Module 2:
Representing Process and Disturbance
Dynamics Using Discrete Time
Transfer Functions
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
2
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
3
Dynamic Models
• dependence on past values
• goal - estimate models of form
y(t  1)  f ( y(t ), y(t  1),, u(t ), u(t  1),, w(t ), w(t  1),)
• example
y(t  1)  y(t )  w(t )
– how can we determine how many lagged inputs,
outputs, disturbances to use?
– correlation analysis - auto/cross-correlations
chee825 - Winter 2004
J. McLellan
4
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
5
Impulse Response
Process
time
y(k)
output
impulse weights
h(k), k=0,1,2,...
time
chee825 - Winter 2004
J. McLellan
6
Impulse Response as a Weighting Pattern
Given sequence of inputs, we can predict
process output

y ( k )   h( j ) u( k  j )
j 0
impulse response infinitely
long if process returns to
steady state asymptotically
chee825 - Winter 2004
J. McLellan
7
Interpretation
Sum of impulse contributions
y( k )  h(0)u( k )  h(1)u( k  1)  h(2)u( k  2) 
0
impact of input
move 1 time step ago
impact of input
move 2 time steps ago
y(k)
output
(ZOH
time
chee825 - Winter 2004
J. McLellan
8
Impulse Response Model
• impulse response is an example of nonparametric 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
9
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 )   (0)e( k )   (1)e( k  1)   (2)e( k  2) 
impulse response
parameters
chee825 - Winter 2004
J. McLellan
white noise
pulse
10
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  j )
j 0
chee825 - Winter 2004
J. McLellan
11
Difference Equation Models
• recursive definition describing dependence of current
output on previous inputs and outputs
y (t  1)  f ( y (t ), y (t  1), y (t  p),
u(t ), u(t  1),u(t  m),
e(t ), e(t  1),e(t  q ))
• y - output; u - manipulated variable input; e - random
shocks (white noise)
• example - ARMAX(1,1,1) model with time delay of 1
y(t )  a1 y(t  1)  b0u(t  1)  e(t )  c1e(t  1)
chee825 - Winter 2004
J. McLellan
12
The Backshift Operator
• dynamic models represent dependence on past
values - need a method to represent “lag”
• backshift operator q-1:
q
1
y(t )  y(t  1)
• forward shift -- using q:
qy (t )  y (t  1)
• alternate notations -- B, z-1
» z-1 - used in discrete control as argument for Z-transform
chee825 - Winter 2004
J. McLellan
13
Transfer Function Models
• start with difference equation model and introduce
backshift operators relative to current time “t”
y(t )  a1q
1
1
1
y(t )  b0q u(t )  e(t )  c1q e(t )
• “solve” for y(t) in terms of u(t) and e(t)
y (t ) 
b0q 1
1  a1q 1
u( t ) 
process
transfer
function
chee825 - Winter 2004
1  c1q 1
1  a1q 1
e( t )
disturbance
transfer
function
J. McLellan
14
Transfer Function Models
General form - ratios of polynomials in q-1
r
b0 bmq  m
c



c
q
0
r
y (t ) 
q b u ( t ) 
e( t )
1  a1q 1  an q  n
1  d1q 1  d p q  p
Roots of denominator represent poles
» of process input-output relationship
» of disturbance input-output relationship
Roots of numerator represent zeros
chee825 - Winter 2004
J. McLellan
15
A Stability Test
• continuous control - poles must have negative real
part in Laplace domain (complex plane)
• discrete dynamics?
Consider the sum…

2
1  f  f    f i
Geometric
i 0
Series
1

1 f
if
chee825 - Winter 2004
J. McLellan
f 1
16
Stability Test
Now consider 1  aq
1
 (aq

1 2
)    (aq 1 )i
i 0

1
aq
1 .
if
1
1  aq 1
1
2
Impulse response of
1 is {1,a,a ,…} which is
1  aq
stable if a  1
Root of denominator is q=a, or q-1=a-1
chee825 - Winter 2004
J. McLellan
17
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
18
Moving Between Representations
From the preceding argument,

1
1
1 2
 1  aq  (aq )    (aq 1 )i
1  aq 1
i 0
so
1
y (t ) 
u(t )  (1  aq 1  (aq 1 ) 2  )u(t )
1  aq 1

2
 u(t )  au(t  1)  a u(t  2)    a i u(t  i )
i 0
chee825 - Winter 2004
J. McLellan
19
Moving Between Representations
1
1  aq 1
 1  aq
transfer
function
1
 (aq
1 2

)    (aq 1 )i
i 0
impulse response
The transformation can be achieved by solving for the
impulse response of the discrete transfer function, or by
“long division”.
chee825 - Winter 2004
J. McLellan
20
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
y(t )  (1  q 1)u(t )
as
u( t ) 
chee825 - Winter 2004
1
1  q

y (t )    i (q 1 )i y (t )
1
i 0
J. McLellan
21
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
22
Invertibility
One use: for some input u …
y(t )  (1  q 1)u(t )
Write
as

y (t )  u(t )    i (q 1 )i y (t )
i 1
 u(t )  y (t  1)   2 y (t  2) 
current input
move
chee825 - Winter 2004
past outputs
(inertia of process)
J. McLellan
23
Invertibility
Importance?
» particularly in estimation, where we will use this to form
residuals
y (t )  a (t )   a (t  1)
given
model
What are the values of a(t)’s?
Reformulate
2
y(t )  a (t )   y(t  1)   y(t  2) 
white
noise
chee825 - Winter 2004
y(t)’s - measured quantities
J. McLellan
24
Representing Time Delays
Using the backshift operator, a delay of “f” steps
corresponds to:
q f
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
chee825 - Winter 2004
J. McLellan
25