular systems having one delay in state only.
Now, the following observations can be made from the above discussions
on the historical developments on the optimal control of time-delay systems
via orthogonal functions :

The BPF approach reported in [4] is restricted to only time-varying
systems containing one delay in state and one delay in control. It is
not yet available for multi-delay systems.

2


The systems considered in [6] are time-varying with multi delays in both
state and control. Such systems are studied only via SLPs. BPF ap-
proach is not yet reported.

Therefore, in this paper, an attempt is made to introduce a unified ap-
proach for computing optimal control of linear time-varying systems with
multiple delays in both state and control via orthogonal functions (BPFs
and SLPs). The proposed approach is different from the one in [4, 6] in the
following two ways :
In [4, 6], x(t) was expressed in terms of orthogonal functions first, the
˙
spectrum of x(t) was expressed in terms of x(t) spectrum, and then u(t) was
˙
finally calculated. In the proposed approach the unknown x(t) is directly
expressed in terms of orthogonal functions, so that the state feedback control
law u(t) can be expressed in terms of the spectrum of x(t). Thus the proposed
approach is straightforward.
Moreover, the manner in which the final cost term in performance index is
handled in terms of orthogonal functions in the proposed approach is different
from that in [4, 6].
The paper is organized as follows: The next section deals with orthogonal
functions (BPFs and SLPs) and their properties. Optimal control of time-
delay, delay free, time-invariant and time-varying systems via BPFs and SLPs
is considered in Section 3. Section 4 contains some numerical examples. The
last section concludes the paper.

2 Orthogonal functions and their properties
We consider two classes of orthogonal functions, namely BPFs and SLPs,
and discuss their properties.

2.1 BPFs and their properties [3]
A set of m BPFs, orthogonal over t ∈ [t0 , tf ), is defined as

1, t0 + iT ≤ t < t0 + (i + 1)T
Bi (t) = (1)
0, otherwise

for i = 0, 1, 2, . . . , m − 1 where
tf − t0
T = , the block-pulse width (2)
m

3


A square integrable function f (t) on t0 ≤ t ≤ tf can be approximately
represented in terms of BPFs as
m−1
f (t) ≈ fi Bi (t) = f T B(t) (3)
i=0

where
T
f= f0 , f1 , . . . , fm−1 (4)
is an m - dimensional block-pulse spectrum of f (t), and
T
B(t) = B0 (t), B1 (t), . . . , Bm−1 (t) (5)

an m - dimensional BPF vector. fi in Eq. (3) is given by
t0 +(i+1)T
1
fi = f (t)dt (6)
T t0 +iT

which is the average value of f (t) over t0 + iT ≤ t ≤ t0 + (i + 1)T.

2.1.1 Integration of B(t) [3]
Integrating B(t) once with respect to t and expressing the result in m - set
of BPFs, we have
t
B(τ )dτ ≈ HB(t) (7)
t0

where  
1
1 1 ... 1
2
 0 1 1 ... 1 
 2 
 0 0 1 ... 1 
H=T 2  (8)
 . . .
.
. . . . 
. 
 . . .
1
0 0 0 ... 2
is called the integration operational matrix of BPFs and it is an m×m upper
triangular matrix.

2.1.2 Representation of a time-delay vector function in BPFs [4]
Assume that f (t) is n - dimensional, and

f (t) = ζ(t) for t ≤ t0 (9)

4


The delayed vector function f (t − τ ) over t ∈ [t0 , tf ] may be approximated
in terms of BPFs as
m−1
f (t − τ ) fi (τ )Bi (t) = F (τ )B(t) (10)
i=0

where
t0 +(i+1)T
1 ζ i (τ ) for i < µ
fi (τ ) = f (t − τ )dt = (11)
T t0 +iT fi−µ for i ≥ µ

t0 +(i+1)T
1
ζ i (τ ) = ζ(t − τ )dt for i < µ , (12)
T t0 +iT

µ is the number of BPFs considered over t0 ≤ t ≤ t0 + τ , and
F (τ ) = f0 (τ ), f1 (τ ), . . . , fm−1 (τ ) (13)
Let
m−1
f (t) fi Bi (t) = F B(t) (14)
i=0

with
F = f0 , f1 , . . . , fm−1 (15)
Then by letting
     
f0 (τ ) f0 ζ 0 (τ )
 f (τ )   f1   ζ 1 (τ ) 
ˆ (τ ) =  1 .
f 

; ˆ = 
f  .
 ˆ ) = 
 ; and ζ(τ  .

