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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 30
COMPUTATIONAL METHODS FOR ENGINEERING SCIENCE Edited by: B.H.V. Topping
Chapter 9
Stochastic Optimal OpenLoop Feedback Control K. Marti
Aerospace Engineering and Technology, Federal Armed Forces University Munich, Neubiberg/Munich, Germany K. Marti, "Stochastic Optimal OpenLoop Feedback Control", in B.H.V. Topping, (Editor), "Computational Methods for Engineering Science", SaxeCoburg Publications, Stirlingshire, UK, Chapter 9, pp 211235, 2012. doi:10.4203/csets.30.9
Keywords: optimal regulators under stochastic uncertainty, stochastic Hamiltonian, Hminimal control, stochastic optimal openloop control, stochastic optimal openloop feedback control, twopoint boundary value problem with random parameters, numerical solution of the twopoint boundary value problem, reduction of the boundary value problem to a fixed point condition, discretization of the fixed point condition.
Summary
The aim of this chapter is to determine (approximate) feedback control laws for
control systems taken under stochastic uncertainty. Stochastic parameter variations are
always present due to the following types of stochastic uncertainties: physical
uncertainty (variability of physical quantities, like material, loads, dimensions, etc.);
manufacturing uncertainty (manufacturing errors, tolerances, etc.); economic uncertainty
(costs, trade, demand, etc.); statistical uncertainty (e.g. estimation errors due to limited
a priori and, or sample data); model and environmental uncertainty uncertainty (e.g. initial
and terminal conditions, model errors or model inaccuracies).
In order to obtain the most insensitive optimal controls with respect to random parameter variations, hence, robust optimal controls, cf. [1], the problem is modeled in the framework of optimal control under stochastic uncertainty: Minimize the expected total costs arising along the trajectory, at the terminal time or point and from the control input subject to the dynamic equation and possible control constraints. As is well known, e.g., from model predictive control [2], optimal feedback controls can be approximated very efficiently by optimal openloop feedback controls. Optimal openloop feedback controls are based on a certain family of optimal openloop controls. Hence, for practical purposes it is sufficient to determine optimal openloop controls only. Extending this construction, stochastic optimal openloop feedback control laws are constructed by taking into account the random parameter variations of the control system. Thus, stochastic optimal openloop feedback controls are obtained by first computing stochastic optimal openloop controls on the "remaining" time intervals. Then by evaluating these controls at the corresponding intermediate starting time points only, a stochastic optimal openloop feedback control law is obtained. For the computation of stochastic optimal openloop controls at each intermediate starting time point, the stochastic Hamilton function H of the optimal control problem under stochastic uncertainty is introduced. Then an Hminimal control law can be determined by solving a finitedimensional stochastic optimization problem [3] for minimizing the conditional expectation of the stochastic Hamiltonian subject to the remaining deterministic control constraints at the current time point. Having a Hminimal control, the related twopoint boundary value problem with random parameters is formulated for the computation of the stochastic optimal state and adjoint state trajectory. In the case of a linearquadratic control problem, which arises often in engineering practice, the state and adjoint state trajectory can be determined analytically to a large extent. Inserting then these trajectories into the Hminimal control, stochastic optimal openloop controls are found. For solving the twopoint boundary value problem (BVP) with random parameters, instead of using the matrix Riccati differential equation, the (BVP) is reduced to a fixed point equation for the conditional mean adjoint trajectory. Moreover, approximating the integrals occurring in this fixed point condition, approximate systems of linear equations are obtained for the values of the conditional mean adjoint trajectory needed at the intermediate starting time points. Thus, the calculations can be done in real time. References
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