Witold O.

Zbadać stabilność układu zakmniętego

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

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

$$G_{z}(s) = \frac{\frac{\frac{3}{s + 1}}{1 + \frac{3}{s + 1}}*(\frac{2}{s + 1} + \frac{\frac{2}{s + 2}}{\frac{3}{s + 1}})}{1 + \left\lbrack \frac{\frac{3}{s + 1}}{1 + \frac{3}{s + 1}}*(\frac{2}{s + 1} + \frac{\frac{2}{s + 2}}{\frac{3}{s + 1}}) \right\rbrack*\frac{3}{s}} = \frac{\frac{\frac{3}{s + 1}}{\frac{s + 4}{s + 1}}*(\frac{2}{s + 1} + \frac{2(s + 1)}{3(s + 2)})}{1 + \frac{\frac{3}{s + 1}}{\frac{s + 4}{s + 1}}*(\frac{2}{s + 1} + \frac{2(s + 1)}{3(s + 2)})*\frac{3}{s}} = \frac{\frac{3}{s + 4}*\frac{6\left( ss + 3 \right) + 2({s + 1)}^{2}\text{\ \ }s(}{3\left( s + 1 \right)*(s + 3)}}{1 + \frac{3}{s + 4}*\frac{6\left( ss + 3 \right) + 2({s + 1)}^{2}\text{\ \ }s(}{3\left( s + 1 \right)*(s + 3)}*\frac{3}{s}} = \frac{\frac{6\left( ss + 3 \right) + 2({s + 1)}^{2}\text{\ \ }s(}{\left( s + 1 \right)*\left( s + 3 \right)*(s + 4)}}{1 + \frac{18\left( ss + 3 \right) + 6({s + 1)}^{2}\text{\ \ }s(}{s*\left( s + 1 \right)*\left( s + 3 \right)*(s + 4)}} = \frac{\frac{6\left( ss + 3 \right) + 2({s + 1)}^{2}\text{\ \ }s(}{\left( s + 1 \right)*\left( s + 3 \right)*(s + 4)}}{\frac{s^{4} + 8s^{3} + 25s^{2} + 42s + 60}{s*\left( s + 1 \right)*\left( s + 4 \right)*(s + 3)}} = \frac{6s\left( ss + 3 \right) + 2s({s + 1)}^{2}\ }{s^{4} + 8s^{3} + 25s^{2} + 42s + 60}\ $$

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 + 8s³ + 25s² + 42s + 60

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

42	60	0
8	25	42
0	1	8

Δ₁=a₁=42



Δ₂=

a1	a0
a3	a4

42	60
8	25

Δ₂= 570

Δ₃=

42	60	0
8	25	42
0	1	8

Δ₃=2796
Δ_i >0 zatem układ jest stabilny


Zbadać własności dynamiczne układu nieliniowego przedstawionego na rys.

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