 (16)
 .
.   .
.   .
. 
fm−1 (τ ) fm−1 ζ µ−1 (τ )
ˆ (τ ) can be expressed in terms of ˆ and ζ(τ ) as follows :
f f ˆ
ˆ (τ ) = E(n, µ)ζ(τ ) + D(n, µ)ˆ
f ˆ f (17)
where E and D are called shift operational matrices, given by
 
  .
.
Inµ×nµ .
 nµ×nd .
nµ×nµ

E(n, µ) =  · · · · · · · · ·  ; D(n, µ) =  · · · · · · .
. ······ (18)
 
nd×nµ .
.
Ind×nd . nd×nµ

with d = m − µ.

5


2.1.3 Representation of C(t)f (t) in terms of BPFs [3]
Let C(t) be an n × n matrix and f (t) be an n × 1 vector. Assume that the
elements of C(t) and f (t) are square integrable over t0 ≤ t ≤ tf . Then
m−1 m−1 m−1
C(t)f (t) Ci Bi (t) fj Bj (t) = Ci fi Bi (t) (19)
i=0 j =0 i=0

Let
m−1
C(t)f (t) = g(t) gi Bi (t) = GB(t) (20)
i=0

where
G = g0 , g1 , . . . , gm−1 (21)
Upon comparing Eqs. (19) and (20), we have
gi = Ci fi (22)

2.2 SLPs and their properties [11]
SLPs satisfy the recurrence relation
(2i + 1) i
Li+1 (t) = ϕ(t) Li (t) − Li−1 (t) (23)
(i + 1) (i + 1)

for i = 1, 2, 3, . . . . . . with
2(t − t0 )
ϕ(t) = −1 (24)
(tf − t0 )

L0 (t) = 1, and L1 (t) = ϕ(t) (25)
A function f (t) that is square integrable on t ∈ [t0 , tf ] can be represented in
terms of SLPs as
m−1
f (t) ≈ fi Li (t) = f T L(t) (26)
i=0

Here f is called Legendre spectrum of f (t), and L(t) is called SLP vector. fi
in Eq. (26) is given by
tf
(2i + 1)
fi = f (t)Li (t)dt (27)
(tf − t0 ) t0

6


2.2.1 Integration of L(t) [11]
Integrating L0 (t) once with respect to t, and expressing the result in terms
of SLPs, we have
t
(tf − t0 )
L0 (τ )dτ = [L0 (t) + L1 (t)] (28)
t0 2

Moreover, for i = 1, 2, 3, . . . . . .
t
(tf − t0 )
Li (τ )dτ = [−Li−1 (t) + Li+1 (t)] (29)
t0 2 (2i + 1)

Eqs. (28) and (29) can be written in the form of Eq. (7) with B(t) replaced
by L(t) and
 
1 1 0 0 ... 0 0
 −1 0 1 0 . . . 0 0 
 3 −1 3 1 
(tf − t0 )  0
 5
0 5 ... 0 0  
H=  . . . . . .  (30)
2  .. . . .
. . . .
. . 
. 
 1
 0 0 0 0 ... 0 2m−3

−1
0 0 0 0 . . . 2m−1 0

which is called integration operational matrix of SLPs.

2.2.2 Representation of a time-delay vector function in SLPs [6]
The SLP representation of f (t − τ ) is given by
m−1
f (t − τ ) fi (τ )Li (t) = F (τ )L(t) (31)
i=0

where
(2i + 1) tf
fi (τ ) = f (t − τ )Li (t)dt
(tf − t0 ) t0
(2i + 1) tf −τ
= ζ i (τ ) + f (t)Li (t + τ )dt (32)
(tf − t0 ) t0

with
t0 +τ
(2i + 1)
ζ i (τ ) = ζ(t − τ )Li (t)dt (33)
(tf − t0 ) t0

7


Now we express Li (t + τ ) in terms of SLPs as
i
Li (t + τ ) = λij (τ )Lj (t) = λT (τ )L(t)
i (34)
j =0

with

λij (τ ) = 0 for j > i. (35)

Since
m−1
f (t) ≈ fi Li (t) = F L(t) = LT (t) ⊗ In ˆ
f (36)
i=0

where ⊗ is the Kronecker product [1], substituting Eqs. (34) and (36) into
Eq. (32), we have
tf −τ
(2i + 1) T
fi (τ ) = ζ i (τ ) + λ (τ ) L(t) LT (t) ⊗ In dt ˆ
f
(tf − t0 ) i t0
tf −τ
(2i + 1) T
= ζ i (τ ) + λ (τ ) L(t)LT (t) ⊗ In dt ˆ
f (37)
(tf − t0 ) i t0

