Slovník Laplaceovej transformácie vo formáte PDF

### Slovník originálov a ich Laplaceových obrazov

$f(t)$ - Originál $F(s)$ - Obraz
$$\delta (t)$$ $$1$$
$$1 (t)$$ $${1}\over{s}$$
$$A \cdot 1 (t)$$ $${A}\over{s}$$
$$e^{-at} \cdot 1(t)$$ $${1}\over{s+a}$$
$$e^{at} \cdot 1(t)$$ $${1}\over{s-a}$$
$$A t \cdot 1 (t)$$ $${A}\over{s^2}$$
$$A {{1} \over {(n-1)!}} t^{n-1} \cdot 1 (t), n>1$$ $${A}\over{s^n}$$
$$t e^{-at} \cdot 1 (t)$$ $${1}\over{(s+a)^2}$$
$${{1} \over {(n-1)!}} t^{n-1} \cdot 1 (t), n \geq 1$$ $${1}\over{(s+a)^n}$$
$$\sin (\omega t ) \cdot 1(t)$$ $${\omega}\over{s^2 + \omega^2}$$
$$\cos (\omega t ) \cdot 1(t)$$ $${s}\over{s^2 + \omega^2}$$
$$e^{-at}\sin (\omega t ) \cdot 1(t)$$ $${\omega}\over{(s+a)^2 + \omega^2}$$
$$e^{-at} \cos (\omega t ) \cdot 1(t)$$ $${s+a}\over{(s+a)^2 + \omega^2}$$
$$A \cdot f(t)$$ $$A \cdot F(s)$$
$$A_1 \cdot f_1 (t) + A_2 \cdot f_2 (t)$$ $$A_1 \cdot F_1 (s) + A_2 \cdot F_2 (s)$$
$$f'(t)$$ $$sF(s)-f(0)$$
$$f^{(n)}(t)$$ $$s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) -\\ \dots - f^{(n-1)}(0)$$
$$\int_{0}^{t} f(\tau) d\tau$$ $${F(s)}\over{s}$$
$$\text{lim}_{t\rightarrow \infty} f(t)$$ $$\text{lim}_{s\rightarrow 0} [sF(s)]$$
$$\text{lim}_{t\rightarrow 0} f(t)$$ $$\text{lim}_{s\rightarrow \infty} [sF(s)]$$
$$\text{lim}_{t\rightarrow \infty} f'(t)$$ $$\text{lim}_{s\rightarrow 0} [s^2 F(s)]$$
$$\text{lim}_{t\rightarrow 0} f'(t)$$ $$\text{lim}_{s\rightarrow \infty} [s^2 F(s)]$$

## Tabuľky pre syntézu regulátorov

### Naslinova metóda syntézy

$$\delta_{max} [\%]$$ $$16$$ $$12$$ $$8$$ $$5$$ $$3$$ $$1$$
$$\alpha [-]$$ $$1.75$$ $$1.8$$ $$1.9$$ $$2$$ $$2.2$$ $$2.4$$
kde $\delta_{max} [\%]$ je maximálne preregulovanie.

### Metóda sysntézy: Zigler-Nichols

$$Typ$$ $$r_0$$ $$r_{-1}$$ $$r_{1}$$
$$P$$ $$0.5 r_{0KR}$$ $$-$$ $$-$$
$$PI$$ $$0.45 r_{0KR}$$ $$\frac{r_{0}}{0.85 T_{K}}$$ $$-$$
$$PID$$ $$0.6 r_{0KR}$$ $$\frac{r_{0}}{0.5 T_{K}}$$ $$0.125 T_{K} r_{0}$$
kde $r_{0KR}$ je kritické zosilnenie a $T_{K}$ je perióda kritických kmitov.