Control System¶
tachometer
is derivative feedback
type
type is defined for OLTF of unity feedback systems
dominant pole
if \(\frac{p_1}{p_2}\ge5\) → \(\text{TF}=\frac{\text{DC Gain}}{\text{significant pole}}\)
eliminate poles at origin and significant poles → \(\text{DC Gain}=\lim \limits_{s\rightarrow 0} \text{TF}\)
sensitivity
\(\displaystyle S^T_G = \dfrac{\frac{\partial T}{T}}{\frac{\partial G}{G}}\)
\(\displaystyle S^T_H = \dfrac{\frac{\partial T}{T}}{\frac{\partial H}{H}}\)
mason gain
\(\boxed{\text{TF} = \frac{{ \sum _{k = 1}^n {M_k}{{\rm{\Delta }}_k}}}{{\rm{\Delta }}}}\)
\(\Delta\) = 1 - (sum of loop gains of individual loops) + (sum of product of loop gain of two non-touching loops) - (sum of product of loop gain of three non-touching loops) + \(\dots\)
\(\Delta_k\) = 1 - (sum of loop gains of individual loops which are not common with given path) + (sum of product of loop gain of two non-touching loops which are not common with given path) - (sum of product of loop gain of three non-touching loops which are not common with given path) + \(\dots\)
first order
\(\text{TF}=\dfrac{k}{1+s\tau}\)
impulse response
\(c(t) = \dfrac{1}{\tau}e^{-\frac{t}{\tau}}\)
step response
\(c(t)=u(t)\lbrack1-e^{\frac{-t}{\tau}}\rbrack\)
ramp response
\(c(t)=tu(t)-\tau\lbrack1-e^{\frac{-t}{\tau}}\rbrack u(t)\)
steady state error
\(e(t)=r(t)-c(t)\)
non-unity
if \(e(t)=r(t)-c(t)\) then → \(G'(s) = \dfrac{G(s)}{1+G(s)H(s)-G(s)}\)
\(K_p = \lim \limits_{s\rightarrow 0} G(s)\)
\(K_v = \lim \limits_{s\rightarrow 0} s\cdot G(s)\)
\(K_a = \lim \limits_{s\rightarrow 0} s^2\cdot G(s)\)
\(\boxed{\def\arraystretch{1.5}\begin{array}{c:c:c} \frac{1}{s} & u(t) & E_{ss} = \frac{1}{1+K_p} \\ \hdashline \frac{1}{s^2} & t \cdot u(t) & E_{ss} = \frac{1}{K_v} \\ \hdashline \frac{1}{s^3} & \frac{t^2}{2} u(t)& E_{ss} = \frac{1}{K_a} \end{array}}\)
time domain analysis
first order
\(\text{CLTF}=\dfrac{1}{s\tau +1}\)
time domain parameters
50% of final value: delay → \(t_d =0.693\tau =\ln(2) \tau\)
10% → 90% : rise-time → \(t_r = 2.2\tau\)
settling time (\(t_s)\)
5% error → \(t_s \approx 3 \tau\)
2% error → \(t_s \approx 4 \tau\)
0% error → \(t_s \approx 5\tau\)
second order
CLTF \(= \dfrac{\omega_n^2}{s^2+2\zeta \omega_n s + \omega_n ^2}\)
\(\boxed{s=-\zeta \omega_n \pm \omega_n \sqrt{\zeta^2-1}=-\alpha\pm j\omega_d}\)
damping frequency: \(\omega_d\)
damping factor: \(\alpha\)
damping ratio: \(\zeta\)
\(\tau =\dfrac{1}{\zeta \omega_n}=\dfrac{1}{\alpha}\)
\(cos(\theta) = \zeta\)
impulse response
\(c(t) = \omega_n\dfrac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}}sin(\omega_n t)\) for \(0<\zeta<1\)
step response
\(c(t) = 1- \dfrac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}}sin(\omega_n t+\theta)\) for \(0<\zeta<1\)
time domain parameters
50% of final value: delay → \(\boxed{t_d =\dfrac{0.7\zeta +1}{\omega_n}}\)
10% → 90% : rise-time → \(\boxed{t_r = \dfrac{n\pi-\theta}{\omega_d}}\)
peak time → \(\boxed{t_p =\dfrac{n\pi}{\omega _d}} \begin{cases} n=odd &\text{maxima} \\ n=even &\text{minima} \end{cases}\)
first time-period: \(T_p = t_{p_3} - t_{p_1}=\dfrac{2\pi}{\omega _d}\)
settling time (\(t_s)\)
5% error → \(t_s \approx 3 \tau\)
2% error → \(t_s \approx 4 \tau\)
0% error → \(t_s \approx 5\tau\)
no of cycles → \(\dfrac{t_s}{T_{p}} \approx \dfrac{4\tau}{T_p}\)
routh hurwitz
third order equation
\(\boxed{\begin{array}{c:c} stable & IP>OP \\ \hdashline marginal & IP=OP \\ \hdashline unstable & IP<OP \end{array}}\)
auxiliary equation → coefficients of \(s^2\) and \(s^0\)
\(\epsilon \rightarrow 0^+\)
ROZ → poles symmetrical about origin
A → auxiliary equation from row above the ROZ
roots of A are symmetrical poles
replace ROZ by \(\frac{dA}{ds}\)
two ROZ → repeated symmetrical poles
roots of A are repeated symmetrical poles
gain margin and phase crossover
last row of routh-hurwitz table must not be zero for \(0<k_1<\infty\)
→ make first element of odd row zero