Let ∆ = diag 1, 3, . . . , (2m − 1) /(tf − t0 ) (38)
 
λ00 (τ ) 0 0 ... 0
 λ10 (τ ) λ11 (τ ) 0 ... 0 
 
 λ20 (τ ) λ21 (τ ) λ22 (τ ) ... 0 
Λ(τ ) =   (39)
 .
. .
. .
. .
. 
 . . . . 
λm−1,0 (τ ) λm−1,1 (τ ) λm−1,2 (τ ) . . . λm−1,m−1 (τ )

is the time-advanced matrix of order m × m, and
tf −τ
P (τ ) = L(t)LT (t)dt (40)
t0

Then Eq. (37) can be written in vector-matrix form as
ˆi (τ ) =
f ζ(τ ) + ∆ Λ(τ ) (P (τ ) ⊗ In ) ˆ
ˆ f
= ˆ ) + (D(τ ) ⊗ In ) ˆ
ζ(τ f (41)

where

D(τ ) = ∆ Λ(τ )P (τ ) (42)

8


is called the delay operational matrix of SLPs, and
 
ζ 0 (τ )
 ζ (τ ) 
ˆ  1 
ζ(τ ) =  .
.  (43)
 . 
ζ m−1 (τ )

To evaluate D(τ ) we need to know Λ(τ ) and P (τ ).

2.2.3 Time-advanced operational matrix of SLPs [6]

Let L(t + τ ) = Λ(τ )L(t) (44)

Here we present a recursive algorithm to generate the elements of matrix
Λ(τ ).

λ00 (τ ) = 1 (45)
2τ
λ10 (τ ) = ; λ11 (τ ) = 1 (46)
(tf − t0 )

−i (2i + 1)j
λi+1,j (τ ) = λi−1,j (τ ) + λi,j−1 (τ )
(i + 1) (i + 1)(2j − 1)
2(2i + 1)τ (2i + 1)(j + 1)
+ λi,j (τ ) + λi,j+1 (τ ) (47)
(tf − t0 )(i + 1) (i + 1)(2j + 3)
for i = 1, 2, 3, . . . . . . and j = 0, 1, 2, . . . . . .

2.2.4 An algorithm for evaluating the integral in Eq. (40) [6]
tf −τ
Let pij (τ ) = Li (t)Lj (t)dt (48)
t0

tf −τ
•
qij (τ ) = Li(t)Lj (t)dt (49)
t0

and

sij (τ ) = Li (tf − τ )Lj (tf − τ ) − (−1)i+j (50)

Then
2(2i + 1)
pij (τ ) = −qi−1,j (τ ) + qi+1,j (τ ) (51)
(tf − t0 )

9


2(2i + 1)
qi+1,j (τ ) = qi−1,j (τ ) + pi,j (τ ) (52)
(tf − t0 )

qij (τ ) = sij (τ ) − qji (τ ) (53)


(tf −t0 )

 [L0 (tf − τ ) + L1 (tf − τ )] , for j =0
2
(tf −t0 )
p0j (τ ) = pj 0 (τ ) = [−Lj−1 (tf − τ ) + Lj+1 (tf − τ )] , (54)


2(2j+1)
for j = 1, 2, 3 . . . . . .

q0j (τ ) = 0,
for j = 0, 1, 2 . . . . . . (55)
2
and q1j (τ ) = p0j (τ ) (56)
(tf − t0 )

The algorithm for evaluating the elements of matrix P (τ ) is as follows :

step 1: Compute p0j (τ ) and pj0 (τ ) for j = 0, 1, . . . , 2(m − 1) using Eq. (54)

step 2: Set q0j (τ ) = 0 for j = 0, 1, . . . , 2(m − 1)

step 3: Compute q1j (τ ) for j = 0, 1, . . . , 2(m − 1) using Eq. (56)

step 4: Compute qi0 (τ ) and qi1 (τ ) for i = 0, 1, . . . , 2(m − 1) using Eq. (53)

step 5: Set i = 1.

step 6: Compute pj i (τ ) and then pij (τ ) = pji (τ ) for j = i, i + 1, . . . . . . ,
2(m − 1) − i using Eq. (51)

step 7: If i = m − 1 then stop, else proceed.

step 8: Compute qi+1,j (τ ) for j = i + 1, i + 2, . . . , 2(m − 1) − i using
Eq. (52)

step 9: Compute qj, i+1 (τ ) for j = i + 2, i + 3, . . . , 2(m − 1) − i using
Eq. (53)

step 10: Set i = i + 1 and go to step 6.

10


2.2.5 Representation of C(t)f (t) in terms of SLPs [6]
The product of two SLPs Li (t) and Lj (t) can be expressed as
m−1
Li (t)Lj (t) ψijk Lk (t) (57)
k=0

where
tf
(2k + 1)
ψijk = Li (t)Lj (t)Lk (t)dt (58)
(tf − t0 ) t0

Let
tf
πijk = Li (t)Lj (t)Lk (t)dt (59)
t0

then
(2k + 1)
ψijk = πijk (60)
(tf − t0 )
Notice that

πijk = πikj = πjik = πjki = πkji = πkij (61)

Also,
j
al aj−l ai−j+l 2(i − j + 2l) + 1
Li (t)Lj (t) = Li−j+2l (t) (62)
l=0
ai+l 2(i + l) + 1

where i ≥ j, and
(2l + 1)
a0 = 1, al+1 = al , for l = 0, 1, 2, . . . . . . (63)
(l + 1)
Moreover, for i ≥ j
al aj−l ai−j+l (tf −t0 )
if k = i − j + 2l
πijk = ai+l 2(i+l)+1 (64)
0 if k = i − j + 2l

Assuming that C(t) is square-integrable over t ∈ [t0 , tf ], it can be expressed
in terms of SLPs as
m−1
C(t) Ci Li (t) (65)
i=0

11


Then
m−1
C(t)f (t) = g(t) gi Li (t) = GL(t) (66)
i=0

Also
m−1 m−1
C(t)f (t) Cj Lj (t) fk Lk (t) (67)
j =0 k=0

Therefore
tf m−1 m−1
(2i + 1)
gi = Cj fk Li (t)Lj (t)Lk (t)dt
(tf − t0 ) t0 j =0 k=0
m−1 m−1
(2i + 1)
= πijk Cj fk (68)
(tf − t0 ) j = 0 k = 0

3 Optimal control of time-delay systems
Consider an nth order linear time-varying system with multiple delays
α β
x(t) =
˙ Al (t)x(t − τl ) + Bl (t)u(t − θl ) (69)
l=0 l=0
x(t) = ζ(t) for t ≤ t0 (70)
u(t) = ν(t) for t ≤ t0 (71)

where x(t) is an n dimensional state vector, u(t) is an r dimensional control
vector, Al (t) and Bl (t) are n×n and n×r time-varying matrices, and τl , θl ≥ 0
are constant time-delays in state and control respectively.
The objective is to find the control vector u(t) that minimizes the quadratic
cost function
tf
1 T 1
J = x (tf )Sx(tf ) + xT (t)Q(t)x(t) + uT (t)R(t)u(t) dt (72)
2 2 t0

where S and Q(t) are n × n symmetric positive semidefinite matrices, and
R(t) is an r × r symmetric positive definite matrix.
Integrating Eq. (69) once with respect to t, we obtain
t α β
x(t) − x(t0 ) = Al (σ)x(σ − τl ) + Bl (σ)u(σ − θl ) dσ (73)
t0 l=0 l=0

12


We express x(t), u(t), x(t0 ), x(t − τl ), u(t − θl ), Al (t) and Bl (t) in terms of
orthogonal functions {φi (t)} (BPFs or SLPs) as follows :
m−1
x(t) ≈ xi φi (t) = Xφ(t) (74)
i=0
m−1
u(t) ≈ ui φi (t) = U φ(t) (75)
i=0

x(t0 ) = V φ(t) (76)
x(t − τl ) ≈ X (τl )φ(t) (77)
u(t − θl ) ≈ U (θl )φ(t) (78)
m−1
Al (t) ≈ Ali φi (t) (79)
i=0
m−1
Bl (t) ≈ Bli φi (t) (80)
i=0

m−1
Al (t)x(t − τl ) ≈ yi (τl )φi (t) = Y (τl )φ(t) (81)
i=0
m−1
Bl (t)u(t − θl ) ≈ zi (θl )φi (t) = Z (θl )φ(t) (82)
i=0

where
T
φ(t) = φ0 (t), φ1 (t), . . . , φm−1 (t)

an m dimensional orthogonal functions vector, i.e. φ(t) is B(t) or L(t). Sub-
stituting Eqs. (74), (76), (81) and (82) into Eq. (73), and using integration
operational property in Eq. (7) of orthogonal functions yield
α β
X −V = Y (τl ) + Z (θl ) H
l=0 l=0
α β
T
⇒ ˆ ˆ
x = v + H ⊗ In ˆ
y (τl ) + ˆ
z (θl ) (83)
l=0 l=0

