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 $(f_\text{p} / f_\text{z})^2$ .