Frequency response

Frequency response is the steady-state output of a linear time-invariant (LTI) system when driven by a sinusoidal input, expressed as a function of the input frequency. It is the bridge between the time-domain description of dynamical systems — differential equations and state-space models — and their frequency-domain characterization — transfer functions and spectral analysis. For an LTI system with transfer function \(G(s)\), the frequency response is \(G(j\omega)\), a complex-valued function of frequency whose magnitude gives the gain and whose argument gives the phase shift.

The power of frequency response lies in its experimental accessibility. Where a state-space model requires knowledge of internal states and parameters, a frequency response can be measured directly: apply a sinusoid at frequency \(\omega\), wait for transients to decay, and measure the output amplitude and phase. Repeat across frequencies, and the result is a complete input-output description of the system's linear dynamics.

Bode and Nyquist Representations

The frequency response is most commonly visualized through two complementary representations. The Bode plot displays magnitude (in decibels) and phase (in degrees) as functions of frequency on logarithmic axes. The logarithmic scale reveals structure across decades of frequency, and the decibel representation converts multiplicative gain into additive magnitude — a visual convenience that mirrors the algebraic convenience of working with transfer functions in the Laplace domain.

The Nyquist plot displays the frequency response as a parametric curve in the complex plane, tracing \(G(j\omega)\) as \(\omega\) varies from zero to infinity. The Nyquist plot is the geometric foundation of the Nyquist stability criterion, which determines closed-loop stability from the open-loop frequency response without computing closed-loop poles. The criterion states that the number of clockwise encirclements of the point \(-1\) by the Nyquist plot equals the number of unstable closed-loop poles minus the number of unstable open-loop poles — a remarkable result that extracts stability information from a single curve.

Bandwidth and Roll-Off

Two parameters extracted from the frequency response dominate control design. The bandwidth is the frequency range over which the system tracks its input with acceptable fidelity — typically defined as the frequency where the magnitude drops to \(-3\) dB (gain of \(1/\sqrt{2}\)). Higher bandwidth means faster response to commands and better disturbance rejection, but also greater sensitivity to high-frequency noise and unmodeled dynamics.

The roll-off rate is the slope of the magnitude curve at high frequencies, typically measured in dB per decade. A steep roll-off attenuates sensor noise and prevents amplification of high-frequency disturbances, but it also limits bandwidth and introduces phase lag that can destabilize the feedback loop. The trade-off between bandwidth and roll-off is one of the fundamental tensions of control design.

Frequency Response in System Identification and Design

In system identification, frequency response data is often the starting point for model building. Nonparametric methods estimate the frequency response directly from input-output data using spectral analysis — the ratio of output power spectrum to input power spectrum at each frequency. Parametric methods fit a rational transfer function to the estimated frequency response, yielding a compact model suitable for controller design.

In controller design, frequency response methods are the lingua franca of practicing engineers. The loop-shaping methodology — shaping the open-loop transfer function to achieve desired sensitivity and complementary sensitivity profiles — is conducted entirely in the frequency domain. The PID controller can be tuned by examining the frequency response of the plant: the proportional gain sets the crossover frequency where the loop gain is unity, the integral term boosts low-frequency gain for disturbance rejection, and the derivative term adds phase lead for stability margin.

The Limitations of Frequency-Domain Thinking

Frequency response methods apply strictly to linear time-invariant systems. For nonlinear systems, the frequency response is not uniquely defined — a sinusoidal input may produce a non-sinusoidal output with harmonics. For time-varying systems, the frequency response changes with time. The extensions — describing functions for nonlinearities, time-frequency analysis for nonstationary systems — are approximations that recover some of the power of the LTI framework at the cost of exactness.

Even within the LTI domain, frequency response has blind spots. It describes steady-state behavior but says nothing about transients — the step response, the settling time, the overshoot. These time-domain properties are related to the frequency response through Fourier and Laplace transforms, but the relationship is not always intuitive. A system with smooth frequency response can exhibit resonant peaks in its step response, and vice versa.

Frequency response is the Fourier transform's gift to engineering: it replaces differential equations with curves, and curves can be drawn, measured, and reasoned about intuitively. But the intuition is seductive. The Bode plot is not the system; it is a projection of the system onto the frequency axis. Projections lose information. The engineer who designs only in the frequency domain forgets that real systems start from rest, encounter disturbances at unpredictable times, and operate in a world where transients matter as much as steady states. Frequency response is indispensable. It is also incomplete.