THE SEQUENTIAL LINEAR-QUADRATIC PROGRAMMING ALGORITHM FOR SOLVING DYNAMIC OPTIMIZATION PROBLEMS - A REVIEW

The optimum principle for dynamic systems as formulated by Pontryagin in 1962 may be used for development of numerical algorithms to solve dynamic optimization problems. This as opposed to the well known methods which discretize controls (and states) to transform the problem into a NLP framework. An obstacle for its use has been the extensive symbolic manipulations needed to derive the optimality equations for a specific problem, and the difficulty of solving the resulting nonlinear two point boundary value problem. There are methods which make use of the optimality conditions for dynamic systems (Pontryagin Minimum Principle) just as SQP methods use the Kuhn-Tucker conditions. As in SQP, a problem with linear constraints and quadratic objective function is solved iteratively. Such a method is presented in this work. This is closely related to the dynamic optimization method based on a combination of a SQP solver and total discretization of the dynamic system. The dynamic linear-quadratic model has a single analytical optimal control solution, acid is thus accurately and effectively solved. Thus, at each iteration, the optimal solution is found for the linear-quadratic approximate model. This gives a search direction which can be used in a iterative scheme to ensure good agreement between the linear-quadratic and the nonlinear model.