Solution of optimal control problem by the daubechies wavelet function (Db 1)

Introduction

Wavelet analysis is similar to Fourier Analysis in that it allows a target function over an interval to be represented in terms of an Orthonormal basis. The Haar sequence was proposed in 1909 by Alfred Haar. He used these functions to give an example of an orthonormal system for the space of quadrically integrable function on the interval [0, 1]. The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support, the Haar wavelet is also known as Db1. The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be advantageous for the analysis of signals with sudden transitions (discrete signals), such as monitoring of tool failure in machines.

Overview of the Haar Wavelets

A Haar Scaling Function is a set of shifted and scaled square summable function used for defining scaling and wavelet functions in multiresolution signal analysis. It satisfies orthonormality conditions and is a solution of a specific refinement equation with two nonzero coefficients. It reconstructs the approximation of the signal at different scales. It normally captures the low frequency signal. It has a relatively smooth shape and compact support. The scaling function is defined as;

The wavelet function is derived from the scaling function through a specific process that involves scaling and translating the scaling function. Its defined as;

The wavelet function often has a more localized and oscillating shape than the scaling function.

These integrals can be evaluated analytically to obtain;

So that the collocation points become;

The symbol t denotes the th grid point ; t A t    . Eqns. ( 1)-( 4 1)

Then we need to integrate by the Haar wavelet method of the equation;

so that we seek a solution in the form;

x aH  (7) and by integration we obtain;

x aP c  (8) where c is the vector , an integration constant. Putting eqns. ( 7) and ( 8) into ( 6) we obtain a system of 2m equations which will enable us obtain the wavelet coefficient ()

where we interpret x as control state variables;

The state equations are given by; (0) (0) (0) 0, (2) 0, (2) 1

with the Hamiltonian given by;

The adjoint system will now take the form;

Firstly, we integrate Eqns. ( 14) with the assumption that;

Matrix R is calculated from Eqn. ( 6), now, substituting Eqn. (15) into Eqn. (14), we obtain; The state variables are then developed into the Haar series using Eqns. ( 10) and (11);

These results show that a small number of points (J = 4; 32 grid points) guarantees sufficient accuracy.

Brysan and Ho (1975) solved the following problem analytically.

(1) 0, (0) (1) 1 , (0) 0, (0) 1, ( ) , ( ) 0

From Eqns. ( 10) and (11), we seek the wavelet solution in matrix form thus;

The matrix H and P is gotten from Eqns. ( 1 0

The second equation of Eqn. ( 21) can be integrated to obtain; ( ) , ( ) 0 x t x t  from which we get;

Which is actually the same result as the one obtained by Bryson and Ho (1975) [10] Finally, suppose we now consider an optimal control problem with control inequality constraints. Let;

be the optimal control problem with the assumption that the control is smooth, implying that the function () ut is continuous.

With the Hamiltonian Let the subinterval be divided into 2 m equal parts and we calculate the matrices H, P and R using Eqns. (11), ( 3) and ( 4) with the assumption that;

11 , 1 A t B t    . We need a solution in the form;

The matrix R is gotten via Eqn.

Eqn. ( 26) is numerically solved. The fact that () ut must be continuous at tt   , however, in the case of an arbitrary chosen value, t  the requirement is not met. This requires the estimation by the function;

The value of t  was varied until the condition 0  was fulfilled with the necessary exactness. Simulation was done for 0 0.

Conclusion

From the three different case solutions above, it's clear that the Haar Wavelet method proves to be an effective and reliable approach for solving optimal control problems. Its piecewise constant structure enables efficient handling of complex boundary conditions and discontinuities, making it particularly well-suited for problems with non-smooth solutions. Furthermore, the method offers significant computational advantages due to its sparse matrix representation and fast convergence properties. Numerical results demonstrate that the Haar Wavelet method achieves high accuracy with relatively low computational cost, validating its applicability to a wide range of control problems in engineering and applied sciences. The method provides accurate solutions, even with a relatively small number of collocation points. This is actually in concurrence with Sengeta, Singh and Kumar (2014) [9] who also noted that the Haar wavelet method presented simple and straight forward numerical technique in solving Differential equations

References

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