Another Ways to Find Response of the System

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Another Ways to Find Response of the System

• Differential equation and simulation diagram
• State variable

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Continuous-Time Domain Analysis

• Differential systems
• Constants a0, a1, , an-1 and b0, b1, , bm
• Linear constant-coefficient differential equation
• Using short-hand notation,above equation becomes

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Continuous-Time Domain Analysis

• For a linear system,
• We use method of undetermined coefficient to
  solve differential equation. (please revise)

Let differential eqn. governing the system
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Impulse response from systems differential
equation
We can determine the impulse response of systems
from their set of differential equation.
Remember - Impulse response h(t), as the input
signal x(t)?(t) and y(t) 0, when -? lt t lt 0
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Practice
Consider,
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Basic System Components
x(t)
y(t)
The integrator
m(t)
m(t)n(t)
The adder

n(t)
m(t)
m(t)-n(t)
The subtractor
-
n(t)
x(t)
y(t)
The multiplier
K
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Simulation Diagram for Cont.-Time System
First canonical form, procedure Let
1.
2. Bring all the term except y to the right side
of the equation
3. Draw diagram (i.e. using Simulink)
Integrating N times
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• State-Variable Representation
• another way to represent a system

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How to develop state equation ?
Consider this differential equation,
Let
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A
b
then
c
d
In general
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Matlab script
gtgt A 0 1 -4 -3 b 0 2 c 1 0
d0 gtgt system ss(A,b,c,d)
To test your system with step input
gtgt step(system)
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To find impulse response for your system
gtgt impulse(system)
To find system response for your system Let say
input x(t)sin(t)
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gtgt t00.110 gtgt insin(t) gtgt lsim(sys,in,t)

• As a rule of thumb, you can generalized any
  input signal for
• the system

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Frequency Response Function of LTI System

• If the input to LTI system is a sinusoid of
  frequency, ?, the steady state response is a
  sinusoid of the same frequency, but with
  amplitude multiplied by a factor A(?) and phase
  shifted by ?(?) radians.
• Frequency response function
• Let input x(t)ej ?t
• where
• H(?) is the Fourier transform of h(t)

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Stability of Linear System

• Many possible definitions
• Our definition
• The system is BIBO stable if and only if every
  bounded input results in bounded output

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Response to Bounded Inputs

• Stable system a bounded input (in amplitude)
  should give a bounded response (in amplitude)
• Linear-time-invariant (LTI) system