Module 2: Representing Process and Disturbance Dynamics Using Discrete Time Transfer Functions.
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Transcript of Module 2: Representing Process and Disturbance Dynamics Using Discrete Time Transfer Functions.
chee825 - Winter 2004 J. McLellan 2
Dynamic Models - a First Pass
• establish linkage between process dynamic representations and possible disturbance representations
• key concept - dynamic element, represented by a transfer function, driven by random shock sequence
» IID Normal - white noise
chee825 - Winter 2004 J. McLellan 3
Dynamic Process Relationships
• dependence of current output on present and past values of – manipulated variable inputs– disturbance inputs
• process transfer function» “deterministic” trends between u and y
• disturbance component» relationship between (possibly stochastic) disturbance n
and y
chee825 - Winter 2004 J. McLellan 4
Dynamic Models
• dependence on past values• goal - estimate models of form
• example
– how can we determine how many lagged inputs, outputs, disturbances to use?
– correlation analysis - auto/cross-correlations
y t f y t y t u t u t w t w t( ) ( ( ), ( ), , ( ), ( ), , ( ), ( ), )+ = − − −1 1 1 1L L L
y t y t w t( ) ( ) ( )+ = +1 φ
chee825 - Winter 2004 J. McLellan 5
Impulse Response
• processes have inertia » no instantaneous jumps» when perturbed, require time to reach steady state
• one characterization – impulse response– pulse at time zero enters the process
chee825 - Winter 2004 J. McLellan 6
Impulse Response
Process
Inp
ut
u(k
)
ou
tput
y(k
)
time
time
impulse weightsh(k), k=0,1,2,...
chee825 - Winter 2004 J. McLellan 7
Impulse Response as a Weighting Pattern
Given sequence of inputs, we can predict process output
y k h j u k jj
( ) ( ) ( )= −∑=
∞
0
impulse response infinitely long if process returns to steady state asymptotically
chee825 - Winter 2004 J. McLellan 8
Interpretation
Sum of impulse contributions
y k h u k h u k h u k( ) ( ) ( ) ( ) ( ) ( ) ( )= + − + − +0 1 1 2 2 K
0impact of inputmove 1 time step ago
impact of inputmove 2 time steps ago
ou
tput
y(k
)
time
(ZOH
chee825 - Winter 2004 J. McLellan 9
Impulse Response Model
• impulse response is an example of non-parametric model
» practically - truncate and use finite impulse response (FIR) form
• impulse response model can be considered in– control modeling
» model predictive control (e.g., DMC)
– disturbance modeling» time series -- moving average representation
chee825 - Winter 2004 J. McLellan 10
Disturbance Models in Impulse Response Form
• inputs are random “shocks”» white noise fluctuations - random pulses
• impulse response weights describe how fluctuations in past affect present measurement
y k e k e k e k( ) ( ) ( ) ( ) ( ) ( ) ( )= + − + − +φ φ φ0 1 1 2 2 K
white noisepulse
impulse responseparameters
chee825 - Winter 2004 J. McLellan 11
Disturbance Models in Impulse Response Form
• also referred to as a moving average representation– moving average of present and past random shocks
entering process
y k j e k jj
( ) ( ) ( )= −∑=
∞φ0
chee825 - Winter 2004 J. McLellan 12
Difference Equation Models
• recursive definition describing dependence of current output on previous inputs and outputs
• y - output; u - manipulated variable input; e - random shocks (white noise)
• example - ARMAX(1,1,1) model with time delay of 1
y t f y t y t y t p
u t u t u t m
e t e t e t q
( ) ( ( ), ( ), ( ),
( ), ( ), ( ),
( ), ( ), ( ))
+ = − −− −− −
1 111
L
L
L
y t a y t b u t e t c e t( ) ( ) ( ) ( ) ( )= − + − + + −1 0 11 1 1
chee825 - Winter 2004 J. McLellan 13
The Backshift Operator
• dynamic models represent dependence on past values - need a method to represent “lag”
• backshift operator q-1:
• forward shift -- using q:
• alternate notations -- B, z-1
» z-1 - used in discrete control as argument for Z-transform
q y t y t− = −1 1( ) ( )
qy t y t( ) ( )= +1
chee825 - Winter 2004 J. McLellan 14
Transfer Function Models
• start with difference equation model and introduce backshift operators relative to current time “t”
• “solve” for y(t) in terms of u(t) and e(t)
y t a q y t b q u t e t c q e t( ) ( ) ( ) ( ) ( )= + + +− − −1
10
11
1
y tb q
a qu t
c q
a qe t( ) ( ) ( )=
−+ +
−
−
−
−
−0
1
11
11
111
1
1processtransfer function
disturbance transfer function
chee825 - Winter 2004 J. McLellan 15
Transfer Function Models
General form - ratios of polynomials in q-1
Roots of denominator represent poles» of process input-output relationship» of disturbance input-output relationship
Roots of numerator represent zeros
y tb b q
a q a qq u t
c c q
d q d qe tm
m
nn
b rr
pp
( ) ( ) ( )= + +
− − −+ + +
− − −
−
− −−
−
− −0
11
0
111 1
L
L
L
L
chee825 - Winter 2004 J. McLellan 16
A Stability Test
• continuous control - poles must have negative real part in Laplace domain (complex plane)
• discrete dynamics?
