The z-domain transfer function and the Bilinear transformation.

Understanding mapping

The concept of mapping is used to transform one domain into another. It can be thought of as a value living in one space being transformed into another space. For example, the mapping of a point in the Cartesian plane to a point in the polar plane is a transformation from one space to another. In the context of control systems, the mapping of a transfer function from the s-domain to the z-domain is a transformation from the continuous-time domain to the discrete-time domain.

This warping of the space is non-linear and the Bilinear transformation is used to account for this non-linearity.

Bilinear transformation from the s-domain to the z-domain

Assume we have a first-order system with a transfer function of the form:

\begin{equation} G(s) = \frac{1}{s + 1} \end{equation}

The z-domain transfer function is obtained by replacing \(s\) with \(\left(\frac{z - 1}{T}\right)\), where \(T\) is the sampling time. Let’s asume that a sampling time of \(1\) KHz was and the cut-off frequency \(\omega_c\) of \(10\) Hz`.

Then, the time period \(T\) is given by: \begin{equation} T = \frac{1}{f_s} = \frac{1}{1000} = 0.001 \end{equation}

The transfer function is given as follows: \begin{equation} H(j\omega) = \frac{\omega_c}{s + \omega_c} \end{equation}

The transformation from s-domain to the z-domain is non-linear. To account for this non-linearity, the cut-off frequency is pre-warped and the resulting frequency is denoted by \(\omega_p\).

\begin{equation} \omega_p = \frac{2}{T} \tan\left(\frac{\omega_c T}{2}\right) \end{equation}

Once the frequency has been pre-warped, the system can undergo Bilinear transformation using the relationship between \(s\) and \(z\).

Here, we evaluate the transfer function at \(s = \frac{2}{T} \frac{z-1}{z+1}\). In other words, we replace \(s\) with \(\frac{2}{T} \frac{z-1}{z+1}\).

Here, the \(z^{-1}\) operator is the delay operator. The equation can be rewritten as:

where \(Y[n]\) is the output at time \(n\) and \(U[n]\) is the input at time \(n\). The equation can be rewritten as:

\begin{equation} \label{eq:diff_eq} Y[n] = 0.939 Y[n - 1] + 0.031 (U[n] + U[n - 1]) \end{equation}

Equation \eqref{eq:diff_eq} is called the difference equation. The difference equation is used to implement the system in a digital computer. now the system can be implemented on a digital device such as a microcontroller.

Here is an example implementation in Python:

1
2
3
4
5
6
7
8
9
10
11
12
13
 
    N = 10  # Number of samples

    # Initialize arrays for Y and U
    Y = [0] * N  # List to store Y values, initialized to 0
    U = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]  # Example input list U

    # Calculate Y values based on the given equation
    for n in range(1, N):
        Y[n] = 0.939 * Y[n - 1] + 0.031 * (U[n] + U[n - 1])

    # Print the resulting Y values
    for n in range(N):
        print(f"Y[{n}] = {Y[n]}")