Control System & PID controller

Control systems, as it is being taught on many universities in the world, regulates the behavior of a system using feedback. Systems may be mechanical, electrical, biological, economical or any other domain. Feedback as in many aspects of life is crucial. A target value $r$ for an essential variable of the system, the process value $y$ , is set. The target value is also referred as setpoint or reference value, whereas the process value is referred as measurement value. The control system calculates the difference between the setpoint and the process value, called the error $e$ of the control system.

From this point we will focus mainly on the structure, design and implementation of the linear feedback controller. The figure below shows the structure of the control system with a linear feedback controller. We neglect any possible source of disturbance.

The linear feedback controller is often the combination a controller and multiple filters. The controller of the control system computes a control value $u$ which is applied to the process and tries to regulate the error $e$ to zero. A filter also often referred as compensator, is tries to improve the characteristics and performance.

The most famous feedback controller is the Proportional–Integral–Derivative (PID) controller is defined by (1). Based on the error it calculates a correction to be applied to the process based on proportional, integral, and derivative terms, respectively. Heuristically, the terms of a PID controller can be interpreted as corresponding to time: the proportional term depends on the present error, the integral term on the accumulation of past errors, and the derivative term is a prediction of future error, based on current rate of change.

(1) $\begin{equation*} H(s) = K_d \cdot s + K_p + K_i \frac{1}{s} \end{equation*}$

Herein, $K_d = K_p / (2\pi f_d)$ and $K_i = K_p 2 \pi f_i$ for which $f_d$ and $f_i$ denote the frequency at which the derivative and integral action start to become active. In addition $K_p$ denotes the proportional gain of the controller. An ideal derivative is not causal – its output depends on future inputs see (2) – therefore, the PID controller is often combined with either a first or a second order low-pass filter (3). This makes the PID controller causal, depending only on past and current inputs but not future inputs.

(2) $\begin{equation*} \frac{\mathrm{d}f(t)}{\mathrm{d}t} \triangleq \lim_{h \rightarrow 0} \frac{f(t + h) - f(t)}{h} \end{equation*}$

(3) $\begin{equation*} H(s) = \left(K_d \cdot s + K_p + K_i \frac{1}{s}\right) \frac{\omega_{\text{lp}}^2}{s^2 + 2\beta\omega_{\text{lp}}s + \omega_{\text{lp}}^2} = \frac{\left(K_d s^2 + K_p s + K_i\right) \omega_{\text{lp}}^2}{s^3 + 2\beta\omega_{\text{lp}}s^2 + \omega_{\text{lp}}^2s} \end{equation*}$

Note that the equation (3) results in a third order polynomial. With some algebra this can be reduced into a first order and second order polynomial. That is to say;

(4) $\begin{equation*} H(s) = \frac{K_i}{s} + \frac{\left(\omega^2_\text{lp} K_d - K_i\right)s + \left(\omega^2_\text{lp} K_p - 2\beta\omega_\text{lp} K_i\right)}{s^2 + 2 \beta \omega_\text{lp} s + \omega_\text{lp}^2}\end{equation*}$

As such, the integral action is decomposed from the derivative and gain action along with the second order low-pass filter. With the integrator in parallel, it is possible to perform a saturation or anti-windup on the integrator signal.

Below you see an interactive bode diagram of a PID controller combined with a second order low-pass filter with some typical values. You clearly see the integrator action up to 10 [Hz], followed by the proportional gain action from 10 to 25 [Hz]. The derivative term is visible from 25 to 300 [Hz] and eventually the low-pass filter from 300 [Hz] and further.

Tags: Continuous-Time, Control Systems, Feedback Controller, Frequency Response

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