Operational Matrix Basic Spline Wavelets of Derivative for linear Optimal Control Problem

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ORIGINAL ARTICLE

Operational Matrix Basic Spline Wavelets of Derivative for Linear

Optimal Control Problem

Maha Delphi, Suha SHIHAB*

Applied Sciences Department, University of Technology, Baghdad, Iraq Email: [email protected] or [email protected]

Abstract: The main importance goal in this paper is studying the interesting properties of basic spline wavelets

functions (BSWFs) and derived some new basic formulations of them. The important operational matrix is devoted in two ways, the first one is the derivative of BSWFs in terms of the lower order of BSWFs while the second is the derivative of BSWFs in terms of the same order of BSWFs. The expression formula for the operational matrix is determined for different orders. In addition an useful formulas concerning the power function and BSMSFs are also presented. The polynomials and wavelets expansions together with operational matrices can be employed to solve problems in applied science and other fields of approximation theory. In this work, two optimal control problem are tested with the aid of operational matrix of derivative for BSWFs with satisfactory results.

Keywords: Basic spline wavelets; Operational matrix; Linear optimal control problem

1. Introduction

Wavelets have been successfully utilized in scientific and engineering problems. Basic spline scaling and wavelets functions play an important role in mathematics; they have been utilized in the solution of differential equations, integral equations and approximation theory. In particular basic spline wavelets have been applied in the approximation of linear and nonlinear Volterra and Fredholm integral equations[1-2]_.

One of the most important algorithms for treating many problems approximately is based on using operational matrix of derivatives. The advantages of operational matrices are to convert the original problem to a system of algebraic equations and then the differentiation will be eliminating with the aide of operational matrix of derivative. In a result, the complexity reduction can be obtained. About the applications of operational matrices, there are some papers for example[3-6]_{. In particular, some}

researchers applied different basis polynomials and functions for solving optimal control problems, such as, shifted Chebyshev polynomials[7]_{, Chebyshev wavelets}[8]_,

Legendre orthonormal basis[9]_{, Tayler wavelets}[10]_,

interpolating scaling functions[11]_{, third kind Chebyshev}

wavelets[12]_{, Bernstein and orthonormal Bernstein}

polynomials[13-19]_.

In this paper, novel approach based on BSMSFs operational matrices with their properties is applied for approximate solution of linear optimal control problem.

2. Basic spline Wavelets Functions

The basic spline wavelets functions _ࣈ 詨ህ can be constructed on the interval 詨詨 ህ as

ࣈ 詨詨ህ 詨詨 ࣈ ࣈ

詨詨ࣈ詨ህ 詨 ݄ ݁ ݎ݄

(1) where ህ 詨ህ _{and the four arguments}

ࣈ 詨 are

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(1) The translation argument ࣈ ህ

(2) The number of partitions on [ ህ] , is any positive integer

(3) The normalized time 詨

(4) The order of orthonormal B-spline function on [ ህ] is . For ህ, ህ, ࣈ ህ and ህ ህ _{ህ 詨詨ህ} ህ ህ _{詨詨 ህህ ,} _t ህ and ህ ህ _{詨詨ህ} ህህ ህ _{詨詨䀀ህ ,} ህ _{t ህ} For ህ ࣈ ህ and ህ ህ ህ 詨詨ህ ህ 詨詨詨 ህ 詨詨 ህህ , t ህ 䀀詨詨 ህ 詨詨 ህህ and ህ ህ 詨詨ህ δ_ህህ 詨 t 詨 3 t 詨 ህ ህ , ህ 詨 ህ δ_ህ 䀀 t 詨 5 t 詨 ህ9ህ For ህ, 3 ࣈ ህ 3 and ህ 3 _{ህ䀀 ህ 詨詨ህ}3 ህ 3 _{ህ ህ 詨詨ህ ህ䀀詨詨 ህህ ,} _詨 ህ _詨 ህ 3 _{ህ 詨詨ህ ݐ䀀詨詨䀀詨詨 ህህ} 3 3 _{ݐ 詨}3_{詨 ህݐ 詨詨 3 詨詨 ህህ} and ህ 3 _{ህ䀀詨詨ህ}3 ህህ 3 _ህ _{詨詨 ህ䀀詨詨 ݐህ} _, ህ _{詨 ህ} ህ 3 _{詨詨ህ ݐ䀀詨詨 ህ ݐ詨詨 3䀀ህ} ህ3 3 _{ݐ 詨}3_詨 _{詨詨䀀詨詨 9 ህ} For ህ, , ࣈ ህ and ህ 䀀 ₃ _{ህ 詨詨ህ}䀀 ህ 䀀 _{ህ䀀 ህ 詨詨ህ}3 _{ህݐ詨詨 ህህ} 䀀 _{ህ ህ 詨詨ህ ህ䀀䀀詨詨 3 詨詨 ህህ} _, 詨 ህ 3 䀀 _{ህ 詨詨ህ} _詨3_{詨 33 詨詨䀀詨詨 ህህ} 䀀䀀 _{ህ 詨}䀀_{詨 ህ 9 詨}3_{詨 5 詨詨䀀ݐ詨詨 ህህ} and ህ 䀀 ₃ _{詨詨ህ}䀀 ህህ 䀀 _{ህ䀀詨詨ህ}3 _{ህݐ詨詨 ህ ህ} ህ 䀀 _ህ _{詨詨ህ ህ䀀䀀詨詨 ህ 詨詨 53ህ ,} ህ 詨 ህ ህ3 䀀 _{ህ 詨詨ህ} _詨3_{詨 ህ3䀀䀀詨詨 ݐݐ 詨詨 ህ9 ህ} 䀀䀀 _{ህ 詨}䀀_{詨 5ݐ 䀀詨}3_{詨 59 詨詨} _{5 詨詨} 䀀3ݐህ

