Keywords: optimal control under stochastic uncertainty, optimal feedback control, optimal open-loop feedback control, stochastic optimal open-loop feedback, stochastic Hamiltonian, two-point boundary value problems, approximation of the fundamental matrix.

Optimal control problems for technical systems depend on several random system and environmental parameters. In order to obtain optimal feedback controls most insensitive with respect to random parameter variations, hence robust optimal controls, the problem is modelled [1] in the framework of optimal control with stochastic uncertainty: Here, the expected total costs F arising along the trajectory, from the control input and at the terminal time/point are minimized subject to the plant differential equation and possible control constraints.

Optimal feedback controls of PD- or PID-type can be approximated very efficiently by optimal open-loop feedback controls based on a certain family of optimal open-loop controls. Extending the standard construction, stochastic optimal open-loop feedback controls are constructed by taking into account still the random parameter variations in the control system. Hence, corresponding to the standard deterministic case, stochastic optimal open-loop feedback controls are obtained by computing first stochastic optimal open-loop controls on the remaining time intervals with an arbitrary intermediate starting time point. Evaluating these controls at the corresponding intermediate starting time points only, a stochastic optimal open-loop feedback control law is obtained.

Using a stochastic Hamiltonian approach, the stochastic optimal open-loop controls for each intermediate starting time point are obtained by computing the "H-minimal controls", based on a finite dimensional stochastic optimization problem, and then solving the related linear two-point boundary value problem with random parameters, the stochastic "Hamiltonian system" for the state and adjoint state variables. Inserting then these trajectories into the H-minimal control, stochastic optimal open-loop controls are found. The remaining problem is then the computation of the H-minimal controls and the solution of the related Hamiltonian system:

In the convex, linear-quadratic case, often present in practice, the H-minimal controls can be determined explicitly. Moreover, in case of fixed system matrices, using the matrix exponential function, solutions of the Hamiltonian system can be found explicitly and in real-time. Extending this method to time-dependent random system matrices, approximations of the required fundamental matrix are constructed as described in the full paper.

1

K. Marti, "Continuous-Time Control under Stochastic Uncertainty", In J.J. Cochran et al., (Editors), "Encyclopedia of Operations Research and Management Science (EORMS)", Wiley, Hoboken, 2010. doi:10.1002/9780470400531.eorms0839

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £130 +P&P)