Transcript Slide 1

Nonlinear Control:
Local Linearization: This involves linearizing the modeling equations around a
steady-state operating condition and applying linear control systems design
results. It is obvious that the controller performance will deteriorate as the process
moves further away from the steady state around which the model was linearized,
but quite often, adequate controllers can be designed this way.
While there may be an extensive understanding of the behavior of nonlinear
processes, satisfactory methods for their control are still evolving. The prevalent
approach to date has been to use a model of the process linearised about a
steady-state operating point to design a linear controller such as the classical PID
algorithm. This is only an approximate solution.
The General Linearization Problem:
Consider the general nonlinear process model:
dx
 f ( x, u )
dt
y  h( x )
(E.1)
1
Kalyana Veluvolu
where f(*,*) is an arbitrary nonlinear function of two variables, x, the process state
variable, and u the process input; h(*) is another nonlinear function relating the
process output, y to the process state variable x.
The linearized approximation of this very general nonlinear model (E.1) may now
be obtained by carrying out a Taylor series expansion of the nonlinear functions
around the point (xs, us), this gives:
dx
 f 
x  xs    f  u  us  + higher order terms
 f ( xs , us )   
dt
 x ( xs ,us )
 u ( xs ,us )
 h 
y  h( xs )    x  xs  + higher order terms
 x ( xs )
Ignoring the higher order terms now gives the linear approximation:
2
dx
 f ( xs , us )  axs , u s x  xs   bxs , u s u  u s 
dt
y  h ( x s )  c ( x s ) x  x s 
where
(E.2)
 f 
a ( xs , u s )   
 dx  ( xs ,u s )
 f 
b( xs , u s )   
 du  ( xs ,u s )
 h 
c ( xs )   
 dx  ( xs )
It is customary to express the equation in terms of deviation variables:
~
x   x  xs 
u~  u  u 
s
~
y   y  y s   y  h( xs )
3
In addition to this, the linearization point (xs,us) is choosen to be a steadystate operating condition, then observe from the definition of a steady-state
that boty dxs/dt and f(xs,us) will be zero (E.2) the becomes
d~
x
 a~
x  bu~
dt
~
y  c~
x
(E.3)
Where for simplicity, the arguments have been dropped from a, b and c. A
transform-domain transfer function model may now be obtained by the
usual procedure, the results is:
 c ( xs ) b( xs , u s )  ~
~
y ( s)  
 u ( s)
 s  a  xs , u s  
(E.4)
with the transfer function as indicated in the square brackets. This transfer
function should provide an approximate linear model valid in a region close
4
to (xs,us)
The principles involved in obtaining approximate linear models by
linearization may now be sumarized as follows:
Identify the functions responsible for the nonlinearity in the system model.
Expand the nonlinear function as a Taylor series around a steady-state.
Reintroduce the linearized function into the model; simplify, and express
the resulting model in terms of deviation variables.
Example:
linearization of a nonlinear model involving a nonlinear function of a single variable
Consider a Liquid Level System shown as in the figure.
Fi
Material balance
equation yields
A
h
c
F
Liquid Level System
dh
A
 Fi  F
dt
F c h
c is the flow
resistance
5
Combining two equations yields
dh
A
 Fi  c h
dt
dh
A  c h  Fi
dt
This is a nonlinear equation. To linearise this equation, one can use Taylor’s
series. That is, around h=hs.
f ( h)  h  h
0.5
1
0.5
 0.5




h  hs 
 hs  hs
2
+ higher order terms
The approximation is shown as in the figure.
y
f(x)=f(xs)+f‘(xs)(x-xs)
y=f(x)
f(xs)
xs
x
6
The steady-state flow is
A* 0  c h  Fis
Fis  c hs
and we have
dy  0.5
1

A  c hs 
0.5
dt 
2hs
2hs
A
c
0.5
2hs
dy
y
dt
c

y   Fi

0.5
u
dy
1
0.5
A c
y

F

ch
u
i
s
0.5
dt
2hs
dy

 y  Ku
dt
where
2 hs
A
c
2 hs
K
c
u  Fi  Fis
y  h  hs
7
The linearized system is given as
K
 s 1
U(s)
Y(s)
Linearized Liquid Level System (about h = hs)
Y(s)
R(s) +
Gc(s)
G(s)
-
This figure shows a feedback loop where a Proportional and Integral (PI)
controller controls the linearized liquid level system. Notice that the
linearized system is an approximation since it is derived for a particular
level h = hs. If the level changes, K and τ will change with it as well.
8
GcG
C ( s)

R( s) 1  GcG
1  GcG  0
where
K
G
 s 1
Gc  K p 
Kp
Ti s
The characteristic equation can be put as
s
2

1 K K  K
s

p

p
K
 i
0
Assuming the closed loop poles
to be at the locations -2±2i, which
corresponds to the roots of the
characteristic equation, for a
linearized system with K = 2.38
and τ=0.59, we have Kp=0.58 and
τi=0.29.
9
0.1
Im
ζ=cos()=2/2.8284=0.707
2

-2
Re
cmax  css
O V
 e  /
css
cmax  0.10432
1  2
 0.0432
10
Response to step reference with magnitude 0.1
11