3. Operational Matrix of Derivative

for BSWFs

This section gives the constructing operational matrix of derivative for BSWFs.

For 詨䀀 5 3 ህ ህ ݐ ህ 詨䀀 3 ህህ , 詨 ህ 詨䀀 3 ህ 詨 ህ ህህ and ህ 詨䀀 5 3 ህህ ህህ ݐ ህህ 詨䀀 3 ህህህ , ህ 詨 ህ ህ 詨䀀3 ህህ 詨 ህ ህህህ

One can write the above equations as δ t δህ _tህ where 詨ህ [ _ህ _ህ _ህህ _ህ ] ህ _詨 _[ ህ ህ ህ ህ ህ ህህ ህ _]

and the operational matrix is a × 䀀 matrix D=

ህ ህ where ህ 詨䀀 5₃ ݐ 詨䀀詨䀀₃ ህ For 3 _詨 5 ህ 3 _ህ _詨 5 3 ህ , 詨 ህ 3 _詨 ݐ 3 3 5 詨 ህ ህ詨 3 3 3 3 ህ 5 詨 and ህ 3 _詨 5 ህ ህህ3 ህ ህ 詨 3 ህህ5 , ህ _{詨 ህ} ህ 3 _詨 ݐ 3 3 5 ህ 詨 ህ ህህ詨 3 3 ህ ህ3 3 ህ 5 ህ 詨 ህ

One can write the above equations as

3 _詨 _詨ህ

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3 _詨ህ [ 3 ህ 3 3 3 3 ህ 3 ህህ 3 ህ 3 ህ3 3 _] 詨 [ _ህ _ህ _ህህ _ህ ] In this case ህ 詨 ₅ ህ 詨 ₃5 詨 _{3 5}ݐ 3 ህ 詨 ₃3 ህ 5 䀀 and ህ 䀀 _詨䀀 ህ 3 ህህ 䀀 _ህ ህ 3 _詨ݐ 5 ህህ3 ህ 䀀 _詨 ݐ 5 5 ህ3 詨 ህ䀀 ህህ3 詨䀀 5 5 3 ህ3 , ህ _{詨 ህ} ህ3 䀀 ህ䀀 3 ህ 3 _詨ݐ 3 5 ህህ3 詨 ህ ህ3 詨 ህ 3 ህ33 ህ䀀䀀 _詨5ህህ 5 ህ3 詨 ህ 9 5 ህህ3 詨 ݐ9 5 3 ህ3 詨 33 ህ33 One can write the above equations as

䀀 _詨 3 _詨ህ where 䀀 _詨ህ [ 䀀 ህ 䀀䀀 3 䀀䀀䀀 ህ 䀀 ህህ 䀀 ህ 䀀 ህ3 䀀 ህ䀀䀀 _] 3 _詨 _[ 3 ህ 3 3 3 3 ህ 3 ህህ 3 ህ 3 ህ3 3 _]

