control-systems

name	control-systems
description	Control systems analysis and PID design with python-control and scipy.signal: transfer functions, Bode/Nyquist plots, root locus, stability margins, and step-response simulation for SISO and discrete-time systems.
tags	["control-systems","python-control","pid-design","stability-analysis","engineering","dynamical-systems"]
version	1.0.0
authors	[{"name":"awesome-rosetta-skills contributors","github":"@xjtulyc"}]
license	MIT
platforms	["claude-code","codex","gemini-cli","cursor"]
dependencies	{"python":["control>=0.9.4","scipy>=1.11.0","numpy>=1.24.0","matplotlib>=3.7.0"]}
last_updated	2026-03-17
status	stable

Control Systems

One-line summary: This Skill helps engineers model, analyse, and design feedback control systems using python-control and scipy.signal, covering everything from transfer-function definition through PID synthesis, frequency-domain plots, and discrete-time simulation.

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When to Use This Skill

• When you need to model a plant as a transfer function or state-space representation
• When you need to assess stability via gain/phase margin, Nyquist criterion, or pole locations
• When you need to design a PID or lead-lag compensator using Ziegler-Nichols or analytical methods
• When you need to visualise frequency response (Bode plot, Nyquist diagram, root locus)
• When you need to simulate closed-loop step response and compute time-domain specifications
• When you need to discretise a continuous controller for embedded implementation (ZOH, Tustin)

Trigger keywords: transfer function, state space, Bode plot, Nyquist plot, root locus, PID tuning, Ziegler-Nichols, gain margin, phase margin, step response, lead compensator, lag compensator, ZOH discretization, stability analysis, python-control

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Background & Key Concepts

Transfer Function Representation

A linear time-invariant (LTI) system is described in the Laplace domain by its transfer function:

$$ G(s) = \frac{Y(s)}{U(s)} = \frac{b_m s^m + b_{m-1} s^{m-1} + \cdots + b_0}{a_n s^n + a_{n-1} s^{n-1} + \cdots + a_0} $$

Where $Y(s)$ is the output Laplace transform and $U(s)$ is the input. The roots of the denominator are the poles; roots of the numerator are zeros.

State-Space Representation

The equivalent time-domain form uses four matrices:

$$ \dot{x}(t) = A x(t) + B u(t), \quad y(t) = C x(t) + D u(t) $$

Where $x \in \mathbb{R}^n$ is the state vector. Eigenvalues of $A$ are the system poles.

PID Controller

The PID control law combines proportional, integral, and derivative action:

$$ u(t) = K_p e(t) + K_i \int_0^t e(\tau),d\tau + K_d \frac{de(t)}{dt} $$

Ziegler-Nichols ultimate-gain method: bring the system to sustained oscillation with a P-only controller to find the ultimate gain $K_u$ and ultimate period $T_u$, then use the ZN table to compute $K_p$, $K_i$, $K_d$.

Stability Margins

The gain margin (GM) and phase margin (PM) quantify robustness against model uncertainty: