Keywords: stochastic optimal control, robust optimal control, inner linearization, convex approximation, stochastic Hamiltonian, H-minimum control, stochastic nonlinear programming, Hamiltonian system of differential equations, stochastic nonlinear model predictive control (SNMPC).

Optimal regulator problems arising in different technical (mechanical, electrical, thermodynamical, chemical, etc.) plants are modelled by dynamical control problems obtained from physical measurements and/or known physical or technical laws. In many practical cases the basic control system (input-output system) can be described mathematically by a system of first order nonlinear differential equations for the plant state vector z=z(t) involving for example displacements, stresses, voltages, currents, pressures, concentration of chemicals, etc. These differential equations depend on the vector u(t) of input or control variables and a vector a(.) of certain random model parameters. Moreover, the vector z₀ of initial values of the plant state vector z=z(t) may be subject to random variations. While the actual realization of the random model parameters and initial values are not known at the planning stage, we may assume that the probability distribution or at least the occurring moments, such as expectations, variances, etc., are known in advance. Concerning the performance function of the stochastic dynamic system we may assume that the costs L along the trajectory and the terminal costs G are convex functions with respect to the pair (u,z) of control and state variables u, z, the final state x(t_f), respectively. The problem [1,4] is then to determine an open-loop, closed-loop or an intermediate open-loop feedback control law u* minimizing the expected total costs arising along the trajectory and at the terminal state.

Approximative solutions [3] are obtained by "inner linearization" of the problem, hence, by sequential linearization of the process differential equation, which yields a convex approximation of the basic stochastic control problem. Moreover, based on this convex approximation, necessary and sufficient optimality conditions for approximate optimal controls can be obtained. In addition, necessary and sufficient optimality conditions for the approximate optimal control problem may be formulated then in terms of the stochastic Hamiltonian of the control problem. Canonical (Hamiltonian) systems of first order differential equations, thus, two-point boundary value problems, are obtained then for the computation of approximate optimal controls, as well as for stationary controls being candidates for optimal controls, respectively. Based on the concept of H-minimum control, the computation of optimal controls may be reduced to some extent to certain finite dimensional stochastic programming problems. Furthermore, the expected Hamiltonians can be computed approximatively by means of Taylor expansions yielding an approximate canonical two-point boundary value problem for the state, costate variables and their sensitivities with respect to the total parameter vector at its mean value. Applications to nonlinear model predictive control (NMPC) [2] are given.

1

Aström K.J., Introduction to stochastic control theory, New York: Academic Press, 1970.

2

Findeisen R., Raff T., Allgöwer F., Sampled-Data Nonlinear Model Predictive Control for Constrained Continuous Time Systems. In: S. Tarbouriech et al. (Eds), Adaptive strategies in control systems with input and output constraints, Berlin: Springer-Verlag, pp. 207-235, 2007. doi:10.1007/978-3-540-37010-9_7

3

Marti K., Stochastic Optimization Problems, Berlin-Heidelberg: Springer-Verlag, 2005.

4

Stengel R.F., Stochastic Optimal Control - Theory and Application, New York [etc.]: J. Wiley, 1986.

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