Trasformata Z e sistemi campionati

Trasformata Z

$$F(z)=\sum_{t=0}^{\infty}f(t)\,z^{-t}$$

Sistema discreto

$$\begin{gathered} \Phi(z)=z(zI-A)^{-1}\\ W(z)=C(zI-A)^{-1}B+D \end{gathered}$$

Impulso e gradino

$$\mathcal Z[\delta(t)]=1$$ $$\mathcal Z[\delta_{-1}(t)]=\frac{z}{z-1}$$

Modo naturale

$$\mathcal Z[\lambda^t]=\frac{z}{z-\lambda}$$

con $|z|>|\lambda|$

Seno e coseno, $|z|>1$

$$\mathcal Z[\sin\theta t]= \frac{z\sin\theta}{z^2-2z\cos\theta+1}$$ $$\mathcal Z[\cos\theta t]= \frac{z(z-\cos\theta)}{z^2-2z\cos\theta+1}$$

Ritardo e anticipo

$$\begin{gathered} \mathcal Z[f(t-1)]=z^{-1}F(z)\\ \mathcal Z[f(t+1)]=zF(z)-zf(0) \end{gathered}$$

Moltiplicare per $\rho^t$ o per $t$

$$\mathcal Z[\rho^t f(t)]=F(z/\rho)$$ $$\mathcal Z[t\,f(t)]=-z\,F'(z)$$

Forma polinomiale discreta

$$t^{[k]}=t(t-1)\cdots(t-k+1)$$ $$\mathcal Z\Bigl[\frac{t^{[k]}}{k!}\Bigr]=\frac{z}{(z-1)^{k+1}}$$

Antitrasformata

$$\begin{aligned} \frac{F(z)}{z}&=\frac{R_1}{z}\\ &\quad+\sum_{i=1}^{n}\frac{R_{i+1}}{z-z_i} \end{aligned}$$ $$f(t)=R_1\,\delta(t)+\sum_{i=1}^{n}R_{i+1}\,{z_i}^{t}$$

Sistema campionato

$$\begin{aligned} W_d(z)&=\frac{z-1}{z}\\ &\quad\cdot\mathcal Z\Bigl[\mathcal L^{-1}\Bigl[\frac{W(s)}{s}\Bigr]_{t=kT}\Bigr] \end{aligned}$$

Ricavare $W(s)$

$$W(s)=\frac{Y_F(s)}{U(s)}$$

$p$ ingressi: $p$ prove indipendenti

Matrici e passaggio diretto

$$\begin{gathered} A=\lim_{t\to0}\frac{d\Phi(t)}{dt}\\ B=H(0)\\ C=\Psi(0)\\ D=\lim_{s\to\infty}W(s) \end{gathered}$$