ME375 Dynamic System Modeling and Control

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MESB374 System Modeling and AnalysisTransfer
Function Analysis
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Transfer Function Analysis

• Dynamic Response of Linear Time-Invariant (LTI)
  Systems
• Free (or Natural) Responses
• Forced Responses
• Transfer Function for Forced Response Analysis
• Poles
• Zeros
• General Form of Free Response
• Effect of Pole Locations
• Effect of Initial Conditions (ICs)
• Obtain I/O Model based on Transfer Function
  Concept

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Dynamic Responses of LTI Systems

• Ex Lets look at a stable first order
  system

• Take LT of the I/O model and remember to keep
  tracks of the ICs

• Rearrange terms s.t. the output Y(s) terms are on
  one side and the input U(s) and IC terms are on
  the other

• Solve for the output

Free Response
Forced Response
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Free Forced Responses

• Free Response (u(t) 0 nonzero ICs)
• The response of a system to zero input and
  nonzero initial conditions.
• Can be obtained by
• Let u(t) 0 and use LT and ILT to solve for the
  free response.
• Forced Response (zero ICs nonzero u(t))
• The response of a system to nonzero input and
  zero initial conditions.
• Can be obtained by
• Assume zero ICs and use LT and ILT to solve for
  the forced response (replace differentiation with
  s in the I/O ODE model).

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In Class Exercise

• Find the free and forced responses of the car
  suspension system without tire model

• Take LT of the I/O model and remember to keep
  tracks of the ICs

• Rearrange terms s.t. the output Y(s) terms are on
  one side and the input U(s) and IC terms are on
  the other

• Solve for the output

Free Response
Forced Response
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Forced Response Transfer Function

• Given a general n-th order system model
• The forced response (zero ICs) of the system due
  to input u(t) is
• Taking the LT of the ODE

Forced Response
Transfer Function
Inputs
Inputs


Transfer Function
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Transfer Function

• Given a general nth order system
• The transfer function of the system is
• The transfer function can be interpreted as

u(t) Input
y(t) Output
U(s) Input
Y(s) Output
Differential Equation
G(s)
Time Domain
s - Domain
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Transfer Function Matrix

• For Multiple-Input-Multiple-Output (MIMO) System
  with m inputs and p outputs

Inputs
Outputs
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Poles and Zeros
Given a transfer function (TF) of a system

• Poles
• The roots of the denominator of the TF, i.e. the
  roots of the characteristic equation.

n poles of TF
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Examples

• (1) Recall the first order system
• Find TF and poles/zeros of the system.

• (2) For car suspension system
• Find TF and poles/zeros of the system.

• Pole

• Pole

• Zero

• Zero

• No Zero

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System Connections

• Cascaded System

Output
Input
Parallel System
Feedback System

-
Output
Input
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General Form of Free Response

• Given a general nth order system model
• The free response (zero input) of the system due
  to ICs is
• Taking the LT of the model with zero input
• (i.e.,
  )

A Polynominal of s that depends on ICs
Free Response (Natural Response)

Same Denominator as TF G(s)
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Free Response (Examples)

• Ex Find the free response of the car suspension
  system without tire model (slinker toy)
• Ex Perform partial fraction expansion (PFE) of
  the above free response when
• 
  (what does this set of ICs means physically)?

phase initial conditions
Decaying rate damping, mass
Frequency damping, spring, mass
Q Is the solution consistent with your physical
intuition?
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Free Response and Pole Locations

• The free response of a system can be represented
  by

exponential decrease
constant
exponential increase
decaying oscillation
Oscillation with constant magnitude
increasing oscillation
t
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Complete Response
Y(s) Output
U(s) Input

• Complete Response
• Q Which part of the system affects both the free
  and forced response ?
• Q When will free response converges to zero for
  all non-zero I.C.s ?

Denominator D(s)
All the poles have negative real parts.
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Obtaining I/O Model Using TF Concept (Laplace
Transformation Method)

• Noting the one-one correspondence between the
  transfer function and the I/O model of a system,
  one idea to obtain I/O model is to
• Use LT to transform all time-domain differential
  equations into s-domain algebraic equations
  assuming zero ICs (why?)
• Solve for output in terms of inputs in s-domain
  to obtain TFs (algebraic manipulations)
• Write down the I/O model based on the TFs obtained

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Example Car Suspension System

• Step 1 LT of differential equations assuming
  zero ICs

• Step 2 Solve for output using algebraic
  elimination method

• of unknown variables equations ?

2. Eliminate intermediate variables one by one.
To eliminate one intermediate variable, solve for
the variable from one of the equations and
substitute it into ALL the rest of equations
make sure that the variable is completely
eliminated from the remaining equations
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Example (Cont.)
from first equation
Substitute it into the second equation

• Step 3 write down I/O model from TFs