Poles & Zeros
Effect of poles on impulse responses 1st and 2nd order systems
Motivation
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It is very important for analysis/design to understand how poles/zeros affect impulse/step responses.
Step response design specifications can be related to locations of poles and zeros
E.g. zero steady-state error tracking a step function with unity feedbackOpen-loop pole at s=0
It is possible to get a model of an LTI system by measuring its step response
Poles and zeros
Consider a proper transfer function
If are real, then are either real or
complex conjugate of the form
are the zeros of the transfer function. are the poles of the transfer function.
Visualising Poles and Zeros in s-Domain
Source: Swarthmore College, US
Example 1:
Example 2:
Effect of poles (and zeros) on transient response
Qualitative- oscillatory or smooth; stable or unstable
Quantitative – rate of decay/explosion, oscillation frequency, rise time, settling time,
Partial fractional expansion:
We can write any rational function using its partial fractional expansion (different poles):
or for repeated poles:
Solutions in time domain
Taking inverse Laplace transform of the previous expressions leads to a set of possible time-domain functions that arise:
– Polynomial functions
– Exponentials in time
– Sinusoidal functions
– Their products
Impulse response of LTI systems
Each such term is called a mode of the system.
• The growth rate σ, freq. ω and powers n depend only each corresponding pole
• But the linear coefficients in front of each mode, and the phase φ, also depend on other poles and zeros
Consider the transfer function:
Its impulse response is obtained via its partial fraction expansion ( ):
When we have repeated poles, we get terms of this form in partial fraction expansion:
They lead to response terms of the form
Note that response is still dominated by exponential terms provided .
1st and 2nd order systems
Impulse response terms with real pole
Impulse response with complex pole
E.g. recall:
2nd order systems
Consider a generic 2nd order system: Suppose then the poles are
Terminology
natural (undamped) frequency
damped frequency (imaginary part)
damping factor
real part
Graphic representation of poles
Response of 2nd order systems
Important relationships:
Time constant increases as damping decreases.
Peak of oscillation increases as damping decreases.
Frequency of oscillation increases when damping decreases.
General systems – summary:
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