🎛️Control Theory Unit 4 – Frequency-Domain Analysis & Design

Key Concepts

• Frequency-domain analysis involves analyzing the behavior of systems in terms of sinusoidal input signals and their corresponding steady-state outputs
• Transfer functions represent the input-output relationship of a linear time-invariant (LTI) system in the frequency domain using Laplace transforms
• Frequency response describes how a system responds to sinusoidal inputs of different frequencies (gain and phase shift)
• Stability determines whether a system's output remains bounded for bounded inputs and can be assessed using frequency-domain techniques
• Gain margin indicates the amount of gain increase that can be tolerated before a system becomes unstable (measured in decibels)
• Phase margin represents the additional phase lag that can be introduced before a system becomes unstable (measured in degrees)
• Controller design in the frequency domain aims to shape the system's frequency response to meet performance specifications and ensure stability
• Frequency-domain techniques provide insights into system behavior, stability, and robustness, complementing time-domain analysis

Mathematical Foundations

• Laplace transforms convert time-domain functions into frequency-domain representations, enabling the analysis of LTI systems
  • The Laplace transform of a function $f(t)$ is given by: $F(s) = \int_0^{\infty} f(t)e^{-st} dt$
• Transfer functions are obtained by taking the Laplace transform of the system's differential equation and assuming zero initial conditions
  • For a system with input $U(s)$ and output $Y(s)$, the transfer function is: $G(s) = \frac{Y(s)}{U(s)}$
• Complex numbers are used to represent sinusoidal signals in the frequency domain, with the real part corresponding to the cosine component and the imaginary part corresponding to the sine component
• Euler's formula relates complex exponentials to sinusoidal functions: $e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)$
• Frequency response is obtained by evaluating the transfer function at $s = j\omega$, where $\omega$ is the angular frequency in radians per second
• Poles and zeros of a transfer function determine its frequency response characteristics and stability properties
  • Poles are values of $s$ that make the denominator of the transfer function zero, while zeros are values of $s$ that make the numerator zero

Frequency Response Methods

• Frequency response methods analyze the steady-state behavior of a system subjected to sinusoidal inputs of varying frequencies
• Magnitude response represents the ratio of the output amplitude to the input amplitude as a function of frequency, often expressed in decibels (dB)
  • Magnitude in dB is calculated as: $20\log_{10}(|G(j\omega)|)$
• Phase response represents the phase shift between the output and input sinusoids as a function of frequency, expressed in degrees or radians
• Bandwidth is the range of frequencies over which the system's magnitude response remains within a specified tolerance (commonly -3 dB)
• Resonant frequency is the frequency at which the system exhibits maximum magnitude response, often associated with a sharp peak in the frequency response plot
• Corner frequency (or break frequency) is the frequency at which the magnitude response changes its slope, typically by -20 dB per decade for each pole or +20 dB per decade for each zero
• Frequency response can be experimentally determined by applying sinusoidal inputs to the system and measuring the steady-state output amplitude and phase shift

Bode Plots

• Bode plots provide a graphical representation of a system's frequency response, consisting of a magnitude plot (in dB) and a phase plot (in degrees) versus frequency (in logarithmic scale)
• Magnitude plot shows the magnitude response of the system, with the vertical axis representing the magnitude in dB and the horizontal axis representing the frequency in logarithmic scale
  • Logarithmic scale allows for a wide range of frequencies to be displayed and emphasizes the system's behavior at different frequency ranges
• Phase plot shows the phase response of the system, with the vertical axis representing the phase shift in degrees and the horizontal axis representing the frequency in logarithmic scale
• Asymptotic approximation is used to sketch Bode plots by approximating the magnitude and phase responses with straight-line segments
  • Each pole contributes a -20 dB/decade slope and a -90° phase shift, while each zero contributes a +20 dB/decade slope and a +90° phase shift
• Bode plots help identify important frequency-domain characteristics such as bandwidth, gain crossover frequency (0 dB crossing), phase crossover frequency (-180° crossing), gain margin, and phase margin
• Stability margins (gain margin and phase margin) can be determined from Bode plots by measuring the distances between the magnitude and phase plots at specific frequencies
• Bode plots are useful for designing controllers and compensators to shape the system's frequency response and ensure desired performance and stability