In this case, the matrix ህ is equal to

24 ₀ ₀ ₀ 7 8 7 16 0 0 5 68 5 ₁₄ 42 5 ₀ 5 7 5 3 14 3 8 3 ₁₂ _{10 3} 7 5 511 179 789 ₃₃ 5 7 5 5 3                                            

4. Operational Matrix of Derivative

for BSWFs in terms of the Same

Order of BSWFs

ህ ህ _{詨 3} ህ _{詨 3} ህ ህ ህ ህ _{3 3} ህ _{詨 3} ህ ህ _, _詨 ህ and ህ ህ _{詨 3} ህ ህ _{詨 3} ህህ ህ ህህ ህ _{3 3} ህ ህ _{詨 3} ህህ ህ _, ህ _{詨 ህ}

ህ _詨 ህ _{詨 where} ህ 詨 3 詨 3 3 3 3 詨 5 詨 5 3 ህ ህ 35 3 3 5 詨 3 ህ詨 ݐ 3 3 , t ህ 詨ህ₅ 詨ህ䀀 _ህ詨 ݐ and ህ 詨 5 ህ 詨 5 3 ህህ ህህ 35 3 3 5 ህ 詨 3 ህህ詨 ݐ 3 3 ህ , ህ 詨 ህ ህ 詨 ህ 5 ህ 詨 ህ䀀 ህህ詨 ݐ ህ

One can write the above equations as 詨詨 , where

5

0

3 35 3

₃

8 3

1 _{3 5}

₃

10

14 ₈

5

3 D







































3 3 _詨 3 _詨 5 3ህ ህ 3 5 5 3 詨 5 3ህ詨ህ 55 3 3 , 詨 ህ 3 _詨ህ䀀 3 3 _詨ህ䀀 3 5 3ህ詨 3 3 詨 5 3 33 3 3 ህ䀀 3 _詨ህ 5 3ህ詨 ህ3 3 詨 ህ5 3 and ህ 3 _詨 ህ 3 _詨 5 ህህ3 ህህ 3 5 5 ህ3 詨 5 ህህ3 詨ህ 55 3 ህ3 , 詨 ህ ህ 3 _詨ህ䀀 3 ህ 3 _詨ህ䀀 3 5 ህህ3 詨 3 ህ3 詨 5 3 ህ33 ህ3 3 ህ䀀 ህ 3 _詨ህ 5 ህህ3 詨 ህ 3 ህ3 詨 ህ5 ህ3

3 _詨 3 _{詨 , where} 7 7 0 0 5 77 5 ₅ 12 5 ₀ 2 7 5 3 1 14 3 14 3 ₃ _{5 3} 7 5 14 10 21 ₁₅ 7 5 3 D        









































䀀䀀 _{詨 9} 䀀 _詨 3 ህ 䀀

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ህ 䀀䀀5 䀀 _詨 ህ 䀀 _詨ህ 5 䀀䀀 _詨 ₅ 䀀 _詨ݐ 5 5 䀀ህ詨 5 䀀詨 5 3ህ 5 䀀3, 詨 ህ 䀀䀀 _詨 ݐ 9 䀀詨 ህ ህݐ ህ䀀5 䀀ህ詨3䀀9䀀9 5 䀀詨 535 ህ䀀5 3 䀀3詨 9 䀀䀀 and ህ 䀀 _{詨 9} ህ 䀀 _詨 3 ህህ 䀀 ህህ 䀀䀀5 ህ 䀀 _詨 ህህ 䀀 _詨ህ 5 ህ䀀 ህ 䀀 _詨 ₅ ህ 䀀 _詨ݐ 5 5 ህህ䀀詨 5 ህ䀀詨 5 3 ህ3ህ 5 䀀詨 ህ ህ䀀䀀 _詨 ݐ 9 ህ䀀詨 ህ ህݐ ህ䀀5 ህህ䀀詨 3䀀9䀀 9 5 ህ䀀詨 535 ህ䀀5 3 ህ3䀀詨 9 ህ䀀䀀

䀀 _詨䀀 _{詨 , where} 3 9 0 0 0 7 45 7 ₇ 16 7 ₀ ₀ 7 7 5 86 5 21 5 6 5 5 0 5 7 5 3 786 16618 3494 5352 66 29 145 7 29 5 145 3 29  _ _                          _ _ _ _     

5. Powers in terms of (BSWFs)

The power of 詨 can be rewritten in terms of BSWFs as follows × 詨 when ህ where 詨 ህ

