Usually, a precomputed feedback control is based on exactly known model parameters. However, in practice, often the exact information about model parameters and initial values is not given. Hence, having an inital point, which differs slightly from the nominal values, a standard precomputed controller may produce bad results. Supposing now that the probability distribution of the random parameter variations is known, in the following stochastic optimisation methods will be taken into account in order to obtain robust optimal feedback controls.

The assumption that we do not exactly know the deviations of the model parameters and initial state from the reference values, but can describe them by stochastic properties, results in a new form of a robust controller.

Taking into account stochastic parameter variations at the initial point, the method works with expected cost functions evaluating the primary control expenses and the tracking error [1,2]. Furthermore, the free regulator parameters are selected then such that the expected total costs are minimized.

After a Taylor expansion to calculate expected cost functions and a few transformations an approximate deterministic substitute control problem follows. Here, retaining only linear terms, approximation of expectations and variances of the expected cost functions can be calculated explicitly. By means of splines, numerical approximations of the objective function and the differential equations are then obtained. The resulting, deterministic substitute problem can be solved by using appropriate software [3].

Using stochastic optimization methods, random parameter variations are incorporated into the optimal control process. Hence, robust optimal feedback controls are obtained.

The functionality of this controller is shown in an example, which is based on a DOF 3 robot [4]. The method works well and shows the basic behaviour, that increasing tracking error costs lead to decreasing deviations of the actual trajectory to the reference trajectory. Furthermore, the most important instant for a control correction is at the initial time interval.

1

Marti K., "Adaptive Optimal Stochastic Trajectory Planning and Control (AOSTPC) for Robots", in "Dynamic Stochastic Optimization", K. Marti, Y. Ermoliev, G. Pflug, (eds.), LNEMS 532, Springer-Verlag, Berlin, 155-206, 2004.

2

Marti K., "Optimale Regelung von Robotern", Mathematisches Seminar, Universität der Bundeswehr München, 2005-07.

3

Stryk O., "User's Guide for DIRCOL - A Direct Collocation Method for the Numerical Solution of Optimal Control Problems", Version 2.1, TU Darmstadt, Fachgebiet Simulation und Systemoptimierung, 2002.

4

Türk S., "Dynamische Robotermodelle am Beispiel des Manutec R3, Technischer Bericht 88-16, DVFLR, Institut für Dynamik der Flugsysteme, Oberpfaffenhofen, 1988.

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