• Nyquist Stability Criterion tells us the stability of the closed-loop system depends on the open-loop transfer function \(C(s)G(s)\).
• Loop shaping means to choose a compensator(controller) \(C(s)\) to give the open-loop transfer function \(C(s)G(s)\) a desired shape.
• Commonly used controller includes PID controller, lag/lead compensator, and combined lag-lead compensator.

• In the low frequency, the gain should be high to have good steady state accuracy.
  • For type-0 system, the position error constant is \(G(0)\).
  • For type-1 system, the position error constant is \(0\) and the velocity error constant is \(\lim_{s\rightarrow 0} sG(s)\).
  • For type-2 system, the position and velocity error constant is \(0\) and the acceleration error constant is \(\lim_{s\rightarrow 0} s^2G(s)\).

• Bode's Relation For a minimum phase system, if the magnitude plot is a straight line with slope \(20N\), then the phase will be \(90N^\circ\).
• Around the gain crossover frequency, the slope of the magnitude plot should be about \(-20\) and extend over a sufficient range of frequency
• The phase will then be \(\approx -90^\circ\) with a stability margin \(\approx 90^\circ\)

• In the high frequency, the gain must be small in order to have good noise attenuation and robustness against model uncertainty.
• The choice of gain crossover frequency is a compromise among steady-state error, rejection of noise and robustness.

Consider the following open-loop transfer function \[G(s) = \frac{10(s+1)}{s^2\left[(s/100)^2+2(s/100)+1\right]}.\]

The closed-loop transfer function is \[\frac{G(s)}{1+G(s)} = \frac{10(s+1)}{10^{-4}s^4+2\times 10^{-2}s^3+s^2+10s+10}.\]

• The low frequency gain is changed by \(20\log K\).
  • \(K\) can be chosen to satisfy specification on steady-state error.
• The phase is always positive, which is why it is called a lead compensator
  • The phase margin will be increased
  • Smaller overshot
• The system's gain crossover frequency will also be increased, resulting in faster response (but the system will be more susceptible to noise)

• The phase response of the lead compensator is given by \[\angle C(j\omega) = \tan^{-1}(\omega T) - \tan^{-1}(\alpha\omega T).\]
• The maximum phase lead occurs when \[ \frac{d\angle C(j\omega)}{d\omega} = \frac{T}{1+\omega^2T^2}-\frac{\alpha T}{1+\alpha^2\omega^2T^2} = 0,\]which implies that \(\omega = 1/\sqrt{\alpha}T\).
• The maximum lead is \[\tan^{-1}\left(1/\sqrt{\alpha} \right)- \tan^{-1}\sqrt{\alpha} = \sin^{-1}\frac{1-\alpha}{1+\alpha}.\]
• Therefore, to achieve \(\phi\) maximum phase lead, we need to choose \[\alpha = \frac{1-\sin \phi}{1+\sin\phi}.\]
• The corresponding magnitude increase is \[K\left|\frac{jT\omega+1}{j\alpha T\omega + 1}\right| = K\frac{\sqrt{1+1/\alpha}}{\sqrt{1+\alpha}} = K\alpha^{-1/2} = 20\log K-10\log \alpha (\text{dB}).\]

• Determine the gain \(K\) to satisfy the steady state error requirement. For example, for type-0 system: \[e_{ss} = \frac{1}{1+K_p},\text{ and }K_p = \lim_{s\rightarrow 0} C(s)G(s) = KG(0).\]
• Draw the Bode plots of \(KG(s)\), derive the phase margin \(\phi_0\).
• From \(\phi_0\) and the required phase margin \(PM\), determine the phase lead \(\phi\): \[\phi = PM - \phi_0 + 5^\circ\sim 10^\circ.\]
• Derive \[\alpha = \frac{1-\sin \phi}{1+\sin \phi}.\]
• To achieve the maximum phase lead, place the gain crossover frequency \(\omega_g\) at \(1/\sqrt{\alpha}T\). Therefore, \[20\log|G(j\omega_g)| + 20\log K = 10\log \alpha,\] and we can derive \(\omega_g\).
• Choose \[T = \frac{1}{\sqrt{\alpha}\omega_g}.\]
• Verify the result using Matlab.

In a solar tracking system, the control objective is to drive the solar panel which is attached to motor shaft in order to track the light source. The transfer function of the system is given by \[G(s) = \frac{10}{s(s+1)}.\] Design a lead compensator such that \[K_v \geq 100,\,\phi_m \geq 40^\circ,\, K_g \geq 10\text{dB}.\]

• The lead compensator improves the phase margin and thus the transient performance (overshoot) of the system.
• The gain crossover frequency is increased. Thus, the speed of the system response is improved.
• However, the lead compensator increases the high frequency gain of the system. This makes the system more susceptible to noise signals.