Kamil Z.

Zbadać stabilność układu zakmniętego

$$G_{1}(s) = \frac{5}{s + 1}\text{\ \ \ \ \ \ \ }G_{2}(s) = \frac{1}{s + 4}\text{\ \ \ \ \ \ \ \ \ \ \ }G_{3}(s) = \frac{2}{s + 2}\text{\ \ \ \ \ \ \ \ \ }G_{4}(s) = \frac{1}{s}$$

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

$$G_{z}(s) = \frac{\frac{\frac{5}{s + 1}}{1 + \frac{5}{s + 1}}*(\frac{1}{s + 4} + \frac{\frac{2}{s + 2}}{\frac{5}{s + 1}})}{1 + \left\lbrack \frac{\frac{5}{s + 1}}{1 + \frac{5}{s + 1}}*(\frac{1}{s + 4} + \frac{\frac{2}{s + 2}}{\frac{5}{s + 1}}) \right\rbrack*\frac{1}{s}} = \frac{\frac{\frac{5}{s + 1}}{\frac{s + 6}{s + 1}}*(\frac{1}{s + 4} + \frac{2(s + 1)}{5(s + 2)})}{1 + \frac{\frac{1}{s + 1}}{\frac{s + 2}{s + 1}}*\left( \frac{1}{s + 4} + \frac{2(s + 1)}{5(s + 2)} \right)*\frac{1}{s}} = \frac{\frac{5}{s + 6}*\frac{5\left( ss + 2 \right) + 2(s\text{sk} + 1\hat{})111qz)*(s + 4)s(}{5\left( s + 2 \right)*(s + 4)}}{1 + \frac{5}{s + 6}*\frac{5\left( ss + 2 \right) + 2(s\text{sk} + 1\hat{})111qz)*(s + 4)s(}{5\left( s + 2 \right)*(s + 4)}*\frac{3}{s}} = \frac{\frac{5\left( ss + 2 \right) + 2(s\text{sk} + 1\hat{})111qz)*(s + 4)s(}{\left( s + 2 \right)*\left( s + 4 \right)*(s + 6)}}{1 + \frac{5\left( ss + 2 \right) + 2(s\text{sk} + 1\hat{})111qz)*(s + 4)s(}{\left( s + 2 \right)*\left( s + 4 \right)*(s + 6)}} = \frac{\frac{5\left( ss + 2 \right) + 2(s\text{sk} + 1\hat{})111qz)*(s + 4)s(}{\left( s + 2 \right)*\left( s + 4 \right)*(s + 6)}}{\frac{s^{4} + 13s^{3} + 56s^{2} + 87s + 18}{s*\left( s + 2 \right)*\left( s + 4 \right)*(s + 6)}} = \frac{5s\left( ss + 2 \right) + 2s(s\text{sk} + 1\hat{})111qz)*(s + 4)\ }{s^{4} + 13s^{3} + 56s^{2} + 87s + 18}\ $$

Badamy stabilność układu

$$G_{z}\left( s \right) = \frac{L\left( s \right)}{M(s)} = \frac{L(s)}{s^{4} + 13s^{3} + 56s^{2} + 87s + 18}$$

M(s) = s⁴ + 13s³ + 56s² + 87s + 18

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

87	18	0
13	56	13
0	1	13

Δ₁=a₁=87



Δ₂=

a1	a0
a3	a4

87	18
13	56

Δ₂= 4638

Δ₃=

87	18	0
13	56	13
0	1	13

Δ₃=59163
Δ_i >0 zatem układ jest stabilny


Zbadać własności dynamiczne układu nieliniowego przedstawionego na rys.

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