Solution of Multi-Delay Dynamic Systems by Using Hybrid Functions ()
1. Introduction
Delays occur frequently in biological, chemical, transportation, electronic, communication, manufacturing and power systems [1] . Time-delay and multi-delay systems are therefore very important classes of systems whose control and optimization have been of interest to many investigators [2] -[5] . While modeling such phenomena naturally requires the use of various systems, in many problems, such systems can not be solved explicitly. Therefore, it is important to find their approximate solutions by using some numerical methods. In recent years, the hybrid functions consisting of the combination of the Block-Pulse functions with the Chebyshev polynomials [6] , the Legendre polynomials [7] [8] , or the Taylor series [9] [10] have been shown to be a mathematical power tool for discretization of selected problems. Among these three hybrid functions, hybrids of the BlockPulse functions with the Legendre polynomials have been shown to be computationally more effective.
Recently a new hybrid function consisting of the combination of the Block-Pulse functions with the Bernoulli polynomials is presented [11] [12] . The advantages of the Bernoulli polynomials
, over shifted the Legendre polynomials are:
• The operational matrix
, in the Bernoulli polynomials, has less errors than 
for shifted the Legendre polynomials for
. This is because for 
in 
we ignore the term 
while for 
in 
we ignore the term
.
• The Bernoulli polynomials have less terms than shifted the Legendre polynomials. For example
, has 5 terms while
, has 7 terms, and this difference will increase by increasing m. Hence for approximating an arbitrary function we use less CPU time by applying the Bernoulli polynomials as compared to the shifted Legendre polynomials.
• The coefficient of individual terms in the Bernoulli polynomials
, is smaller than the coefficient of individual terms in the shifted Legendre polynomials
. Since the computational errors in the product are related to the coefficients of individual terms, the computational errors are less by using the Bernoulli polynomials.
In the present paper, we use the hybrid functions consisting of the combination of the Block-Pulse functions and the Bernoulli polynomials to solve the MDS. The method is based on converting the MDS into a system of multi-delay integral equations through integration. To eliminate integral operations, the unknown functions and various functions involved in the equations are approximated by the hybrid function and the operational matrices are used. To this end, operational matrices of multi-delay systems for the hybrid function are given. It will be seen that the operational matrices have many zero elements and are more sparse than the Legendre polynomials. These matrices are used to reduce the solution of MDS to the solution of a system of linear algebraic equations.
The paper is organized as follows: In Section 2, we describe the basic properties of the hybrid functions of the Block-Pulse and the Bernoulli polynomials required for our subsequent development. Section 3 is devoted to the formulation of linear time-varying multi-delay systems and the proposed numerical method is applied to the MDS. And in Section 4, we report our numerical findings and demonstrate the accuracy of the proposed scheme by considering some numerical examples. Finally, Section 5 gives some brief conclusions.
2. Hybrid of the Block-Pulse Functions and the Bernoulli Polynomials
Hybrid functions
, are defined on the interval 
as [11]
(1)
where 
and 
are the order of the Block-Pulse functions and the Bernoulli polynomials, respectively. The Bernoulli polynomials of order 
are defined in [13] by
where
, are the Bernoulli numbers. These numbers are a sequence of signed rational numbers that arise in the series expansion of trigonometric functions [14] and can be defined by the identity
Let 
be an arbitrary element in
, there exist unique coefficients 
such that [11]
where
and
By using Equation (2) we obtain
where 
, and 
denotes the inner product. So we get
with
where 
is a matrix of order 
and is given by
(2)
Integration of the vector 
defined in Equation (4) can be approximated by
(3)
where 
is the 
operational matrix for integration and is given by [11]
(4)
where 
and 
are the 
identity and zero matrices, respectively, and
(5)
The following property of the product of two hybrid function vectors will also be used. Let
(6)
where 
is a 
product operational matrix. To illustrate the calculation procedure see [11] .
Multi-Delay Operational Matrix
The delay functions
, 
are the shift of the function 
defined in Equation (4), along the time axis by
, where 
are rational numbers in
. It is assumed without loss of generality that
. If we expand 
in terms of
, we find
where 
is the 
delay operational matrix of hybrid functions corresponding to 
and is given by
(7)
where elements of the delay matrix are the 
matrix 
given by
(8)
It is noted that the first 1 in the first row is located at the 
th column where
We define 
as the smallest positive integer number for which 
for 
and 
is the greatest common divisor of the integers
,
.
