V π ( S t ) {\displaystyle V^{\pi }(S_{t})} R ( τ ) {\displaystyle R(\tau )} P ( τ | π ) {\displaystyle P(\tau |\pi )} J ( π ) {\displaystyle J(\pi )} π θ ( a t | s t ) {\displaystyle \pi _{\theta }(a_{t}|s_{t})} σ = ξ ω 0 {\displaystyle \sigma =\xi {\omega }_{0}} v g ( t ) = u ( t ) {\displaystyle v_{g}(t)=u(t)} ω 0 = 1 L C {\displaystyle {\omega }_{0}={\frac {1}{\sqrt {LC}}}} d 2 v c ( t ) d t 2 + R L d v c ( t ) d t + 1 L C v c ( t ) = v g ( t ) {\displaystyle {\frac {d^{2}v_{c}(t)}{dt^{2}}}+{\frac {R}{L}}{\frac {dv_{c}(t)}{dt}}+{\frac {1}{LC}}v_{c}(t)=v_{g}(t)} ω d = 1 − ξ 2 ω 0 {\displaystyle {\omega }_{d}={\sqrt {1-{\xi }^{2}}}{\omega }_{0}} x ( t ) = 1 − ( 1 + ω 0 t ) e − ω 0 t {\displaystyle x(t)=1-(1+\omega _{0}t)e^{-\omega _{0}t}} x ( t ) = 1 − e − ξ ω n t 1 − ξ 2 ( 1 − ξ 2 cos ω d t + ξ sin ω d t ) {\displaystyle x(t)=1-{\frac {e^{-{\xi }\omega _{n}t}}{\sqrt {1-\xi ^{2}}}}({\sqrt {1-\xi ^{2}}}\cos \omega _{d}t+{\xi }\sin \omega _{d}t)} x ( t ) = 1 − e − ξ ω 0 t ( cos ω d t + ξ 1 − ξ 2 sin ω d t ) {\displaystyle x(t)=1-{e^{-{\xi }\omega _{0}t}}(\cos \omega _{d}t+{\frac {\xi }{\sqrt {1-\xi ^{2}}}}\sin \omega _{d}t)} x ( t ) = 1 − e − ξ ω n t 1 − ξ 2 sin ( ω d t + ϕ ) {\displaystyle x(t)=1-{\frac {e^{-{\xi }\omega _{n}t}}{\sqrt {1-\xi ^{2}}}}\sin(\omega _{d}t+\phi )} x ( t ) = 1 + a 1 ∗ e s 1 ∗ t + a 2 ∗ e s 2 ∗ t {\displaystyle x(t)=1+a_{1}*e^{s_{1}*t}+a_{2}*e^{s_{2}*t}} a 1 = ω 0 2 s 1 ( s 1 − s 2 ) {\displaystyle a_{1}={\frac {{\omega }_{0}^{2}}{s_{1}(s_{1}-s_{2})}}} a 2 = ω 0 2 s 2 ( s 2 − s 1 ) {\displaystyle a_{2}={\frac {{\omega }_{0}^{2}}{s_{2}(s_{2}-s_{1})}}} s 1 = − ξ ω 0 + ω 0 ξ 2 − 1 {\displaystyle s_{1}=-{\xi }{\omega }_{0}+\omega _{0}{\sqrt {{\xi }^{2}-1}}} s 2 = − ξ ω 0 − ω 0 ξ 2 − 1 {\displaystyle s_{2}=-{\xi }{\omega }_{0}-\omega _{0}{\sqrt {{\xi }^{2}-1}}} ω d = ω 0 1 − ξ 2 {\displaystyle {\omega }_{d}=\omega _{0}{\sqrt {1-{\xi }^{2}}}} s 1 = − R 2 L + ( R 2 L ) 2 − 1 L C {\displaystyle s_{1}=-{\frac {R}{2L}}+{\sqrt {({\frac {R}{2L}})^{2}-{\frac {1}{LC}}}}} s 2 = − R 2 L − ( R 2 L ) 2 − 1 L C {\displaystyle s_{2}=-{\frac {R}{2L}}-{\sqrt {({\frac {R}{2L}})^{2}-{\frac {1}{LC}}}}} s 1 = − ξ ω 0 + ω 0 ξ 2 − 1 {\displaystyle s_{1}=-{\xi }{\omega }_{0}+\omega _{0}{\sqrt {{\xi }^{2}-1}}} s 2 = − ξ ω 0 − ω 0 ξ 2 − 1 {\displaystyle s_{2}=-{\xi }{\omega }_{0}-\omega _{0}{\sqrt {{\xi }^{2}-1}}} s 2 = − R 2 