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.

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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) = 10(s + 4) / [ s(s² + 4s + 5) ]
and a controller
K(s) = (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.

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System Construction

• Open-loop transfer function
  L(s) = K(s)G(s)

• Closed-loop transfer function
  T(s) = y / r = L(s) / (1 + L(s))

• Sensitivity function
  S(s) = 1 / (1 + L(s))

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

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Analysis Performed

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Closed-Loop Properties

• Closed-loop transfer function 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

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Sensitivity Function

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

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Time-Domain Responses

• Unit step response of T(s) plotted over 0–10 s
• Axes and limits chosen to match the practical specification
• Sinusoidal response for
  r(t) = sin(ωt), with ω = 1.7 rad/s
• Direct comparison of input and output using lsim

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Frequency-Domain Phase Interpretation

• Evaluation of T(jω) at ω = 1.7 rad/s
• Extraction of magnitude and phase
• Conversion of phase lag into an equivalent time delay
  t_d = −φ / ω
• Interpretation of the sinusoidal steady-state response as
  y(t) ≈ |T(jω)| sin(ω(t − t_d))

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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 in polynomial form
• Poles and zeros of T(s) and S(s)
• DC gain of the closed-loop system
• Phase, magnitude, and time-delay estimates
• Characteristic equation verification using both:
  • Closed-loop denominator
  • poly(poles(T))

This mirrors the format expected in a written coursework submission.

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Plot Outputs

The script automatically generates:

• Unit step response of the closed-loop system
• Sinusoidal input/output comparison plot
• Clearly labelled figures corresponding to the practical questions

All plots are produced programmatically and require no manual interaction.

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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.