13


where ⊗ is the Kronecker product [1] of matrices, say A and B, given by
 
a11 B a12 B . . . a1q B
 a21 B a22 B . . . a2q B 
 
A⊗B =  . . .  (84)
 . . .
. . 
.
ap1 B ap2 B . . . apq B
       
x0 v0 y0 (τl ) z0 (θl )
  x1   v1   
y1 (τl )  z1 (θl )
       
x=
ˆ  ; v=
ˆ
.
.  ; y (τl ) = 
ˆ .
.  ; ˆ (θl ) = 
z .
.  (85) .
.
  .   .   .  .
xm−1 vm−1 ym−1 (τl ) zm−1 (θl )
It is possible to write y (τl ) and ˆ (θl ) as
ˆ z
ˆ ˆˆ z ˆˆ
y (τl ) = Al x (τl ) and ˆ (θl ) = Bl u (θl ) (86)
where
   
x0 (τl ) u0 (θl )
 x1 (τl )   u1 (θl ) 
   
x (τl ) = 
ˆ .
. ; u (θl ) = 
ˆ .
.  (87)
 .   . 
xm−1 (τl ) um−1 (θl )
ˆ ˆ
and Al and Bl are defined in the following subsections. Eq. (83) can be
rewritten in the form
ˆ ˆ ˆ
x = Mu + w (88)
ˆ ˆ ˆ
where u is similar to x in Eq. (85), and M and w are defined in the following
subsections.
Now the final cost term in Eq. (72) can be written as
xT (tf )Sx(tf ) = [x(t0 ) + Ω x]T S [x(t0 ) + Ω x]
ˆ ˆ
where
Ω = 2 bT ⊗ I n (89)
and b is defined in the following subsections. Expressing Q(t) and R(t) in
terms of orthogonal functions, we have
m−1
Q(t) ≈ Qi φi (t) (90)
i=0
m−1
R(t) ≈ Ri φi (t) (91)
i=0

14


tf
and xT (t)Q(t)x(t) + uT (t)R(t)u(t) dt ˆ ˆx ˆ û
xT Qˆ + uT Rˆ
t0

ˆ ˆ
where Q and R are defined in the following subsections. So Eq. (72) becomes
1 1 Tˆ
J = [x(t0 ) + Ω x]T S [x(t0 ) + Ω x] +
ˆ ˆ ˆ x ˆ û
x Qˆ + uT Rˆ (92)
2 2
Substituting Eq. (88) into Eq. (92) and setting the optimization condition
∂J
= 0T yield the optimal control law
ˆ
∂u
−1
ˆ ˆ
u = − M T ΩT SΩ + Q M + R
ˆ ˆ ˆ
M T ΩT Sx(t0 ) + M T ΩT SΩ + Q w (93)

3.1 Using BPFs
In Eq. (85)
T
ˆ
v = xT (t0 ), xT (t0 ), . . . , xT (t0 ) (94)
In Eq. (86)
ˆ
Al = diag Al0 , Al1 , . . . , Al,m−1 (95)
ˆ
Bl = diag Bl0 , Bl1 , . . . , Bl,m−1 (96)
ˆ ˆ
x (τl ) = E (n, µl ) ζ(τl ) + D (n, µl ) x ˆ (97)
ˆ ˆ
u (θl ) = E (r, δl ) ν (θl ) + D (r, δl ) u
ˆ (98)
where
τl = µl T and θl = δl T (99)
µl and δl represent number of BPFs on t0 ≤ t ≤ t0 + τl and t0 ≤ t ≤ t0 + θl
respectively,
   
ζ 0 (τl ) ν 0 (θl )
 ζ (τl )   ν 1 (θl ) 
ˆ l) = 
ζ(τ 
1
.

ˆ

 ; ν (θl ) =  .

