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Closed-Loop-Feedback-Analysis-Practical-2-MATLAB

MATLAB implementation of ENG3018 Practical 2, analysing a negative unity feedback control system using classical control techniques.
The script performs transfer-function construction, stability analysis, time-domain simulation, and frequency-domain phase interpretation using standard Control System Toolbox functions.

Overview

This repository contains a single, linear MATLAB script implementing Tasks (a)–(g) from ENG3018 Practical 2 exactly as specified in the practical sheet.

The system consists of a plant:

$$G(s)=\frac{10(s+4)}{s(s^2+4s+5)}$$

and a controller:

$$K(s)=\frac{s+1}{s+3}$$

connected in a negative unity feedback loop.

The code is structured as an academic submission rather than a modular software library, prioritising transparency, traceability, and analytical clarity.

System Construction

  • Open-loop transfer function:
$$L(s)=K(s)G(s)$$
  • Closed-loop transfer function:
$$T(s)=\frac{y(s)}{r(s)}=\frac{L(s)}{1+L(s)}$$
  • Sensitivity function:
$$S(s)=\frac{1}{1+L(s)}$$

Both toolbox-based construction (series, feedback) and explicit polynomial convolution (conv) are used, as required.

Analysis Performed

Closed-Loop Properties

  • Closed-loop transfer function $T(s)$ derived using feedback
  • Manual reconstruction of the closed-loop denominator using polynomial addition
  • Extraction of poles and zeros of $T(s)$
  • DC gain evaluation of the closed-loop system $T(0)$

Sensitivity Function

  • Construction of the sensitivity function $S(s)$
  • Pole and zero analysis of $S(s)$
  • Interpretation of disturbance rejection characteristics via $S(s)$

Time-Domain Responses

  • Unit step response of $T(s)$ plotted over $0,\text{s}$ to $10,\text{s}$
  • Axes and limits chosen to match the practical specification
  • Sinusoidal response for:
$$r(t)=\sin(\omega t),\quad \omega=1.7\ \text{rad/s}$$
  • Direct comparison of input $r(t)$ and output $y(t)$ using lsim

Frequency-Domain Phase Interpretation

  • Evaluation of $T(j\omega)$ at $\omega=1.7\ \text{rad/s}$
  • Extraction of magnitude $\lvert T(j\omega)\rvert$ and phase $\angle T(j\omega)$
  • Conversion of phase lag into an equivalent time delay:
$$t_d=-\frac{\phi}{\omega}$$
  • Interpretation of the sinusoidal steady-state response as:
$$y(t)\approx \lvert T(j\omega)\rvert\,\sin\big(\omega(t-t_d)\big)$$

Output and Reporting

A large PRINT BLOCK at the end of the script provides a clean summary of all required results, including:

  • Open-loop and closed-loop transfer functions ($L(s)$ and $T(s)$) in polynomial form
  • Poles and zeros of $T(s)$ and $S(s)$
  • DC gain of the closed-loop system $T(0)$
  • Phase, magnitude, and time-delay estimates at $\omega=1.7\ \text{rad/s}$
  • Characteristic equation verification using both:
    • Closed-loop denominator polynomial
    • poly(poles(T))

This mirrors the format expected in a written coursework submission.

Plot Outputs

The script automatically generates:

  • Unit step response of the closed-loop system $T(s)$
  • Sinusoidal input/output comparison plot for $r(t)=\sin(\omega t)$
  • Clearly labelled figures corresponding to the practical questions

All plots are produced programmatically and require no manual interaction.

Context

This repository is maintained as an academic reference and portfolio artefact.
It demonstrates closed-loop control analysis, feedback theory, and careful numerical reporting using MATLAB’s Control System Toolbox, rather than serving as a general-purpose control design package.