Linear systems and control

1
Linear systems and control

• Review of Cambridge 2nd year material
• Hayden Taylorhkt_at_mit.edu
• web/hkt/www/control.ppt
• 23 September 2005

2
Background

• 14 lectures in Michaelmas of 2nd year
• 6 examples papers
• They spend 1 hour answering exam questions in
  June
• They arent necessarily fluent in these concepts
• Dont worry if any of this is unfamiliarThere is
  time to studyThe point is that tricks are key
• This is all continuous time they go back and
  cover discrete

3
Outline

• Linear, time-invariant systems
• How can you describe an LTI system?
• Feedback uses
• Characterizing a feedback system
• Tools for studying feedback systems
• Types of feedback

4
Signals, systems
x(t)
y(t)
x(t)
y(t)
t
t
voltage pressure position stock price
time space
5
LTI systems
x(t)
y(t)
x(t)
y(t)
t
t
Linear
x(t)
y(t)
t
t
6
LTI systems
x(t)
y(t)
Linear
7
LTI systems
x(t)
y(t)
x(t)
y(t)
t
t
Time-invariant
x(t)
y(t)
t
t
8
LTI systems
x(t)
y(t)
x(t)
t
t
LTI
x(t)
y(t)
t
t
9
Causal systems
x(t)
y(t)
x(t)
y(t)
Causal
t
t
x(t)
y(t)
Non-causal
t
t
10
LTI, causal systems

• A system can be LTI without being causal
• and vice versa
• Causal
• linear circuits response over time
• Non-causal
• blurring filters in image processing (2D)
• speech processing looking ahead into stored
  data

11
The impulse response completely characterizes an
LTI system.
Impulse response
d(t)
h(t)
d(t)
h(t)
t
t
12
Integrate the input integrate the output
d(t)
h(t)
d(t)
h(t)
t
t
u(t)
s(t)
1
t
t
13
The step response is the integral of the impulse
response.
Integrate the input integrate the output
u(t)
s(t)
d(t)
h(t)
14
Convolution integral
For when you cant use tricks
x(t)
y(t)
d(t)
h(t)
Commutative!
15
Flip and shift
d(t)
h(t)
t
t
16
Flip and shift
d(t)
h(t)
t
t
x(t)
t
17
Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
t
18
Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
x(t)
h(0-t)
t
t
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Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
x(t)
h(1-t)
t
t
20
Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
x(t)
h(2-t)
t
t
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Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
x(t)
h(3-t)
t
t
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Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
h(4-t)
t
t
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Flip and shift
d(t)
h(t)
t
t
x(t)
t
x(t)
h(t-t)
t
y(t)
h(8-t)
t
t
24
Frequency domain

• Often easier to use the frequency domain
• Laplace transforms allow you to switch
• s is a complex variable the magnitude of whose
  imaginary part is frequency

25
Laplace transforms

• Differentiation in time ? multiplication by
  sIntegration in time ? division by s
• Convolution in time ? multiplication in s
• Time shift, frequency shift identities
• Duality (remember 2p factors)
• Look at the tables in Mathematics and Elec. Inf.
  Databooks. Know what tricks are out there.

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Transfer functions

• The Laplace Transform of the impulse response is
  the transfer function of the system. Easier than
  writing DEs.
• The frequency response is given by s j?.

u(t)
s(t)
d(t)
h(t)
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Adding feedback

• What if you wrap feedback around the
  plant?Feedback usually negative.

28
Adding feedback

• What if you wrap feedback around the
  plant?Feedback usually negative.

Blacks rule
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Why use feedback?

• To get (close) to the final output you
  want(reject disturbances friction, headwind,
  leakage)
• To make an unstable plant stable more soon.
• To increase bandwidth (at the cost of gain)
• To remove the effect of nonlinearities. For
  H(s) gtgt1, plant can be nonlinear

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Poles and zeroes

• For a transfer functionPoles at roots of
  denominatorZeroes at roots of numerator
• Open loop zero at s- 1 poles at 1 and 3
• Closed loop zero at -1 poles depend on K
• Routh-Hurwitz stability criterion (databook)

31
Pole-zero plots
Im(s)
Re(s)
X
X
-1
1
3
32
Root-locus plots (not in IB syllabus)
Im(s)
Kgt0, K increasing
Re(s)
X
X
-1
1
3
33
Changing K moves the closed-loop poles. When
theyre in the left half of the s-plane, a causal
system is stable
Root-locus plots (not in IB syllabus)
Im(s)
LHP
Kgt0, K increasing
Re(s)
X
X
-1
1
3
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This is a different plane from the Root-locus.
The Nyquist plot shows the frequency response of
the open loop system
Nyquist plot
Im(H(j?))
?0
?-8
Re(H(j?))
?8
-1/K
35
Number of CCW encirclements of (-1,0) must equal
number of RHP open loop poles for stability
Nyquist stability criterion
Im(H(j?))
?0
?-8
Re(H(j?))
?8
-1/K
-1?
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Gain margin aK
Gain margin
Im(H(j?))
a
?0
?-8
Re(H(j?))
-1/K
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Phase margin ?
Phase margin
Im(H(j?))
1/K
?0
?-8
?
Re(H(j?))
-1/K
38
Margins

• Measure how close you are to being unstable
• Typical phase margins
• chemical industry, 35
• aircraft 60

39
Bode plots (for open- or closed-loop systems)

• Log-log plot of transfer functions magnitude
  lin-log plot of its phase. s j?

0
H(s)(dB)
-20
? /rps
1
10
100
1000
? /rps
0
ltH(s)
-90
-3dB point, breakpoint, half power frequency
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Initial and final value theorems

• Very useful
• Multiply RHS by s and evaluate to find
  initial/final slopes

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Tools for analysing systems

• Open loop
• Bode plots
• Nyquist plots
• Pole-zero diagram
• Gain and phase margin
• Closed loop
• Bode plots
• Root locus diagram
• Initial and final value theorems

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Compensators

• K may be K(s) a compensator
• K(s) K proportional
• K(s) K1s derivativereduces overshoot of step
  response, but slows system
• K(s) K2/s integralreduces steady-state error,
  but increases risk of overshoot and ringing
• PID common. Design 6.302 or part IIA courses!

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Resources

• Part IB paper 6 lecture noteswww-control.eng.cam.
  ac.uk/gv/p6/index.html(from within .cam.ac.uk
  domain)
• Signals and SystemsOppenheim and Willsky
• Signals and Systems Made Ridiculously SimpleZZ
  Karu, Barker stacks TK5102.9.K37 1995
• CUED and college libraries are good