 (100)
 .
.   .
. 
ζ µl −1 (τl ) ν δl −1 (θl )

t0 +(i+1)T
1
ζ i (τl ) = ζ(t − τl )dt for i = 0, 1, 2, . . . , µl − 1 (101)
T t0 +iT
t0 +(i+1)T
1
and ν i (θl ) = ν(t − θl )dt for i = 0, 1, 2, . . . , δl − 1 (102)
T t0 +iT

15


In Eq. (88)
β
M = N −1 T
H ⊗ In ˆ
Bl D (r, δl ) (103)
l=0

α β
ˆ
w=N −1
ˆ T
v + H ⊗ In ˆ ˆ
Al E (n, µl ) ζ(τl ) + ˆ ˆ
Bl E (r, δl ) ν (θl ) (104)
l=0 l=0

where
α
N = Imn − H T ⊗ In ˆ
Al D (n, µl ) (105)
l=0

In Eq. (89)
T
b = −1, 1, −1, 1, . . . , −1, 1 (106)
an m dimensional vector. In Eq. (92)
ˆ
Q = T × diag Q0 , Q1 , . . . , Qm−1 (107)
ˆ
and R = T × diag R0 , R1 , . . . , Rm−1 (108)

3.2 Using SLPs
In Eq. (85)
T
ˆ
v = xT (t0 ), 0T , . . . , 0T (109)
m−1 m−1
(2i + 1)
yi (τl ) =
ˆ πijk Alj xk (τl )
ˆ (110)
(tf − t0 ) j = 0 k = 0
m−1 m−1
(2i + 1)
ˆi (θl ) =
z πijk Blj uk (θl )
ˆ (111)
(tf − t0 ) j = 0 k = 0

In Eq. (86)
 m−1 m−1 
 π0j0 Alj ... π0j,m−1 Alj 
 j =0 j =0 
 m−1 m−1 
ˆ 1   3 π1j 0 Alj ... 3 π1j,m−1 Alj 

Al =  j =0 j =0 
(tf − t0 ) .
. .
.

 . . 
 
 m−1 m−1 
(2m − 1) πm−1,j 0 Alj . . . (2m − 1) πm−1,j,m−1 Alj
j =0 j =0
(112)

16


 m−1 m−1 
 π0j0 Blj ... π0j,m−1 Blj 
 j =0 j =0 
 m−1 m−1 
ˆ 1   3 π1j 0 Blj ... 3 π1j,m−1 Blj 

Bl =  j =0 j =0 
(tf − t0 ) .
. .
.

 . . 
 
 m−1 m−1 
(2m − 1) πm−1,j 0 Blj . . . (2m − 1) πm−1,j,m−1 Blj
j =0 j =0
(113)

ˆ ˆ
x (τl ) = ζ(τl ) + (D(τl ) ⊗ In ) xˆ (114)
ˆ ˆ
u (θl ) = ν (θl ) + (D(θl ) ⊗ Ir ) u
ˆ (115)
where
   
ζ 0 (τl ) ν 0 (θl )
 ζ 1 (τl )   ν 1 (θl ) 
ˆ l) = 
ζ(τ  .

; ν

ˆ (θl ) =  .

 (116)
 .
.   .
. 
ζ m−1 (τl ) ν m−1 (θl )

t0 +τl
(2i + 1)
ζ i (τl ) = ζ(t − τl )Li (t)dt (117)
(tf − t0 ) t0
t0 +θl
(2i + 1)
ν i (θl ) = ν(t − θl )Li (t)dt (118)
(tf − t0 ) t0

for i = 0, 1, 2, . . . , m − 1. In Eq. (88)
β
M = N −1 T
H ⊗ In ˆ
Bl (D(θl ) ⊗ Ir ) (119)
l=0

α β
w = N −1
ˆ v + H T ⊗ In
ˆ ˆˆ
Al ζ(τl ) + ˆˆ
Bl ν (θl ) (120)
l=0 l=0

where
α
N = Imn − H ⊗ In T ˆ
Al (D(τl ) ⊗ In ) (121)
l=0

In Eq. (89)
T
b = 0, 1, 0, 1, . . . , 0, 1 (122)

17


an m dimensional vector. In Eq. (92)
 m−1 m−1 m−1 
 π0j 0 Qj π0j1 Qj ... π0j,m−1 Qj 
 j =0 j =0 j =0 
 m−1 m−1 m−1 
 π1j 0 Qj π1j1 Qj ... π1j,m−1 Qj 
ˆ 
Q= j =0 j =0 j =0

 (123)
 . . . 
 .
. .
. .
. 
 
 m−1 m−1 m−1 
πm−1,j 0 Qj πm−1,j 1 Qj . . . πm−1,j,m−1 Qj
j =0 j =0 j =0

 m−1 m−1 m−1 
 π0j 0 Rj π0j1 Rj ... π0j,m−1 Rj 
 j =0 j =0 j =0 
 m−1 m−1 m−1 
 π1j 0 Rj π1j1 Rj ... π1j,m−1 Rj 
ˆ=
R  j =0 j =0 j =0

