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ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Chapter 4 describes five key areas

1. How we define and characterize stochastic process

2. The need and a framework for developing stochastic

linear system models

3. Forming models of linear differential or difference

equations

4. A way for describing measured systems outputs

5. Ways of building ‘practical system models’ to reflect

reality

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Agenda – Class Meeting 04

• Stochastic processes (Section 4.2) - pp. 133-139 (7p)

• Stationary stochastic processes and power-spectral density (Section 4.3) - pp.

139-145 (7p)

• System modeling (Section 4.4) - pp. 145-147 (3p)

• Frequency analysis and domain techniques - to be provided

• Response of lightly damped systems - to be provided

• Derivatives of random processes (from Bendat and Piersol handout)

In a 2nd meeting

• White Gaussian noise and Brownian motion (Section 4.5) - pp. 147-156

(10p)

• Sections 4.6, 4.7, 4.8 and 4.9

• Sections 4.10, 4.11 and 4.12

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Definitions

• Stochastic process

• Mean value function

• Covariance and cross-covariance

• Correlation and cross-correlation

• Independent and uncorrelated

• Strictly and wide-sense stationary

• Fourier transform

• Power-spectral density and cross-power

• Ergodic processes

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Stochastic process

: ( , )A tω ω ξ= ∈Ω ≤x

for any t an element of T and nR∈ξ are in the underling σ algebra.

is a stochastic process if sets of the form, ( , )⋅ ⋅x

For each point in the sample

space, there is an associated

‘sample’ stochastic process.

Discrete or continouous

1 1( , )t ωx

2 1( , )t ωx

3 1( , )t ωx

4 1( , )t ωx

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

The collection of all possible sample functions that a random

phenomenon might have produced is called a random or stochastic

process.

One sample record of data may be thought of as one physical

realization.Bendat and Piersol [3]

Maybeck [1]

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Stochastic process (cont.)

e.g., road profile

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

To characterize a CT stochastic process completely, you need to

know the joint probability distribution function or density function.

However, you’d

need 1st and 2nd order

moments, as well as

all higher order

functions for all t

values of interest.

This is not practical.

For Gaussian

processes, the first

two moments are

sufficient.

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Mean value function

The mean value function is defined for all t in T, being an average

value taken over the entire ensemble of samples from the process.

The covariance matrix gives an indication of the spread of values

from the mean at time t,

Covariance kernel – measure of how fast x(t) samples change over

time

Correlation matrix

Correlation kernel

Correlation matrix

( ) ( )x t E tm ≜ x

[ ][ ] ( ) ( ) ( ) ( ) ( )T

xx x xt E t t t t− −P m m≜ x x

[ ][ ] 1 2 1 1 2 2( , ) ( ) ( ) ( ) ( )T

xx x xt t E t t t t− −P m m≜ x x

( ) ( , )xx xx

t t tP P≜

1 2 1 2 1 2 1 2( , ) ( ) ( ) ( , ) ( ) ( )T T

xx xx x xt t E t t t t t t= +Ψ P m m≜ x x

( ) ( , ) ( ) ( )T

xx xxt t t E t t=Ψ Ψ≜ x x

Characterizing Stochastic Processes

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Cross-covariance kernel – relating two processes x(t) and y(t)

Cross-covariance matrix

Cross-correlation kernel

[ ] 1 2 1 1 2 2( , ) ( ) ( ) ( ) ( )T

xy x yt t E t t t t − − P m m≜ x y

1 2 1 2 1 2 1 2( , ) ( ) ( ) ( , ) ( ) ( )T T

xy xy x yt t E t t t t t t= +Ψ P m m≜ x y

( ) ( , )xy xy

t t tP P≜

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Independent, white, and uncorrelated processes Discussion(these notes need to be re-worked/cleaned up)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Discussion

(these notes need to be re-worked/cleaned up)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

A strictly stationary process is WSS iff it has finite second order

moments, and WSS does not imply SSS. A Gaussian WSS is SSS.

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

For WSS, diagonal terms of the correlation matrix are even

functions only of τ with a maximum value at τ = 0.

(add more here later)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

For WSS processes, we can use Fourier transforms to generate

frequency domain characterizations of stochastic processes.

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Good question: what is it about stationary processes that enable

application of Fourier transforms?

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Mean-square value from PSD

2 1( ) (0) ( ) ( )

2xx xx xx

E t d f dfω ωπ

∞ ∞

−∞ −∞

= Ψ = Ψ = Ψ∫ ∫x

2(quantity)

PSD has units Hz

(0) mean-square value of ( )xx tΨ ≜ x

(a lot of practical use comes from this)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

White

Exponentially

time-correlated

Random bias

From Maybeck Example 4.2

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

From Bendat and Piersol [3]

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

“A process is ergodic if any statistic calculated by averaging over all

members of the ensemble of samples at a fixed time can be

calculated equivalently by time-averaging over any single

representative member of the ensemble, except possibly a single

member out of a set of probability zero.”

