$$\dot{x}\left( t \right)=f\left( x\left( t \right),u\left( t \right) \right),$$
linearizirati oko referetnog stanja i dobiti linearnie
diferencijalne jednadžbe koje opisuju dinamiku malih perturbacija.
Pretpostavimo da imamo referentno stanje
i odgovarajući referentni upravljački vektor
U slučaju bez pertrubacija, referentni vektori
zadovoljavaju nelinearnau jednadžbu stanja
$$\dot{\bar{x}}\left( t \right)=f\left( \bar{x}\left( t \right),\bar{u}\left( t \right) \right),$$
$$\dot{\bar{x}}\left( t \right)=f\left( \bar{x}\left( t \right),\bar{u}\left( t \right) \right),$$
U perturbiranom stanju, vektor stanja i upravljanja možemo
prikazati na sljedeći način
$$\begin{align} & x\left( t \right)=\bar{x}\left( t \right)+\delta x\left( t \right), \\ & u\left( t \right)=\bar{u}\left( t \right)+\delta u\left( t \right), \\ \end{align}$$
$$\begin{align} & x\left( t \right)=\bar{x}\left( t \right)+\delta x\left( t \right), \\ & u\left( t \right)=\bar{u}\left( t \right)+\delta u\left( t \right), \\ \end{align}$$
gdje su
i
male varijacije
oko referentnog vektora stanja i upravljanja, respektivno. Uvrstimo li sada u
dobivamo
$$\dot{\bar{x}}\left( t \right)+\delta \dot{x}\left( t \right)=f\left( \bar{x}\left( t \right)+\delta x\left( t \right),\bar{u}\left( t \right)+\delta u\left( t \right) \right).$$
$$\dot{\bar{x}}\left( t \right)+\delta \dot{x}\left( t \right)=f\left( \bar{x}\left( t \right)+\delta x\left( t \right),\bar{u}\left( t \right)+\delta u\left( t \right) \right).$$
Prethodni izraz možemo razviti u Taylorov reda za i-tu
komponentu (i = 1,....,n).
$${{\dot{\bar{x}}}_{i}}+\delta {{\dot{x}}_{i}}\left( t \right)={{f}_{i}}\left( \bar{x}\left( t \right),\bar{u}\left( t \right) \right)+\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{x}_{j}}}\delta {{x}_{j}}\left( t \right)+}\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{u}_{j}}}\delta {{u}_{j}}\left( t \right)}$$
$${{\dot{\bar{x}}}_{i}}+\delta {{\dot{x}}_{i}}\left( t \right)={{f}_{i}}\left( \bar{x}\left( t \right),\bar{u}\left( t \right) \right)+\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{x}_{j}}}\delta {{x}_{j}}\left( t \right)+}\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{u}_{j}}}\delta {{u}_{j}}\left( t \right)}$$
Gdje smo zanemarili članove drugog i viših redova zbog
pretpostavke da su varijacije oko referentnog stanja dovoljno male. Usporedimo
li izraz prethodnu jednadžbu s nelinearnom jednadžbom stanja, dobivamo
$$\delta {{\dot{x}}_{i}}\left( t \right)=\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{x}_{j}}}\delta {{x}_{j}}\left( t \right)+}\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{u}_{j}}}\delta {{u}_{j}}\left( t \right)},\text{ }i=1,2,...,n,$$
$$\delta {{\dot{x}}_{i}}\left( t \right)=\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{x}_{j}}}\delta {{x}_{j}}\left( t \right)+}\sum\limits_{j=1}^{n}{\frac{\partial {{f}_{i}}}{\partial {{u}_{j}}}\delta {{u}_{j}}\left( t \right)},\text{ }i=1,2,...,n,$$
