delay_sys

A Python library for discretizing continuous-time transfer functions with time delays.

Overview

delay_sys provides utilities for converting continuous-time control systems to discrete-time while incorporating deadtime delays. The library features a DelayTransferFunction class that extends python-control's TransferFunction to natively handle time delays.

Installation

Using pip

pip install delay_sys

Using uv (recommended for development)

git clone https://github.com/jamestjsp/delay_sys.git
cd delay_sys
uv sync

For development installation with pip:

git clone https://github.com/jamestjsp/delay_sys.git
cd delay_sys
pip install -e .

Command-line Interface

The package includes an educational demonstration showing the impact of time delays on feedback control systems:

# Run the demonstration
uv run delay_sys

This will:

• Test multiple delay scenarios (0s, 0.2s, 0.5s, 1.0s) with a first-order plant
• Show how system order increases with delay: Order = Original + (Delay/Sampling_time)
• Demonstrate stability challenges as delays increase
• Plot comprehensive analysis including:
  • Step responses for different delays
  • System order vs delay time
  • Maximum pole magnitude vs delay
  • Relationship between sampling time and delay complexity
• Provide practical guidance for controlling delayed systems

Key Learning Points:

• Large delays create very high-order discrete systems
• Stability becomes more challenging with many delay poles
• Proper sampling time selection is crucial for delayed systems

Quick Start

Using DelayTransferFunction directly

import control as ct
import delay_sys

# Create a continuous-time transfer function and discretize
G = ct.tf([1], [1, 2, 1])  # 1/(s^2 + 2s + 1)
G_discrete = ct.sample_system(G, 0.1, method='zoh')

# Create with deadtime
Gd = delay_sys.DelayTransferFunction(
    num=G_discrete.num[0][0],
    den=G_discrete.den[0][0],
    dt=0.1,
    deadtime=0.3
)

print(Gd)

Using helper functions

from delay_sys.delayed_tf import create_fopdt

# Create FOPDT system directly
Gd = create_fopdt(
    gain=2.0,
    time_constant=1.5, 
    dt=0.1,
    deadtime=0.3
)

print(Gd)  # Shows transfer function with deadtime info

API Reference

DelayTransferFunction

A transfer function class with explicit time delay support.

Parameters:

• num (list of float): Numerator coefficients of the base transfer function
• den (list of float): Denominator coefficients of the base transfer function
• dt (float): The sampling period for the discrete-time system. Must be > 0
• deadtime (float, optional): The deadtime in the same units as dt. Defaults to 0.0

Attributes:

• deadtime (float): The deadtime value used in the system

Examples

Basic Discretization

import control as ct
import delay_sys

# First-order system
G = ct.tf([2], [1, 3])
G_discrete = ct.sample_system(G, 0.1, method='zoh')
Gd = delay_sys.DelayTransferFunction(
    G_discrete.num[0][0], 
    G_discrete.den[0][0], 
    dt=0.1
)
print(f"Deadtime: {Gd.deadtime}")  # Access deadtime attribute

With Deadtime

# Add 0.5 second deadtime
Gd_delayed = delay_sys.DelayTransferFunction(
    G_discrete.num[0][0], 
    G_discrete.den[0][0], 
    dt=0.1, 
    deadtime=0.5
)
print(Gd_delayed)  # Shows transfer function with deadtime info

Different Discretization Methods

# Using Tustin (bilinear) method
G_tustin = ct.sample_system(G, 0.1, method='tustin')
Gd_tustin = delay_sys.DelayTransferFunction(
    G_tustin.num[0][0], 
    G_tustin.den[0][0], 
    dt=0.1
)

Direct DelayTransferFunction Usage

# Create a first-order system with deadtime
fopdt_system = delay_sys.DelayTransferFunction(
    num=[0.095],             # Numerator coefficients
    den=[1, -0.819],         # Denominator coefficients
    dt=0.1,                  # 100ms sampling
    deadtime=0.25            # 250ms deadtime
)

