Section 3 Optimal Control

**Chapter 3**

*Yossi Peretz*

**Abstract**

Algorithms for LQR via Static

Randomized and deterministic algorithms for the problem of LQR optimal control via static-output-feedback (SOF) for discrete-time systems are suggested in this chapter. The randomized algorithm is based on a recently introduced randomized optimization method named the Ray-Shooting Method that efficiently solves the global minimization problem of continuous functions over compact non-convex unconnected regions. The randomized algorithm presented here has a proof of convergence in probability to the global optimum. The suggested deterministic algorithm is based on the gradient method and thus can be proved to converge to local optimum only. A comparison between the algorithms is provided as well as

Discrete-Time LTI Systems

**Keywords:** control systems, optimal control, discrete-time systems, state-space models, NP-hard control problems, randomized algorithms,

The application of static-output-feedbacks (SOFs) for linear-quadratic regulators (LQR) is very attractive, since they are cheap and reliable and their implementation is simple and direct, because their components has direct physical interpretation in terms of sensors amplification rates and actuator activation power. Moreover, the long-term memory of dynamic feedbacks is useless for systems subject to random disturbances, to fast dynamic loadings or to random bursts and impulses, and the application of state feedbacks is not always possible due to unavailability of full-state measurements (see, e.g., [1]). Also, the use of SOF avoids the need to reconstruct the state by Kalman filter or by any other state

On the other hand, in practical applications, the entries of the needed SOFs are bounded, and since the problem of SOFs with interval constrained entries is NPhard (see [2, 3]), one cannot expect the existence of a deterministic efficient (i.e., polynomial-time) algorithm to solve the problem. Randomized algorithms are thus natural solutions to the problem. The probabilistic and randomized methods for the constrained SOF problem and robust stabilization via SOFs (among other hard problems) are discussed in [4–7]. For a survey of the SOF problem see [8], and for a

Output Feedback for

the performance of the hybrid algorithm.

recent survey of the robust SOF problem see [9].

deterministic algorithms

**1. Introduction**

reconstructor.

**55**
