• A transfer function \(G(s)\) is minimum phase if both \(G(s)\) and \(1/G(s)\) are causal and stable.
• Roughly speaking it means that the system does not have zeros or poles on the right-half plane. Moreover, it does not have delay.
• Bode discovered that the phase can be uniquely derived from the slope of the magnitude for minimum-phase system. Bode's Relation

Basic Factor	Mag Slope(Low Freq)	Phase(Low Freq)	Mag Slope(High Freq)	Phase (High Freq)
K	0	0	0	0
\(s^N\)	20N	90N	20N	90N
\(1/(\tau s+1)\)	0	0	-20	-90
\(1/((s/\omega_n)^2+2\zeta(s/\omega_n)+1)\)	0	0	-40	-180

Consider two transfer functions: \[G_1(s) = 1,\,G_2(s) = \frac{1-s}{1+s}\]

• \(G_1(s)\) is minimum phase since it does not have unstable zeros/poles.
• The magnitude of \(G_1(s)\) is 0dB, and the phase is \(0^\circ\).
• \(G_2(s)\) is non-minimum phase, since \(G_2(1) = 0\).
• One can check that \(1/G_2(s) = \frac{1+s}{1-s}\) is unstable.

\[|G_2(j\omega)| = \frac{\sqrt{1+\omega^2}}{\sqrt{1+\omega^2}} = 1,\,\angle G_2(j\omega) = -\tan^{-1}\omega -\tan^{-1}\omega = -2\tan^{-1}\omega.\]

• Time delay often exists in communication systems, chemical and manufacturing processes, etc.
• Time delay systems are of non-minimum phase behavior.
• The transfer function of the time delay factor is \(G_3(s) = \exp(-\tau s)\), where \(\tau\) is the delay.

\[G_3(j\omega) = \exp(-j\omega\tau) = 1\angle -\omega\tau\times\frac{180^\circ}{\pi}.\]

Minimum phase system has minimum phase change amongst all transfer functions that have the same magnitude plot.

• Non-minimum phase systems are much more difficult to control than minimum phase system
• Unfortunately, non-minimum phase is quite common in practice
  • Flexible Structure
  • Circuits
  • Back a car into the parking lot
• The unstable zeros can be changed by reallocating sensors and actuators, or by introducing new sensors and actuators.
• You will learn this in EE4273 Digital Control Systems where we teach state space model.

\begin{align} G(s) = \frac{2000(s+0.5)}{s(s+10)(s+50)} = \frac{2(2s+1)}{s(0.1s+1)(0.02s+1)}. \end{align}

num = [2000,1000];
den = [1,60,500,0];
sys = tf(num, den);
bode(sys);

\begin{align} G(s) = \frac{10}{s(s^2+0.4s+4)} = \frac{2.5}{s((s/2)^2+2\times 0.1s/2+1)}. \end{align}

num = [10];
den = [1,0.4,4,0];
sys = tf(num, den);
bode(sys);

\begin{align} G(s) &= \frac{2500(s+10)}{s(s+2)(s^2+30+2500)}\\ &=\frac{5(0.1s+1)}{s(0.5s+1)\left[(s/50)^2+0.6s/50+1\right]}. \end{align}

Factor	Corner Frequency
5
\(1/s\)
\(1/(0.5s+1)\)	2
\(0.1s+1\)	10
\(1/\left[(s/50)^2+0.6s/50+1\right]\)	50

• Magnitude
  • Low Frequency: Draw a straight line passing through \((1,0dB)\) with slope -20. Shift this line up by \(20\log(5)=14dB\).
  • Adding the first and second order term. Add \(-20\log(0.6) = 4.4dB\) correction at 50.

Frequency	Low	2	10	50
Slope Change		-20	+20	-40
Slope	-20	-40	-20	-60

• Phase
  • Phase starts at \(-90^\circ\) due to the integrator.
  • Phase ends at \(-90-90+90-180 = -270^\circ\).
  • Use calculator to draw the phase plot.