Pawlik

Zbadać stabilność układu zakmniętego

$$G_{1}(s) = \frac{3}{s + 2}\text{\ \ \ \ \ \ \ }G_{2}(s) = \frac{3}{s + 1}\text{\ \ \ \ \ \ \ \ \ \ \ }G_{3}(s) = \frac{2}{s + 4}\text{\ \ \ \ \ \ \ \ \ }G_{4}(s) = \frac{1}{s}$$

$$G_{z}(s) = \frac{\frac{G_{1}(s)}{1 + G_{1}(s)}*\left( G_{2}(s) + \frac{G_{3}(s)}{G_{1}(s)} \right)}{1 + \left\lbrack \frac{G_{1}(s)}{1 + G_{1}(s)}*\left( G_{2}(s) + \frac{G_{3}(s)}{G_{1}(s)} \right) \right\rbrack*G_{4}(s)}$$

$$G_{z}(s) = \frac{\frac{\frac{3}{s + 2}}{1 + \frac{3}{s + 2}}*(\frac{3}{s + 1} + \frac{\frac{2}{s + 4}}{\frac{3}{s + 2}})}{1 + \left\lbrack \frac{\frac{3}{s + 2}}{1 + \frac{3}{s + 2}}*(\frac{3}{s + 1} + \frac{\frac{2}{s + 4}}{\frac{3}{s + 2}}) \right\rbrack*\frac{1}{s}} = \frac{\frac{\frac{3}{s + 2}}{\frac{s + 5}{s + 1}}*(\frac{3}{s + 1} + \frac{2(s + 2)}{3(s + 4)})}{1 + \frac{\frac{3}{s + 2}}{\frac{s + 5}{s + 1}}*(\frac{3}{s + 1} + \frac{2(s + 2)}{3(s + 4)})*\frac{1}{s}} = \frac{\frac{3}{s + 5}*\frac{9\left( ss + 4 \right) + 2({s + 2)}^{2}\text{\ \ }s(}{3\left( s + 1 \right)*(s + 4)}}{1 + \frac{3}{s + 5}*\frac{9\left( ss + 4 \right) + 2({s + 2)}^{2}\text{\ \ }s(}{3\left( s + 1 \right)*(s + 4)}*\frac{1}{s}} = \frac{\frac{9\left( ss + 4 \right) + 2({s + 2)}^{2}\text{\ \ }s(}{\left( s + 1 \right)*\left( s + 4 \right)*(s + 5)}}{1 + \frac{9\left( ss + 4 \right) + 2({s + 2)}^{2}\text{\ \ }s(}{s\left( s + 1 \right)*\left( s + 4 \right)*(s + 5)}} = \frac{\frac{9\left( ss + 4 \right) + 2({s + 2)}^{2}\text{\ \ }s(}{\left( s + 1 \right)*\left( s + 4 \right)*(s + 5)}}{\frac{s^{4} + 10s^{3} + 31s^{2} + 33s + 42}{s*\left( s + 1 \right)*\left( s + 4 \right)*(s + 5)}} = \frac{9s\left( ss + 4 \right) + 2s({s + 2)}^{2}\ }{s^{4} + 10s^{3} + 31s^{2} + 33s + 42}\ $$

Badamy stabilność układu

$$G_{z}\left( s \right) = \frac{L\left( s \right)}{M(s)} = \frac{L(s)}{s^{4} + 8s^{3} + 25s^{2} + 42s + 60}$$

M(s) = s⁴ + 10s³ + 31s² + 33s + 42

2. Wszystkie wyznaczniki Δ­_i muszą być większe od zera aby układ był stabilny

33	42	0
10	31	33
0	1	10

Δ₁=a₁=42

Δ₂= 603

Δ₃=4941
Δ_i >0 zatem układ jest stabilny


Zbadać własności dynamiczne układu nieliniowego przedstawionego na rys.

$${x\left( t \right) = Asin\omega t\backslash n}{B = 1\ \ \ \ \ \ \ \ \ \ G\left( s \right) = \frac{2}{s*\left( s + 1 \right)^{2}} = \frac{2}{s^{3} + 2s^{2} + s}\backslash n}{I = \frac{4B}{\text{πA}}\backslash n}{k\left( s \right) = G\left( s \right)*I\left( A \right)\backslash n}{G_{z} = \frac{I\left( A \right)G\left( s \right)}{1 + I\left( A \right)G\left( s \right)}\backslash n}$$