1ME 233 Advanced Control II
Lecture 12
Stationary
Linear Quadratic Gaussian (LQG)
Optimal Control(ME233 Class Notes pp.LQG1-LQG7)

2Outline• Stationary LQG• Relationship to H2 optimal control

3

Stationary random inputs
Linear system contaminated by noise:

W
Bw
U

V+B

(zI-A ) -1

X+
C

Assume that both•w(k) and v(k) are WSS, zero-meanY

4

Stationary LQG
We want to regulate the state

underWSS zero-mean
white GaussianNoise5

Stationary LQG

Define the “incremental” cost

The control that minimizes

also minimizes

6

Stationary LQG“Incremental” cost:

Under the stationarity assumptions:

7

Stationary LQG
Obtain the optimal control that minimizes:

under•

w(k) and v(k) are WSS

8Optimal stationary LQG
Theorem:a)

The optimal control is given by

Such thatis SchurStandard deterministic infinite-horizon LQR solution!

9

Optimal stationary LQG
Theorem (cont’d):A-posteriori state observer structure:

Such thatis Schur

10State space form of LQG controller

Kalmanfilter
LQR

Eliminatingfrom the expression for uo(k) yields

Plugging this expression for uo(k) into the expression foryields the state space model on the next slide11State space form of LQG controller

K is the standard deterministic LQR gain
F and L are the standard Kalman filter gains

The closed-loop poles are the eigenvalues ofA – BK and the eigenvalues of A – LC

12

Optimal stationary LQGTheorem (cont’d):b)

The optimal cost is(see the derivation of this result at the end)

13

Conditions for existence• Existence of infinite-horizon LQR solution– (A,B) stabilizable– (CQ ,A) has no unobservable modes on theunit circle• Existence of stationary KF solution– (C,A) detectable– (A, BWW1/2) has no uncontrollable modeson the unit circle

14

H2 norm• Let G(z) be a stable discrete-time transferfunction• The H2 norm of G(z) is defined byAverage overfrequency

Squared Frobenius
norm of15

H2 norm

Suppose U(k) is WSS and zero-mean,Then

16

H2 norm

LQG cost function canbe written in this form

17

Plant dynamics

define18

Noise covariance19

Stationary LQG cost function

factor asdefine

20Plant dynamics and LQG costController

21

H2 optimal control problem• For any given stabilizing LTI controller, the squared H2norm of the closed-loop system isThis is equal to the stationary LQG cost!
Minimizing the closed-loop H2 norm is equivalent
to minimizing the stationary LQG costController

22

One way to choose an LQG cost function
Each pi(k) is a signal you would like to keep “small” in theclosed-loop systeme.g. position error, control effort, actuator displacement

Always include control effort!

Controller

23

One way to choose an LQG cost function

For any chosen nonzero values of,you can perform an optimal control design and thenfind the values ofChoose nonzero values ofvalues ofThis requires iteration

so that theare reasonableAdditional material(you are not responsible for this)• Derivation of optimal stationary LQG cost

24

25

Stationary LQG
Optimal cost (derivation)
The incremental optimal cost isThus

26

Stationary LQG
Optimal cost (derivation)

Note:(least squares)

27

Stationary LQG
Optimal cost (derivation)
last term:

first term:

28

Stationary LQG
Optimal cost (derivation)