Filters in Control Systems

In this blog I want to elaborate on the various types of filters that are being used in Control Systems. These filters are often used in combination with a PI, PD or PID controller to obtain a robust controller. The filters which we will discuss are:

  • Low-Pass filter
  • High-Pass filter
  • Lead-Lag filter
  • Notch filter

Using these four filters we can create other filter types, such as a Band-Stop or Band-Pass filter. The behavior of each filter can be captured by a transfer function in the continuous-time using the Laplace domain s of either first and/or second order. We use the transfer functions to describes the filter H(s) = Y(s) / U(s) and as such the relation between the input U(s) and the output Y(s). Throughout this blog we will write the equations in the form of their angular frequency \omega in [rad/s], also known as radian frequency. However, we will specify filters using their frequency f in [Hz]. Using the radial frequency notation results in a more visual compact formula. Additionally, it is also possible to specify the filters in form of their time constant \tau in [s]. The following relation holds between the angular frequency, frequency and time constant.

    \begin{equation*} \omega \triangleq 2\pi f \triangleq \frac{1}{\tau} \end{equation*}

Many forms are used within literature, one book will use angular frequencies, the other will use time-constants. Finally, for second order filters, the only filter with possibly complex poles or zeros, can be written in various ways. We will specify second order filters in terms of the damping \beta of the corresponding frequency. It is also possible to describe these formulas using the Quality factor Q. Whereas \beta describes how oscillations decay in a system after a disturbance, Q describes how underdamped the system is. The following relation holds between \beta and Q

    \begin{equation*} \beta \triangleq \frac{1}{2Q} \end{equation*}

Low-pass filter

A low-pass filter is used to pass signals with a frequency lower than a certain cut-off frequency f_\text{lp}. Below the formulas for both the first- and second-order low-pass filter is given.

    \begin{equation*} H(s) = K\cdot\frac{\omega_{\text{lp}}}{s + \omega_{\text{lp}}} \quad\quad H(s) = K\cdot\frac{\omega_{\text{lp}}^2}{s^2 + 2\beta\omega_{\text{lp}}s + \omega_{\text{lp}}^2} \end{equation*}

Herein, K denotes the gain, f_\text{lp} denotes the low-pass cut-off frequency and \beta_\text{lp} denotes the damping. Whereas the first-order supresses with 20 [dB/dec], the second-order supresses with 40 [dB/dec]. Low frequent the filter gain is K.

High-pass filter

The complement of a low-pass filter is a high-pass filter. This filter is used to pass signals with a frequency higher than a certain cut-off frequency f_\text{hp}. Below the formulas for both the first- and second-order high-pass filter is given.

    \begin{equation*} H(s) = K\cdot\frac{s}{s + \omega_{\text{hp}}} \quad\quad H(s) = K\cdot\frac{s^2}{s^2 + 2\beta\omega_{\text{hp}}s + \omega_{\text{hp}}^2} \end{equation*}

Herein, K denotes the gain, f_\text{hp} denotes the high-pass cut-off frequency and \beta_\text{hp} denotes the damping. Likewise as the low-pass filter the first-order supresses frequencies with 20 [dB/dec] and the second-order with 40 [dB/dec].

Lead-lag filter

A lead-lag filter, also known as a lead-lag compensator, is often mainly used for phase compensation rather then magnitude. Below the formula for a lead or lag filter is shown.

    \begin{equation*} H(s) = K\cdot\frac{\omega_\text{p}}{\omega_\text{z}}\cdot\frac{s + \omega_\text{z}}{s + \omega_\text{p}} \end{equation*}

Herein, f_\text{p} and f_\text{z} denote the frequency of the pole and zero, respectively. The filter functions as a lead filter if f_\text{p} > f_\text{z}  and otherwise as a lag filter. The filter has its maximum or minimum phase at \sqrt{f_\text{p}f_\text{z}}. Finally, at f = \infty the filter has a gain of Kf_\text{p} / f_\text{z} or Kf_\text{z} / f_\text{p} in case of a lead or lag filter, respectively. Naturally, the filter can be cascaded with itself by which a the filter can be a lead and lag filter simultaneously.

Notch filter

A notch filter is often used to filter undesired resonance peaks. Below the formula for a notch filter is shown.

    \begin{equation*} H(s) = K\cdot\frac{\omega_\text{p}}{\omega_\text{z}}\cdot\frac{s^2 + 2\beta_\text{z}\omega_\text{z}s + \omega_\text{z}^2}{s^2 + 2\beta_\text{p}\omega_\text{p}s + \omega_\text{p}^2} \end{equation*}

Herein, f_\text{p} and f_\text{z} denote the frequency of the pole and zero, respectively. Likewise, \beta_\text{p} and \beta_\text{z} denote the damping of the pole and zero and K is as usual the gain. When f_\text{z} = f_\text{z} the notch will target one specific frequency. The gain at that frequency is given by \beta_\text{z} / \beta_\text{p}. When f_1 \neq f_2 the notch filter is also referred as a skewed notch and the difference between gain at low and high frequencies is given by (f_\text{p} / f_\text{z})^2.