ENGiVAULTLuqman Ismat

Control Systems

Comprehensive guide to control systems theory, analysis, and design techniques.

Transfer Functions

A transfer function is a mathematical representation of the relationship between the input and output of a linear time-invariant (LTI) system:

G(s) = Y(s) / X(s)

Where:

  • G(s) = transfer function
  • Y(s) = Laplace transform of the output
  • X(s) = Laplace transform of the input
  • s = complex frequency variable

For a general nth-order system, the transfer function has the form:

G(s) = (b_m s^m + b_(m-1) s^(m-1) + ... + b_1 s + b_0) / (a_n s^n + a_(n-1) s^(n-1) + ... + a_1 s + a_0)

Where m ≤ n for a proper system.

Stability Analysis

Stability is a fundamental property of control systems. A system is stable if its output remains bounded for any bounded input.

1. Routh-Hurwitz Criterion:

A system is stable if all poles of its transfer function have negative real parts. The Routh-Hurwitz criterion provides a method to determine if all roots of a polynomial have negative real parts without actually finding the roots.

2. Nyquist Stability Criterion:

For a closed-loop system with open-loop transfer function G(s)H(s), the Nyquist stability criterion states that the system is stable if and only if:

  • The number of clockwise encirclements of the (-1, 0) point by the Nyquist plot of G(s)H(s) equals the number of poles of G(s)H(s) with positive real parts.

3. Bode Stability Criterion:

For a minimum-phase system, the closed-loop system is stable if the phase margin and gain margin are positive.

  • Phase Margin: The additional phase lag required to reach -180° at the gain crossover frequency.
  • Gain Margin: The factor by which the gain can be increased before the system becomes unstable.

PID Control

Proportional-Integral-Derivative (PID) control is a widely used control algorithm. The control signal is:

u(t) = K_p × e(t) + K_i × ∫e(t)dt + K_d × de(t)/dt

Where:

  • u(t) = control signal
  • e(t) = error signal (difference between setpoint and measured output)
  • K_p = proportional gain
  • K_i = integral gain
  • K_d = derivative gain

The transfer function of a PID controller is:

C(s) = K_p + K_i/s + K_d×s

Effects of each term:

  • Proportional (P): Reduces rise time, increases overshoot, reduces steady-state error.
  • Integral (I): Eliminates steady-state error, may increase overshoot and settling time.
  • Derivative (D): Reduces overshoot and settling time, improves stability.

PID tuning methods include:

  • Ziegler-Nichols method
  • Cohen-Coon method
  • Relay method
  • Internal Model Control (IMC)