Table of Contents

1 Minimum Phase and Non-Minimum Phase System

1.1 Constant K

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1.2 Integrator \(1/s\)

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1.3 First Order \(1/(\tau s+1)\)

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1.4 Second Order \(1/((s/\omega_n)^2+2\zeta(s/\omega_n)+1)\)

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1.5 Minimum Phase System

  • 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

1.6 Non-Minimum Phase System: Unstable Zeros

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.

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

1.7 Non-Minimum Phase System: Delay

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  • 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.

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

1.8 Bode Plot of Minimum Phase and Non-Minimum Phase System

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

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1.9 Implications of Non-Minimum Phase

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  • 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.

2 Bode Plots of Complex Transfer Functions

Bode plots of a transfer function (system) can be obtained using Matlab (experiment). However, to interpret the Bode plots, we shall take a look at how basic factors of the transfer function affect the Bode plots.

2.1 Example 1

\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}

2.1.1 Step 1: Decompose the Transfer Function into Basic Factors

Factor Corner Frequency
2  
\(1/s\)  
\(2s+1\) 0.5
\(1/(0.1s+1)\) 10
\(1/(0.02s+1)\) 50

By adding the Bode plots of the above factors together, we can get the Bode plot of \(G(s)\)

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2.1.2 Step 2: Determine the Plot at the Low Frequency

  • For low frequency, only need to look at the constant \(K\) and \(s^N\) term.
  • Draw the plot of \(s^N\) first
    • Magnitude: a straight line passes through \((1,0dB)\) with a slope of \(20NdB\).
    • Phase: a horizontal line at \(90N^\circ\).
  • Adding the constant term \(K\)
    • Magnitude: add \(20\log|K|\)
    • Phase: If \(K > 0\), do nothing. If \(K < 0\), subtract \(180^\circ\).

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2.1.3 Step 3: Add First and Second Order Terms

Factor Corner Frequency
\(2s+1\) 0.5
\(1/(0.1s+1)\) 10
\(1/(0.02s+1)\) 50
  • Magnitude

For \((\tau s+1)^N\), change the slope at corner frequency \(1/\tau\) by \(20N\).

Frequency Low 0.5 10 50
Slope Change   +20 -20 -20
Slope -20 0 -20 -40
  • Phase

For \((\tau s+1)^N\)

  • change the slope at \(0.1/\tau\) by \(45N\).
  • change the slope at \(10/\tau\) by \(-45N\).
Frequency Low 0.05 1 5 100 500
Slope Change   +45 -45 -45 -45 +45 +45
Slope 0 45 0 -90 -45 0

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2.1.4 Step 4 (Optional): Check your result in Matlab

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

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2.2 Example 2

\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}

2.2.1 Step 1: Decompose the Transfer Function into Basic Factors

Factor Corner Frequency
2.5  
\(1/s\)  
\((s/2)^2+0.2s/2+1\) 2

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2.2.2 Step 2: Determine the Plot at the Low Frequency

The low frequency factors are \(2.5\) and \(1/s\).

  • Magnitude
    • Draw a straight line passing through \((1,0dB)\) with slope -20.
    • Shift the magnitude plot of \(1/s\) up by \(20\log(2.5)\).
  • Phase
    • Draw a constant line at \(-90^\circ\).

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2.2.3 Step 3: Add First and Second Order Terms

Consider a second order term:

\[\left[\left(\frac{s}{\omega_n}\right)^2+2\zeta\frac{s}{\omega_n}+1\right]^{\pm 1}.\]

  • Magnitude
    • change the slope at the corner frequency \(\omega_n\) by \(\pm 40dB\);
    • add \(\pm 20\log(2\zeta)\) correction at the corner frequency \(\omega_n\).
  • Phase
    • draw the phase plot using calculators.

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2.2.4 Step 4: Check your result in Matlab

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

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2.3 Example 3

\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.

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