Wavelets Series Used to Solve Dynamic Optimization Problems

Lizandro S. Santos, Argimiro R. Secchi, Evaristo. C. Biscaia Jr.

Programa de Engenharia Química/COPPE, Universidade Federal do Rio de Janeiro, Cidade Universitária, CP 68502, Rio de Janeiro, RJ, Brazil 21945-970

Introduction

Dynamic optimization is usually used for developing optimal operation policies for new operational situations such as startup procedures and load changes [1]. This technique is currently gaining attention in the design, synthesis and operation of industrial processes. Because of the complexity of most chemical processes, it is difficult to guarantee an optimal dynamic operation [2]. In recent years, there has been considerable effort to develop highly efficient dynamic optimization algorithms; however most of these works focuses on a fixed parameterization of either control or both control (independent) and state (dependent) variables. These methods are known as direct methods, where the dynamic system is transformed into a nonlinear programming (NLP) problem [3].

Recent works [1-3] have been focusing in the adaptive parameterization of the control variables for solving dynamic optimization problems by direct sequential methods. In this procedure, after each iteration of the optimization algorithm, the discretization is changed in order to improve the accuracy of the approximation. In [3] the wavelets were used as a mathematical procedure to eliminate and/or add discrete points in a non-uniform mesh.

Our work addresses possible improvements of the numerical technique for dynamic optimization problems in the context of an adaptive sequential direct method using wavelets. The aim of this study is to improve this technique, also studied by SCHLEGEL (2004) [3], by applying a different procedure to choose the thresholding on the level of detail coefficients of the wavelet transform, based on DONOHO (1995) [4] work, in order to eliminate unnecessary points on the wavelet domain mesh. We compared this methodology with the same problem studied by SCHLEGEL (2004) who used an user-specified fixed threshold: 10-7.

Mathematical Model

The illustrative example considered is a dynamic isothermal semi-batch reactor described by the following equations:

Here, index j denotes the scale, which corresponds to the level of resolution, whereas k denotes translation index on a specific scale.

Considering the control variable uj transformed in the wavelets domain as:

where are the wavelet coefficients. The proposed method starts with an initial guess for the control profile parameterized in ns constant linear stages. After the NLP optimization, points are inserted in regions where the gradient function is high, while points are removed according to an universal threshold level, DONOHO (1995) [4]:

where is the estimated standard deviation of the noise and ns is the control vector length.

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Results

The evolution of optimal control profile with time is plotted in the following figures:

Conclusion

The present work suggests a improvement in wavelet refinement strategy in order to approximate the control profile in dynamic optimization problems. The Universal threshold rule was capable to solve the present problem with a coarser mesh compared with fixed threshold.

References

[1] Binder T., Cruse, A., Villar C.A.C, Marquardt, W. Computers and Chemical Engineering, 24, 1201 (2000).[2] Souza D.F., Vieira R.C., Biscaia, E.C. Computer Aided Engineering, 21, 333 (2006).[3] Schlegel, M. Adaptive discretization methods for the efficient solution of dynamic optimization problems. PhD Thesis; (2004). [4] Donoho D.L. De-Noising by Soft-Thresholding. IEEE Transactions on Information Theory, 41, (1995).

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The results show that in both methodologies there is a high number of points closed to the discontinuities of the control profile. This behavior is expected because the wavelets are capable to capture the most relevant information of a signal.

The recursive algorithm adds points according to the wavelet policy for refine or coarsen the mesh. It can be seen that if Universal threshold is considered, the problem could be solved with 30 points, while in Fixed threshold 44 points were necessary. It means that the choice of the refinement procedure may be an important aspect in order to reduce the computational time. According to the Universal threshold rule, the threshold changes with the number of points and the estimative of the noise. The detail matrix, after the refinement procedure, is shown in the figure below, where dark points show more representative details:

In which the horizontal axis shows the number of detail coefficients and vertical axis the wavelet level of decomposition.

Wavelet Analysis

Wavelet analysis is a relatively new numerical concept that allows one to represent a function in terms of a set of basis functions, called wavelets, which are localized both in location and scale. These functions are generated from one single function called the mother wavelet by the simple operations of dilation and translation [4]:

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