Transient response analysis of first order and second order systems - - PowerPoint PPT Presentation

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Transient response analysis of first order and second order systems - - PowerPoint PPT Presentation

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STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Transient response analysis of first order and second order systems Lecture 9 Systems and Control Theory STADIUS - Center for Dynamical Systems, Signal Processing

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Transient response analysis of first order and second order systems

Lecture 9

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Transient Response

The time response of a control system may be written as: Where ytr(t) is the transient response and yss(t) is the steady state response.

  • Most important characteristic of dynamic system is absolute stability.
  • System is stable when returns to equilibrium if subject to initial

condition

  • System is critically stable when oscillations of the output continue

forever

  • System is unstable if unstable when output diverges without bound

from equilibrium if subject to initial condition

  • Transient response: when input of system changes, output does not

change immediately but takes time to go to steady state

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

First-order systems

  • E.g. RC circuit, thermal system, …
  • Transfer function is given by
  • Unit step response
  • Laplace of unit-step is 1/s  substituting Y(s) = 1/s into equation
  • Expanding into partial fractions gives
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Unit step transient response

  • Taking the inverse Laplace transform
  • At t=0, the output c(t) = 0
  • At t=T, the output c(t) = 0.632, or c(t) has reached 63.2% of its total change
  • Slope at time t = 0 is 1/T
  • Where T is called the system’s time constant
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Unit step transient response

y(t)

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Unit ramp transient response

  • Laplace transform of unit ramp is 1/s2
  • Expanding into partial fractions gives
  • Taking the inverse Laplace transform gives
  • The error signal e(t) is then
  • For t approaching infinity, e(t) approaches T
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Unit ramp transient response

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Unit-Impulse Response

  • For a unit-impulse input, U(s)=1 and the output is
  • The inverse Laplace transform gives
  • For t  +∞ , y(t)  0
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Unit-Impulse Response

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Second order systems

  • A second order system can generally be written as:
  • A system where the closed-loop transfer function possesses two poles is called a

second-order system

  • If the transfer function has two real poles, the frequency response can be found by

combining the effects of both poles

  • Sometimes the transfer function has two complex conjugate poles. In that case we

have to find a different solution for finding the frequency response.

  • In order to study the transient behaviour, let us first consider the following

simplified example of a second order system

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Step response second order system

  • The transfer function can be rewritten as:
  • The poles are complex conjugates if
  • The poles are real if
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

  • To simplify the transient analysis, it is convenient to write
  • Where
  • The transfer function can now be rewritten as

Which is called the standard form of the second-order system.

  • The dynamic behavior of the second-order system can then be described in terms
  • f only two parameters ζ and ωn

Step response second order system

is the attenuation is the undamped natural frequency is the damping ratio

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

  • If 0 < ζ < 1 , the poles are complex conjugates and lie in the left-half s plane
  • The system is then called underdamped
  • The transient response is oscillatory
  • If ζ = 0, the transient response does not die out
  • If ζ = 1, the system is called critically damped
  • If ζ > 1, the system is called overdamped
  • We will now look at the unit step response for each of these cases

Step response second order system

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Underdamped system

  • For the underdamped case (0 < ζ < 1 ), the transfer function can be written as:
  • Where ωd is called the damped natural frequency
  • For a unit-step input we can write
  • Which can be rewritten as
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Underdamped system

  • It can be shown that
  • Therefore:
  • It can be seen that the frequency of the transient oscillation is the damped natural

frequency ωd and thus varies with the damping ratio ζ

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Underdamped system

  • The error signal is the difference between input and output
  • The error signal exhibits a damped sinusoidal oscillation
  • At steady state, or at t =∞ the error goes to zero
  • If damping ζ = 0, the response becomes undamped
  • Oscillations continue indefinitely
  • Filling in ζ = 0 into the equation for y(t) gives us
  • We see that the system now oscillates at the natural frequency ωn
  • If a linear system has any amount of damping, the undamped natural

frequency cannot be observed experimentally, only ωd can be observed

  • ωd is always lower than ωn
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Underdamped system

  • The error signal is the difference between input and output
  • The error signal exhibits a damped sinusoidal oscillation
  • At steady state, or at t =∞ the error goes to zero
  • If damping ζ = 0, the response becomes undamped
  • Oscillations continue indefinitely
  • Filling in ζ = 0 into the equation for y(t) gives us
  • We see that the system now oscillates at the natural frequency ωn
  • If a linear system has any amount of damping, the undamped natural

frequency cannot be observed experimentally, only ωd can be observed

  • ωd is always lower than ωn
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Critically damped system

  • If the two poles of the system are equal, the system is critically damped and ζ = 1
  • For a unit-step, R(s)=1/s we can write
  • The inverse Laplace transform gives us
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Overdamped system

  • A system is overdamped (ζ > 1) when the two poles are negative, real and unequal
  • For a unit-step R(s)=1/s, Y(s) can be written as
  • The inverse Laplace transform is
  • Where
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Overdamped system

  • Thus y(t) includes two decaying exponential terms
  • When ζ >>1, one of the two decreases much faster than the other, and then

the faster decaying exponential may be neglected

  • Thus if –s2 is located much closer to the jω axis than –s1 (|s2|>>|s1|), then –s1

may be neglected

  • Once the faster decaying exponential term has disappearedm the response is

similar to that of a first-order system

  • In that case, H(s) can be approximated by
  • With the approximate transfer function, the unit-step response becomes
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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

  • With the approximate transfer function, the unit-step response becomes
  • The time response for the approximate transfer function is then given as

Overdamped system

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Second order systems – unit step response curves

  • Response on a step function

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Second order systems - characteristics

  • Overshoot: Highest amplitude

above steady state.

  • Rise Time: Time needed to

reach the steady state for the first time.

  • Peak Time: Time to reach
  • vershoot.
  • Settling Time: Time needed to

approximate the steady state.

  • For:
  • We find:

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Second order systems - resonace

  • The resonance frequency is the frequency at which the systems
  • utput has a larger amplitude than at other frequencies. This

happens when underdamped functions oscillate at a greater magnitude than the input.

  • An input with this frequency can sometime have catastrophic

effects.

  • A different view on the Tacoma bridge disaster:

https://www.youtube.com/watch?v=6ai2QFxStxo

  • In fact the collapse was a

result of a number of effects like Aerodynamic flutter and

  • vortices. Read the full

article here: http://www.ketchum.org/bi llah/Billah-Scanlan.pdf

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Second order systems - resonance

  • The resonance frequency is:
  • Systems with a damping > 0.707 do not resonate.
  • The resonance frequency can be found as a local maximum in a bode plot of the

system.

  • The resonance frequency and the natural frequency are equal when a system has

no damping.

  • Another phenomenon with bridges and resonance is that many people marching

with the same rhythm can cause a bridge to start resonating like the Angers bridge in 1850. A more recent example is the Millennium bridge in London who started resonating (see video lecture 2). Source: http://en.wikipedia.org/wiki/Angers_Bridge

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Second order systems - damping

  • When we want a system with no

resonance, we choose one with damping <0.707. This means a pole between 135° and 225°:

  • We mostly also want a short settling

time ( < 4s). This results in another restriction on the poles of the system:

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