Review

• \(a+bi = r\angle \phi\).
• \(r = \sqrt{a^2+b^2}\) and \[\phi=\begin{cases}\tan^{-1}(b/a)&a > 0\\ \tan^{-1}(b/a) + 180^\circ &a <0\\ 90^\circ& a=0,\,b>0\\ -90^\circ &a=0,b<0 \\indeterminate &a=b=0\end{cases}.\]
• \((a+bi)^n = r^n\angle n\phi\). (This is true even when \(n\) is negative.)
• \[\frac{1}{a+bi} = \frac{a-bi}{a^2+b^2}.\]

The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input.

It is also a sinusoidal signal, where

• The frequency is the same.
• The amplitude is \(A\times |G(j\omega)|\).
• The phase is \(\phi + \angle G(j\omega)\).

• Bode plots of a transfer function \(G(s)\) include two graphs:
  • Magnitude plot: \(20\log_{10}|G(j\omega)|\) (dB) in linear scale v.s. \(\omega\) in log scale.
  • Phase plot: \(\angle G(j\omega)\) (degree) in linear scale v.s. \(\omega\) in log-scale.

• A complex polynomial can always be factorized in to basic terms
  • \(K\)
  • \(s\)
  • \(\tau s + 1\)
  • \[\left(\frac{s}{\omega_n}\right)^2 + 2\zeta \left(\frac{s}{\omega_n}\right) + 1.\]

• Decompose the Transfer Function into Basic Factors
• Determine the Plot at the Low Frequency
• Add First and Second Order Terms
• Verify the results in Matlab

• Determine an asymptotic magnitude plot by approximating the actual gain plot by a sequence of straight lines with slopes of \(20N\).
• From the low frequency plot, determine the number of integrator \(1/s^N\) and the constant term \(K\).
• Determine the corner frequency from the intersection of two consecutive lines.
• From the change of slopes of the consecutive lines, determine the associated basic factor.
• Determine the damping ratio for second order zeros and poles.
• The phase plot can be used to verify your results.

A feedback system is stable if and only if \(N=-P\), i.e. the number of the counterclockwise encirclements of \(–1\) point by the Nyquist plot in the \(G\)-plane is equal to the number of the unstable poles of the open-loop transfer function.

• We need to find 4 types of point from the Bode plots:
  • \(\omega = 0\);
  • Real intersection: Phase = \(180N^\circ\);
  • Imaginary intersection: Phase = \(180N+90^\circ\);
  • \(\omega = +\infty\);
• We can also deduce the trend of the plot around those points:
  • If the phase is decreasing, the plot goes clockwise
  • If the phase is increasing, the plot goes counterclockwise
• Plot those points on the \(G\)-plane and draw a smooth line to connect them.

• Segment 2 is the origin point for strictly proper function. It is a constant for proper function.
• Segment 3 is the mirror reflection of segment 1 around the real axis.

• Define \(\omega_{\phi}\) to be the smallest frequency, where the phase of the open-loop transfer function is \(-180^\circ\).
• \(\omega_{\phi}\) is called the phase crossover frequency.
• For most minimum phase system, the gain margin for the system is given by \[K_g = \frac{1}{|G(j\omega_\phi)H(\omega_\phi)|}.\]
• In terms of decibles,\[K_g(dB) = -20\log|G(j\omega_\phi)H(\omega_\phi)|.\]

• Define \(\omega_{g}\) to be the smallest frequency, where the magnitude of the open-loop transfer function is \(1\) or \(0dB\).
• \(\omega_{g}\) is called the gain crossover frequency.
• For most minimum phase system, the phase margin for the system is given by \[\phi_m = 180^\circ+\angle G(j\omega_g)H(\omega_g).\]

• In the low frequency, the gain should be high to have good steady state accuracy.
• Around the gain crossover frequency, the slope of the magnitude plot should be about \(-20\) and extend over a sufficient range of frequency
• In the high frequency, the gain must be small in order to have good noise attenuation and robustness against model uncertainty.

• The transfer function of a lead compensator is \[C(s) = K\frac{Ts+1}{\alpha Ts+1},\,0<\alpha<1.\]
• The transfer function of a lag compensator is \[C(s) = K\frac{Ts+1}{\beta Ts+1},\,\beta > 1.\]

• Determine the gain \(K\) to satisfy the steady state error requirement.
• 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 \[\alpha = \frac{1-\sin \phi}{1+\sin \phi}.\]

• Determine the compensator gain \(K\) to meet the steady state error requirement.
• Draw the Bode plots of \(KG(s)\).
• From the Bode plots, find the frequency \(\omega_g\) at which the phase of \(KG(s)\) is \[\angle KG(\omega_g) = PM -180^\circ + 5^\circ\sim 10^\circ.\]
• Calculate \(\beta\) to make \(\omega_g\) the gain crossover frequency, \[20\log\beta = 20\log K + 20\log |G(j\omega_g)|.\]
• Choose \(T\) to be much greater than \(1/\omega_g\), for example, \(T = 10/\omega_g\).
• Verify the results using matlab.

The standard form of PID controller is \[C(s) = K_p\left(1+\frac{1}{T_is}+T_ds\right) = K_p+\frac{K_I}{s}+K_Ds.\]

• Find the maximum slop tangent line
• Read \(K\), \(T\) and \(L\) from the plot
• Compute \(K_p\), \(T_i\) and \(T_d\) from the table

• The critical gain \(K_{cr}\) is the gain margin
• The critical period \(P_{cr} = 2\pi/\omega_\phi\)
• Compute \(K_p\), \(T_i\) and \(T_d\) from the table