3

1 0 0

2 6 2 2

1

1 0 0

6 2 2

T















_

_













_where ህ _{詨 ህ} where ህ 詨 and 詨 ህ ህ ህ ህ ህ ህህ ህ when

5

1

1 _{0 0 0}

3 10

6 3 2

5

3

1 _{0 0 0}

24 10 8 6

6 2

1

1 _{0 0 0}

24 10 8 6 12 2

                    where 詨 ህ

2

3

1 0 0 0

3 10

4 6 3 2

25

7

1 0 0 0

24 10 8 6 3 2

5

1

1 0 0 0

3 10

6 3 2

                    where ህ 詨 ህ where ህ 詨詨 and 詨 ህ ህ ህህ ህ when 3

7

5

3

1 _{0 0 0 0}

4 14

4 10

4 6

4 2

7

3

13

1 _{0 0 0 0}

40 14

8 10

40 6

8 2

7

5

11

1 _{0 0 0 0}

240 14 48 10 80 6 16 2

1

9

1 _{0 0 0 0}

160 14 32 10 160 6 32 2

                          where 詨 ህ

7

5

3

1 0 0 0 0

4 14

4 10

4 6

4 2

21

1

7

1 0 0 0 0

20 14

10 10 6 4 2

77

19

13

1 0 0 0 0

120 14 24 10 20 6 4 2

2

5

3

1 0 0 0 0

5 14

8 10

5 6

4 2

                          where ህ 詨 ህ where ህ 詨詨詨3 and 詨 3 ህ 3 3 3 3 ህ 3 ህህ 3 ህ 3 ህ3 3

6. Numerical Examples

The application problems in this work are

Example 1

This example clarifies the following concepts Find the optimal state and optimal control based on

3

1 _{0 0}

2 6 2 2

1

1 _{0 0}

4 6 4 2

T















_

_













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minimizing the performance index = ህ 詨詨

ህ _詨

詨, 詨 ህ

subject to 詨詨ህ 詨詨ህ with the condition ህ , ህህ ህ ህ 詨ህ_݄ህ

The exact solution for the state 詨ህ and the control 詨ህ is

詨 ህ 詨 .5݄詨詨ህ_{詨 .ݐህ} _3݄詨詨

詨 ህ 詨 ݄詨詨ህ

and ݄ ܽ詨 . ݐ䀀䀀5

Example 2

Consider the linear control system, which consists of minimizing ህ ህ 3 詨ህ 詨 tህ 詨

subject to 詨ህ 詨ህ 詨詨ህ , ህ , ህ and ݄ ܽ詨 .ህ5ݐ .

Example 1 is solved using BSWFs as follows

Let the initial approximation of x(t) is

ህ _詨 _詨 ህ_詨 ህ 䀀 ህ 詨 ህ 䀀 ህህ詨 ህ ህህ 詨 ህ ህህ ህ That is, ህ _詨 .ህ99 ݐݐ ህ ህ _詨 . 99ݐ9䀀 ህህ ህ _or ህ _詨 ህ ህ 詨 ህ _詨 _{. 䀀99䀀} ህ ህ _詨 . 99ݐ9 ህህ ህ _or ህ _詨 ህ ህ 詨 ህ _詨 _{. ݐ3 䀀5} ህ ህ _詨 .ህ99 ݐݐ ህህ ህ _or ህ _詨 ݁ህ ህ 詨詨 ህ 詨 where ህ [ .ህ99 ݐݐ . 99ݐ9䀀] , ህ [ . 䀀99䀀 . 99ݐ9] ݁ህ [ . ݐ3 䀀5 .ህ99 ݐݐ] and ህ _詨 _[ ህ ህ ህ ህ ህ ህህ ህ _] ህ _詨 _[ ህ ህ ህ ህ ህ ህህ ህ _]

