ME 233 Advanced Control II Continuous time results 1 Random processes (ME233 Class Notes pp. PR6-PR13) 1
A random processes is a continuous function of time
Such that for any time  to ,
Is a random variable defined over the same probability space
2リqwəsuコ3 4 Random process
be a random process
is the joint PDF of
This is often a huge amount of redundant information 5 2nd order statistics
be a random vector process Expected value or mean of X(t),
Auto-covariance function:
Auto-covariance function Define:
7 Strict Sense Stationary random sequence A random process   X(t) is Strict Sense Stationary (SSS) if the joint probability,  is invariant with time
for any time shift T, Ergodicity A Strict Sense Stationary  random process
is  ergodic if we can recover an ensemble average from the time average of any realization:
with probability 1 (almost surely)
10 Wide Sense Stationarity A random sequence is Wide Sense Stationary (WSS) if: 1) Its mean is time invariant 11
A random sequence is Wide Sense Stationary (WSS) if: 2)  Its covariance only depends on the correlation shift τ
12
The auto-covariance function can be defined only as a function of the correlation time shift  T
Notice that:
13 Cross-covariance function Let       X(t) ∈ R      and      Y(t) ∈ Rm be two  WSS random vector processes The cross-covariance function:
for any time t Cross-covariance function
15
For WSS random process, the power spectral density function is the Fourier transform of the auto- covariance function:
Since,
17
A WSS random process W ( t ) ∈ R is white if:
Where δ(t) is the Dirac delta impulse white noise is zero mean if
18
The power spectral density function for white noise is: Proof:
19
0
Infinite bandwidth 20
A WSS random vector sequence     W(t) ∈ R n white if:
where
and
21
be the impulse response of an LTI SISO system with transfer function
MIMO Linear Time Invariant Systems U(t) E Rm  be WSS Let Then the forced response (zero initial state)
23 MIMO Linear Time Invariant Systems We will assume that • The WSS random process    U(t) is zero mean, I.e.
Thus, the output random process is also zero mean
MIMO Linear Time Invariant Systems be WSS U(t) Let lf Then: ΛΥυ (τ) = E{Υ(t+τ)Ű™(t)} 25 MIMO Linear Time Invariant Systems U(t) be a  WSS random process Let
Φyu(w):
26 MIMO Linear Time Invariant Systems U(t) be a  WSS random process Let
27 MIMO Linear Time Invariant Systems Let     U ( t )         be a  WSS vector random process lf
Then: L 8 Λγυ(τ) =  J_  G(η) Λυυ(τ − η) dη
28 MIMO Linear Time Invariant Systems l 80 Λγυ(τ) =  /_  G(η) Λυυ(τ — η) dη Proof:
Then:
29 MIMO Linear Time Invariant Systems U(t) be WSS Let lf Then: E{Û(t+r)ÝT(t)}= MIMO Linear Time Invariant Systems U(t) be a  WSS random process Let
31
MIMO Linear Time Invariant Systems Let     U(t)     be WSS
Then: Φγγ(ω) = G(jω) Φυυ(ω) GT(—jω)
33 MIMO Linear Time Invariant Systems Proof: Use
and
White noise driven state space systems Consider a LTI system driven by white noise:
35 White noise driven state space systems
Assume that
36 White noise driven state space systems
White noise driven state space systems
Taking expectations on the equations above, we obtain:
White noise driven state space systems Subtracting the means,
d ¯^xx(t,0) = A^xx(t,0)+^xx(t,0)AT with ^xx(t,0) = E { X(t) XT(t)} White noise driven covariance propagation Also, ^xx(t,τ) = eAτ Nxx(t,0) T ＞ 0 where:
41 White noise driven covariance propagation Also, where:
42 Stationary  covariance equation For W(t) WSS,
and A Hurwitz,
43 Stationary  covariance equation For W(t) WSS,    and A Hurwitz,
Satisfies: A ¯ X X (0) + ¯ X X (0) A ¯ = – B ¯ W B ¯ Λ χχ(τ) = e Ατ Λ χχ(0) T ＞ 0 The next section contains some Proofs of the CT Please go over them by yourselves... Proof of continuous time results – Method 1 We first prove that:
By starting from the Discrete Time (DT) results 46 Proof of continuous time results – Method 1 Approximate the state equation ODE
using the Euler numerical integration method.
•  We have to be careful in dealing with white noise W(t) 47 Approximate W(t) 1. Define W(k) as the time average of W(t)
Similarly, taking expectations
48 Approximate  Λ W (k, 0)    for  W(t) white
Approximate  Λ W (k,0)    for  W(t) white
since for  W(t) white
50 Approximate  Λ W (k, 0)    for  W(t) white
51 Approximate  Λ W (k, 0)    for  W(t) white
52 Approximate  Λ W (k, 0)    for  W(t) white
Where   Σww (k) is the time average of  Σww (t)
53 Numerical Integration The state equation
By the discrete time state equation
where
54 Proof of continuous time results – Method 1 1. Obtain DT state equations by approximating the CT state equation solution:
Thus,
where
Proof of continuous time results – M1 2. Obtain the CT covariance propagation equation from from the DT covariance propagation, using the approximated DT state equation:
≈ (I + ∆t A) ∧ x x (k, 0) (I + ∆t A) T + ∆t B Σ w (k) B T ~ ∧ x x (k, 0) + ∆ t A∧ x x (k, 0) + ∆ t ∧ x x (k, 0) A T +(∆t)² AΛχχ(k,0)A™ + ∆t B Σww(k) B™ Proof of continuous time results – M1 3. Take the limit as    Δt → 0      of ^xx((k + 1)∆t,0) − ^xx(k∆t,0) At }} ANxx(kAt,0) + Λ xx(kΔt,0) AT + B Eww(k) BT +△t AΛ xx (k△t,0) AT and noticing that
57 Proof of continuous time results – M1 3. Take the limit as    Δt → 0      of
58 Proof of continuous time results – Method 2 We now proof that:
Directly from continuous time (CT) results Proof of continuous time results – M2 1)  Lets calculate using
= ANxx(t,0)+ Axx(t,0)AT +B E{W(t)XT(t)} + E{X(t)WT(t)}BT Proof of continuous time results – M2 2)  We now need to calculate
using
61
(notice that the Dirac impulse occurs at the edge t ) 62
2) Continuing,
0 Proof of continuous time results – M2 2) A similar calculation for    E{X(t)WT(t)} BT yields
(notice that the Dirac impulse occurs at the edge t )
2) Continuing,
(make integral symmetrical w/r  0)
Proof of continuous time results – M2 2) Thus B E{W(t)XT(t)} + E{X(t)WT(t)}BT = B Eww(t)BT and d 
= ^xx(t,0)  =  A ^xx(t,0) + ^xx(t,0) A T
dt + B Σww(t) BT 66 Proof of continuous time results – M2 Now we proof that:
Notice that:
where,
Proof of continuous time results – M2 Therefore,
▼()
Notice that W(η) and 文(t)  are uncorrelated for η > t
Proof of continuous time results – M2 Thus,