Transient%20Response

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Transient Response

• First order system transient response
• Step response specs and relationship to pole
  location
• Second order system transient response
• Step response specs and relationship to pole
  location
• Effects of additional poles and zeros

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Simple first order system
1 ts
E
Y(s)
U(s)

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First order system step resp
Normalized time t/t
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Simple first order system

• No overshoot, tpinf, Mp 0
• Yss1, ess0
• Settling time ts -ln(tol)/p
• Delay time td -ln(0.5)/p
• Rise time tr ln(0.9) ln(0.1)/p
• All times proportional to 1/p t
• Larger p means faster response

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The error signal e(t) 1-y(t)e-ptus(t)
Normalized time t/t
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In every t seconds, the error is reduced by 63.2
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General First-order system
We know how this responds to input
Step response starts at y(0)k, final value
kz/p 1/p t is still time constant in every t,
y(t) moves 63.2 closer to final value
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Step response by MATLAB
gtgt p . .
gtgt n b1 b0
gtgt d 1 p
gtgt step ( n , d )
Other MATLAB commands to explore plot, hold,
axis, xlabel, ylabel, title, text, gtext,
semilogx, semilogy, loglog, subplot
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Unit ramp response
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Note In step response, the steady-state
tracking error zero.
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Unit impulse response
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Prototype 2nd order system
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Unit step response
1) Under damped, 0 lt ? lt 1
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cosq z q cos-1z
d
s
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To find y(t) max
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For 5 tolerance Ts 3/zwn
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• Delay time is not used very much
• For delay time, solve y(t)0.5 and solve for t
• For rise time, set y(t) 0.1 0.9, solve for t
• This is very difficult
• Based on numerical simulation

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Useful Range Td(0.80.9z)/wn
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Useful Range Tr4.5(z-0.2)/wn
Or about 2/wn
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Putting all things together
Settling time
(3 or 4 or 5)/s
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2) When ? 1, ?d 0
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The tracking error
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3) Over damped ? gt 1
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Transient Response
Recall 1st order system step response
2nd order
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Pole location determines transient
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• All closed-loop poles must be strictly in the
  left half planes
• Transient dies away
• Dominant poles those which contribute the most
  to the transient
• Typically have dominant pole pair
• (complex conjugate)
• Closest to j?-axis (i.e. the least negative)
• Slowest to die away

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Typical design specifications

• Steady-state
• ess to step
• ts
• Speed (responsiveness)
• tr
• td
• Relative stability
• Mp

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These specs translate into requirements on ?, ?n
or on closed-loop pole location
Find ranges for ? and ?n so that all 3 are
satisfied.
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Find conditions on s and ?d.
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In the complex plane
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Constant s vertical lines s gt is half
plane
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Constant ?d horizontal line ?d lt is a
band ?d gt is the plane excluding band
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Constant ?n circles ?n lt inside of a
circle ?n gt outside of a circle
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Constant ? f cos-1? constant Constant ?
ray from the origin ? gt is the cone ?
lt is the other part
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If more than one requirement, get the common
(overlapped) area e.g. ? gt 0.5, s gt 2, ?n gt 3
gives
Sometimes meeting two will also meet the third,
but not always.
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Try to remember these
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Example

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When given unit step input, the output looks like
Q estimate k and t.
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Effects of additional zeros
Suppose we originally have
i.e. step response
Now introduce a zero at s -z
The new step response
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Effects

• Increased speed,
• Larger overshoot,
• Might increase ts

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When z lt 0, the zero s -z is gt 0, is in the
right half plane. Such a zero is called a
nonminimum phase zero. A system with nonminimum
phase zeros is called a nonminimum phase system.
Nonminimum phase zero should be avoided in
design. i.e. Do not introduce such a zero in
your controller.
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Effects of additional pole
Suppose, instead of a zero, we introduce a pole
at s -p, i.e.
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L.P.F. has smoothing effect, or averaging effect
Effects

• Slower,
• Reduced overshoot,
• May increase or decrease ts