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
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.