ELEC 3200 - System Modeling, Analysis and Control
This cheat sheet summarizes key concepts in System Modeling, Analysis and Control.
Transfer Function
For an LTI system, we can take the Laplace transform with zero initial conditions
Using linear algebra, we can express
The transfer function of an LTI system is defined as the ratio of the Laplace transform of the output to that of the input when the initial conditions are zero:
Different state space representations can be derived for a particular system due to the infinite number of linear transformations, but the transfer function remains unique.
Routh Stability Criterion
Given the characteristic polynomial:
Kharitonov Theorem
Let
All members of
Prototype 2nd-Order System
The transfer function for a second-order system is given by:
Time Constants
The following approximations hold:
Percent Overshoot
The percent overshoot (PO) is defined as:
Final Value Theorem
The final value theorem states that:
Bode's Sensitivity
In the nominal situation, we have the motor with DC gain =
Perturbations
Let:
Then, the sensitivity can be approximated as:
Sensitivity Function
The sensitivity
Root Locus
The standard form of the root locus is:
Transformation to Standard Form
Change to standard form:
This can be expressed as:
Root Locus Rules
Rule A | n branches |
Rule B | starts at s = x, x, ... |
Rule C | ends at s = x, x, ... |
Rule D | Real locus: (-xx,-xx) U (-xx,-xx) |
Rule E | n - m =xx , l = 0,1,...,xx-1 Asymptotes = xxx°, xxx° |
Rule F | a(s)+Kb(s)=0 Routh Table => K∈(xx,xx) j·w? , w=? |