Control Systems Engineering and PID Controller Design

Posted on Feb 27, 2026 in Design and Engineering

Control Systems, Signals, and Loop Architectures

A system transforms an input u(t) into an output y(t). It can be classified as continuous or discrete, linear or nonlinear, and time-invariant or time-variant.

• Signals: Can be continuous or discrete, periodic or aperiodic, deterministic or random, and energy or power signals.
• Open loop: Operates without feedback; the control action does not depend on the output. This is simple but sensitive to disturbances and parameter changes.
• Closed loop: Utilizes feedback by comparing a reference r with the output y to determine the error e = r − y. This provides better accuracy and disturbance rejection.

Main elements: Reference, comparator (error), controller, actuator, plant/process, sensor, feedback path, and disturbance/noise.

Requirements: Stability, small steady-state error, fast response (rise/settling time), low overshoot, good disturbance rejection, and robustness.

Linear Models and System Response Analysis

A Linear Time-Invariant (LTI) system obeys the principle of superposition and possesses constant parameters. The input-output model can be expressed as a differential equation: a_n y⁽ⁿ⁾ + … + a₀ y = b_m u^(m) + … + b₀ u.

• Test signals: Impulse δ(t), step 1(t), ramp, and sine; these are used to identify system behavior.
• Impulse response h(t): The output in response to δ(t); it fully describes an LTI system.
• Step response: The output in response to a unit step; it illustrates speed, overshoot, and steady-state behavior.
• Transfer function: Defined as G(s) = Y(s) / U(s) under zero initial conditions.

The transition between the time domain and the operator domain (s-domain) simplifies differential equations into algebraic equations.

Fourier and Laplace Transform Applications

The Fourier series represents periodic signals as a sum of harmonics (sines and cosines), while the Fourier integral/transform represents aperiodic signals as a continuous spectrum.

The Laplace transform generalizes Fourier analysis and is highly effective for analyzing systems, Ordinary Differential Equations (ODEs), transients, and stability: F(s) = ∫ f(t)e^−st dt.

• Key theorems: Linearity, time shift, frequency shift, differentiation, integration, and convolution.
• Inverse transforms: Used to return the signal to the time domain (Inverse Fourier / Inverse Laplace).

Laplace transforms are essential for deriving transfer functions, analyzing poles and zeros, and solving control loops.

Dynamic Elements in Time and S-Domain

• Zero-order element: Defined by gain K, where G(s) = K. The step response jumps immediately to K.
• First-order element: G(s) = K / (Ts + 1). The step response rises exponentially, where T determines the speed.
• Second-order element: G(s) = Kω_n² / (s² + 2ζω_ns + ω_n²). Here, ζ controls overshoot/oscillation and ω_n controls speed.
• Ideal integrator: G(s) = K / s. A step input results in a ramp output; it removes steady-state error but may cause stability issues.
• First-order integrator (real): K / (s(Ts + 1)), which is more realistic than the ideal version.
• Ideal derivative: G(s) = Ks. This boosts high frequencies but is sensitive to noise.
• Real derivative: Ks / (Ts + 1), which limits high-frequency amplification.

Frequency Response and Stability Plots

Frequency response illustrates the output relative to a sinusoidal input by evaluating the transfer function at s = jω → G(jω). It is represented by magnitude |G(jω)| and phase ∠G(jω).

• Bode plot: Displays magnitude (dB) and phase versus log ω; it is convenient for system design.
• Nyquist plot: A complex curve of G(jω) in the complex plane used to determine stability.

Typical shapes:

• Gain K: Flat magnitude, 0° phase.
• 1st order: −20 dB/dec after 1/T, phase approaches −90°.
• Integrator 1/s: Constant −20 dB/dec slope, −90° phase.
• Derivative s: Constant +20 dB/dec slope, +90° phase (real derivatives flatten at high frequencies).

Stability Definitions and Performance Margins

A stable system returns to equilibrium after a disturbance. For Bounded-Input Bounded-Output (BIBO) stability, every bounded input must produce a bounded output. In continuous systems, stability depends on the closed-loop poles, which must reside in the left half-plane.

• Nyquist criterion: Checks the encirclement of the −1 point by the open-loop L(jω) to infer closed-loop stability.
• Bode stability: Utilizes gain margin and phase margin.
• Gain margin: The amount the gain can increase before instability occurs (measured at −180° phase).
• Phase margin: The additional phase lag allowed before instability (measured at 0 dB crossover).

Higher margins indicate a more robust system, though often at the cost of a slower response.

Closed-Loop Transfer Functions and PID Roles

In a closed loop with controller C(s) and plant G(s), the open loop is L(s) = C(s)G(s).

• Closed-loop transfer: T(s) = L(s) / (1 + L(s)).
• Error transfer: E(s) = 1 / (1 + L(s)).
• Type number: The number of integrators (poles at 0) in the open loop. A higher type number results in smaller steady-state errors for step and ramp inputs.

Role of PID terms:

• P (Proportional): Increases speed but high values cause overshoot or instability.
• I (Integral): Eliminates steady-state error but slows the system and can increase overshoot.
• D (Derivative): Adds damping and reduces overshoot but amplifies high-frequency noise.

PID Controller Tuning and Parameter Effects

• P: C(s) = K_p. Simple speed increase, but steady-state error remains.
• PI: C(s) = K_p(1 + 1 / (T_i s)). Removes steady-state error but may increase overshoot.
• PD: C(s) = K_p(1 + T_d s). Improves damping and reduces overshoot; however, it is noise-sensitive.
• PID: C(s) = K_p(1 + 1 / (T_i s) + T_d s). Combines all benefits: fast response, zero steady-state error, and better damping.

Tuning methods: Ziegler-Nichols, trial-and-error, frequency-domain tuning (phase margin), and relay auto-tuning.

Sampling Theory and the Z-Transform

Sampling converts a continuous signal into a discrete sequence: x[k] = x(kT). Physical sampling uses a sample-and-hold mechanism.

• Shannon Theorem: The sampling frequency must be f_s > 2 * f_max to avoid aliasing (the Nyquist rate).
• Z-transform: X(z) = Σ x[k] z^−k. It is the discrete equivalent of the Laplace transform.
• Properties: Linearity and time shifting (multiplication by z⁻ⁿ).
• Stability: For discrete systems, poles must be located inside the unit circle in the z-plane.

Discrete PID Implementation and Digital Control

To implement a PID controller digitally, the integral and derivative terms must be discretized (e.g., using Euler or Tustin methods).

• Position algorithm: Outputs u[k] directly from error terms; it can be sensitive to integral windup.
• Velocity (incremental) algorithm: Computes the change Δu[k]. This is numerically more stable and suitable for actuators.
• Digital closed loop: Consists of a sampler, digital controller, D/A converter, hold (usually Zero-Order Hold or ZOH), and the plant.

The sampling time T significantly affects performance: a T that is too large leads to instability, while a T that is too small increases the computational load and noise sensitivity.