L − ( R 2 L ) 2 − 1 L C {\displaystyle s_{2}=-{\frac {R}{2L}}-{\sqrt {({\frac {R}{2L}})^{2}-{\frac {1}{LC}}}}} A 1 = s 2 s 2 − s 1 v 0 {\displaystyle A_{1}={\frac {s_{2}}{s_{2}-s_{1}}}v_{0}} A 2 = s 1 s 1 − s 2 v 0 {\displaystyle A_{2}={\frac {s_{1}}{s_{1}-s_{2}}}v_{0}} v c ( t ) = v 0 ( s 2 s 2 − s 1 e s 1 t + s 1 s 1 − s 2 e s 2 t ) {\displaystyle v_{c}(t)=v_{0}({\frac {s_{2}}{s_{2}-s_{1}}}e^{s_{1}t}+{\frac {s_{1}}{s_{1}-s_{2}}}e^{s_{2}t})} A 1 + A 2 = 0 {\displaystyle A_{1}+A_{2}=0} s 1 A 1 + s 2 A 2 = 0 {\displaystyle s_{1}A_{1}+s_{2}A_{2}=0} x ( 0 ) = v 0 {\displaystyle x(0)=v_{0}} d x ( 0 ) d t = 0 {\displaystyle {\frac {dx(0)}{dt}}=0} τ = L R {\displaystyle \tau ={\frac {L}{R}}} v c ( t ) = A 1 e s 1 t + A 2 e s 2 t {\displaystyle v_{c}(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}} ξ = R 2 C L {\displaystyle {\xi }={\frac {R}{2}}{\sqrt {\frac {C}{L}}}} i ( t ) = C d v c ( t ) d t {\displaystyle i(t)=C{\frac {dv_{c}(t)}{dt}}} v c ( t ) + R i ( t ) + L d i ( t ) d t = 0 {\displaystyle v_{c}(t)+Ri(t)+L{\frac {di(t)}{dt}}=0} v c ( t ) + R C d v c ( t ) d t + L C d 2 v c ( t ) d t 2 = 0 {\displaystyle v_{c}(t)+RC{\frac {dv_{c}(t)}{dt}}+LC{\frac {d^{2}v_{c}(t)}{dt^{2}}}=0} d 2 v c ( t ) d t 2 + R L d v c ( t ) d t + 1 L C v c ( t ) = 0 {\displaystyle {\frac {d^{2}v_{c}(t)}{dt^{2}}}+{\frac {R}{L}}{\frac {dv_{c}(t)}{dt}}+{\frac {1}{LC}}v_{c}(t)=0} s 2 + R L s + 1 L C = 0 {\displaystyle s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0} − v 1 + v i n + v R A = 0 {\displaystyle -v_{1}+v_{in}+v_{R_{A}}=0} Y ( s ) R ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle {\frac {Y(s)}{R(s)}}={\frac {G(s)}{1+G(s)H(s)}}} y = e − 0.2 t s i n ( t ) {\displaystyle y=e^{-0.2t}sin(t)} v R A = R A R A + R F v 2 {\displaystyle v_{R_{A}}={\frac {R_{A}}{R_{A}+R_{F}}}v_{2}} v 2 = A v i n {\displaystyle v_{2}=Av_{in}} v 1 = v 2 A + R A R A + R F v 2 {\displaystyle v_{1}={\frac {v_{2}}{A}}+{\frac {R_{A}}{R_{A}+R_{F}}}v_{2}} v 2 = 1 1 A + R A R A + R F v 1 {\displaystyle v_{2}={\frac {1}{{\frac {1}{A}}+{\frac {R_{A}}{R_{A}+R_{F}}}}}v_{1}} v 2 = ( 1 + R F R A ) v 1 {\displaystyle v_{2}=(1+{\frac {R_{F}}{R_{A}}})v_{1}} v o = ( 1 + R F R A ) v T {\displaystyle v_{o}=(1+{\frac {R_{F}}{R_{A}}})v_{T}} v o = ( 1 + R F R A ) ( R T R 1 v 1 + R T R 2 v 2 + R T R 3 v 3 ) {\displaystyle v_{o}=(1+{\frac {R_{F}}{R_{A}}})({\frac {R_{T}}{R_{1}}}v_{1}+{\frac {R_{T}}{R_{2}}}v_{2}+{\frac {R_{T}}{R_{3}}}v_{3})} R T = R 1 / / R 2 / / R 3 = 1 1 R 1 + 1 R 2 + 1 R 3 {\displaystyle R_{T}=R_{1}//R_{2}//R_{3}={\frac {1}{{\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+{\frac {1}{R_{3}}}}}} v T = R T ( v 1 R 1 + v 2 R 2 + v 3 R 3 ) {\displaystyle v_{T}=R_{T}({\frac {v_{1}}{R_{1}}}+{\frac {v_{2}}{R_{2}}}+{\frac {v_{3}}{R_{3}}})} v o = − R F R T v T {\displaystyle v_{o}=-{\frac {R_{F}}{R_{T}}}v_{T}}
v o = − R F ( v 1 R 1 + v 2 R 2 + v 3 R 3 ) {\displaystyle v_{o}=-{R_{F}}({\frac {v_{1}}{R_{1}}}+{\frac {v_{2}}{R_{2}}}+{\frac {v_{3}}{R_{3}}})} v i n + R F i + A v i n = 0 {\displaystyle v_{in}+R_{F}i+Av_{in}=0} − v 1 + R A i + R F i + v 2 = 0 {\displaystyle -v_{1}+R_{A}i+R_{F}i+v_{2}=0} v 2 = A v i n {\displaystyle v_{2}=Av_{in}} i = − A + 1 R F v i n = − A + 1 A v 2 R F {\displaystyle i=-{\frac {A+1}{R_{F}}}v_{in}=-{\frac {A+1}{A}}{\frac {v_{2}}{R_{F}}}} v 2 = − R F R A + 1 A ( R A + R B ) v 1 = − R F R A v 1 {\displaystyle v_{2}={\frac {-R_{F}}{R_{A}+{\frac {1}{A}}(R_{A}+R_{B})}}v_{1}=-{\frac {R_{F}}{R_{A}}}v_{1}} v 2 = v 1 − ( R A + R F ) i {\displaystyle v_{2}=v_{1}-(R_{A}+R_{F})i} = v 1 + R A + R F R F A + 1 A v 2 {\displaystyle =v_{1}+{\frac {R_{A}+R_{F}}{R_{F}}}{\frac {A+1}{A}}v_{2}} − v 1 + R A i + R F i + A v 1 = 0 {\displaystyle -v_{1}+R_{A}i+R_{F}i+Av_{1}=0} ( R A + R F ) i = ( 1 − A ) v 1 {\displaystyle (R_{A}+R_{F})i=(1-A)v_{1}} v i n = 1 A + 1 v i {\displaystyle v_{in}={\frac {1}{A+1}}v_{i}} v 2 = A v i n = A A + 1 v 1 = v 1 {\displaystyle v_{2}=Av_{in}={\frac {A}{A+1}}v_{1}=v_{1}} − v 1 + v i n + A v i n = 0 {\displaystyle -v_{1}+v_{in}+Av_{in}=0} v = 1 C ∫ i d t {\displaystyle v={\frac {1}{C}}\int {idt}} c = a 2 + b 2 {\displaystyle c={\sqrt {a^{2}+b^{2}}}} − E 1 + R 1 I 1 + R 3 ( I 1 − I 2 ) = 0 {\displaystyle -E_{1}+R_{1}I_{1}+R_{3}(I_{1}-I_{2})=0} R 3 ( I 2 − I 1 ) + R 2 I 2 + E 2 = 0 {\displaystyle R_{3}(I_{2}-I_{1})+R_{2}I_{2}+E_{2}=0} ( R 1 + R 3 ) I 1 − R 3 I 2 = E 1 {\displaystyle (R_{1}+R_{3})I_{1}-R_{3}I_{2}=E_{1}} − R 3 I 1 + ( R 2 + R 3 ) I 2 = − E 2 {\displaystyle -R_{3}I_{1}+(R_{2}+R_{3})I_{2}=-E_{2}} R 11 1 1 + R 12 I 2 = V 1 {\displaystyle R_{11}1_{1}+R_{12}I_{2}=V_{1}} R 21 1 1 + R 22 I 2 = V 2 {\displaystyle R_{21}1_{1}+R_{22}I_{2}=V_{2}} R 2 I 1 + R 4 ( I 1 − I 2 ) + R 3 ( I 1 − I 3 ) = 0 {\displaystyle R_{2}I_{1}+R_{4}(I_{1}-I_{2})+R_{3}(I_{1}-I_{3})=0} R 