Himanish Agarwal ❚
Notes on software, systems, and the science of making machines learn

Control Systems

[Sat Jan 14, 2023]
— Himanish

Mathematical Models of Systems #

Laplace Transform #

Time Frequency
\(tf(t)\) \(-F’(s)\)
\(t^nf(t)\) \((-1)^nF^{(n)}(s)\)
\(f(t-a)u(t-a)\) \(e^{-as}F(s)\)
\(f’(t)\) \(sF(s)-f(0^-)\)
\(\frac{1}{t}f(t)\) \(\int_s^{\infty}F(\sigma)d\sigma\)
\(e^{at}f(t)\) \(F(s-a)\)
\(u(t-a)\) \(e^{-as}/s\)
\(\delta(t)\) 1
\(t^n\) \(\frac{n!}{s^{n+1}}\)
\(e^{-at}\) \(\frac{1}{s+a}\)
\(\cos(\omega_0t) u(t)\) \(\frac{s}{s^2+\omega_0^2}\)
\(\sin(\omega_0t) u(t)\) \(\frac{\omega_0}{s^2+\omega_0^2}\)
  • The Laplace variable s can be considered as the differential operator, i.e.

\[s \equiv \frac{d}{dt}\] \[\frac{1}{s} = \int_{0^-}^tdt\]

  • Used to solve differential equations, which arise when describing system behaviour.
  • Coefficient A of partial fraction decomposition of \(Y(s)\), corresponding to pole \(p_1\) is given by

\[A = \lim_{s\to p_1}(s-p_1)Y(s)\]

Transfer Function #

\[G(s) = \frac{Y(s)}{R(s)}\]

Block Diagram #

Time Domain Analysis #

  • DC Gain, \(K = G(s=0)\)

Response of a System #

  • Time Response is for aperiodic inputs:
    • Impulse
    • Step
    • Ramp
  • Whereas frequency response is for periodic inputs (e.g. sinusoids)
Input R(s)
Impulse 1
Step \(\frac{1}{s}\)
Ramp \(\frac{1}{s^2}\)
Parabola (\(t^2/2\)) \(\frac{1}{s^3}\)
  • To determine the steady state of response \(y(t)\), we may use the final value theorem: \[ y(t\to\infty) = \lim_{s\to0}sY(s)\]

Feedback Control System #

  • A closed-loop system uses a measurement of the output signal and a comparison with the desired output to generate an error signal that is used by the controller to adjust the actuator.
  • Error: \(E(s) = Y_{des}(s) - Y(s) = R(s) - Y(s)\)
  • For a unity feedback system, \[E(s) = \frac{R(s)}{1+G(s)}\]
    • So, steady state error: \[e_{ss}=\lim_{s \to 0} \frac{sR(s)}{1+G(s)}\]

First Order Systems #

\[G(s) = \frac{K}{\tau s + 1}\]

  • Settling time = \(4\tau\)

Second Order Systems #

\[G(s) = \frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}\]

Unit Step Response #

\[s_{1,2} = -\xi\omega_n\pm j\omega_d\] \(\omega_d = \omega_n\sqrt{1-\xi^2}\) \[\cos \theta = \xi, \ \sin\theta = \sqrt{1-\xi^2} \]

  • Peak time \(t_p = \pi/\omega_d\) at which
    • Peak overshoot: \(M_p = \exp(-\pi\xi/\sqrt{1-\xi^2})\cdot 100\%\)
  • Rise time \[t_r = \frac{\pi-\theta}{\omega_d}\]
  • Settling time

\[t_s = \begin{cases} \frac{3}{\xi\omega_n} & 5\% \text{ tolerance} \\ \frac{4}{\xi\omega_n} & 2\% \text{ tolerance} \end{cases} \]

Stability #

  • A stable system is a dynamic system with a bounded response to a bounded input.
  • Impulse response \(g(t)\) must be absolutely integrable.
  • A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.
    • A system is stable if all the poles of the transfer function are in the left- hand s-plane
    • A system is not stable if not all the roots are in the left-hand plane.
  • Marginally stable if simple roots on imaginary axis, with all other roots in left-hand plane.

