Linear Quadratic Design of Structure-Constrained Controllers

Linear quadratic design of parameter-optimized linear controllers is considered for discrete time stochastic systems. In the design method the feedback gains, or parameters, of a linear controller, for instance of a discrete PID-regulator, are tuned so as to minimize a quadratic loss function. Linear descent methods for the solution of the optimal feedback gains are discussed. The concept of partial line search is found useful in constructing effective linear descent methods. Several different algorithms can be obtained in this way. An algorithm utilizing the Goldstein step length rule is shown to generate a sequence of feedback gains converging to a stationary point of the loss function. The convergence result is based on convergence properties of descent gradient methods for function minimization. The rate of convergence of the proposed algorithm is considerably faster than that of five other algorithm for the test problems presented in the paper. The extension of the optimal structure-constrained control problem to include explicit variance inequality restrictions in the control signals and state variables is discussed in detail. Numerical methods to solve the control problem are discussed. A reliable algorithm utilizing a monotonicity property of the Lagrangian function of the control problem is proposed for the case of a single explicit variance restriction.