 (124)
 . . . 
 .
. .
. .
. 
 
 m−1 m−1 m−1 
πm−1,j 0 Rj πm−1,j 1 Rj . . . πm−1,j,m−1 Rj
j =0 j =0 j =0

3.3 Time-invariant systems
The algorithms discussed above are applicable to time-varying systems. With
the following changes, they are also applicable to time-invariant systems as
the time-varying matrices Al (t), Bl (t), Q(t) and R(t) are constant for such
systems. Therefore,
ˆ
Al = Im ⊗ Al , ˆ
Bl = Im ⊗ Bl (125)

3.3.1 Via BPFs

ˆ
Q = T (Im ⊗ Q) , ˆ
R = T (Im ⊗ R) (126)

3.3.2 Via SLPs

ˆ
Q = ⊗ Q, ˆ
R = ⊗R (127)

where
1 1
= (tf − t0 ) diag 1, 3
, ..., 2m−1 (128)

18


3.4 Delay free systems
In this case α = β = 0, τl = θl = 0, and x(t) = x(t0 ) at t = t0 . Therefore
ˆ ˆ
ζ(τl ) = ν (θl ) = 0 (129)

3.4.1 Via BPFs

D (n, µl ) = Imn and D (r, δl ) = Imr (130)

3.4.2 Via SLPs

D (τl ) = D (θl ) = Im (131)

4 Illustrative Examples
Example 1 :
Consider the system [6]

x(t) = x(t − 1) + u(t)
˙
x(t) = 1 for − 1 ≤ t ≤ 0

with the cost function
2
1
J = 105 x2 (2) + u2 (t)dt
2 0

The exact solution is given by

−2.1 + 1.05t for 0 ≤ t ≤ 1
u(t) =
−1.05 for 1 ≤ t ≤ 2
1 − 1.1t + 0.525t2 for 0 ≤ t ≤ 1
x(t) = 2 3
−0.25 + 1.575t − 1.075t + 0.175t for 1 ≤ t ≤ 2

Figure 1 shows u(t) and x(t) obtained via the proposed BPF and SLP
approaches with m = 8, and the analytical method. The value of J is shown
in Table 1. The results match well with the exact results in each case.

19


Example 2 :
Consider the following delay system [12]
1 1 2
x(t) = −x(t) + x t −
˙ + u(t) − u t −
3 2 3
1
x(t) = 1 for − ≤t≤0
3
2
u(t) = 0 for − ≤ t ≤ 0
3
with the performance index
1
1 1
J = x2 (t) + u2 (t) dt
2 0 2
We solve this problem using BPFs and SLPs with m = 12. The values of
u(t) and x(t) are calculated and shown in Fig. 2. The J value obtained by
the proposed BPF and SLP approaches, and the hybrid functions method
(# of BPFs = 3 and # of SLPs = 4) is presented in Table 2.

Example 3 :
Consider the time-varying and multi-delay system [6]
x1 (t)
˙ 0 1 x1 (t) 0 0 x1 (t − 0.8)
= +
x2 (t)
˙ t 0 x2 (t) 0 1 x2 (t − 0.8)
1 0 x1 (t − 1) 0 0
+ + u(t) + u(t − 0.5)
2 0 x2 (t − 1) 1 −1
x1 (t) 1
= for − 1 ≤ t ≤ 0
x2 (t) 1
u(t) = 5(t + 1) for − 0.5 ≤ t ≤ 0

with the cost function
1 1 0 x1 (3)
J = x1 (3) x2 (3)
2 0 2 x2 (3)
3
1 2 1 x1 (t) 1 2
+ x1 (t) x2 (t) + u (t) dt
2 0 1 1 x2 (t) 2+t
The optimal control law u(t) and the state variables x1 (t) and x2 (t) are
computed with m = 30 in the BPF approach and m = 12 in the SLP
approach, and the results are shown in Fig. 3. The value of J in each case
is given in Table 3.

20


5 Conclusion
A unified approach for computing optimal control law of linear time-invariant/
time-varying systems with/without time delays in control and state has been
proposed. The proposed method is based on using two classes of orthogonal
functions, namely BPFs and SLPs. The nature of orthogonal functions used
is reflected in the final solution, i.e. the solution is always piecewise constant
if BPFs (piecewise constant functions) are used while it is smooth with SLPs
(polynomial functions). As the orthogonal functions approach, in general,
helps us reduce differential/integral calculus to algebra (in the sense of least
squares), the proposed approach is computationally attractive.