4 ( I 2 − I 1 ) + R 5 I 2 + R 6 ( I 2 − I 3 ) = 0 {\displaystyle R_{4}(I_{2}-I_{1})+R_{5}I_{2}+R_{6}(I_{2}-I_{3})=0} R 1 I 3 + R 3 ( I 3 − I 1 ) + R 6 ( I 3 − I 2 ) − V = 0 {\displaystyle R_{1}I_{3}+R_{3}(I_{3}-I_{1})+R_{6}(I_{3}-I_{2})-V=0} ( R 2 + R 3 + R 4 ) I 1 − R 4 I 2 − R 3 I 3 = 0 {\displaystyle (R_{2}+R_{3}+R_{4})I_{1}-R_{4}I_{2}-R_{3}I_{3}=0} − R 4 I 1 + ( R 2 + R 4 + R 6 ) I 2 − R 6 I 3 = 0 {\displaystyle -R_{4}I_{1}+(R_{2}+R_{4}+R_{6})I_{2}-R_{6}I_{3}=0} − R 3 I 1 − R 6 I 2 + ( R 1 + R 3 + R 6 ) I 3 = V {\displaystyle -R_{3}I_{1}-R_{6}I_{2}+(R_{1}+R_{3}+R_{6})I_{3}=V}
I a = G 1 V 1 + G 2 ( V 1 − V 2 ) + G 3 ( V 1 − V 3 ) {\displaystyle I_{a}=G_{1}V_{1}+G_{2}(V_{1}-V_{2})+G_{3}(V_{1}-V_{3})} G 2 ( V 1 − V 2 ) = G 4 V 2 + G 5 ( V 2 − V 3 ) {\displaystyle G_{2}(V_{1}-V_{2})=G_{4}V_{2}+G_{5}(V_{2}-V_{3})} G 2 ( V 1 − V 3 ) + G 5 ( V 2 − V 3 ) = G 6 V 3 + I b {\displaystyle G_{2}(V_{1}-V_{3})+G_{5}(V_{2}-V_{3})=G_{6}V_{3}+I_{b}}
( 2 + 2 + 1 ) V 1 − 2 V 2 − 2 V 3 = − 5 {\displaystyle (2+2+1)V_{1}-2V_{2}-2V_{3}=-5} − 2 V 1 + ( 2 + 1 + 3 ) V 2 − 3 V 3 = 0 {\displaystyle -2V_{1}+(2+1+3)V_{2}-3V_{3}=0} − 2 V 1 − 3 V 2 + ( 1 + 3 + 1 ) V 3 = 7 {\displaystyle -2V_{1}-3V_{2}+(1+3+1)V_{3}=7}
( 1 + 1 2 + 1 3 ) V 1 − 1 2 V 2 − 1 3 V 3 = 10 {\displaystyle (1+{\frac {1}{2}}+{\frac {1}{3}})V_{1}-{\frac {1}{2}}V_{2}-{\frac {1}{3}}V_{3}=10} − 1 2 V 1 + ( 1 2 + 1 5 + 1 4 ) V 2 − 1 4 V 3 = 0 {\displaystyle -{\frac {1}{2}}V_{1}+({\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{4}})V_{2}-{\frac {1}{4}}V_{3}=0} − 1 3 V 1 − 1 4 V 2 + ( 1 3 + 1 4 + 1 ) V 3 = 0 {\displaystyle -{\frac {1}{3}}V_{1}-{\frac {1}{4}}V_{2}+({\frac {1}{3}}+{\frac {1}{4}}+1)V_{3}=0}
( 2 + 3 ) I 1 − 3 I 2 = 10 {\displaystyle (2+3)I_{1}-3I_{2}=10} − 3 I 1 + ( 3 + 3 ) I 2 = − 5 {\displaystyle -3I_{1}+(3+3)I_{2}=-5}
( 5 + 4 ) I 1 − 4 I 2 − 2 I 3 = 0 {\displaystyle (5+4)I_{1}-4I_{2}-2I_{3}=0} − 4 I 1 + ( 1 + 4 + 5 ) I 2 − 5 I 3 = 0 {\displaystyle -4I_{1}+(1+4+5)I_{2}-5I_{3}=0} − 2 I 1 − 5 I 2 + ( 1 + 2 + 5 ) I 3 = − 10 {\displaystyle -2I_{1}-5I_{2}+(1+2+5)I_{3}=-10}
( 2 + 3 + 4 ) I 1 − 4 I 2 − 2 I 3 = 0 {\displaystyle (2+3+4)I_{1}-4I_{2}-2I_{3}=0} − 4 I 1 + ( 1 + 4 + 5 ) I 2 − 5 I 3 = 0 {\displaystyle -4I_{1}+(1+4+5)I_{2}-5I_{3}=0} − 2 I 1 − 5 I 2 + ( 1 + 2 + 5 ) I 3 = 5 {\displaystyle -2I_{1}-5I_{2}+(1+2+5)I_{3}=5}
