4.1 Frequency response analysis

Frequency response analysis is a powerful tool in control systems theory. It examines how systems respond to sinusoidal inputs across different frequencies, providing insights into stability and performance.

This topic covers key concepts like Bode plots, transfer functions, and system characteristics. Understanding frequency response helps engineers design filters, analyze stability, and create effective controllers for various applications.

Frequency response definition

• Frequency response describes how the steady-state output of a system changes in both magnitude and phase as the frequency of a sinusoidal input varies
• Provides insights into system behavior, stability, and performance across a range of input frequencies
• Foundational concept in control systems theory for analyzing and designing feedback systems

Sinusoidal input vs output

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• Consider applying a sinusoidal input signal $u(t) = A \sin(\omega t)$ to a linear time-invariant (LTI) system
• The steady-state output will also be a sinusoid $y(t) = B \sin(\omega t + \phi)$ at the same frequency $\omega$ but with a different amplitude $B$ and phase shift $\phi$
• The frequency response captures how the system modifies the amplitude and phase of the input signal

Magnitude vs phase

• Magnitude response $M(\omega)$ represents the ratio of the output amplitude to the input amplitude as a function of frequency: $M(\omega) = \frac{B}{A}$
• Phase response $\phi(\omega)$ represents the phase shift between the output and input sinusoids as a function of frequency
• Together, the magnitude and phase responses provide a complete characterization of the system's frequency-domain behavior

Bode plots

• Bode plots are graphical representations of a system's frequency response, consisting of two separate plots: magnitude and phase
• Named after Hendrik Wade Bode, who introduced the concept in the 1930s
• Provide a intuitive way to visualize and analyze the frequency-domain characteristics of a system

Magnitude plot

• The magnitude plot displays the magnitude response $M(\omega)$ in decibels (dB) versus frequency on a logarithmic scale
• Magnitude in dB is calculated as $20 \log_{10}(M(\omega))$, which helps to compress the vertical scale and emphasize relative changes
• Logarithmic frequency scale allows for a wide range of frequencies to be displayed and highlights system behavior at both low and high frequencies

Phase plot

• The phase plot displays the phase response $\phi(\omega)$ in degrees versus frequency on a logarithmic scale
• Phase shifts are typically wrapped to the range $[-180^\circ, 180^\circ]$ to avoid discontinuities in the plot
• The phase plot provides information about the relative timing of the output with respect to the input at different frequencies

Logarithmic frequency scale

• Bode plots use a logarithmic frequency scale, typically expressed in radians per second or hertz
• Logarithmic scale compresses the frequency axis, making it easier to visualize system behavior over a wide range of frequencies
• Decades (factors of 10) and octaves (factors of 2) are often used to divide the frequency axis, facilitating the identification of key frequency regions

Frequency response of LTI systems

• The frequency response of an LTI system is completely determined by its transfer function $G(s)$
• The transfer function is a mathematical representation of the system's input-output relationship in the complex frequency domain
• Evaluating the transfer function along the imaginary axis ($s = j\omega$) yields the frequency response $G(j\omega)$

Transfer function representation

• The transfer function $G(s)$ is defined as the ratio of the Laplace transform of the output $Y(s)$ to the Laplace transform of the input $U(s)$, assuming zero initial conditions: $G(s) = \frac{Y(s)}{U(s)}$
• For a rational transfer function, $G(s)$ can be expressed as a ratio of polynomials in $s$: $G(s) = \frac{N(s)}{D(s)}$
• The roots of the numerator $N(s)$ are called zeros, while the roots of the denominator $D(s)$ are called poles

Poles vs zeros

• Poles are the values of $s$ that cause the transfer function to become infinite, corresponding to the roots of the denominator polynomial $D(s)$
• Zeros are the values of $s$ that cause the transfer function to become zero, corresponding to the roots of the numerator polynomial $N(s)$
• The locations of poles and zeros in the complex plane determine the system's frequency response and stability characteristics

Stability implications

• A system is stable if all of its poles lie in the left half of the complex plane (i.e., have negative real parts)
• Poles on the imaginary axis indicate marginal stability, while poles in the right half-plane indicate instability
• The frequency response can provide insights into the stability margins of the system, such as gain and phase margins

Frequency domain specifications

• Frequency domain specifications define desired performance characteristics of a system in terms of its frequency response
• These specifications are often used in the design of filters, controllers, and other systems to ensure desired behavior across a range of frequencies
• Common frequency domain specifications include bandwidth, resonant peak, and gain and phase margins

Bandwidth

• Bandwidth is a measure of the frequency range over which a system can effectively operate or transmit signals
• For a low-pass system, bandwidth is typically defined as the frequency at which the magnitude response drops by 3 dB from its maximum value
• A larger bandwidth indicates that the system can respond to higher-frequency inputs without significant attenuation

