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Adaptive Shooting Methods for Dynamic Optimization - Concepts,. Algorithms and Applications. Ralf Hannemann, Arndt Hartwich, Wolfgang Marquardt and Lynn ...
Adaptive Shooting Methods for Dynamic Optimization - Concepts, Algorithms and Applications Ralf Hannemann, Arndt Hartwich, Wolfgang Marquardt and Lynn Wurth ¨ Process Systems Engineering, Center for Computational Engineering Science RTWH Aachen University, D-52064 Aachen, Germany e-mails: [email protected], [email protected], [email protected], [email protected] Keywords: optimal control, single shooting, adaption, structure detection

1. INTRODUCTION This contribution will review some of the work on the development of adaptive shooting methods for the solution of optimal control problems carried out in our research group in recent years. The basic feature of the shooting method is an adaptive choice of the control vector parameterization in a shooting type of solution strategy. The following class of multi-stage dynamic optimization problems is considered: min

(uk (t),pk ,tk )k=1,...,S

S X

Φ(tk , x(tk ), pk )

k=1

s. t. Mk x˙ k (t) = fk (t, xk (t), uk (t), pk ) , x1 (t0 ) = x0 (p1 ) ∈ Rnxk , xk (tk ) = Bk xk−1 (tk ) , gk (t, xk (t), uk (t), pk ) ≤ 0 , hk (tk , xk (tk ), pk ) ≤ 0 , uk (t) ∈ Uk ,

pk ∈ Pk ,

for tk−1 < t ≤ tk , k = 1, . . . , S.

optimization problem with a small number of basis functions is solved. Depending on the wavelet coefficients of the optimal solution, some basis functions are deleted and a number of additional basis functions of the next resolution level are added. Details of the adaption algorithm and numerical case studies are given by Schlegel et al. [8]. 3. STRUCTURE DETECTION Furthermore, for single-stage problems (S = 1) the control switching structure of the solution is automatically detected during the refinement process of the adaptation of the control vector parameterization which gives insight into the solution features facilitating the interpretation of the result. The such detected structure is exploited to reparameterize the single-stage into a multi-stage problem with a close to minimal number of control vector parameters (Schlegel and Marquardt [6, 5]). 4. SENSITIVITIES

The dynamic optimization problem comprises S stages, each with possibly different differentialalgebraic equations of index less than or equal to one. 2. ADAPTION The control variables are adaptively discretized by multi-scale basis functions to resolve local detail with an appropriate number of parameters. In detail, the basis functions are wavelets generated from the Haar basis or the hat basis, respectively. At the beginning of the adaption procedure, an

First and second order derivatives are computed by novel and highly efficient numerical algorithms exploiting forward as well as backward mode differentiation. Schlegel et al. [7] provide an efficient numerical algorithm based on the extrapolated linear-implicit Euler’s method for the computation of first order sensitivities. Hannemann and Marquardt [1] modify the secondorder adjoint sensitivity analysis (Haug and Ehle [3]) to efficiently compute the Hessian of the Lagrangian for path-constrained optimal control problems in shooting algorithms.

5. ONLINE APPLICATIONS Kadam and Marquardt [4] introduce a two-level strategy for the dynamic real-time optimization of industrial processes. A parametric sensitivitybased technique is used to calculate optimal firstorder updates to a nominal reference solution. The technique does not assume that the active constraint set remains the same after changes in uncertain parameters. The structure detection algorithm (cf. section 3) is adapted for nonlinear model predictive control (Hartwich et al. [2]). The achieved reduction in terms of degrees of freedom in the concerned nonlinear program decreases the computational time. 6. CONCLUSIONS Our numerical method conceptually links singleshooting and multiple type shooting on the one hand as well as direct and indirect methods on the other. The robustness and performance of the algorithms will be illustrated by different kinds of examples from chemical engineering of different complexity. The implementation has been proven to be very robust and highly efficient for largescale optimal control problems with up to 15 000 differential-algebraic equations with a number of control variables and many inequality path and endpoint constraints. Some extensions of the algorithm to cover real-time applications in nonlinear model-predictive and neighboring extremal control will be briefly discussed together with illustrating examples. REFERENCES [1] R. Hannemann and W. Marquardt. Fast computation of the hessian of the lagrangian in shooting algorithms for dynamic optimization. accepted for: DYCOPS-2007, June 6-8, Cancun, Mexiko. [2] A. Hartwich, M. Schlegel, L. W¨urth, and W. Marquardt. Adaptive control vector parameterization for nonlinear model-predictive control. submitted to: International Journal of Robust and Nonlinear Control. [3] E. J. Haug and P. E. Ehle. Second-order design sensitivity analysis of mechanical system dynamics. Internat. J. Numer. Methods Engrg., 18:1699–1717, 1982. [4] J. Kadam and W. Marquardt. Sensitivity-based solution updates in closed-loop dynamic optimization. In Proceedings of the DYCOPS 7 conference, July 2004.

[5] M. Schlegel and W. Marquardt. Adaptive switching structure detection for the solution of dynamic optimization problems. Erscheint in Industrial & Engineering Chemistry Research, 2006. [6] M. Schlegel and W. Marquardt. Detection and exploitation of the control switching structure in the solution of dynamic optimization problems. Journal of Process Control, 16(3):275–290, 2006. [7] M. Schlegel, W. Marquardt, R. Ehrig, and U. Nowak. Sensitivity analysis of linearlyimplicit differential-algebraic systems by onestep extrapolation. Appl. Numer. Math., 48(1):83–102, 2004. [8] M. Schlegel, K. Stockmann, T. Binder, and W. Marquardt. Dynamic optimization using adaptive control vector parameterization. Computers & Chemical Engineering, 29(8), 2005.