Linear time-invariant (LTI) systems are widely used in the field of signal processing. Scipy contains powerful tools for simulating LTI systems in the scipy.signal package, but they are not well documented. I will provide a simple example that demonstrates how to use a few of the core classes and functions in scipy.signal for simulating LTI systems with Python.
Define an LTI system
You will need to have Scipy installed, and you will need to have Matplotlib as well to make plots. We will start with an example of a first-order LTI system, which is characterized by a single parameter known as the time constant. scipy.signal defines the class lti to represent a linear system in Python.
#!/usr/bin/env python from numpy import * from matplotlib import pyplot as plt from scipy import signal tau = 5.0*60 # 5 minutes h_times = arange(0.0, 10*tau, 0.1) sys = signal.lti(1,[1,1.0/tau]) step_response = sys.step(T=h_times) plt.plot(h_times, step_response/step_response.max()) # normalized plt.axhline(0.63, color='red') plt.axvline(tau, color='red') plt.xlabel('t') plt.title('Step response') plt.show()
An lti object can be defined in three ways:
- If two arguments are provided, they are assumed to be the numerator and denominator of the transfer function of the system.
- If three arguments are provided, they represent poles, zeros, and gain.
- If four arguments are given, they must represent the state space (A,B,C,D) of the system.
In this case, we used the transfer function representation. The transfer function is 1/(s+1/tau) where tau is the delay coefficient and s is the independent variable in the Laplace domain. In Python, the denominator is represented as a list of coefficients, starting with the highest-order coefficient. In our example, the coefficient of s to the first power is one, and the coefficient of s to the zero power is two.
We can use the step_response method of an lti object to get the response of the system to a step function:
The blue curve shows the step response, while the red lines illustrate that the time constant is the amount of time required for the step response to reach a value of 1-1/e, or about 63.2% of its final value.
In a future post, I will cover some of the other things you can do with the tools in scipy.signal.
- Scipy.signal documentation
- Feedback Systems online text by Astrom and Murray
- Linear Time Invariant (LTI) system theory
- Transfer function
- Time constant
- Laplace transform