EE/ME/AE324:Dynamical Systems

Chapter 9: Developing Linear Models

From there to here.From here to there.

Nonlinearities are everywhere.

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Linearization of Nonlinear ElementsMost 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:

ˆ ( ) ( )operating point small signal response

x t x x t

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Linearization of Nonlinear Elements

operating point small signal response

Given this, the response of a nonlinear element ( ) can be

ˆ ˆdecomposed as shown: ( ( )) ( ) ( ( ))

Since the Taylor-series expansion (TSE) of ( ) about th

f x

f x t f x f x t

f x

2

2

2

e

operating point , is given by:

1( ) ( )

2!

A linear approximation for ( ) can be formed by noting

ˆthat and neglecting the higher order TSE terms:

ˆ ˆ( ) ( )

x x

x

f x

df d ff x f x x x x x

dx dx

f x

x x x

dff x f x x f kx

dx

• Linearization of Nonlinear Elements

ˆGiven this, we can treat as a small signal, linear approx.

of that is valid only for sufficiently small variations

ˆˆ ˆ ˆ about the operating ponint : ( ) ( ) ( )

f

f

x x f x f x f x kx

Note: is the slope

of the tangent line

at the operating point

k

Ex.9.1: Nonlinear Spring•

2

2

A nonlinear spring has the following force-displacement

relationship:

, for 0 ( )

, for 0

The slope of the tangent at the operating point is:

x xf x x x

x x

x

k

2 , for 0

2 , for 0

2 , for

The linearized model of the spring for small variations

about the operating point can now be expressed as:

x

x xdf

x xdx

x x

x

ˆ ˆ ˆ ˆ ( ) 2

f x kx x x

Ex.9.1: Nonlinear Spring• Evaluating the linear spring model at various operating

points results in the following graphical representation:x

Ex.9.2: Nonlinear Torque•

•

M

A nonlinear torque-displacement relation is given by:

sin

The first two terms in the TSE at the operating point is:

ˆsin sin sin cos

ˆ

t

M

M

D

x

dD D D D

d

k

M

he linearized model of the spring for small variations

about the operating point can now be expressed as:

ˆ ˆˆ , where cos

M M

x

k k D

Ex.9.2: Nonlinear Torque• Evaluating the linear torque model at various operating

points results in the following graphical representation:x

, etc.

• 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

Linearization of Nonlinear Systems

Ex.9.8: Nonlinear Resistor•

3

3 3

Develop a linear I/O eqn. for the circuit with nonlinear resistor

defined as 2 with input ( ) 18 cos( )

KCL at Node O: 1 1

( ) 2 0 2 ( )2 2

o o i

o o i o o o o i

i e e t A t

e e e t e e e e e t

124

Ex.9.8: Nonlinear Resistor•

3 3

The system OP occurs at the EQ pt. where ( ) is replaced by

18, replaced by , and replaced by 0 to obtain:

2 18 2 2 16

i

i o o o

o o o o o

e t

e e e e

e e e i e

Ex.9.8: Nonlinear Resistor•

•

3

To develop a linear I/O model, replace the linear terms:

ˆ ( ) ( ) 18 cos( )

ˆ ˆ ˆ ˆ 2

and the nonlinear term 2 with its first two TS

i i i

o o o o o o o o

o o

e t e e t A t

e e e e e e e e

i e

2 2

E terms:

ˆ ˆ ˆ ˆ 6 16 6 2 16 24

ˆ ˆThis implies that at the OP 24 , as shown

on prior figure, and thus the incremental resistance is:

oo o o o o o o

o o

o o o o

e

dii e i e e e e

de

i i i e

ˆ 1ˆ

ˆ 24o

oo

er

i

Ex.9.8: Nonlinear Resistor•

Substituting the linearized terms into the nonlinear I/O:

1ˆ ˆ ˆ 16 24 2 ( ) 18 cos( )

2Canceling the constant terms in the above eqn. yields:

1ˆ ˆ ˆ ˆ 25 cos( ) 50 2 cos

2

o o o

o o o o

e e e t A t

e e A t e e A

st

( )

linearized system is 1 order with 0.02 [s]

t

Ex.9.9: Nonlinear Resistor•

32

Develop a linear I/O eqn. for the circuit with nonlinear resistor

1defined as with input ( ) 2 sin( )

8 C ii e e t A t

Ex.9.8: Nonlinear Resistor•

•

3

Choosing and as the system states, we note by KVL that

( ) and by KCL at the upper right node : 1

2 08

Combining the above with the element laws for

C L

L i C

L L C C

e i

e e t e

i e e e

3

and yields the nonlinear state eqns.:

1 1 ( )

2 81

( )3

C L

C i L C C

Li C

e i

e e t i e e

die t e

dt

Ex.9.8: Nonlinear Resistor•

3

3

3

2

OPs of the system occur at EQ pts. of the state eqn., i.e.,

where the state derivatives are zero 0 and1

21

18

8

8

0

11

C i

C

L C

i C

i L C

C

C

i

e e

e i e e

e e V

i e A

i e e e A

Ex.9.9: Nonlinear Resistor•

•

•

32

2 2 2 2 2 2 2

The incremental variables are now defined by:

ˆˆ 2 , 1 , ( ) 2 sin( )

1The first two TSE terms of the nonlinear resistor are:

83 3ˆ ˆˆ ˆ 1 12 2

C C L L i

C

C C

e e i i e t A t

i e

i i i e i i i i e

22

This implies that the small signal resistor model at the OP is:ˆ 2

, as shown on the prior figureˆ 3Ceri

Ex.9.9: Nonlinear Resistor•

Substituting the prior expressions into the state eqn. and

cancelling the constant terms yields the linearized state eqns.

1 5ˆˆ ˆ ˆabout the OPs: ( ) 2 2

ˆ

C i L C

L

e e t i e

di

1ˆ ˆ( )

3 i Ce t edt

Linear and Nonlinear

responses with A=0.1V

Ex.9.9: Nonlinear ResistorLinear and Nonlinear

responses with A=1.0VLinear 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

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The Wide Variety of Nonlinear EffectsAnalyzing 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 f

2 3

3

10

orms, e.g.:

1. Trigonomic: sin( ), cos( ), tan( )

2. Hyperbolic: sinh( ), cosh( ), tanh( )

3. Polynomial: , , etc.

4. Radical: , , etc.

5. Exponential: , 10 , etc.

6. Logarithmic: ln( ), log ( ),

x x

x x x

x x x

x x

x x

e

x x etc.

7. Special functions: , sgn( ), etc.x x

Mechanical Friction Has Many Models

Mechanical Friction

Mechanical Friction

Static (stiction) friction

Mechanical Friction

Stick-Slip

Mechanical Friction

Mechanical Friction

Mechanical Friction

Backlash, e.g., Gears

Elastic Hysteresis, e.g., Rubber Band

The relationship between magnetic field strength (H) and magnetic flux density (B) is nonlinear

Magnetic Hysteresis, e.g., Transformer

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 CircuitsA 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 thresholdThe contacts remain closed until the voltage diminishes to a turn-off value, at which point the switch contacts open

CCV

Saturation, e.g., Op-Amp Circuits

Asymmetric Nonlinearity, e.g., Diode

Voltage

Current

v

Di

Questions?