3. Problem Statement and Approximation Using Hybrid Functions
Consider the following linear time multi-delay dynamic systems:
(9)
(10)
(11)
where
, 
, 
and
, 
, are matrices of appropriate dimensions, 
is a constant specified vector, and 
is an arbitrary known function. The problem is to find
, 
satisfying Equations (13) and (14). Let
(12)
(13)
where 
and 
are the 
and 
dimensional identity matrices, 
is 
vector and 
denotes the Kronecker product [15] . Using the property of the Kronecker product, 
and 
are matrices of order 
and
, respectively. Assume that each 
and each of
, 
, 
, can be written in terms of hybrid functions as
Then, using Equations (15) and (16), we have
(14)
where 
and 
are vectors of order 
and
, respectively, given by
Similarly, we have
(15)
where 
and
, 
, are vectors of order 
given by
Let approximate 
and
, 
, by Equations (2)-(4) as follows
(16)
where
, 
are of dimensions
, and 
is of dimension
.
We can also write
, 
, in terms of the hybrid functions as
where
and 
is the delay operational matrix. Moreover
(17)
(18)
where
and
where
, 
, are constant matrices of order
. Note 
is 
product operational matrix that to illustrate the calculation procedure we choose 
and
. Thus we have
(19)
where 
and
, are the 
matrices given by
By integrating Equation (12) from 
to 
and using Equations (15)-(22), we have
(20)
simplifying Equation (23) we obtain
(21)
by solving the set of linear algebraic equations Equation (24), we obtain the coefficients vector
.
4. Numerical Implementation
In this section, to give a clear overview of the analysis method presented and to demonstrate the applicability and accuracy of the method three examples are given.
Example 1. Consider the multi-delay dynamic system from [7] described by
(22)
with
(23)
and
(24)
The exact solutions are
To solve this problem by the hybrid functions, we select 
and
. Let
(25)
where
, 
and 
can be obtained similarly to Equations (3) and (4). By expanding 
and 
in terms of the hybrid functions we get
(26)
(27)
Therefore, we have
(28)
(29)
where 
and 
the 24 × 24 matrices, can be calculated as Equation (19). Also D1 and D2 are the 24 ×24 delay operational matrices given by
where
Integrating Equation (25) from 
to 
and using Equations (26)-(27) and substituting Equations (28)-(32) we get
(30)
where 
is the operational matrix of integration given in Equation (7). By solving Equation (33) the values of 
and 
can be found as
To define 
and 
for 
in the interval 
we map 
into 
by mapping 
into
, and for 
in the interval 
we map this interval into 
by mapping 
into
, and similarly for the other intervals. From Equation (28) we get
After simplifying the same value as the exact 
and 
would be obtained.
Example 2. Consider the delay dynamic system described by
(31)
with
(32)
and
(33)
The exact solutions are
To solve this problem by using of the hybrid functions, we select 
and
. Let
(34)
where
, 
and 
can be obtained similarly to Equations (3)-(4). Using Equation (37) we get
where 
is the 
delay operational matrix given by
where
(35)
By expanding 
in terms of hybrid functions we obtain
(36)
Integrating Equation (34) from 
to 
and using Equations (35)-(36) and substituting Equations (37)-(39), we get
(37)
where 
is the operational matrix of integration given in Equation (7). By solving Equation (40) the values of 
and 
can be found. By using from Equation (37) and simplifying the same value as the exact 
and 
would be obtained.
Example 3. Consider the following multi-delay system with delay in both control and state described by
(38)
with
(39)
and
(40)
The exact solutions are
To solve this problem by using the hybrid functions, we select 
and
. Let
(41)
where
, 
and 
can be obtained similarly to Equations (3) and (4). By expanding 
and 
in terms of hybrid functions we get
(42)
Using Equation (44) and (45) we obtain
(43)
where 
and 
are the 
delay operational matrices given by
(44)
where
By integrating Equation (41) from 0 to t and using Equations (42) and (43) and substituting Equations (44) and (47), we get
(45)
where 
is the operational matrix of integration given in Equation (7). By solving Equation (48) the values of 
and 
can be found. By using from Equation (44) and simplifying the same value as the exact 
and 
would be obtained. In Table 1 a comparison is made between the exact solution and the approximation solution of 
and 
for
. The approximation value of 
and 
on
, is the same as the exact solution.
Table 1. Approximate solutions and exact solutions of Example 3.
5. Conclusion
The hybrid of the Block-Pulse functions and the Bernoulli polynomials and the associated operational matrices of integration and delay are applied to solve the linear multi-delay dynamic systems. The method is computationally very attractive, at the same time keeping the accuracy of the solution. It is also shown that the hybrid functions provide exact solutions in each subintervals for Examples 1, 2 and 3. The presented method reduces multi-delay systems to the solution a system of algebraic equations, and so the calculation is easy. The matrices
, 
and 
in Equations (5), (7) and (10) are sparse, hence the present method is very attractive and reduces the CPU time and computer memory.
NOTES
*Corresponding author.