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

CHAPTER 12 SOME ANALYTIC SOLUTIONS

so thatJlqr= limt z(t)Tz(t)

E

!1

the mean-square deviation of z. Since w is a white noise, we have (see section 5.2.2)

Jlqr = H 22

k

k

the square of the 2 norm of the closed-loop transfer matrix.

H

In our framework, the plant for the LQR regulator problem is given by

AP = A

Bu = B

Bw = I

Cz = 0Q12

Cy = I

Dzw = 0

Dzu = R120

Dyw = 0

Dyu = 0

(the matrices on left-hand side refer to the state-space equations from section 2.5).

This is shown in gure 12.1.

P

w

o

1 2

=

R

z

+

q

r

(sI A) 1 x

B

q

1 2

;

=

;

Q

+

u

y

K

The LQR cost is

22.

Figure

12.1

kH

k

The speci cations that we consider are realizability and the functional inequality

speci cation

H 2

:

(12.1)

k

k

12.1 LINEAR QUADRATIC REGULATOR

277

Standard assumptions are that (Q A) is observable, (A B) is controllable, and

R > 0, in which case the speci cation (12.1) is stronger than (i.e., implies) internal

stability. (Recall our comment in chapter 7 that internal stability is often a redun-

dant addition to a sensible set of speci cations.) With these standard assumptions,

there is actually a controller that achieves the smallest achievable LQR cost, and it

turns out to be a constant state-feedback,

Klqr(s) = Ksfb

;

which can be found as follows.

Let Xlqr denote the unique positive de nite solution of the algebraic Riccati

equationATXlqr+XlqrA XlqrBR 1BTX

;

lqr + Q = 0:

(12.2)

;

One method of nding this Xlqr is to form the associated Hamiltonian matrix

M = A

BR 1BT

;

;

Q

AT

(12.3)

;

;

and then compute any matrix T such that

T 1MT = ~A11 ~A12

;

0 ~A22

where ~A11 is stable. (One good choice is to compute an ordered Schur form of M

see the Notes and References in chapter 5.) We then partition T as

T = T11 T12

T21 T22

and the solution Xlqr is given by

Xlqr = T21T 1

;

11 :

(We encountered a similar ARE in section 5.6.3 this solution method is analogous

to the one described there.)

Once we have found Xlqr, we have

Ksfb = R 1BTX

;

lqr

which achieves LQR cost

Jlqr = Xlqr:

T

r

In particular, the speci cation (12.1) (along with realizability) is achievable if and

only if

p

Xlqr, in which case the LQR-optimal controller Klqr achieves the

T

r

speci cations.

In practice, this analytic solution is not used to solve the feasibility problem for

the one-dimensional family of speci cations indexed by rather it is used to solve

multicriterion optimization problems involving actuator e ort and state excursion,

by solving the LQR problem for various weights R and Q. This is explained further

in section 12.2.1.

278