## Tesi etd-01132010-122342 |

Thesis type

Tesi di laurea specialistica

Author

BINI, ENRICO

email address

e.bini@sssup.it

URN

etd-01132010-122342

Title

Design of Optimal Control Systems

Struttura

SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di studi

MATEMATICA

Supervisors

**relatore**Prof. Buttazzo, Giuseppe

**controrelatore**Prof. Buttazzo, Giorgio C.

Parole chiave

- controllo ottimo
- equazioni di riccati
- campionamento
- campionamento non periodico
- tempo reale
- controllori

Data inizio appello

29/01/2010;

Consultabilità

Completa

Riassunto analitico

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.

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.

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