# Use with python-control functions
t, y = ct.step_response(fopdt_system)

System Analysis

import numpy as np
import matplotlib.pyplot as plt
from delay_sys.delayed_tf import create_fopdt

# Create system with deadtime using helper
Gd = create_fopdt(gain=1.0, time_constant=1.0, dt=0.1, deadtime=0.3)

# Analyze stability margins
gm, pm, wg, wp = ct.margin(Gd)
print(f"Gain margin: {gm:.2f}, Phase margin: {pm:.2f}°")

# Plot frequency response
w = np.logspace(-2, 2, 100)
ct.bode(Gd, w)
plt.show()

PID Control with Delayed Systems

import control as ct
from delay_sys.delayed_tf import create_fopdt

# Create delayed plant
plant = create_fopdt(gain=2.0, time_constant=1.5, dt=0.1, deadtime=0.3)

# Create PID controller (discrete form)
# PID: Kp + Ki*dt/(z-1) + Kd*(z-1)/z
Kp, Ki, Kd = 1.0, 0.2, 0.05
pid_num = [Kp + Ki*0.1 + Kd, -Kp + Ki*0.1 - 2*Kd, Kd]
pid_den = [1, -1, 0]
pid = ct.tf(pid_num, pid_den, dt=0.1)

# Create closed-loop system
open_loop = pid * plant
closed_loop = ct.feedback(open_loop, 1)

# Analyze performance
t, y = ct.step_response(closed_loop)
gm, pm, _, _ = ct.margin(open_loop)
print(f"Gain margin: {gm:.2f}, Phase margin: {pm:.1f}°")

Features

• Native deadtime support: DelayTransferFunction class with explicit deadtime handling
• Multiple discretization methods: Supports all methods available in python-control
• Integer delay implementation: Automatically rounds deadtime to nearest integer number of samples
• Full compatibility: Works seamlessly with python-control functions
• Input validation: Comprehensive error checking for invalid parameters
• SISO systems: Currently supports Single-Input Single-Output systems

Requirements

• Python >= 3.8
• control >= 0.9.0
• numpy >= 1.19.0
• matplotlib >= 3.0.0

Development

Using uv (recommended)

# Clone and setup
git clone https://github.com/jamestjsp/delay_sys.git
cd delay_sys

# Install dependencies and run
uv run delay_sys

# Run tests
uv run pytest

# Create FOPDT system example
uv run python -c "from delay_sys.delayed_tf import create_fopdt; sys = create_fopdt(gain=2.0, time_constant=1.0, dt=0.1, deadtime=0.3); print(sys)"

Traditional development setup

pip install -e .
python -m delay_sys

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

1. Fork the repository
2. Create a feature branch (git checkout -b feature/amazing-feature)
3. Commit your changes (git commit -m 'Add amazing feature')
4. Push to the branch (git push origin feature/amazing-feature)
5. Open a Pull Request

License

This project is licensed under the MIT License - see the LICENSE file for details.

Delay Considerations

When working with systems that have significant delays:

Sampling Time Selection

# For a system with 1s delay:
# Too fine sampling (1ms) → 1000 delay poles → Very high order
# Better sampling (50ms) → 20 delay poles → Manageable

delay = 1.0  # seconds
dt_fine = 0.001    # 1000 poles!
dt_good = 0.05     # 20 poles - much better

Managing Large Delays

from delay_sys.delayed_tf import create_fopdt

# Example: Plant with significant delay
plant = create_fopdt(gain=1.0, time_constant=2.0, dt=0.1, deadtime=1.5)

# Check system complexity
delay_samples = int(1.5 / 0.1)  # 15 additional poles
print(f"System has {delay_samples} delay poles")

# For very large delays, consider:
# - Smith Predictor control
# - Model Predictive Control (MPC)  
# - Feedforward compensation