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Zbadać stabilność układu zakmniętego

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

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

$$G_{z}(s) = \frac{\frac{\frac{1}{s + 2}}{1 + \frac{1}{s + 2}}*(\frac{2}{s + 3} + \frac{\frac{1}{s}}{\frac{1}{s + 2}})}{1 + \left\lbrack \frac{\frac{1}{s + 2}}{1 + \frac{1}{s + 2}}*(\frac{2}{s + 3} + \frac{\frac{1}{s}}{\frac{1}{s + 2}}) \right\rbrack*\frac{2}{s}} = \frac{\frac{\frac{1}{s + 2}}{\frac{s + 3}{s + 2}}*(\frac{2}{s + 3} + \frac{s + 2}{s})}{1 + \frac{\frac{1}{s + 2}}{\frac{s + 3}{s + 2}}*(\frac{2}{s + 3} + \frac{s + 2}{s})*\frac{2}{s}} = \frac{\frac{1}{s + 3}*\frac{2s + \left( ss + 2 \right)*(s + 3)\text{\ \ }s(}{s*(s + 3)}}{1 + \frac{1}{s + 3}*\frac{2s + \left( ss + 2 \right)*(s + 3)\ \ s(}{s*\left( s + 3 \right)^{2}}*\frac{2}{s}} = \frac{\frac{2s + \left( ss + 2 \right)*(s + 3)\ \ s(}{s*\left( s + 3 \right)^{2}}}{1 + \frac{4s + 2\left( ss + 2 \right)*(s + 3)\ \ s(}{s^{2}*\left( s + 3 \right)^{2}}} = \frac{\frac{2s + \left( ss + 2 \right)*(s + 3)\ \ s(}{s*\left( s + 3 \right)^{2}}}{\frac{s^{4} + 6s^{3} + 12s^{2} + 14s + 12}{s^{2}*\left( s + 3 \right)^{2}}} = \frac{2s + \left( ss + 2 \right)*(s + 3\ }{s^{4} + 6s^{3} + 12s^{2} + 14s + 12}\ $$

Badamy stabilność układu

$$G_{z}\left( s \right) = \frac{L\left( s \right)}{M(s)} = \frac{L(s)}{s^{4} + 6s^{3} + 12s^{2} + 14s + 12}$$

M(s) = s⁴ + 6s³ + 12s² + 14s + 12

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

14	12	0
6	12	14
0	1	6

Δ₁=a₁=14

Δ₂= 96

Δ₃=380
Δ_i >0 zatem układ jest stabilny


Zbadać własności dynamiczne układu nieliniowego przedstawionego na rys.

$${x\left( t \right) = Asin\omega t\backslash n}{B = 1\ \ \ \ \ \ \ \ \ \ G\left( s \right) = \frac{2}{s*\left( 0,5s + 1 \right)^{*}(2s + 10)} = \frac{2}{s^{3} + 7s^{2} + 10s}\backslash n}{I = \frac{4B}{\text{πA}}\backslash n}{k\left( s \right) = G\left( s \right)*I\left( A \right)\backslash n}{G_{z} = \frac{I\left( A \right)G\left( s \right)}{1 + I\left( A \right)G\left( s \right)}\backslash n}$$