1ME 233 Advanced Control II

Lecture 5
Random Vector Sequences(ME233 Class Notes pp. PR6-PR10)

2Outline• Random vector sequences– Mean, auto-covariance, cross-covariance• MIMO Linear Time Invariant Systems• State space systems driven by white noise• Lyapunov equation for covariancepropagation

3Random vector sequences
A two-sided random vector sequence is acollection of random vectors

each

is itself a random vector

defined over the same probability space4Random vector sequences
We either will useorShorthand(sloppy) notation
to denote the two-sided random vector sequence.
Each elementof the sequence is a random vector:

5Random vector sequences

A sample sequencecorresponds to the value ofobtained after performing an experiment6

2nd order statistics
For a two-sided Random Vector Sequence (RVS)

Expected value or mean of X(k),7Auto-covariance
Define:8

Cross-covariance
Define:9Wide Sense Stationary (WSS)
A two-sided random vector sequenceis WSS if:

1)(time invariant)2)(only depends on l)

10Auto-covariance function
For WSS RVS, the auto-covariance is only
a function of the correlation index jfor any index k

11Auto-covariance function Z-transform

Z-transform12

Auto-covariance function

Proof:

Define13

Auto-covariance function Z-transform

Proof:

Define14

Cross-covariance function
andare two WSS random vector sequences

for any index k
Notice that:

15

Auto-covariance function Z-transform

Z-transform

16

Cross-covariance function

Proof:

Define17

Auto-covariance function Z-transform

Proof:

Define18Ergodicity
A Wide Sense Stationary random sequenceis ergodic
if its ensemble average = time average (constant)with probability 1(almost surely)

sample sequence19

Ergodicity
For any WSS ergodic random sequence
we can approximate the covariance as a “time average”

with probability 1(almost surely)

sample sequence20Power Spectral Density Function
Fourier transform of the auto-covariance function:

Complex-valued matrix

Note:l: correlation index

The power spectral density function is periodic,
with period

21Power Spectral Density Function
Using the inverse Fourier transform we obtain:

22Power Spectral Density Function
Properties of the power spectral density function:1.

2.

3.

4.

23

Power Spectral Density Function
1.
Proof:

Define24Power Spectral Density Function
2.
Proof:

Define25

Power Spectral Density Function
Properties of the power spectral density function:(scalar case)

1.

2.

3.

4.

26

White noise vector sequence
A WSS random vector sequence
white if:whereis27

White noise vector sequence
Given the white WSS random sequencewith

Its power spectral density (Fourier transform)

is

28

White noise illustration (scalar case)• zero-mean white noise
Matlab commands:
2
1.5

w = randn(N,1);

10.5w

0-0.5-1-1.5-2-2.5
0510k

first 20 samples

1520

29

MIMO Linear Time Invariant Systems
Letwith

be the pulse response of an asymptotically stable
MIMO LTI system
Transfer function

30

MIMO Linear Time Invariant Systems
Letbe WSS

The forced response (zero initial state)is also WSS

31

MIMO Linear Time Invariant Systems
Let

be WSSg(k)32MIMO Linear Time Invariant Systems
We will assume
is zero mean, I.e.

Thus, the forced response output is also zero mean

33

MIMO Linear Time Invariant Systems
LetbeWSS

Ifg(k)

Then:g(l)34MIMO Linear Time Invariant Systems

Proof:Then:

35

MIMO Linear Time Invariant Systems
Letbe WSS

G(z)

36

MIMO Linear Time Invariant Systems
Letbe WSS

Ifg(k)
Then:g(l)37MIMO Linear Time Invariant Systems
Proof:Then:

38

MIMO Linear Time Invariant Systems
Letbe WSS

G(z)

39

MIMO Linear Time Invariant Systems

This is a consequence of the fact that

40

MIMO Linear Time Invariant Systems
Proof:

Define41

MIMO Linear Time Invariant Systems
If

Then:

42

MIMO Linear Time Invariant Systems
Proof:

43

MIMO Linear Time Invariant Systems
If

Then:

44

MIMO Linear Time Invariant Systems
Proof:

