${\displaystyle \angle e^{-jwT}=-wT}$

${\displaystyle |e^{-jwT}|=1\,}$

${\displaystyle 1+K\left[1+{\frac {1}{T_{i}s}}\right]{\frac {0.4}{280s+9.81}}e^{-sT}=0}$

${\displaystyle {\frac {0.4KT_{i}s+0.4K}{280T_{i}s^{2}+9.81T_{i}s}}e^{-sT}}$

${\displaystyle }$

${\displaystyle }$

${\displaystyle G_{R}(s)=K\left[1+{\frac {1}{T_{i}s}}\right]}$

${\displaystyle {\frac {G_{R}(s)G(s)}{1+G_{R}(s)G(s)}}={\frac {K\left[1+{\frac {1}{T_{i}s}}\right]{\frac {0.4}{280s+9.81}}}{1+K\left[1+{\frac {1}{T_{i}s}}\right]{\frac {0.4}{280s+9.81}}}}=}$

${\displaystyle ={\frac {0.4KT_{i}s+0.4K}{280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K}}}$

Karakteristisk ekv: ${\displaystyle 280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K\,}$

Krav: ${\displaystyle K>0\,}$

${\displaystyle D_{1}={\begin{vmatrix}(9.81+0.4K)T_{i}\end{vmatrix}}=(9.81+0.4K)T_{i}>0\Rightarrow T_{i}>0}$

${\displaystyle E(s)=R(s)-Y(s)\,}$

${\displaystyle Y(s)=G(s)\cdot G_{R}(s)\cdot E(s)}$

${\displaystyle E(s)=R(s)-G(s)\cdot G_{R}(s)\cdot E(s)}$

${\displaystyle E(s)={\frac {1}{1+G_{R}(s)\cdot G(s)}}R(s)={\frac {1}{1+K\left[1+{\frac {1}{T_{i}s}}\right]\cdot {\frac {0.4}{280s+9.81}}}}R(s)=}$

${\displaystyle ={\frac {(280s+9.81)T_{i}s}{280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K}}R(s)}$

Stegstörning: ${\displaystyle L[steg]={\frac {1}{s}}}$

${\displaystyle E(s)={\frac {(280s+9.81)T_{i}s}{280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K}}{\frac {1}{s}}}$

${\displaystyle \lim _{t\to \infty }E(t)=\lim _{s\to 0}sE(s)=\lim _{s\to 0}{\frac {(280s+9.81)T_{i}s}{280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K}}=}$

${\displaystyle ={\frac {0}{0.4K}}=0}$

Impuls: ${\displaystyle L[impuls]=1\,}$

${\displaystyle E(s)={\frac {(280s+9.81)T_{i}s}{280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K}}\cdot 1}$

${\displaystyle \lim _{t\to \infty }E(t)=\lim _{s\to 0}sE(s)=0\cdot {\frac {0}{0.4K}}=0}$

${\displaystyle {\frac {G_{R}(s)G(s)}{1+G_{R}(s)G(s)}}={\frac {B(s)}{s^{2}+2\xi w_{n}s+w_{n}^{2}}}}$

${\displaystyle {\frac {0.4KT_{i}s+0.4K}{280T_{i}s^{2}+(9.81+0.4K)T_{i}s+0.4K}}={\frac {\frac {0.4KT_{i}s+0.4K}{280T_{i}}}{s^{2}+{\frac {(9.81+0.4K)}{280}}s+{\frac {0.4K}{280T_{i}}}}}}$

${\displaystyle \Rightarrow w_{n}^{2}={\frac {0.4K}{280T_{i}}}\Rightarrow w_{n}={\sqrt {\frac {0.4K}{280T_{i}}}}}$

${\displaystyle \Rightarrow 2\xi w_{n}s={\frac {(9.81+0.4K)}{280}}s\Rightarrow \xi ={\frac {(9.81+0.4K)}{280}}\cdot {\frac {1}{\sqrt {\frac {0.4K}{280T_{i}}}}}\cdot {\frac {1}{2}}}$

${\displaystyle \xi ={\frac {0.0472456\cdot {\sqrt {T_{i}}}\cdot (0.4K+9.81)}{\sqrt {K}}}}$

${\displaystyle {\frac {dV}{dt}}=q_{in}(t)-q_{ut}(t)=\alpha u-\beta {\sqrt {h}}}$

${\displaystyle q_{in}(t)=4u\cdot 10^{-6}\ m^{3}/s}$ (från lab 1)

${\displaystyle q_{ut}(t)=a{\sqrt {2gh}}}$

${\displaystyle a=0.14\cdot 10^{-4}\ m^{2}}$ arbetspunkt (5 V, 0.1 m)

${\displaystyle \alpha =4\cdot 10^{-6}\ m^{3}/s}$

${\displaystyle \beta =0.14\cdot 10^{-4}{\sqrt {2g}}}$

${\displaystyle A{\frac {d\Delta h}{dt}}=\alpha \Delta u-\gamma \Delta h}$

${\displaystyle A=2.8\cdot 10^{-3}\ m^{2}}$

${\displaystyle \gamma =a{\sqrt {\frac {g}{2h_{0}}}}=9.81\cdot 10^{-5}}$

${\displaystyle L\left[A{\frac {d\Delta h}{dt}}=\alpha \Delta u-\gamma \Delta h\right]\Rightarrow AsH(s)=\alpha U(s)-\gamma H(s)}$

${\displaystyle H(s)={\frac {\alpha }{As+\gamma }}U(s)}$

${\displaystyle G(s)={\frac {\alpha }{As+\gamma }}={\frac {4\cdot 10^{-6}}{2.8\cdot 10^{-3}s+9.81\cdot 10^{-5}}}={\frac {0.4}{280s+9.81}}}$

${\displaystyle E(s)=R(s)-Y(s)\,}$

${\displaystyle Y(s)=G(s)\cdot K\cdot E(s)}$

${\displaystyle E(s)=R(s)-G(s)\cdot K\cdot E(s)}$

${\displaystyle E(s)={\frac {1}{1+K\cdot G(s)}}R(s)={\frac {1}{1+K\cdot {\frac {0.4}{280s+9.81}}}}R(s)=}$

${\displaystyle ={\frac {280s+9.81}{280s+9.81+0.4K}}R(s)}$

Karakteristisk ekv: ${\displaystyle 280s+9.81+0.4K\,}$

Hurwitz: ${\displaystyle D_{1}={\begin{vmatrix}9.81+0.4K\end{vmatrix}}=9.81+0.4K>0\Rightarrow K>-{\frac {981}{40}}}$

${\displaystyle E(s)=G_{2}(s)Q(s)\,}$

${\displaystyle E(s)={\frac {280s+9.81}{280s+9.81+0.4K}}Q(s)}$

Stegstörning: ${\displaystyle L[steg]={\frac {1}{s}}}$

${\displaystyle E(s)={\frac {280s+9.81}{280s+9.81+0.4K}}{\frac {1}{s}}}$

${\displaystyle \lim _{t\to \infty }E(t)=\lim _{s\to 0}sE(s)=\lim _{s\to 0}{\frac {280s+9.81}{280s+9.81+0.4K}}={\frac {9.81}{9.81+0.4K}}}$

Impuls: ${\displaystyle L[impuls]=1\,}$

${\displaystyle E(s)={\frac {280s+9.81}{280s+9.81+0.4K}}\cdot 1}$

${\displaystyle \lim _{t\to \infty }E(t)=\lim _{s\to 0}sE(s)=0\cdot {\frac {9.81}{9.81+0.4K}}=0}$