The properties of differential-algebraic equations representing optimal control problems

Abstract

A procedure is described for transforming a general optimal control problem to a system of Differential-Algebraic Equations (DAEs). The Kuhn-Tucker conditions consist of differential equations, complementarity conditions and corresponding inequalities. The latter are converted to equalities by adding a new variable combining the slack variable and the corresponding Lagrange multiplier.We investigate the properties of the resulting DAEs. The index of a system of DAEs determines the well-conditioning of the problem. The concept of the tractability index is used to investigate the index in a systematic way, and during this process, it indicates which components of the system of equations must be differentiated to reduce the index. For an index-3 problem, the index is reduced without increasing the number of equations, and a numerical procedure is used to determine the index.In the examples used here, the DAEs can be solved analytically. The examples are tested by the numerical determination of the index, and the results confirm the previously known properties of these examples.The reformulation proposed here, as well as the index determination, might be used in the future, to develop a methodology to solve optimal control problems.

The advantages of robust control of pulse amplitude modulation series-parallel load induction heating inverters

“…Feedback control system [12]. practice, the minimum is usually replaced by an 'infimum', which must, in general, aim to provide the system designer with a stabilizing controller K, such that [14]…”

“…For good disturbance attenuation and good settling time, for the present case study, Wu and Wp has the form Wp ¼ 10 s þ 50 Wu ¼ 0:001 (14) Following the previously presented design method, the resulting H∞ controllers are as follows: 1) Controller for plant with additive uncertainty Using these controllers, a number of simulations were conducted in order to test the system for robustness.…”

“…However, it is by no means clear that such a controller achieves this minimum exists. In practice, the minimum is usually replaced by an ‘infimum’, which must, in general, aim to provide the system designer with a stabilizing controller K , such that where γ is the suboptimal upper bound (positive number).…”