Optimal feedback solution of a constrained stochastic one-storage model

We describe a method for obtaining the optimal feedback solution to a constrained discrete time linear–quadratic optimal control problem. The dimensions of the state and control vectors are one and the stochasticity is represented by a finite number of possible outcomes at each stage. The method is based on dynamic programming and exploits that the optimal value is piecewise quadratic and that the optimal feedback solution is piecewise linear. Also the special case with linear criterion function is considered. Approximation methods which allow a trade-off between computation times and solution accuracy are discussed. The method is applied to two cases, viz., hydropower scheduling and temperature control of a greenhouse. Comparative studies on these two cases were made with a mathematical programming formulation solved by a standard code. On the cases studied the new method was found to be around 100 times faster and to display a better solution stability.