Difference Equation Simulator

One way to figure out how the model behaves would be to compute its response to a given input by hand. However, this can be tedious for complicated models, or even for simple models when dealing with large numbers of timesteps.

As such, we will develop in this section a simulator that can compute the response of arbitrary LTI systems to arbitrary inputs (which you used in Design Lab 2 to simulate the robot using the LTI model you built).

The simulator we will create will have the following form:

def generic_lti_simulate(d_coeffs, c_coeffs, inputs, prev_inp, prev_out):
    assert len(prev_inp) == len(d_coeffs) - 1
    assert len(prev_out) == len(c_coeffs)
    # Your code here

• d_coeffs and c_coeffs are the coefficient lists representing the difference equation we want to simulate, which represent the coefficients associated with the various terms in a difference equation expressed in the following form:

  y[n] = c_0 y[n-1] + c_1 y[n-2] + \ldots + c_{k-1} y[n-k] + d_0 x[n] + d_1 x[n-1] + \ldots + d_j x[n-j]

• inputs is a list of input values we want to give to the system, starting at time 0 ([x[0], x[1], x[2], \ldots])

• prev_inp is a list of input values received before time 0, [x[-1], x[-2], \ldots]. It should have one fewer element than d_coeffs (since d_coeffs includes a term associated with the current input). Note that this list should contain only zeros if the system starts at rest.

• Similarly, prev_out is a list of output values before time 0, [y[-1], y[-2], \ldots]. It should have the same length as c_coeffs. Note that this list should contain only zeros if the system starts at rest.

This function should return a list of the system's output values [y[0], y[1], y[2], \ldots] of the same length as the inputs list.

Write your function in a Python file on your own machine, and run it with a few test cases to make sure it behaves as you would expect. One test case you can use for debugging is the system whose unit sample response (when started from rest) is the canonical Fibonacci sequence: Y = X + {\cal R}Y + {\cal R}^2Y. When this system is instead simulated with prev_out = [2, 3], the first 7 samples of its unit sample response are: [6, 8, 14, 22, 36, 58, 94].

Check Yourself 1:

Test your simulator on your own machine with the examples above, or with another system of your choice.

When you are ready, upload a Python file containing your generic_lti_simulate function below for checking.

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