Design of Optimal Control Systems

In this thesis we investigate the problem of optimal control, in the domain of digital controllers. Digital controllers relies only on some observation of the state that occur at the sampling instants. In Chapter 1 we recall the basic terminology of the system theory. We define the linear systems, that will be the only ones investigated in this work, and we introduce the standard quadratic cost of a control input. Finally, we recall an existence theorem for the problem of finding the input that minimizes the cost. In Chapter 2 we recall the Riccati differential equation that provides the solution of the cost minimization problem in continuous-time control systems. We compute the explicit solution for the simple case of a uni-dimensional state, and we discuss how the parameters affect the solution. In Chapter 3 we examine, instead, discrete-time control systems. In this case an analogous solution is provided by the discrete Riccati recurrent equation. We recall the general results about the convergence of this recurrent definition, and we provide the proof of the convergence for the simple uni-dimensional case. In Chapter 4, we investigate the sampled-time systems. These systems evolve according to a continuous-time dynamics, however the control input is provided only at some predetermined instants, called sampling instants. We show that a sampled-time system can be studied as a discrete-time one. When the sampling instants are all evenly spaced, we say that we are sampling periodically. We show how the cost is affected by the choice of the sampling period. In addition we also show that a lower cost can be achieved by relaxing the constraint of a periodic sampling. Hence we search for the optimal sampling sequence. Finally, we observe that the density of the sampling instants has indeed an effect on the computing device that hosts the controller. For this reason we extract from any sampling sequence, not necessarily periodic, two key features (the asymptotic period and the burstiness) that have an impact on the amount computational resource required by a controller running at those sampling instants. We conclude by evaluating the amount of cost reduction that is possible depending on a period-burstiness constraint