References
[1] J. W. Brewer, “Kronecker products and matrix calculus in system the-
ory,” IEEE Trans. on Circuits and Systems, Vol. CAS-25, No. 9, 772-781,
1978.

[2] K. R. Palanisamy and G. P. Rao, “Optimal control of linear systems with
delays in state and control via Walsh functions,” IEE Proc., pt. D, Vol.
130, No. 6, 300-312, 1983.

[3] G. P. Rao, “Piecewise Constant Orthogonal Functions and Their Appli-
cation to Systems and Control, ”Springer-Verlag, Berlin, 1983.

[4] C. Hwang and Y. P. Shih, “Optimal control of delay systems via block-
pulse functions,” J. of Optimization Theory and Application, Vol. 45, No.
1, 101-112, Jan. 1985.

[5] I. R. Horng and J. H. Chou, “Analysis, parameter estimation and optimal
control of time-delay systems via Chebyshev series,” Int. J. Control, Vol.
41, No. 5, 1221-1234, 1985.

[6] C. Hwang and M. Y. Chen, “Suboptimal control of linear time-varying
multi-delay systems via shifted Legendre polynomials,” Int. J. Systems
Science, Vol. 16, No. 12, 1517-1537, 1985.

[7] M. H. Perng, “Direct approach for the optimal control of linear time-delay
systems via shifted Legendre polynomials,” Int. J. Control, Vol. 43, No.
6, 1897-1904, 1986.

21


[8] M. M. Zavarei and M. Jamshidi, “Time-Delay Systems Analysis, Opti-
mization and Applications,” North-Holland Systems and Control Series,
Vol. 9, Amsterdam, 1987.

[9] S. C. Tsay, I. L. Wu and T. T. Lee, “Optimal control of linear time-delay
systems via general orthogonal polynomials,” Int. J. Systems Science,
Vol. 19, No. 2, 365-376, 1988.

[10] M. Razzaghi, M. F. Habibi and R. Fayzebakhsh, “Subptimal control of
linear delay systems via Legendre series,” Kybernetica, Vol. 31, No. 5,
509-518, 1995.

[11] K. B. Datta and B. M. Mohan, “Orthogonal Functions in Systems and
Control,” World Scientiﬁc, Singapore, 1995.

[12] H. R. Marzban and M. Razzaghi, “Optimal control of linear delay sys-
tems via hybrid of block-pulse and Legendre polynomials,” J. of the
Franklin Inst., Vol. 341, 279-293, 2004.

[13] X. T. Wang, “Numerical solution of optimal control for time-delay sys-
tems by hybrid of block-pulse functions and Legendre polynomials,” Ap-
plied Mathematics and Computation, Vol. 184, 849-856, 2007.

[14] F. Khellat, “Optimal control of linear time-delayed systems by linear
Legendre multiwavelets, J. Optimization Theory Application,” Vol. 143,
No. 1, 107-121, 2009.

[15] M. Razzaghi, “Optimization of time delay systems by hybrid functions,”
Optimization and Engineering, Vol. 10, No. 3, pp: 363-376, 2009.

[16] X. T. Wang, “A numerical approach of optimal control for generalized
delay systems by general Legendre wavelets,” Int. J. Computer Mathe-
matics, Vol. 86, No. 4, 743-752, 2009.

Table 1: Cost function (Example 1)
Method J
Exact 1.8375
BPF 1.8404
SLP 1.8379

22


Method J
BPF 0.3731831
SLP 0.3731102
Hybrid [12] 0.3731129

Method m J
BPF 30 19.6414
SLP 12 19.7079

1
Actual
BPFs
0.5 SLPs

0 x(t)
control & state

−0.5

u(t)
−1

−1.5

−2

−2.5
−1 −0.5 0 0.5 1 1.5 2
time

Figure 1: Actual, BPF and SLP solutions of u(t) and x(t) variables in Ex-
ample 1

23


1

0.8

0.6 x
0.4
control & state

0.2

0

−0.2

−0.4
u
−0.6

−0.8

−1
−1 −0.5 0 0.5 1
time

Figure 2: BPF and SLP solutions of u(t) and x(t) variables in Example 2

6

4

2 x1
0
control & state

−2

−4
x2
−6
u

−8

−10

−12

−14
−1 −0.5 0 0.5 1 1.5 2 2.5 3
time

Figure 3: BPF and SLP solutions of u(t) and x(t) variables in Example 3

24

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