Transcript EE324_Chapter9_Notes_S13x

EE/ME/AE324:
Dynamical Systems
Chapter 9: Developing Linear
Models
From there to here.
From here to there.
Nonlinearities are everywhere.
Linearization of Nonlinear Elements
• Most real world systems have significant nonlinear elements
• The objective of this chapter is to learn how to develop
linearized models about a system's equilibrium point, called
an operating point, that will be similar to the nonlinear system
response for a set of inputs and initial conditions
• For analysis purposes, we shall assume all signals consist of a
constant nominal value (operating point) and a time-varying
incremental variable (small signal response) as shown:
x (t ) 
x

xˆ (t )
operating point small signal response
Linearization of Nonlinear Elements
• Given this, the response of a nonlinear element f ( x) can be
decomposed as shown: f ( x(t )) 
f (x )
fˆ ( xˆ (t ))

operating point
small signal response
• Since the Taylor-series expansion (TSE) of f ( x) about the
operating point  f , x  is given by:
df
f ( x)  f ( x ) 
dx
1 d2 f
x  x
2
2!
dx
x
x  x
2

x
• A linear approximation for f ( x) can be formed by noting
that xˆ  x  x and neglecting the higher order TSE terms:
f ( x)
df
f (x ) 
xˆ  f  kxˆ
dx x
Linearization of Nonlinear Elements
• Given this, we can treat fˆ as a small signal, linear approx.
of f that is valid only for sufficiently small variations
xˆ about the operating ponint x : fˆ ( xˆ )  f ( x)  f ( x )  kxˆ
Note: k is the slope
of the tangent line
at the operating point
Ex.9.1: Nonlinear Spring
• A nonlinear spring has the following force-displacement
relationship:
 x 2 , for x  0
f ( x)  x x   2
 x , for x  0
The slope of the tangent at the operating point x is:
df
k
dx
x
2 x , for x  0

 2 x , for x  0
 2 x , for x
The linearized model of the spring for small variations
about the operating point x can now be expressed as:
fˆ ( xˆ )  kxˆ  2 x xˆ
Ex.9.1: Nonlinear Spring
• Evaluating the linear spring model at various operating
points x results in the following graphical representation:
Ex.9.2: Nonlinear Torque
• A nonlinear torque-displacement relation is given by:
 M  D sin 
• The first two terms in the TSE at the operating point x is:
d
 M D sin  
D sin       D sin    D cos  ˆ
d

   kˆ
M
 the linearized model of the spring for small variations
about the operating point x can now be expressed as:

ˆM ˆ
 M   M  kˆ, where k  D cos 
Ex.9.2: Nonlinear Torque
• Evaluating the linear torque model at various operating
points x results in the following graphical representation:
, etc.
Linearization of Nonlinear Systems
• The procedure for linearizing a nonlinear system is
similar to that used for a single element:
1. Determine desired system OPs by solving the nonlinear
system eqns. at its EQ pts., e.g., State Eqn. = 0
2. Rewrite all nonlinear terms in the model as a sum of their
nominal and incremental values, noting derivatives of
constants, e.g., nominal values, equal zero
3. Replace all nonlinear terms by the first two terms in their
TSE, e.g., constant plus linear terms
4. Cancel constant terms in the resulting diff. eqns. leaving
only linear terms involving the incremental variables
5. Determine the ICs of all incremental variables in terms of
the ICs associated with variables in the nonlinear system
Ex.9.8: Nonlinear Resistor
• Develop a linear I/O eqn. for the circuit with nonlinear resistor
defined as io  2eo 3 with input ei (t )  18  A cos(t )
KCL at Node O:
1
eo   eo  ei (t )   2eo 3  0
2

1
eo  2eo 3  eo  ei (t )
2
Ex.9.8: Nonlinear Resistor
• The system OP occurs at the EQ pt. where ei (t ) is replaced by
ei  18, eo replaced by eo , and eo replaced by 0 to obtain:
2eo3  eo  18

eo  2

io  2eo 3  16
24 1
Ex.9.8: Nonlinear Resistor
• To develop a linear I/O model, replace the linear terms:
ei (t )  ei  eˆi (t )  18  A cos(t )
eo  eo  eˆo  2  eˆo  eo  eo  eˆo  eˆo
and the nonlinear term io  2eo 3 with its first two TSE terms:
dio
io 
deo