No readily applied condition or test for ergodicity.

“In practice, empirical results for stationary processes are often

obtained by time-averaging of a single process sample, under the

assumption of ergodicity, such as:

[ ]

[ ]

[ ]

1( , ) lim ( , )

2

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

2

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

2

T

x iTT

T

xx i iTT

T

xy i iTT

m E t t dtT

E t t t t dtT

E t t t t dtT

ω

τ τ ω τ ω

τ τ ω τ ω

−→∞

−→∞

−→∞

= ⋅ =

Ψ ⋅ + ⋅ = +

Ψ ⋅ + ⋅ = +

∫

∫

∫

≜

≜

x x

x x x x

x y x y

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

This works by (seems obvious now):

1( ) ( )

2

Let:

1( ) ( )

2

So,

1( ) ( ) ( )

2

T R

T

T R

nn

V n t n t dtT

t

V n n dtT

V n n dt RT

τ

τ τ

ρ τ

ρ ρ τ τ

ρ ρ τ τ

∞

−∞

∞

−∞

∞

−∞

= − −

= −

= + −

= + =

∫

∫

∫

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

A random data analysis perspective [3]

Consider the mean (1st moment) of the ensemble computed at t1. This is estimated

by taking value at this time for each sample function and dividing sum by the

number of samples.

1 1

1

1( ) lim ( )

N

x kN

k

t x tN

µ→∞

=

= ∑

1 1 1 1

1

1( , ) lim ( ) ( )

N

xx k kN

k

t t x t x tN

τ τ→∞

=

Ψ + = +∑If both of these vary with time, the data is said to be

nonstationary. If these do not vary as t1 varies, the

process is said to be weakly stationary, or stationary in

the wide sense.

Weakly stationary random processes have constant mean

values and the autocorrelation function depends only on

τ. Verifying weak stationarity usually suffices for

justifying an assumption of strong stationarity (all

possible moments and joint moments are invariant).Bendat and Piersol [3]

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Refer to Bendat and Piersol, Chapter 5.

Good examples of calculating autocorrelation functions for

different types of signals.

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Example: Cross-correlation function for time-delay [3]

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Example: Cross-correlation function for time-delay [3] (cont.)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

These notes need to be

cleaned up and

organized.

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Response of a linear system to stochastic input –

first, review frequency response analysis

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Key results for time and frequency response

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Now, calculating

some response

measures of

interest – time

domain

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Frequency domain response in terms

of PSD

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Getting the mean square response using PSD

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

These results are also used in system identification – to estimate

the system H(ω)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

More on PSD – Another interpretation of its meaning

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

More on PSD – Another interpretation of its meaning (cont.)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

More on PSD – Another interpretation of its meaning (cont.)

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Response of Lightly Damped Systems

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Example: response of a lifting surface

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Insert here additional examples, such as for a vehicle

traversing a terrain with stochastic elevation.

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

For the lifting surface example, the mean square value gives

you a good idea of expected RMS level, for example.

What other kind of response measures might you need if you

were assessing the design or control of this lifting surface?

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

For example, consider this problem:

From [3]

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Derivative random processes [3]

• Section 5.4 (Bendat and Piersol) – see handout

• Why would you care? Fatigue/failure, etc.

• Nice follow on to discuss how you take

derivatives, how you estimate zeros, etc.

• See Bendat and Piersol Problem 5.9 (expected

number of zero crossings)

• Also see Papoulis, Chapter 16, level-crossing

problem

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

1t

( )tξ 1( )tξ

2t

2( )tξ

2( )tx

1( )tx

( )tx

Later: add discussion on level-crossing problem

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Problems

• 4.5 - Brownian motion and stochastic DE

• 4.9 - Modeling and variance response

• 4.10 - Response of linear system to stochastic

input - mean-square response

• 4.11 - Response of linear system to stochastic

input - autocorrelation response

ME 384Q.7 – Prof. R.G. LongoriaStochastic Systems, Estimation and Control

Department of Mechanical EngineeringThe University of Texas at Austin

References

1. P.S. Maybeck, Stochastic Models, Estimation, and Control, Vol. 1,

Academic Press, Orlando, FL, 1979.

2. A. Papoulis, Probability, random variables, and stochastic processes, 3rd

edition, McGraw-Hill, New York, 1991.

3. J.S. Bendat and A.G. Piersol, Random data: analysis and measurement

procedures, Wiley-Interscience, New York, 1986.