Resonant peak

• The resonant peak is the maximum value of the magnitude response, often occurring at the system's natural frequency or resonant frequency
• A large resonant peak indicates a strong amplification of signals near the resonant frequency, which can lead to oscillations or instability
• The height and sharpness of the resonant peak are related to the system's damping ratio and quality factor

Gain vs phase margin

• Gain margin is the amount of additional gain that can be applied to the system before it becomes unstable, expressed in decibels
• Phase margin is the amount of additional phase lag that can be introduced to the system before it becomes unstable, expressed in degrees
• Larger gain and phase margins indicate a more robust system that can tolerate greater variations in gain and phase without becoming unstable

Frequency response of standard systems

• Standard systems, such as first-order, second-order, and higher-order systems, have characteristic frequency response patterns
• Understanding these patterns helps in the analysis and design of more complex systems by decomposing them into simpler, standard components
• The frequency response of standard systems can be derived from their transfer functions and represented using Bode plots

First-order systems

• First-order systems have a transfer function of the form $G(s) = \frac{K}{\tau s + 1}$, where $K$ is the DC gain and $\tau$ is the time constant
• The magnitude response is a low-pass filter, with a constant gain at low frequencies and a -20 dB/decade roll-off at high frequencies
• The phase response shows a maximum phase lag of 90° at high frequencies, with the phase lag increasing monotonically with frequency

Second-order systems

• Second-order systems have a transfer function of the form $G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$, where $\omega_n$ is the natural frequency and $\zeta$ is the damping ratio
• The magnitude response can exhibit a resonant peak near the natural frequency, depending on the damping ratio
• The phase response shows a maximum phase lag of 180°, with the phase lag increasing rapidly near the resonant frequency

Higher-order systems

• Higher-order systems have transfer functions with polynomials of degree three or more in the denominator
• The frequency response of higher-order systems can be complex, with multiple resonant peaks, zeros, and high-frequency roll-off
• Higher-order systems can often be approximated by a combination of first-order and second-order systems, facilitating their analysis and design

Experimental determination of frequency response

• In practice, the frequency response of a system can be determined experimentally by measuring the system's output for a range of input frequencies
• Several methods exist for experimentally determining the frequency response, including sinusoidal testing, correlation methods, and the use of spectrum analyzers
• Experimental methods are particularly useful when the system's mathematical model is unknown or difficult to obtain

Sinusoidal testing

• Sinusoidal testing involves applying a sinusoidal input signal to the system and measuring the steady-state output amplitude and phase
• The test is repeated for a range of frequencies, and the magnitude and phase responses are plotted as a function of frequency
• Automated test equipment, such as frequency response analyzers, can streamline the sinusoidal testing process

Correlation methods

• Correlation methods use broadband input signals, such as white noise or chirp signals, to excite the system over a range of frequencies simultaneously
• The input and output signals are cross-correlated to determine the system's impulse response, which can then be Fourier transformed to obtain the frequency response
• Correlation methods are faster than sinusoidal testing but may require more advanced signal processing techniques

Spectrum analyzers

• Spectrum analyzers are instruments that measure the frequency content of a signal by displaying its power spectral density
• To determine the frequency response, the input and output signals are simultaneously measured using a two-channel spectrum analyzer
• The ratio of the output spectrum to the input spectrum yields the system's frequency response, which can be displayed as a Bode plot

Frequency response applications

• The frequency response is a powerful tool with numerous applications in control systems, signal processing, and system design
• Some key applications include filter design, stability analysis, and controller design
• Understanding the frequency response of a system is essential for optimizing its performance and ensuring its stability

Filter design

• Filters are systems designed to selectively attenuate or amplify specific frequency components of a signal
• The desired frequency response of a filter (e.g., low-pass, high-pass, band-pass, or band-stop) can be achieved by appropriate placement of poles and zeros in the transfer function
• Bode plots are used to visualize the filter's frequency response and to verify that it meets the desired specifications

Stability analysis

• The frequency response can be used to assess the stability of a closed-loop system, such as a feedback control system
• Techniques like the Nyquist stability criterion and Bode plot analysis can determine the stability margins (gain and phase margins) of the system
• A system with sufficient stability margins is more robust to uncertainties and disturbances, ensuring reliable operation

Controller design

• Frequency response methods are used in the design of controllers, such as PID (Proportional-Integral-Derivative) controllers and lead-lag compensators
• Controllers are designed to shape the frequency response of the closed-loop system to achieve desired performance specifications, such as bandwidth, disturbance rejection, and robustness
• Bode plots are used to visualize the effect of the controller on the system's frequency response and to tune the controller parameters for optimal performance