→ compare roots of \(\frac{d(AE)}{ds} =0\) with \(s=\pm j \omega_{pc}\)
otherwise
\(\omega_{pc}\) is not defined
\(\boxed{\begin{array}{c:c:c} CLTF & OLTF & GM (dB) \\ \hdashline \begin{array}{c} stable \\ unstable \end{array} & min\space phase & \begin{array}{c} \infty \\ -\infty \end{array} \\ \hdashline \begin{array}{c} stable \\ unstable \end{array} & non-min\space phase -I & \begin{array}{c} \infty \\ -\infty \end{array} \\ \hdashline \begin{array}{c} stable \\ unstable \end{array} & non-min\space phase -II & \begin{array}{c} -\infty \\ \infty \end{array} \end{array}}\)
root locus
root locus is symmetric about real axis
number of root locus branches is P or Z whichever is greater
root locus exists if their is odd sum of poles and zeroes
asymptotes = \(N = P\sim Z\)
angle = \(\dfrac{(2q+1)180^o}{N}\)
centroid = \(\dfrac{(real \space part \space of \space poles)-(real \space part \space of \space zeroes)}{N}\)
break points \(\dfrac{dK}{ds}=0\) by characteristic equation (valid/invalid breakpoints) and intersection with imaginary axis by RH criteria (auxiliary equation)
angle of departure \(\phi _d = 180 ^{\circ} + \angle GH(s) |_\text{at +ve imaginary complex pole}\)
angle of arrival \(\phi _a = 180 ^{\circ} - \angle GH(s) |_\text{at +ve imaginary complex zero}\)
frequency domain analysis
bode plot (\(\omega\) vs \(dB\) )
\(dB = 20 \cdot log(|G(j\omega)H(j\omega)|)\)
\(\dfrac{20dB}{decade} = \dfrac{6dB}{octave}\)
at pole
slope = \(- 20n \space \frac{dB}{decade}\)
phase = \(-n\frac{\pi}{2}\)
at zero
slope = \(+ 20n \space \frac{dB}{decade}\)
phase = \(+n\frac{\pi}{2}\)
magnitude
eliminate corner frequencies \(\ge\omega_{known}\)
origin frequency \((\omega=0.1)\) is not corner frequency
write TF in time constant format : \(\dfrac{K(\dfrac{s}{\tau_1}+1)(\dfrac{s}{\tau_2}+1)\dots}{(\dfrac{s}{\tau_a}+1)(\dfrac{s}{\tau_b}+1)\dots}\)
\(M = 20 \cdot log(|G(j\omega)H(j\omega)|)\)
polar plot (\(|M|\) vs \(\phi\) )
direction : \(\phi_o - \phi_{\infty}\)
\(\boxed{\begin{array}{c:c:c} \omega & M & \phi \\ \hdashline 0 & M _0 & \phi _0 \\ \hdashline \infty & M _{\infty} & \phi _{\infty} \end{array}}\)
gain cross over frequency (\(\omega_{gc}\)) : \(\boxed{M = |G(s)H(s)| = 1 }\)
phase cross over frequency (\(\omega_{pc}\)) : \(\boxed {\angle G(s)H(s) = \pm 180^{\circ}}\)
gain margin \(= \dfrac{1}{|G(j\omega_{pc})H(j\omega_{pc})|}\)
phase margin \(= 180^{\circ} +\angle G(j\omega_{gc})H(j\omega_{gc})\)
\(\boxed{\begin{array}{c:c} \omega _{pc} > \omega_{gc} & stable \\ \hdashline \omega _{pc} = \omega_{gc} & marginal \space stable \\ \hdashline \omega _{pc} < \omega_{gc} & unstable\end{array}}\)
nyquist plot
\(N=P-Z\)
N = number of encirclement of origin by GH(s) in GH(s) plane. N is +ve if contour and contour in GH(s) plane are opposite in direction
P = number of poles of GH(s) strictly inside contour in s-plane
Z = number of zeroes of GH(s) strictly inside contour in s-plane
NYQUIST contour is CLOCKWISE
compensator and controller
lead compensator
LEAD behaves similar to HIGH-PASS filter (zero dominant)
\(TF = \alpha \Big( \dfrac{1+s\tau}{1+\alpha\tau s}\Big)\) where \(\alpha <1\) and \(\tau = R_1 C\)
\(\alpha=\dfrac{R_2}{R_1+R_2}\)
lag compensator
LAG behaves similar to LOW-PASS filter (pole dominant)
\(TF = \dfrac{1+s\tau}{1+\beta\tau s}\) where \(\beta >1\) and \(\tau = R_2 C\)
\(\beta=\dfrac{R_1+R_2}{R_2}\)
lead-lag and lag-lead
LEAD LAG behaves similar to BAND-PASS filter (zero dominant)
LAG LEAD behaves similar to BAND-REJECT filter (pole dominant)
state space analysis
\(\dot{X } = A X + BU\)
\(Y = CX +DU\)
\(TF = \dfrac{Y(s)}{U(s)}\)
controllability
\(Q_c = \begin{bmatrix} B & AB & A^2B \dots A^{n-1}B \end{bmatrix}\)
\(rank(Q_c)=rank(A)=n\) → completely controllable system
\(|Q_c|\ne 0\) → controllable system
observability
\(Q_o = \begin{bmatrix} C \\ CA \\ CA^2\\ \vdots \\ C A^{n-1} \end{bmatrix}\)
\(|Q_o|\ne 0\) → observable system
\(rank(Q_o)=rank(A)=n\) → completely observable system