Consider the sum…
if
1
11
2
0+ + + = ∑
=−
=
∞f f f
f
i
iK
f <1
Geometric Series
chee825 - Winter 2004 J. McLellan 17
Stability Test
Now consider
if .
Impulse response of is {1,a,a2,…} which is
stable if
Root of denominator is q=a, or q-1=a-1
1
1
1
1 1 2 1
0
1
+ + + = ∑
=−
− − −
=
∞
−
aq aq aq
aq
i
i( ) ( )K
aq− <1 11
1 1− −aqa <1
chee825 - Winter 2004 J. McLellan 18
Stability Test
Dynamic element is STABLE if» root in “q” is less than 1 in magnitude» root in “q-1” is greater than 1 in magnitude
Approach - check roots of denominator» based on argument that higher order denominator can
be factored into sum of first-order terms - Partial Fraction Expansion
» each first-order term corresponds to a elementary response - decaying or exploding
chee825 - Winter 2004 J. McLellan 19
Moving Between Representations
From the preceding argument,
so
1
11
11 1 2 1
0−= + + + = ∑−
− − −
=
∞
aqaq aq aq i
i( ) ( )K
y taq
u t aq aq u t
u t au t a u t a u t ii
i
( ) ( ) ( ( ) ) ( )
( ) ( ) ( ) ( )
=−
= + + +
= + − + − + = −∑
−− −
=
∞
1
11
1 2
11 1 2
2
0
K
K
chee825 - Winter 2004 J. McLellan 20
Moving Between Representations
1
11
11 1 2 1
0−= + + + = ∑−
− − −
=
∞
aqaq aq aq i
i( ) ( )K
transfer function
impulse response
The transformation can be achieved by solving for theimpulse response of the discrete transfer function, or by“long division”.
chee825 - Winter 2004 J. McLellan 21
Inversion
We can express transfer fn. model as impulse response model - infinite sum of past inputs.
Can we do the opposite?» express input as infinite sum of present and past
outputs? » example
asy t q u t( ) ( ) ( )= − −1 1θ
u tq
y t q y ti i
i( ) ( ) ( ) ( )=
−= ∑−
−
=
∞1
1 11
0θθ
chee825 - Winter 2004 J. McLellan 22
Invertibility
Answer - this is the dual problem to stability, and is known as invertibility.
We can invert the moving average term if -- » root in “q” is less than 1 in magnitude» root in “q-1” is greater than 1 in magnitude
Invertibility corresponds to “minimum phase” in control systems, and is a “stability check” of the numerator in a transfer function.
chee825 - Winter 2004 J. McLellan 23
Invertibility
One use: for some input u …
Write
as
y t q u t( ) ( ) ( )= − −1 1θ
y t u t q y t
u t y t y t
i i
i( ) ( ) ( ) ( )
( ) ( ) ( )
= −∑
= − − − − −
−
=
∞θ
θ θ
1
121 2 K
current input move
past outputs(inertia of process)
chee825 - Winter 2004 J. McLellan 24
Invertibility
Importance?» particularly in estimation, where we will use this to form
residuals
y t a t a t( ) ( ) ( )= − −θ 1given model
What are the values of a(t)’s?
Reformulate
y t a t y t y t( ) ( ) ( ) ( )= − − − − −θ θ1 22 Kwhitenoise y(t)’s - measured quantities
chee825 - Winter 2004 J. McLellan 25
Representing Time Delays
Using the backshift operator, a delay of “f” steps corresponds to:
Notes -- » f is at least one for sampled systems because of
sampling and “zero-order hold”» effect of current control move won’t be seen until at least
the next sampling time
q f−