Part 11 - Design of controllers (2)

Given that $u(t) = K_p(e(t) + T_d \dot{e}(t))$

In the $s$ -domain, the PD controller is:

F_{PD}(s) = \dfrac{U(s)}{E(s)} = K_p(1 + T_d s).

PD-controllers are often paired with low-pass filters (LP), so that $\dot{e}(t)$ doesn’t become too large.

\begin{align*} F_{PD}(s) & = K_p\left(1 + \dfrac{T_d s}{1 + T_f s}\right) \newline & = K_p\left(\dfrac{1 + (T_d + T_f) s}{1 + T_f s}\right) \newline & = K_p\left(\dfrac{1 + \tau_d s}{1 + \frac{\tau_d}{b} s}\right), \quad \tau_d = T_d + T_f, \quad b = \frac{T_d + T_f}{T_f}. \end{align*}

This means that:

F(j\omega) = K_p\left(\dfrac{1 + \tau_d j \omega}{1 + \frac{\tau_d}{b} j\omega}\right),

\omega_{\max} = \dfrac{\sqrt{b}}{\tau_d}, \quad \text{if we choose } \omega_c = \dfrac{\sqrt{b}}{\tau_d},

\arg(L(j\omega)) = \arg(F_{PD}(j\omega)) + \arg(G(j\omega)).

\begin{align*} \phi_{\max} & = \arg(F_{PD}(j\omega)) \newline & = \arctan\left(\tau_d \cdot \frac{\sqrt{b}}{\tau_d}\right) - \arctan\left(\frac{\tau_d}{b} \cdot \frac{\sqrt{b}}{\tau_d}\right) \newline & = \arctan(\sqrt{b}) - \arctan\left(\frac{1}{\sqrt{b}}\right). \end{align*}

b = \ldots = \dfrac{1 + \sin(\phi_{\max})}{1 - \sin(\phi_{\max})}.

Example 1

G(s) = \dfrac{2 - s}{s(1 + s)^2}, \quad \phi_m = 50^\circ, \quad \omega_c = 0.5.

Design a PD-controller

F_{PD}(s) = K_p \dfrac{1 + \tau_d s}{1 + \frac{\tau_d}{b} s}.

\begin{align*} \phi_{\max} & = 50^\circ - 180^\circ - \arg(G(j\omega_c)) \newline & = 50^\circ - 180^\circ - \left(-\arctan\left(\frac{\omega_c}{2}\right) - 90^\circ - 2 \arctan(\omega_c)\right) \newline & = 27^\circ. \end{align*}

b = \dfrac{1 + \sin(\phi_{\max})}{1 - \sin(\phi_{\max})} = 2.66, \quad \tau_d = \dfrac{\sqrt{b}}{\omega_c} = 3.26.

\begin{align*} |F(j\omega_c)||G(j\omega_c)| & = 1 \newline K_p \cdot \dfrac{\sqrt{1^2 + (\tau_d \cdot \omega_c)^2}}{\sqrt{1^2 + (\frac{\tau_d}{b} \cdot \omega_c)^2}} \cdot \dfrac{\sqrt{2^2 + (-\omega_c)^2}}{\omega_c \left(\sqrt{1^2 + \omega_c^2}\right)^2} & = 1 \newline K_p & = 0.186. \end{align*}

Algorithm 1: Ziegler-Nichols method

Set up the controller as a pure P-controller ( $T_i = \infty, T_d = 0$ ).
Increase $K_p$ until self-oscillation occurs, let $K_p = K_0$ and the period be $T_0$ .

Then, approximately:

Table 1: Ziegler-Nichols controller tuning rules.