formelblad

att ha med:

Laplace transforms

$$\{\begin{array}{ll}f(t)& F(s)\\ 1& \frac{1}{s}\\ e^{at}& \frac{1}{s-a}\\ 1-e^{at}& \frac{a}{s(s+a)}\end{array}$$

$G(s)=\frac{1}{s(s+5)}=\frac{A}{s}+\frac{B}{s+5}$

$A={\frac{1}{s+5}|}_{s=0}=\frac{1}{5}$

$B={\frac{1}{s}|}_{s=-5}=-\frac{1}{5}$

$g(t)=\frac{1}{5}-\frac{1}{5}{e}^{-5t}$

Nyquist

$Z=N+P$

• $N$ - number of clock-wise encirclements of $(-1,0)$
• $P$ - number of unstable poles ($\text{Re}>0$) (open-loop)
• $Z$ - number of unstable poles ($\text{Re}>0$) (closed-loop)

Bode

$K_0(j\omega {)}^n$

Magnitude: $20\cdot {\log }_{10}{K}_0+20\cdot n\cdot {\log }_{10}\omega$

Phase: $90^{\circ }\cdot n$

$(j\omega \tau +1{)}^{\pm 1}$

Breakpoint: $\omega =1/\tau$

Magnitude: $\pm 20\cdot {\log }_{10}\omega$

Phase: $\pm 90^{\circ }$ from $\omega =0.2/\tau$ to $\omega =5/\tau$

${\left[{\left(\frac{j\omega }{{\omega }_n}\right)}^2+2\zeta \frac{j\omega }{{\omega }_n}+1\right]}^{\pm 1}$

Breakpoint: $\omega ={\omega }_n$

Magnitude: $\pm 40\cdot {\log }_{10}\omega$

Small $\zeta <0.5$ peak at breakpoint: $20\cdot {\log }_{10}\frac{1}{2\zeta }$

Phase: $\pm 180^{\circ }$

Linearisering

Example: $\ddot{y}+5\dot{y}+2y^2=F$

$F_0=0$

$x_1=y$

$x_2=\dot{y}={\dot{x}}_1$

$\Rightarrow {\dot{x}}_2=\ddot{y}={\ddot{x}}_1$

${\dot{x}}_2=F-5x_2-2x_1^2$

Equilibrium:

${\dot{x}}_2=0$

${\dot{x}}_1=x_2=0$

$F_0=0\Rightarrow 2x_1^2=F_0=0\Rightarrow x_1=0$

$\ddot{y}+5\dot{y}=F$