Optimal Control

Lecture

Prof. Daniela Iacoviello

Department of Computer, Control, and Management Engineering Antonio Ruberti

Sapienza University of Rome

27/11/2015 Controllo nei sistemi biologici

Lecture 1

Pagina 2

Prof. Daniela Iacoviello

Department of computer, control and management

Engineering Antonio Ruberti

Office: A219 Via Ariosto 25

http://www.dis.uniroma1.it/

Prof.Daniela Iacoviello- Optimal Control

References B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989 C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993 L. Evans, An introduction to mathematical optimal control theory, 1983 H.Kwakernaak , R.Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972 Prof.Daniela Iacoviello- Optimal Control

References

D. E. Kirk, "Optimal Control Theory: An Introduction, New

York, NY: Dover, 2004 D. Liberzon, "Calculus of Variations and Optimal Control

Theory: A Concise Introduction", Princeton University

Press, 2011 How, Jonathan, Principles of optimal control, Spring

2008. (MIT OpenCourseWare: Massachusetts Institute of

Technology). License: Creative Commons BY-NC-SA.

Prof.Daniela Iacoviello- Optimal Control

Harmonic oscillator (from Bruni et al.1993)

Prof.Daniela Iacoviello- Optimal Control

0)()()(

)()(

12

21

tutxtx

txtx

0

0

A

1

0b

jEigenvalues:

When u=0

0)()(

0)()(

22

2

12

1

txtx

txtx

The natural modes are oscillatory

Problem

Prof.Daniela Iacoviello- Optimal Control

1)(

0)()(

tu

txxtx fiiDetermine a control such that

That minimizes the cost index:

singularnot01

0

Abb

There exists a unique, non singular solution and the control

Is bang-bang for any initial condition

iff tttJ )(

Eigenvalues of A: j

of

oo tux ,,

Prof.Daniela Iacoviello- Optimal Control

)()()()()()(1 21221 tuttxttxtH

BUT

Since the eigenvalues are complex we can’t apply the theorem about the maximum

Number of commutation points

Alternative procedure:

Necessary conditions:

Introduce the Hamiltonian:

)()( tAt oTo

1)(:

,)()()()()()(1

)()()()()()(1

2122

2122

1

1

t

tttxttxt

tuttxttxt

ooooo

oooooo

Prof.Daniela Iacoviello- Optimal Control

)()(

)()(

1

11

2 tt

tt

oo

oo

1)(:,)()()()(22

ttttut ooo

)(sin

)()()(2

i

ooo

ttKsign

tsignbtsigntu

)(sin)()()( 22 2 iooo ttKttt

All the switching subintervals have length equal to

With the exception of the first and the last

whose length is less or equal than

Prof.Daniela Iacoviello- Optimal Control

All the switching subintervals have length equal to

With the exception of the first and the last

whose length is less or equal than

Figura 1 (Bruni et al.1993)

Prof.Daniela Iacoviello- Optimal Control

Let’s assume

And integrate the system:

1)( tu

0)()()(

)()(

12

21

tutxtx

txtx

1)(sin)(cos

1)( 211

i

ii

i ttxttxtx

)(cos)(sin1

)( 212 ii

ii ttxttxtx

22

2

12

2

1

1)(

1)(

2

ii xxtxtx

Figura 2. (Bruni et al.1993)

22

2

12

2

1

1)(

1)(

2

ii xxtxtx

The trajectories with control are circumferences with center

passing through the initial condition

The direction is clockwise.

1)( tu

0,

1

ix

Prof.Daniela Iacoviello- Optimal Control

Only one trajectory of this family passes through the origin

1)(

1)( 2

2

1 2

txtx

Figura 3. (Bruni et al.1993)

Prof.Daniela Iacoviello- Optimal Control

For a generic instant t:

1

)(

)()(

1

1 2

tx

txtgt

2

1

1212

2

1

21

)(

)()(1

)()(

1)(

)(1

1)(

2

tx

txtxtxtx

tx

txt

)(t

0

)()()(

)()(

12

21

tutxtx

txtx

The trajectories are traversed

with constant angular velocity

Prof.Daniela Iacoviello- Optimal Control

The trajectories are traversed with constant angular velocity

t

1)( tu

The trajectories are traversed with constant angular velocity

We must reach the origin along arcs with amplitude

There exists a unique way to do it!

Prof.Daniela Iacoviello- Optimal Control

Definitions:

,....2,1,0,0,112

: 22

22

2

12

kxx

kxRxk

,...2,1,0,0,112

: 22

22

2

12

kxx

kxRxk

Figura 4. (from Bruni et al. 1993):

Prof.Daniela Iacoviello- Optimal Control

Let and , k=1,2,…., be the sets in Figure:

And define:

k

k

0k

k

0k

k

0k

k

0k

k

Figura 5. (from Bruni et al. 1993)

Prof.Daniela Iacoviello- Optimal Control

The initial states and may be moved to the origin with a

constant control equal to +1 and -1 respectively and the corresponding arc

is less or equal than π.

The points belonging to are the only ones that can be transfered to the

origin with a constant control, without switching

0ix 0ix

00

1a 1b

Prof.Daniela Iacoviello- Optimal Control

Let’s consider initial states

They may be moved to the origin with the following path:

- First a constant control +1 up to the arc for an interval time

- Switching to control -1 to reach the origin.

1 switching instant

001 \ ix

0

Let’s consider initial states

They may be moved to the origin with the following path:

- First a constant control -1 up to the arc for an interval time

- Switching to control +1 to reach the origin.

1 switching instant

0

001 \ ix

2a

2b

Let’s consider initial states

They may be moved to the origin with the following path:

- First a constant control +1 up to the arc for an interval time

- Note that: apply strategy 2b

- Switch to control -1 to reach and then switch to control +1 to reach the

origin two switching instants

12 \ ix

1

Let’s consider initial states

They may be moved to the origin with the following path:

- First a constant control -1 up to the arc for an interval time

- Note that apply strategy 2a

- Switch to control +1 to reach and then switch to control -1 to reach the

origin: two switching instants

1

12 \ ix

11

0

3a

3b

11

0

Figura 6. (from Bruni et al. 1993)

Figura 7. (from Bruni et al. 1993)

Figura 8. (from Bruni et al. 1993)

Figura 9. (from Bruni et al. 1993)

The same happens for any intial condition:

1\ kkix

1\ kkix

\ixAt the generic state you must associate u=+1

At the generic state you must associate u=-1 \ix

\)(1

\)(1)(

tx

txtxu

o

ooo

The number of switching points is given by the minimum index k among the ones

Characterizing the sets and

Example: Consider the point

Then there is only one switching instant.

kk

321P

Time necessary to get to the origin:

If is the number of switching instants:

foii

of tt 1

If

where:

if

if

i

i

ix

xtg

1

21

1

o

Rotation of the optimal trajectory

to go from the initial point to the

first point of commutation

Rotation of the optimal trajectory

to go from the final point to the

origin

0o

ii

of tt

i

i

ix

xtg

1

21

1

oix

oix