ELEC 3200 Quick Notes
Routh Stability Criterion
\[
c(s)=1s^5+2s^4+3s^3+4s^2+5s^1+6s^0
\]
| \(s^5\) | ==\(1\)== | \(3\) | ==\(5\)== |
| \(s^4\) | ==\(2\)== | \(4\) | ==\(6\)== |
| \(s^3\) | \(\begin{aligned}&-\frac12\begin{vmatrix}1&3\\2&4\end{vmatrix}\\=&-\frac12(1\times4-2\times3)\\=&3-\frac{1\times4}2\\=&1\end{aligned}\) | ==\(\begin{aligned}&-\frac12\begin{vmatrix}1&5\\2&6\end{vmatrix}\\=&-\frac12(1\times6-2\times5)\\=&5-\frac{1\times6}2\\=&2\end{aligned}\)== | |
| \(s^2\) | \(4-\frac{2\times2}1=0\) | \(6\) | |
| \(s^1\) | \(2\) | ||
| \(s^0\) | \(6\) |
Kharitonov Theorem
\[
{\mathcal P}=\set{a(s)=a_0s^n+a_1s^{n-1}+\cdots+a_{n-1}s+a_n\ |\ a_i\in[\underline{a_i}, \overline{a_i}]}
\]
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
\[
H(s)=\frac{k\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}=\frac{k(\sigma^2+\omega_d^2)}{(s+\sigma)^2+\omega_d^2}
\]
\[
\begin{cases}
t_r\approx\frac{1.8}{\omega_n} \\
t_p=\frac\pi{\omega_n\sqrt{1-\zeta^2}} = \frac\pi{\omega_d} \\
t_s\approx\frac3{\zeta\omega_n} = \frac3\sigma
\end{cases}
\]
\[
PO=\left(\frac{y(t_p)}{y(\infty)}-1\right)\times100\%=\exp\left(-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}\right)
\]
\[
FVT=H(0)=k
\]
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 \(T\)).
\[
\hat A=A+\delta A
\]
\[
\hat T=T+\delta T
\]
\[
\delta T\approx\frac{{\rm d}T}{{\rm d}A}\delta A
\]
\[
{\mathcal S}=\frac{\frac{\delta T}T}{\frac{\delta A}A}=\frac{\delta T\cdot A}{\delta A\cdot T}\approx\frac{{\rm d}T}{{\rm d}A}\cdot\frac AT
\]
Root Locus
Standard Form:
\[
1+KL(s)=0
\]
Change to standard form
\[
a(s)+Kb(s)=0
\]
\[
1+K\cdot\frac{b(s)}{a(s)}=0
\]
| 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-1Asymptotes = xxx°, xxx° |
| Rule F | a(s)+Kb(s)=0Routh Table => K∈(xx,xx)j·w?, w=? |