Let z = ejω45

Next Topic• Stable causal LTI systems driven byuncorrelated random vector sequences•Similar to “white”•Definition in 2 slides• State-space• No WSS assumption

46

2nd order statistics of a random sequence
We now consider one-sided random sequence

Expected value or mean of X(k),Auto-covariance function:

47

Uncorrelated random vector sequence
A random vector sequence
uncorrelated if:

whereis48

Subtracting the mean• Define

Auto-covariance49

State space systems
Consider a LTI system driven by an uncorrelated RVS:50

State space systems
W(k) is an uncorrelated RVS

51

State space systems

State Initial Conditions (IC):52

Dynamics of the mean

Taking expectations on the equations above:53

State space systems
Subtracting the means we obtain,

Where now

54

Causality in cross-covariance

Proof: (by induction on k)
1. Base case, k=0 : trivial by assumptions on system
2. Case k>0 :0(by induction hypothesis)

55

Covariance propagation

Notice that:

56

Covariance propagation
Taking expectations to:57

Covariance propagation
Notice that:

00(W(k) is an uncorrelated RVS)

58

Covariance propagation
We obtain the following Lyapunov equation:

59

Covariance propagation
From the output equation

we obtain

60

Covariance propagation
Lets now compute,

Using the solution of the LTI system,

61

Covariance propagation0

62

Covariance propagation
Lets now computedefine63

Covariance propagation

Satisfies:

64

Stationary covariance equation
If W(k) is WSS
and A is Schur (i.e. all eigenvalues inside unit circle):and X(k) and Y(k) will converge to WSS RVS:

65

WSS Stationary covariance equation
For W(k) WSS,

converges toand A Schur,66

WSS Stationary covariance equation
For W(k) WSS,

and A Schur,

Satisfies the Lyapunov equation:

67

WSS Stationary covariance equation
For W(k) WSS,

Satisfies

and A Schur,68

Illustration – first order system• Plant:• Input:• State initial conditions:

69

Matlab simulation: 500 sample sequences
lyy0 = 0.1
lww = 0.2
sys1=ss(.5,1,1,0,1)
N=20;
p=500;
w = sqrt(lww)*randn(N,p)+1;
y = zeros(N,p);y0 = sqrt(lyy0)*randn(1,p);k = (0:1:N-1)';
for j=1:p[y(:,j),k] = lsim(sys1,w(:,j),k,y0(1,j));end
m_y=mean(y')
L_yy=diag(cov(y'));

70

W(k)
w1(k)

21w2(k)
w3(k)

15

20

51015

20

51015

20

510k15

20

100
2
100
4w4(k)

10200kEnsemble5(4 out of 500)

002

71

Y(k)
y1(k)

50y2(k)
y3(k)

15

20

510

15

20

510

15

20

510k15

20

200
4200
4y4(k)

10200kEnsemble5(4 out of 500)

-504

72

Mean Transient Response
Actual:y2(k)510152051015205101520510k1520

2004y3(k)Not using ergodicity
because not WSS!!

0-504

2004y4(k)Matlab calculation:
Ensemble mean
m_y=mean(y');

y1(k)

5200

73

Mean Transient Response
1.5

yActual m (k)

2

1
0.5
00510k15201520

2y

Estimated m (k)

2.5

1.5

Calculated by
m_y=mean(y');

10.500510k74

Covariance Transient Response
Actual:

y2(k)

51015

20

51015205101520510k15202004y3(k)Not using ergodicity
because not WSS!!

0-5042004y4(k)Matlab calculation:
Ensemble covariance
L_yy=diag(cov(y'));

y1(k)

52
00

75

Covariance Transient Response
Actual  yy(0,k)

0.35
0.3
0.25
0.2
0.15
0.1
0510k1520

Estimated  yy(0,k)

0.35
0.3
0.25

Calculated by
L_yy=diag(cov(y'));

0.2
0.15
0.1
0

510
k1520

76

Steady State Covariance
Actual  yy(0,k)

0.35
0.3
0.25
0.2
0.15
0.1
0510k1520Estimated  yy(0,k)

0.35
0.3
0.25

Calculated by
L_yy=diag(cov(y'));

0.2
0.15
0.1
0510
k1520