Classic Control Quick Notes

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

\[ \begin{aligned} sX(s) &= AX(s) + bU(s) \\ Y(s) &= cX(s) + dU(s) \end{aligned} \]

Using linear algebra, we can express \(X(s)\) as:

\[ X(s) = (sI-A)^{-1} b U(s) \]

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:

\[ G(s) = \frac{Y(s)}{U(s)} = c(sI-A)^{-1} b + d \]

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:

\[ c(s)=1s^5+2s^4+3s^3+4s^2+5s^1+6s^0 \]


\(s^5\)	\(1\)	\(3\)	\(5\)
\(s^4\)	\(2\)	\(4\)	\(6\)
\(s^3\)	\(\begin{aligned}&-\frac12\begin{vmatrix}1&3\\2&4\end{vmatrix}\\=&-\frac12(1\times4-2\times3)\\=&3-\frac{1\times4}2\\=&1\end{aligned}\)	\(\begin{aligned}&-\frac12\begin{vmatrix}1&5\\2&6\end{vmatrix}\\=&-\frac12(1\times6-2\times5)\\=&5-\frac{1\times6}2\\=&2\end{aligned}\)
\(s^2\)	\(4-\frac{2\times2}1=0\)	\(6\)
\(s^1\)	\(2\)
\(s^0\)	\(6\)

Kharitonov Theorem

Let

\[ {\mathcal P}=\set{a(s)=a_0s^n+a_1s^{n-1}+\cdots+a_{n-1}s+a_n\ |\ a_i\in[\underline{a_i}, \overline{a_i}]} \]

All members of \(\mathcal P\) are stable if and only if the following four polynomials are stable

\[ \begin{aligned} a_1(s)=\underline{a_0}s^n+\underline{a_1}s^{n-1}+\overline{a_2}s^{n-2}+\overline{a_3}s^{n-3}+\underline{a_4}s^{n-4}+\cdots\\ a_2(s)=\underline{a_0}s^n+\overline{a_1}s^{n-1}+\overline{a_2}s^{n-2}+\underline{a_3}s^{n-3}+\underline{a_4}s^{n-4}+\cdots\\ a_3(s)=\overline{a_0}s^n+\overline{a_1}s^{n-1}+\underline{a_2}s^{n-2}+\underline{a_3}s^{n-3}+\overline{a_4}s^{n-4}+\cdots\\ a_4(s)=\overline{a_0}s^n+\underline{a_1}s^{n-1}+\underline{a_2}s^{n-2}+\overline{a_3}s^{n-3}+\overline{a_4}s^{n-4}+\cdots\\ \end{aligned} \]

Prototype 2nd-Order System

The transfer function for a second-order system is given by:

\[ H(s)=\frac{k\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}=\frac{k(\sigma^2+\omega_d^2)}{(s+\sigma)^2+\omega_d^2} \]

Time Constants

The following approximations hold:

\[ \begin{cases} t_r\approx\frac{1.8}{\omega_n} \\ t_p=\frac\pi{\omega_n\sqrt{1-\zeta^2}} = \frac\pi{\omega_d} \\ t_s\approx\frac3{\zeta\omega_n} = \frac3\sigma \end{cases} \]

Percent Overshoot

The percent overshoot (PO) is defined as:

\[ PO=\left(\frac{y(t_p)}{y(\infty)}-1\right)\times100\%=\exp\left(-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}\right) \]

Final Value Theorem

The final value theorem states that:

\[ FVT=H(0)=k \]

Bode's Sensitivity

In the nominal situation, we have the motor with DC gain = \(A\), and the overall transfer function, either open- or closed-loop, has some other DC gain (call it \(T\)).

Perturbations

Let:

\[ \begin{aligned} \hat A=A+\delta A \hat T=T+\delta T \end{aligned} \]

Then, the sensitivity can be approximated as:

\[ \delta T\approx\frac{{\rm d}T}{{\rm d}A}\delta A \]

Sensitivity Function

The sensitivity \(\mathcal S\) is given by:

\[ {\mathcal S}=\frac{\frac{\delta T}T}{\frac{\delta A}A}=\frac{\delta T\cdot A}{\delta A\cdot T}\approx\frac{{\rm d}T}{{\rm d}A}\cdot\frac AT \]

Root Locus

The standard form of the root locus is:

\[ 1+KL(s)=0 \]

Transformation to Standard Form

Change to standard form:

\[ a(s)+Kb(s)=0 \]

This can be expressed as:

\[ 1+K\cdot\frac{b(s)}{a(s)}=0 \]

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=?`