Implementing PID Controller using Backward Difference Approximation

Figure 5.9 shows Xcos diagram for implementing PID controller.

Figure 5.9: Xcos for PID controller available as pid_bda_virtual.xcos

The PID controller in continuous time is given by

$$u(t) =K \left\{e(t)+\frac{1}{\tau_i}\int_0^t e(t)dt+\tau_d\frac{de(t)}{dt}\right\}$$ (5.37)
On taking the Laplace transform, we obtain

$$u(t) =K\left\{1+\frac 1{\tau_i s}+\tau_d s\right\}e(t)$$ (5.38)
By mapping controller given in equation 5.38 to the discrete time domain using backward difference formula, we get

$$u(n) =K\left\{1+\frac{T_s}{\tau_i}\frac{z}{z-1}+\frac{\tau_d}{T_s}\frac{z-1}{z}\right\}e(n)$$ (5.39)
On cross multiplying, we obtain

$$(z^2-z)u(n) =K\left\{(z^2-z)+\frac{T_s}{\tau_i}z^2+\frac{\tau_d}{T_s}(z-1)^2\right\}e(n)$$ (5.40)

$$z^2$$

$$u(n)-u(n-1) =K\left\{e(n)-e(n-1)+\frac{T_s}{\tau_i}e(n)\right.$$

$$\hspace{1cm} + \left. \frac{\tau_d}{T_s}[e(n)-2e(n-1)+e(n-2)]\right\}$$ (5.41)
The PID controller is usually written as

$$u(n) =u(n-1)+s_0 e(n)+s_1e(n-1)+s_2e(n-2)$$ (5.42)
where

$$s_0 =K\left[1+\frac{T_s}{\tau_i}+\frac{\tau_d}{T_s}\right]$$ (5.43)

$$s_1 =K\left[-1-2\frac{\tau_d}{T_s}\right]$$ (5.44)

$$s_2 =K\left[\frac{\tau_d}{T_s}\right]$$ (5.45)