eˆo  io  6eo 2 eˆo  16  6  22 eˆo  16  24eˆo
eo
• This implies that at the OP iˆo  io  io  24eˆo , as shown
on prior figure, and thus the incremental resistance is:
eˆo
1
rˆo  

iˆo 24
Ex.9.8: Nonlinear Resistor
• Substituting the linearized terms into the nonlinear I/O:
1
eˆo  16  24eˆo    2  eˆo (t )   18  A cos(t )
2
Canceling the constant terms in the above eqn. yields:
1
eˆo  25eˆo  A cos(t )  eˆo  50eˆo  2 A cos(t )
2
 linearized system is 1st order with   0.02 [s]
Ex.9.9: Nonlinear Resistor
• Develop a linear I/O eqn. for the circuit with nonlinear resistor
1 3
defined as i2  eC with input ei (t )  2  A sin(t )
8
Ex.9.8: Nonlinear Resistor
• Choosing eC and iL as the system states, we note by KVL that
eL  ei (t )  eC and by KCL at the upper right node :
1 3
iL  eL  eC  2eC  0
8
• Combining the above with the element laws for eC and iL
yields the nonlinear state eqns.:
1
1 3

eC  ei (t )  iL  eC  eC 
2
8

diL 1
  ei (t )  eC 
dt 3
Ex.9.8: Nonlinear Resistor
• OPs of the system occur at EQ pts. of the state eqn., i.e.,
where the state derivatives are zero  ei  eC  0 and
1 3
ei  iL  eC  eC  0
8
 eC  ei  2V
1 3
 i2  eC  1A
8
1 3
 iL  eC  eC  ei  1A
8
Ex.9.9: Nonlinear Resistor
• The incremental variables are now defined by:
eC  2  eˆC ,
iL  1  iˆL ,
ei (t )  2  A sin(t )
1 3
• The first two TSE terms of the nonlinear resistor i2  eC are:
8
3
3
ˆ
ˆ
i2  i2  i2 1  eˆC  i2  i2  i2 i2  1  eˆC
2
2
• This implies that the small signal resistor model at the OP is:
eˆC 2
r2 
 , as shown on the prior figure
iˆ2 3
Ex.9.9: Nonlinear Resistor
• Substituting the prior expressions into the state eqn. and
cancelling the constant terms yields the linearized state eqns.
about the OPs:
eˆC
diˆL
dt
1
5 
ˆ
 eˆi (t )  iL  eˆC 
2
2 
1
  eˆi (t )  eˆC 
3
Linear and Nonlinear
responses with A=0.1V
Ex.9.9: Nonlinear Resistor
Linear and Nonlinear
responses with A=1.0V
Linear and Nonlinear
responses with A=10.0V
Clearly, the quality of the linear model depends on how much the
system response varies about the OP, i.e., the incremental terms
0.2
The Wide Variety of Nonlinear Effects
• Analyzing nonlinear systems can be extremely difficult
as many properties we use to solve linear systems, e.g.,
superposition in the frequency domain, no loner apply
• System nonlinearities can take many forms, e.g.:
1. Trigonomic: sin( x), cos( x), tan( x)
2. Hyperbolic: sinh( x), cosh( x), tanh( x)
3. Polynomial: x 2 , x3 , etc.
4. Radical:
x,
3
x , etc.
5. Exponential: e x , 10 x , etc.
6. Logarithmic: ln( x), log10 ( x), etc.
7. Special functions: x , sgn( x), etc.
Mechanical Friction Has Many Models
Mechanical Friction
Mechanical Friction
Static (stiction) friction
Mechanical Friction
Mechanical Friction
Stick-Slip
Mechanical Friction
Mechanical Friction
Backlash, e.g., Gears
Elastic Hysteresis, e.g., Rubber Band
Magnetic Hysteresis, e.g., Transformer
The relationship between magnetic field strength (H)
and magnetic flux density (B) is nonlinear
Dead Zone, e.g., Hydraulic Valve
In hydraulic valves, a dead
zone nonlinearity results if
the land width is greater
than the port width when
the spool is at null position
(Merrit, 1967)
Hysteresis + Dead Zone, e.g., Relay Circuits
A relay is a voltage-controlled switch with a coil that creates
a magnetic field that causes the switch contacts to close
when the voltage is greater than a turn-on threshold
The contacts remain closed until the voltage diminishes to a
turn-off value, at which point the switch contacts open
VCC
Saturation, e.g., Op-Amp Circuits
Asymmetric Nonlinearity, e.g., Diode
Current
i 
D
 v 
Voltage
Questions?