Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

CHAPTER 5 NORMS OF SYSTEMS

The objective is then simply ( ) with free. The solution of this free end-point optimal

z

T

T

control problem (which always occurs as

) yields

wc. This optimal control

T

!

1

kH

k

problem has linear dynamics and convex state and control constraints, so essentially all

numerical methods of solution will nd the value

wc (and not some local minimum).

kH

k

See, for example, the books by Pontryagin Pon62], Bryson and Ho BH75, ch.7], and

the survey article by Polak Pol73].

The Figures of Section 5.2.9

The input signals in gure 5.11 were computed by nely discretizing (with rst-order hold)

the optimization problem

10

Z

max

(10 ) ( )

1 0 h ; t w t dt

kw

k

1

_

1

kw

k

1

and solving the resulting linear program.

The input signals in gure 5.12 were computed from (5.6) with = 10.

T

The unit-energy input signals in gure 5.13 give the largest possible square root output en-

ergy for

5. These input signals were computed by nding the nite-time controllability

t

Gramian

5

Z

= 0 5]

5

5

contr =

T

T

At

T

A

t

= contr

A

contr A

W

W

0 e BB e

dt

W

;

e

W

e

:

where ( ) = (

) 1

;

H

s

C

sI

;

A

B

:

If is the largest eigenvalue of 1 2

1 2 and is the corresponding eigenvector,

=

obs

=

W

W

W

z

with

2 = 1, then the input signal

kz

k

( ) =

(5 )

1 2

T

for 0

5

T

A

;t

;

=

B

e

W

z

t

w

t

0

otherwise

has unit energy, and drives the system state to 1 2 at = 5. (It is actually the smallest

=

W

z

t

energy signal that drives the state from the origin to 1 2= in 5 seconds.) The output for

W

z

5,

t

( ) =

( 5) 1 2

A

t;

=

z

t

C

e

W

z

has square root energy

1 2

1 2

1 2

2

=

= ; max( = obs = ) 1=

W

W

W

(c.f. (5.39)).