Routh Criterion #

  • For a stable system, there should be no changes in sign in the first column.
    • The number of roots of the characteristic polynomial \(q(s)\) with positive real parts is equal to the number of sign changes in the first column of the array.

Root Locus #

  • Centroid and angles of asymptote have \((p-z)\) in the denominator

Angle of Departure #

\[\theta_d = 180^{\circ} - (\sum \theta_p - \sum \theta_z)\]

  • \(\theta_p\) is the angle made by the pole/zero whose angle of departure is to be found w.r.t. other poles, i.e. angle made by a vector originating from other poles and ending at the pole in question with real axis.
  • Angle of departure need not be calculated for poles/zeros on real axis as graph of root locus should be symmetric about real axis

Controller Design #

  • The control effort \(U(s)\) is the output given by the compensator i.e. \(U(s) = G_c(s)E(s)\)
  • In general, a PID controller has a transfer function \[G_c(s) = K_p + \frac{K_I}{s}+ K_Ds\]
    • When \(K_D = 0\), we have a proportional plus integral (PI) controller
    • When \(K_I = 0\), we have a proportional plus derivative (PD) controller
    • The PID controller can also be viewed as a cascade of the PI and the PD controllers.
  • We cannot change settling time using a proportional compensator alone
    • We can control \(\zeta\) or \(\omega_n\) but not both parameters

\[\cos\theta = \zeta \] \[\alpha = \zeta\omega_n\]

Frequency Response #

Polar Plot #

  1. Put \(s = j \omega\)
  2. Find magnitude and phase of \(G(j\omega)\)
  3. Find points (mag, phase) at which \(\omega = 0\) and \(\omega = \infty\)
  4. Separate Real and Imaginary parts by multiplying with conjugate
  5. Points of intersection with Real axis and Imaginary axis
  6. Plot these four points (intersections with axes and 0 and \(\infty\)) and connect them

Gain Margin and Phase Crossover #

\[\omega_{pc} \ @ \ \operatorname{Im}{G(j\omega)} = 0 \] \[ GM = \frac{1}{|G(j\omega_{pc})|}\]

Phase Margin and Gain Crossover #

  • Find \(\omega = \omega_{gc}\) for which

\[ |G(j\omega)| = 1\]

  • Then \[PM = 180^{\circ} + \phi\] where \[\phi = \angle{G(j\omega_{gc})}\]

Stability in Frequency Domain #

Contours #

By convention, the area within a contour to the right of the traversal of the contour is considered to be the area enclosed by the contour. Therefore, we will assume clockwise traversal of a contour to be positive and the area enclosed within the contour to be on the right.

Nyquist Plot #

  1. Polar Plot
  2. Inverse Polar Plot
  3. Nyquist Contour

Nyquist Criterion #

  • A transfer function is called a minimum phase transfer function if all its zeros lie in the left-hand s-plane. It is called a nonminimum phase transfer function if it has zeros in the right-hand s-plane.
  • A feedback system is stable if and only if the contour in the L(s)-plane does not encircle the \((-1, 0)\) point when the number of poles in the right hand plane is zero.
    • \(\omega_{pc} > \omega_{gc}\): stable system
    • \(\omega_{pc} < \omega_{gc}\): unstable system
    • \(\omega_{pc} = \omega_{gc}\): critically stable system
  • If the number of poles in the RHP is not zero, the number of counterclockwise encirclements
  • The \((-1, 0)\) point on Nyquist plot is same as the \(0 dB, -180^\circ\) point on Bode plot.
  • The gain margin is the increase in the system gain when phase = −180° that will result in a marginally stable system with intersection of the \(−1 + j0\) point on the Nyquist plot.