Optimal control of input constrained linear systems

Explicit, closed-form formulas of optimal and suboptimal control on finite time interval for stochastic, linear, input-constrained systems are derived. The formulas are given in terms of the state transfer matrix of the system, the input constraint, weights of a quadratic criterion and noise variance. For a general system, only implicit solutions can be derived. However, for systems with positive impulse response, an explicit closed-form solution is derived. The optimal control in this case is the control for an unconstrained system with limiting on the input. The limiting function is the random-input describing function of the saturation function