Introduction to Digital Control

#Control #DigitallControlSystems This note is primarily based on chapter 3 of the book Digital Control of Dynamic Systems by Franklin, Powell, and Workman.

In today’s world, most of the controllers are implemented on digital computers. Digital computers work on samples of data. This requires implementing a few additional parts, which we will study with more detail later. The output of the actual plant, which has continuous dynamics, goes through an Analog to Digital converter (A/D). Then this value is subtracted from the reference signal and the discrete error signal is formed (if the reference is also continuous, that has to pass through an A/D as well). Then some process is done on this signal in the computer and a discrete output is generated. This signal then goes through a Digital to Analog converter (D/A), which is typically a Zero Order Hold (ZOH) and is applied to the actual plant. Note that although the course is often called digital control, we mostly work with discrete signals. This means that we rarely consider the effect of quantization, because today’s processors have very high resolutions.

Forward Euler Approximation

We know that the definition of derivative is \(\dot{x} = \lim_{\delta t \to 0} \frac{\delta x}{\delta t} .\) One simple way to approximate continuous-time dynamics is to use the Euler’s method: \(x(k) \approx \frac{x (k + 1) - x(k)}{T} ,\) where \(T = t_{k + 1} - T_{k}, \quad t_k = k T , \quad x(k) = x (t_k) , \quad x(k + 1) = x(t_{k + 1}) , k \ \text{is an integer} .\) Generally, sample rate should be faster than $30$ times the bandwidth of the closed-loop system.

Effect of Sampling

The most important impact of sampling is the delay that is caused by the ZOH. The input that is being calculated digitally is held constant between samples and then applied to the system. This creates, on average, a delay of $\frac{T}{2}$. The effect of this delay can be approximated using the Pade formula. This formula approximates the delay as a first order dynamics with the same time constant as the delay. Therefore, here we have \(G (s) = \frac{\frac{2}{T}}{s + \frac{2}{T}} .\) Alternatively, the effect of delay can be investigated in the frequency domain. A delay does not change the magnitude, but has a phase decrease of \(\delta_{\phi} = - \frac{\omega T}{2} .\)

PID Control

In the continuous-time domain, a Proportional-Integral-Derivative (PID) controller has three terms: \(\begin{align} \text{proportional control}: & \quad u(t) = K e(t) \\ \text{integral control}: & \quad u(t) = \frac{K}{T_I} \int_0^t e(\eta) \text{d} \eta \\ \text{derivative control}: & \quad u(t) = K T_D \dot{e}(t) \end{align}\) Approximations can be used to derive the discrete-time equivalent of those control gains as \(\begin{align} \text{proportional control}: & \quad u(k) = K e(k) \\ \text{integral control}: & \quad u(k) = u(k - 1) + \frac{K}{T_I} T e(k) \\ \text{derivative control}: & \quad u(k) = \frac{K T_D}{T} [ e(k) - e(k - 1) ] \end{align}\) However, approximating these three terms separately is not a good method! These terms are usually used together as \(C(s) = \frac{U(s)}{E(s)} = K (1 + T_I \frac{1}{s} + T_D s) \Rightarrow \dot{u} = K ( \dot{e} + \frac{1}{T_I} e + T_D \ddot{e} ) .\) Now using Euler’s method results in \(u(k) = u(k - 1) + K [ ( 1 + \frac{T}{T_I} + \frac{T_D}{T} ) e(k) - (1 + 2 \frac{T_D}{T} ) e (k - 1) + \frac{T_D}{T} e (k - 2) ] .\) Again, the sampling rate should be at least $30$ times the bandwidth of the closed-loop system for this approximation to be good. Slower sampling generally leads to an increased overshoot compared to the continuous case. If that high of a sampling frequency is not possible, then we could at least tune the controller gains to get a better result.

You can continue learning about digital control systems here.