gdje su parcijalne derivacije u prethodnom izrazu funkcije
referentnog stanja i upravljanja,
Sustav
jednadžbi možemo prikazati u matričnom obliku
$$\delta \dot{x}\left( t \right)=A\left( t \right)\delta x\left( t \right)+B\left( t \right)\delta u\left( t \right),$$
$$\delta \dot{x}\left( t \right)=A\left( t \right)\delta x\left( t \right)+B\left( t \right)\delta u\left( t \right),$$
gdje su Jakobiani A i B definirani sa
$$A\left( t \right)=\left[ \begin{matrix} \frac{\partial {{f}_{1}}}{\partial {{x}_{1}}} & \frac{\partial {{f}_{1}}}{\partial {{x}_{2}}} & \cdots & \frac{\partial {{f}_{1}}}{\partial {{x}_{n}}} \\ \frac{\partial {{f}_{2}}}{\partial {{x}_{1}}} & \frac{\partial {{f}_{2}}}{\partial {{x}_{2}}} & \cdots & \frac{\partial {{f}_{2}}}{\partial {{x}_{n}}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {{f}_{n}}}{\partial {{x}_{1}}} & \frac{\partial {{f}_{n}}}{\partial {{x}_{2}}} & \cdots & \frac{\partial {{f}_{n}}}{\partial {{x}_{n}}} \\ \end{matrix} \right]\text{ }B\left( t \right)=\left[ \begin{matrix} \frac{\partial {{f}_{1}}}{\partial {{u}_{1}}} & \frac{\partial {{f}_{1}}}{\partial {{u}_{2}}} & \cdots & \frac{\partial {{f}_{1}}}{\partial {{u}_{n}}} \\ \frac{\partial {{f}_{2}}}{\partial {{u}_{1}}} & \frac{\partial {{f}_{2}}}{\partial {{u}_{2}}} & \cdots & \frac{\partial {{f}_{2}}}{\partial {{u}_{n}}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {{f}_{n}}}{\partial {{u}_{1}}} & \frac{\partial {{f}_{n}}}{\partial {{u}_{2}}} & \cdots & \frac{\partial {{f}_{n}}}{\partial {{u}_{n}}} \\ \end{matrix} \right]$$
$$A\left( t \right)=\left[ \begin{matrix} \frac{\partial {{f}_{1}}}{\partial {{x}_{1}}} & \frac{\partial {{f}_{1}}}{\partial {{x}_{2}}} & \cdots & \frac{\partial {{f}_{1}}}{\partial {{x}_{n}}} \\ \frac{\partial {{f}_{2}}}{\partial {{x}_{1}}} & \frac{\partial {{f}_{2}}}{\partial {{x}_{2}}} & \cdots & \frac{\partial {{f}_{2}}}{\partial {{x}_{n}}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {{f}_{n}}}{\partial {{x}_{1}}} & \frac{\partial {{f}_{n}}}{\partial {{x}_{2}}} & \cdots & \frac{\partial {{f}_{n}}}{\partial {{x}_{n}}} \\ \end{matrix} \right]\text{ }B\left( t \right)=\left[ \begin{matrix} \frac{\partial {{f}_{1}}}{\partial {{u}_{1}}} & \frac{\partial {{f}_{1}}}{\partial {{u}_{2}}} & \cdots & \frac{\partial {{f}_{1}}}{\partial {{u}_{n}}} \\ \frac{\partial {{f}_{2}}}{\partial {{u}_{1}}} & \frac{\partial {{f}_{2}}}{\partial {{u}_{2}}} & \cdots & \frac{\partial {{f}_{2}}}{\partial {{u}_{n}}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial {{f}_{n}}}{\partial {{u}_{1}}} & \frac{\partial {{f}_{n}}}{\partial {{u}_{2}}} & \cdots & \frac{\partial {{f}_{n}}}{\partial {{u}_{n}}} \\ \end{matrix} \right]$$
Drugim riječima, dobili smo linearni vremenski-varijabilni
sustav diferencijalnih jednadžbi. U slučaju da je referentno stanje i
upravljanje konstantno, tada su matraice A i B konstantne, odnosno imamo
linearni vremenski – invarijanti sustav.
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