Now the second approximation of and

ህ ህ詨 δ 詨 ህህ

where is the unknown parameter, here 詨 .䀀 9ህ39 , then 詨 and 詨 can be written as

詨 ህδህ 詨詨 δ 詨 where and ݁ህ ህ詨 ህ 詨 ݁ 詨 where ݁ህ . ݐ3 䀀5 .ህ99 ݐݐ ݁ .5 9 ህ35ݐ ህ . 䀀5 95 詨 . ህ 䀀䀀9 ህ . 3䀀 9䀀9 詨 . 55 ህህݐݐ 詨 .ህ3 3 9 Similarly, the third approximation for 詨ህ and 詨ህ can be found 3 _詨 _{詨詨} 3 δ3 詨詨詨 ህ Here 3 =0.011654 and 3 詨 ህδህ 詨詨 δ 詨詨 3δ3 詨 where 3 詨 . 詨 . ݐ䀀9 ህ 詨 . 9䀀 ݐ9 詨 . 3 䀀ህ9 詨 . ݐህ 3ݐ ህ䀀詨 . ህ9䀀 33 ህ 詨 . 5ݐ ݁ህ ህ詨 ህ 詨 ݁ 詨詨 ݁3 3詨 3 where 3 詨 . 33 5 詨 . 59䀀ݐ3ህ ህ . 䀀ህ 9 ህ 詨 . ህ 9 5 詨 . 䀀ݐ55ݐ3 ህ䀀 . 䀀 9 ህ 詨 . 5ݐ

The value of are shown in Table 1.

Iteration Our method Error

1 0.08401526011 3.036×10-5

2 0.08402496173 2.065×10-5

3 0.08402530645 2.031×10-5

Table 1. The results of for example 1 using BSWFs

Example 2 is solved using BSWFs as follows

Let the initial approximation of 詨ህ is

ህ _詨 _詨 ህ_詨 ህ 䀀 ህ 詨 ህ 䀀 ህህ詨 ህ ህህ 詨 ህ ህህ ህ Therefore; ህ _詨 ህ ህ ህ _詨 ህ ህህ ህ _or ህ ህ ህ ህ ህ ህ ህ _詨 ህ ህህ ህ _or ህ ህ ህ ህ 5 ህ ህ _詨 ህህ ህ _or ህ _t _݁ ህ ህ 詨詨 ህ 詨 where ህ [ ህ] , ህ [ ህ ] , ݁ህ [ 5 ] and ህ _詨 _[ ህ ህ ህ ህ ህ ህህ ህ _] _, ህ _詨

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[ ህ ህ ህ ህ ህ ህህ ህ _]

Now the second approximation of and

ህ ህ詨 δ 詨 ህህ

where is the unknown parameter, here ህ.䀀 ݐ , then 詨 and 詨 can be written as

詨 ህδህ t 詨 δ 詨 where 詨 3ݐህ ህ ህ 詨 ህ䀀3 詨 3ݐህ 詨 .535 5 ህ 詨 ህ䀀3 and the control variable can be evaluated to be

詨 ݁ህ ህ詨 ህ 詨 ݁ 詨 where ݁ህ 5 ݁ . 3ݐ5 ህ 詨 ህ䀀3 ህ 詨 3ݐህ 䀀詨 3ݐህ ህ ህ䀀3 ݐ 3ݐህ 5

Similarly, the third approximation for 詨ህ and 詨ህ can be found 3 _詨 _{詨詨} 3 δ3 詨詨詨 ህ Here 3 .3ݐݐݐ and 3 詨 ህδህ 詨詨 δ 詨詨 3δ3 詨 where 3 詨 ݐ9ህ ህ ህ䀀詨 5 ህ 詨 . 3ህ59 詨䀀3 詨 3䀀9 5 ህ䀀詨 ݐህ ህ 5 ህ 詨䀀3 ህ 5 詨 ݁ህ ህ詨 ህ 詨 ݁ 詨詨 ݁3 3詨 3 where ݁3 詨 .ህህ 9 ህ䀀詨 39 9 ህ 詨 .ݐ 䀀33 詨 9 詨 ݐህ 5 ህ䀀 ∓ ݐህ ህ ህ 詨䀀3 ህ 5

For optimal values of the performance index corresponding to ࣈ ህ , ࣈ , ࣈ 3 that is when

詨ህ ህ_, _詨 ህ_, _詨 _, _, 3_, 3 _{respectively, one refers to Table 2. Also, the}

value of is shown in Table 2.

Iteration Our method Error

ህ .ህ9 䀀 ህ9 . 3ህ9

.ህ 5ህ3 ݐ . ህݐ9

3 .ህ 䀀ݐ ህ55 . ህ

Table 2. The results of cost functional of example 2 using BSWFs.

Note the exact value of is .ህ5ݐ .

7. Discussion

The solution of optimal control problems were obtained using basic spline wavelets based on operational matrix of derivative. The algorithm is comparable in terms of accuracy depending on the exact errors if the exact value of the performance index is known.

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