Nyquist Diagrams

• Nyquist diagrams are polar plots that represent the frequency response of a system by plotting the real and imaginary parts of the open-loop transfer function $G(j\omega)H(j\omega)$ as the frequency $\omega$ varies from $-\infty$ to $+\infty$
• Nyquist stability criterion determines the stability of a closed-loop system based on the number of encirclements of the -1 point by the Nyquist plot
  • A stable system has no encirclements of the -1 point when the open-loop transfer function has no poles in the right-half plane (RHP)
• Gain margin can be determined from the Nyquist plot by measuring the reciprocal of the distance from the -1 point to the intersection of the plot with the negative real axis
• Phase margin can be determined from the Nyquist plot by measuring the angle between the negative real axis and the line connecting the -1 point to the intersection of the plot with the unit circle
• Nyquist diagrams provide insights into the stability and robustness of the system, as well as the presence of unstable poles or non-minimum phase zeros
• Contours and indentations in the Nyquist plot indicate the presence of poles or zeros near the imaginary axis, which can affect the system's stability and transient response
• Nyquist plots are particularly useful for analyzing systems with time delays or non-rational transfer functions, as they can handle such systems more easily than Bode plots

Stability Analysis

• Stability analysis in the frequency domain assesses whether a system's output remains bounded for bounded inputs and determines the stability margins
• Routh-Hurwitz criterion determines the stability of a system based on the coefficients of its characteristic equation without explicitly solving for the roots
  • It constructs a Routh table and examines the signs of the entries in the first column to determine the number of roots in the right-half plane (RHP)
• Nyquist stability criterion determines the stability of a closed-loop system based on the number of encirclements of the -1 point by the Nyquist plot of the open-loop transfer function
• Gain margin indicates the amount of gain increase that can be tolerated before the system becomes unstable, determined from Bode plots or Nyquist diagrams
• Phase margin represents the additional phase lag that can be introduced before the system becomes unstable, determined from Bode plots or Nyquist diagrams
• Stability margins provide a measure of the system's robustness to variations in gain and phase, with larger margins indicating greater robustness
• Gain crossover frequency is the frequency at which the magnitude plot crosses the 0 dB line in a Bode plot, indicating a unity gain
• Phase crossover frequency is the frequency at which the phase plot crosses the -180° line in a Bode plot, indicating a potential instability if the magnitude is greater than unity

Controller Design Techniques

• Controller design in the frequency domain aims to shape the system's frequency response to meet performance specifications and ensure stability
• Lead compensators introduce a phase lead (positive phase shift) to improve the system's stability and transient response
  • They have a transfer function of the form: $G_c(s) = K\frac{s+z}{s+p}$, where $z < p$
• Lag compensators introduce a phase lag (negative phase shift) to improve the system's steady-state accuracy and disturbance rejection
  • They have a transfer function of the form: $G_c(s) = K\frac{s+z}{s+p}$, where $z > p$
• Lead-lag compensators combine the benefits of lead and lag compensators by introducing both phase lead and phase lag at different frequency ranges
  • They have a transfer function of the form: $G_c(s) = K\frac{(s+z_1)(s+z_2)}{(s+p_1)(s+p_2)}$, where $z_1 < p_1$ and $z_2 > p_2$
• PID controllers are widely used in industry and consist of proportional (P), integral (I), and derivative (D) terms to improve system performance
  • The transfer function of a PID controller is: $G_c(s) = K_p + \frac{K_i}{s} + K_ds$
• Frequency-domain design techniques, such as loop shaping, involve manipulating the open-loop transfer function to achieve desired closed-loop performance
• Gain and phase margins are used as design specifications to ensure adequate stability margins and robustness
• Sensitivity functions, such as the sensitivity $(S)$ and complementary sensitivity $(T)$, are used to analyze the system's response to disturbances, noise, and model uncertainties in the frequency domain

Practical Applications

• Frequency-domain analysis and design techniques are widely used in various engineering fields, including control systems, signal processing, and communication systems
• In control systems, frequency-domain methods are used to analyze the stability and performance of feedback control loops, such as in process control, automotive systems, and aerospace applications
• Power system stability analysis employs frequency-domain techniques to assess the stability of large-scale electrical power networks and design controllers to maintain stable operation
• Vibration analysis in mechanical systems uses frequency-domain methods to identify resonant frequencies, mode shapes, and damping characteristics, enabling the design of vibration isolation and control strategies
• Audio and speech processing applications rely on frequency-domain techniques for filtering, equalization, and spectral analysis, such as in audio equipment, hearing aids, and speech recognition systems
• Communication systems employ frequency-domain methods for modulation, demodulation, and channel equalization, as well as for analyzing the frequency spectrum of signals and designing filters
• Radar and sonar systems use frequency-domain techniques for target detection, ranging, and Doppler processing, exploiting the frequency content of the transmitted and received signals
• Biomedical engineering applications, such as EEG and ECG signal processing, utilize frequency-domain analysis to extract relevant features and diagnose abnormalities based on the frequency content of the signals