Closed-Loop Control System Modeling & Lead-Lag Compensator Optimization

Tools: MATLAB, Control System Toolbox, SISO Design Tool (sisotool), Simulink
Course: Systems & Control — German University in Cairo (GUC)

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Technical Overview

This repository implements the mathematical modeling, frequency-domain stabilization, and multi-objective Nelder-Mead Simplex (fminsearch) optimization of a Lead-Lag compensator for an unstable Type-2 fourth-order electromechanical plant:

$$G_p(s) = \frac{658.56}{s^2 (s^2 + 56s + 3136)}$$

Compensator Architecture

To simultaneously stabilize the double-integrator plant, shape the transient step/ramp/parabolic response, and satisfy strict steady-state parabolic tracking error requirements, a cascaded Lead-Lag controller $G_c(s)$ is synthesized:

$$G_c(s) = K_c \cdot \underbrace{\left(\frac{s + z_{\text{lead}}}{s + p_{\text{lead}}}\right)}_{G_{\text{lead}}(s),; p_{\text{lead}} > z_{\text{lead}}} \cdot \underbrace{\left(\frac{s + z_{\text{lag}}}{s + p_{\text{lag}}}\right)}_{G_{\text{lag}}(s),; z_{\text{lag}} > p_{\text{lag}}}$$

Multi-Objective Nelder-Mead Simplex Optimization (src/lead_lag_simplex_optimizer.m)

Rather than relying solely on trial-and-error Bode shaping, the 5-dimensional parameter vector $\mathbf{\theta} = [K_c,; z_{\text{lead}},; p_{\text{lead}},; z_{\text{lag}},; p_{\text{lag}}]^T$ is optimized via constrained Nelder-Mead Simplex search (fminsearch) minimizing a weighted quadratic penalty function over the open-loop transfer function $L(s) = G_c(s)G_p(s)$:

$$J(\mathbf{\theta}) = w_1 \left(\text{PM}(\mathbf{\theta}) - 55^\circ\right)^2 + w_2 \left(\text{GM}_{\text{dB}}(\mathbf{\theta}) - 20\text{ dB}\right)^2 + w_3 \left(K_a(\mathbf{\theta}) - 120\right)^2 + \Phi_{\text{stability}}(\mathbf{\theta})$$

where:

• Target Phase Margin ($\text{PM}$): $55^\circ$ (ensuring fast rise time with well-damped transient overshoot)
• Target Gain Margin ($\text{GM}$): $20\text{ dB}$ (robustness against plant parameter variations)
• Target Acceleration Error Constant ($K_a = \lim_{s \to 0} s^2 L(s)$): $120\text{ s}^{-2}$ (bounding steady-state quadratic acceleration tracking error $e_{\text{ss,quad}} = 1/K_a < 1%$)
• Structural & Hurwitz Barrier ($\Phi_{\text{stability}}$): Enforces $\mathbf{\theta} > 0$, $p_{\text{lead}} > z_{\text{lead}}$, $z_{\text{lag}} > p_{\text{lag}}$, and positive stability margins.

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Repository Structure

• src/lead_lag_simplex_optimizer.m — 5-variable Nelder-Mead Simplex optimizer (fminsearch) with structural Lead/Lag pole-zero constraints and automated margin/error verification.
• src/closed_loop_trajectory_analysis.m — Time-domain simulation (lsim) comparing uncompensated vs. compensated closed-loop tracking across Step (Position), Ramp (Velocity), and Quadratic (Acceleration) reference inputs, plus closed-loop pole-zero maps (pzmap).
• src/sisotool_lead_tuning.m — Reduced-order Lead compensator tuning script integrated with MATLAB sisotool root-locus/Bode interactive design.
• src/compensator_verification.m — Baseline unity-feedback closed-loop transfer function construction script.