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Zbadać stabilność układu zakmniętego

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

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

$$G_{z}(s) = \frac{\frac{\frac{1}{s + 1}}{1 + \frac{1}{s + 1}}*(\frac{4}{s + 2} + \frac{\frac{2}{s}}{\frac{1}{s + 1}})}{1 + \left\lbrack \frac{\frac{1}{s + 1}}{1 + \frac{1}{s + 1}}*(\frac{4}{s + 2} + \frac{\frac{2}{s}}{\frac{1}{s + 1}}) \right\rbrack*\frac{3}{s}} = \frac{\frac{\frac{1}{s + 1}}{\frac{s + 2}{s + 1}}*(\frac{4}{s + 2} + \frac{2(s + 1)}{s})}{1 + \frac{\frac{1}{s + 1}}{\frac{s + 2}{s + 1}}*\left( \frac{4}{s + 2} + \frac{2(s + 1)}{s} \right)*\frac{3}{s}} = \frac{\frac{1}{s + 2}*\frac{2\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 4ss(}{\left( s + 2 \right)*s}}{1 + \frac{1}{s + 2}*\frac{2\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 4ss(}{\left( s + 2 \right)*s}*\frac{3}{s}} = \frac{\frac{2\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 4ss(}{\left( s + 2 \right)^{2}*s}}{1 + \frac{6\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 12ss(}{\left( s + 2 \right)^{2}*s^{2}}} = \frac{\frac{2\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 4ss(}{\left( s + 2 \right)^{2}*s}}{\frac{s^{4} + 4s^{3} + 10s^{2} + 30s + 12}{\left( s + 2 \right)^{2}*s^{2}}} = \frac{2s\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 4s^{2}\ }{s^{4} + 4s^{3} + 10s^{2} + 30s + 12}\ $$

$$G_{z}(s) = \frac{2s\left( ss + 1 \right)*(s\text{sk} + \hat{})2) + 4s^{2}\ }{s^{4} + 4s^{3} + 10s^{2} + 30s + 12}$$

Badamy stabilność układu

$$G_{z}\left( s \right) = \frac{L\left( s \right)}{M(s)} = \frac{L(s)}{s^{4} + 4s^{3} + 10s^{2} + 30s + 12}$$

M(s) = s⁴ + 4s³ + 10s² + 30s + 12

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

30	12	0
4	10	30
0	1	6

Δ₁=a₁=30



Δ₂=

a1	a0
a3	a4

30	12
4	10

Δ₂= 252

Δ₃=

30	12	0
4	10	30
0	1	6

Δ₃=612
Δ_i >0 zatem układ jest stabilny


Zbadać własności dynamiczne układu nieliniowego przedstawionego na rys.

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