Transfer functions are powerful tools for analyzing linear time-invariant systems. They simplify complex differential equations into algebraic expressions, making it easier to understand system behavior. This chapter explores how transfer functions relate to Laplace transforms and system modeling.

Poles and zeros of transfer functions provide crucial insights into system stability and response characteristics. By examining their locations in the complex plane, we can predict system behavior, design controllers, and optimize performance. This knowledge is essential for engineers working with dynamic systems.

Transfer functions for LTI systems

Laplace transform and transfer function definition

• The Laplace transform converts a time-domain signal into a complex frequency-domain representation, denoted by the variable "s"
• The transfer function, H(s), is the ratio of the Laplace transform of the output signal to the Laplace transform of the input signal, assuming zero initial conditions
• The transfer function is a compact representation of the system's input-output relationship and provides insight into the system's frequency response and stability

Deriving the transfer function

• Express the system's differential equation in the time domain
• Apply the Laplace transform to both sides of the equation
• Algebraically manipulate the equation to isolate the output signal in terms of the input signal, resulting in the transfer function H(s) = Y(s) / X(s)
  1. Take the Laplace transform of the differential equation
  2. Substitute the Laplace transforms of the input and output signals
  3. Rearrange the equation to express the output in terms of the input
  4. Factor out the input signal to obtain the transfer function H(s)

Poles and Zeros: System Behavior

Definition and interpretation of poles and zeros

• Poles of a transfer function are the values of "s" that cause the denominator to equal zero, while zeros are the values of "s" that cause the numerator to equal zero
• The locations of poles and zeros in the complex plane determine the stability and transient response of the system
  • Poles in the left half-plane (negative real parts) indicate a stable system
  • Poles in the right half-plane indicate an unstable system
  • Poles closer to the imaginary axis (jω-axis) result in slower transient responses and longer settling times
  • Poles further from the imaginary axis lead to faster transient responses and shorter settling times

Effects of pole and zero locations on system response

• Zeros in the left half-plane can cancel out the effect of nearby poles
• Zeros in the right half-plane can cause non-minimum phase behavior (undershoot or overshoot)
• The multiplicity of poles and zeros (repeated poles or zeros) affects the system's response
  • Higher multiplicities lead to more pronounced effects on the transient behavior
  • Example: A double pole at s = -1 results in a slower response compared to a single pole at s = -1

First and Second-Order System Response

First-order systems

• Transfer function: H(s) = K / (τs + 1), where K is the steady-state gain and τ is the time constant
• Step response: Characterized by an exponential rise or decay
  • Time constant τ determines the speed of the response (time to reach 63.2% of the final value)
  • Example: A first-order system with K = 2 and τ = 0.5 has a transfer function H(s) = 2 / (0.5s + 1)

Second-order systems

• Transfer function: H(s) = ω_n^2 / (s^2 + 2ζω_n s + ω_n^2), where ω_n is the natural frequency and ζ is the damping ratio
• Step response depends on the damping ratio ζ:
  • Underdamped (0 < ζ < 1): Oscillatory behavior with overshoot and settling time dependent on ζ and ω_n
  • Critically damped (ζ = 1): Fastest response without overshoot, characterized by a single exponential rise
  • Overdamped (ζ > 1): Slower response without overshoot, characterized by a double exponential rise
• Natural frequency ω_n determines the speed of oscillation (underdamped) or the speed of response (critically damped and overdamped)
• Example: A second-order system with ω_n = 2 and ζ = 0.5 has a transfer function H(s) = 4 / (s^2 + 2s + 4)

System Stability: Pole Location

Stability criteria based on pole locations

• A system is stable if and only if all poles lie in the left half of the complex plane (negative real parts)
• Poles on the imaginary axis (jω-axis) indicate marginal stability
  • The system's response neither decays nor grows exponentially over time
  • May exhibit sustained oscillations
• Poles in the right half-plane indicate an unstable system
  • The response grows exponentially without bound for any bounded input
• The location of zeros does not directly affect stability but can influence the transient response and steady-state behavior

Determining stability using the Routh-Hurwitz criterion

• The Routh-Hurwitz criterion determines the stability of a system without explicitly solving for pole locations
• Analyze the coefficients of the characteristic equation (the denominator of the transfer function)
• Construct the Routh array using the coefficients of the characteristic equation
• Count the number of sign changes in the first column of the Routh array
  • The number of sign changes equals the number of poles in the right half-plane
  • No sign changes indicate a stable system
  • Example: For a characteristic equation s^3 + 2s^2 + 3s + 4 = 0, the Routh array shows no sign changes, indicating a stable system