Backstepping Controller

A Python package for developing controllers for control affine systems using backstepping control laws. This package provides a flexible and easy-to-use interface for designing, simulating, and analyzing control systems with options for trajectory tracking, plotting, and saving results.

Introduction to Backstepping Control

Backstepping is a systematic and recursive control design technique primarily used for stabilizing nonlinear systems. Unlike traditional control methods that might struggle with complex nonlinearities, backstepping breaks down the control problem into smaller, more manageable sub-problems. These sub-problems are then solved sequentially, "stepping back" from the output to the input, hence the name "backstepping."

Key Concepts: Virtual Control: Intermediate control laws are designed for each state, leading to the final control input. Lyapunov Function: A mathematical function that helps ensure system stability. Backstepping uses this function to guide the control design process. Recursive Design: The control input is designed by recursively stabilizing each state in the system.

Worked examples can be found in this paper.

Installation

To install the package, simply run:

pip install backstepping_controller

Usage

1. Importing the Package To use the package, import the necessary functions:

import sympy as sp
import numpy as np

from backstepping_controller import generic_backstepping_controller, simulate_system, plot_responses, save_responses

2. Defining the System First, you need to define your system's state equations and parameters. For example, if you have a 3D system:

Define system parameters

num_states = 3

x1, x2, x3 = sp.symbols('x1 x2 x3') #states
u = sp.Symbol('u') #input
a, b, c = sp.symbols('a b c') #known parameters

# Define the state equations for a 3D system
state_equations = [
    a * x1 + x2,
    b * x2 + x3,
    c * x3 + u
]

# Define the gains
gains_vals = [10.0, 10.0, 15.0] #can modify to change system's response(typically higher values , faster response (but more expensive)

3. Creating the Control Law Use the generic_backstepping_controller function to create the control law for your system:

final_control_law, states, gains = generic_backstepping_controller(num_states, state_equations, 'u', gains_vals, tracking=False)

If you want to include trajectory tracking, you can enable the tracking parameter:

final_control_law, states, gains, refs, errors = generic_backstepping_controller(num_states, state_equations, 'u', gains_vals, tracking=True)

4. Simulating the System Simulate the system using the simulate_system function. You can configure it to print the control law, plot the results (states, inputs), and save the responses as a .json file:

# Simulation parameters
time = np.linspace(0, 10, 1000)  # 10 seconds of simulation with 1000 time steps
initial_conditions = [0.1, 0.0, 0.1]  # Initial conditions for x1, x2, x3, you can change this since region of attraction is the entire R^n. 
params_subs = {a: 1.0, b: 0.5, c: 0.2, 'k1': gains_vals[0], 'k2': gains_vals[1], 'k3': gains_vals[2]} #can change parameter values based on system

# Simulate the system
state_values, control_inputs = simulate_system(
    final_control_law, states, gains_vals, initial_conditions, time, state_equations, params_subs, 
    plot=True, save_path='results.json', print_law=True) #You can change plot to `False` if you dont want to see the plots yet

5. Plotting the Results If you haven't plotted the results directly in the simulation, you can use the plot_responses function to plot the states and control inputs on separate plots:

plot_responses(time, state_values, control_inputs)

6. Saving the Results The results can be saved to a JSON file for later analysis. The simulate_system function allows you to save the results directly during the simulation by specifying the save_path:

simulate_system(final_control_law, states, gains_vals, initial_conditions, time, state_equations, params_subs, save_path='results.json')

Alternatively, you can save the results after the simulation using the save_responses function:

save_responses('results.json', time, state_values, control_inputs)

Example: Full Workflow Here's a complete example that ties everything together:

import numpy as np
import sympy as sp
from backstepping_controller import generic_backstepping_controller, simulate_system, plot_responses, save_responses

# Define system parameters
num_states = 3
x1, x2, x3 = sp.symbols('x1 x2 x3')
u = sp.Symbol('u')
a, b, c = sp.symbols('a b c')

# Define the state equations for a 3D system
state_equations = [
    a * x1 + x2,
    b * x2 + x3,
    c * x3 + u
]

# Define the gains
gains_vals = [10.0, 10.0, 15.0]

# Create the control law
final_control_law, states, gains = generic_backstepping_controller(num_states, state_equations, 'u', gains_vals, tracking=False)

# Simulation parameters
time = np.linspace(0, 10, 1000)  # 10 seconds of simulation with 1000 time steps
initial_conditions = [0.1, 0.0, 0.1]  # Initial conditions for x1, x2, x3
params_subs = {a: 1.0, b: 0.5, c: 0.2, 'k1': gains_vals[0], 'k2': gains_vals[1], 'k3': gains_vals[2]}

# Simulate the system
state_values, control_inputs = simulate_system(
    final_control_law, states, gains_vals, initial_conditions, time, state_equations, params_subs, 
    plot=True, save_path='results.json', print_law=True
)

# Optional: Plot and save the results separately
plot_responses(time, state_values, control_inputs)
save_responses('results.json', time, state_values, control_inputs)