A x = − λ x {\displaystyle Ax=-{\lambda }x} ( A − λ I ) x = 0 {\displaystyle (A-{\lambda }I)x=0} A = ( 2 1 1 2 ) {\displaystyle A={\begin{pmatrix}2&1\\1&2\end{pmatrix}}} d e t ( ( 2 1 1 2 ) − λ ( 1 0 0 1 ) ) = 0 {\displaystyle det({\begin{pmatrix}2&1\\1&2\end{pmatrix}}-{\lambda }{\begin{pmatrix}1&0\\0&1\end{pmatrix}})=0} d e t ( ( 2 − λ 1 1 2 − λ ) ) = 0 {\displaystyle det({\begin{pmatrix}2-{\lambda }&1\\1&2-{\lambda }\end{pmatrix}})=0} ( 2 − λ ) 2 − 1 = 0 {\displaystyle (2-{\lambda })^{2}-1=0} ( λ − 1 ) ( λ − 3 ) = 0 {\displaystyle ({\lambda }-1)({\lambda }-3)=0} λ = 1 {\displaystyle {\lambda }=1} or λ = 3 {\displaystyle {\lambda }=3} λ = 1 {\displaystyle {\lambda }=1} ( 2 1 1 2 ) ( x 1 x 2 ) = 1 ( x 1 x 2 ) {\displaystyle {\begin{pmatrix}2&1\\1&2\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}=1{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}} x 1 + x 2 = 0 {\displaystyle x_{1}+x_{2}=0} ( x 1 x 2 ) = ( 1 2 − 1 2 ) {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}}
( x 1 x 2 ) = ( 1 2 1 2 ) {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}\end{pmatrix}}}
( 2 1 1 2 ) ( x 1 x 2 ) = 3 ( x 1 x 2 ) {\displaystyle {\begin{pmatrix}2&1\\1&2\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}=3{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}} x 1 − x 2 = 0 {\displaystyle x_{1}-x_{2}=0} ( x y z v ) {\displaystyle {\begin{pmatrix}x&y\\z&v\end{pmatrix}}} d e t ( A − λ I ) = 0 {\displaystyle det(A-{\lambda }I)=0} x + y = a {\displaystyle x+y=a} y = a − x {\displaystyle y=a-x} h = c 2 − x 2 {\displaystyle h={\sqrt {c^{2}-x^{2}}}} h 2 = c 2 − x 2 {\displaystyle h^{2}=c^{2}-x^{2}} h 2 = d 2 − x 2 {\displaystyle h^{2}=d^{2}-x^{2}} h 2 = e 2 − y 2 {\displaystyle h^{2}=e^{2}-y^{2}} h 2 = f 2 − z 2 {\displaystyle h^{2}=f^{2}-z^{2}}
a → = [ a , 0 ] {\displaystyle {\vec {a}}=[a,0]} c → = [ c cos θ , c sin θ ] {\displaystyle {\vec {c}}=[c\cos {\theta },c\sin {\theta }]} A = 1 2 | a → × c → | {\displaystyle A={\frac {1}{2}}\left|{\vec {a}}\times {\vec {c}}\right|} A = 1 2 | a → × c → | {\displaystyle A={\frac {1}{2}}{\sqrt {\left|{\vec {a}}\times {\vec {c}}\right|}}} a → = [ a , 0 , 0 ] {\displaystyle {\vec {a}}=[a,0,0]} c → = [ c cos θ 1 , c sin θ 1 , 0 ] {\displaystyle {\vec {c}}=[c\cos {\theta _{1}},c\sin {\theta _{1}},0]} x → = [ x cos θ 2 , x sin θ 2 , 0 ] {\displaystyle {\vec {x}}=[x\cos {\theta _{2}},x\sin {\theta _{2}},0]} d → = [ x cos θ 2 , x sin θ 2 , h ] {\displaystyle {\vec {d}}=[x\cos {\theta _{2}},x\sin {\theta _{2}},h]} y → = a → − x → {\displaystyle {\vec {y}}={\vec {a}}-{\vec {x}}} z → = c → − x → {\displaystyle {\vec {z}}={\vec {c}}-{\vec {x}}} c 2 − x 2 = b 2 − a 2 + 2 a x − x 2 {\displaystyle c^{2}-x^{2}=b^{2}-a^{2}+2ax-x^{2}} h 2 = b 2 − ( a − x ) 2 {\displaystyle h^{2}=b^{2}-(a-x)^{2}} h = b 2 − ( a − x ) 2 {\displaystyle h={\sqrt {b^{2}-(a-x)^{2}}}} h = c 2 − ( a 2 + c 2 − b 2 2 a ) 2 {\displaystyle h={\sqrt {c^{2}-({\frac {a^{2}+c^{2}-b^{2}}{2a}})^{2}}}} A = 1 2 a h = 1 2 a c 2 − ( a 2 + c 2 − b 2 2 a ) 2 {\displaystyle A={\frac {1}{2}}ah={\frac {1}{2}}a{\sqrt {c^{2}-({\frac {a^{2}+c^{2}-b^{2}}{2a}})^{2}}}} A = 1 4 4 a 2 c 2 − ( a 2 + c 2 − b 2 ) 2 {\displaystyle A={\frac {1}{4}}{\sqrt {4a^{2}c^{2}-(a^{2}+c^{2}-b^{2})^{2}}}} A = 1 4 ( 2 a c + a 2 + c 2 − b 2 ) ( 2 a c − a 2 − c 2 + b 2 ) {\displaystyle A={\frac {1}{4}}{\sqrt {(2ac+a^{2}+c^{2}-b^{2})(2ac-a^{2}-c^{2}+b^{2})}}} A = 1 4 ( a + c ) 2 − b 2 ) ( b 2 − ( a − c ) 2 ) {\displaystyle A={\frac {1}{4}}{\sqrt {(a+c)^{2}-b^{2})(b^{2}-(a-c)^{2})}}} A = 1 4 ( a + c + b ) ( a + c − b ) ( b + a − c ) ( b − a + c ) {\displaystyle A={\frac {1}{4}}{\sqrt {(a+c+b)(a+c-b)(b+a-c)(b-a+c)}}} A = 1 4 2 S ( 2 S − 2 a ) ( 2 S − 2 b ) ( 2 S − 2 c ) {\displaystyle A={\frac {1}{4}}{\sqrt {2S(2S-2a)(2S-2b)(2S-2c)}}} A = S ( S − a ) ( S − b ) ( S − c ) {\displaystyle A={\sqrt {S(S-a)(S-b)(S-c)}}} S = 1 2 ( a + b + c ) {\displaystyle S={\frac {1}{2}}(a+b+c)} 1 2 a h = 1 4 2 S ( 2 S − 2 a ) ( 2 S − 2 b ) ( 2 S − 2 c ) {\displaystyle {\frac {1}{2}}ah={\frac {1}{4}}{\sqrt {2S(2S-2a)(2S-2b)(2S-2c)}}} 2 a x = a 2 + c 2 − b 2 {\displaystyle 2ax=a^{2}+c^{2}-b^{2}} c x = a 2 + c 2 − b 2 2 a {\displaystyle c_{x}={\frac {a^{2}+c^{2}-b^{2}}{2a}}} c y = c 2 − ( a 2 + c 2 − b 2 2 a ) 2 {\displaystyle c_{y}={\sqrt {c^{2}-({\frac {a^{2}+c^{2}-b^{2}}{2a}})^{2}}}} d 2 − x 2 = f 2 − z 2 {\displaystyle d^{2}-x^{2}=f^{2}-z^{2}} d 2 − ( d x 2 + d y 2 ) = f 2 − ( ( d x − c x ) 2 + ( d y − c y ) 2 ) {\displaystyle d^{2}-(d_{x}^{2}+d_{y}^{2})=f^{2}-((d_{x}-c_{x})^{2}+(d_{y}-c_{y})^{2})} c o s θ = x c {\displaystyle cos\theta ={\frac {x}{c}}} c o s θ = a 2 + c 2 − b 2 2 a c {\displaystyle cos\theta ={\frac {a^{2}+c^{2}-b^{2}}{2ac}}} θ = c o s − 1 ( a 2 + c 2 − b 2 2 a c ) {\displaystyle \theta =cos^{-1}({\frac {a^{2}+c^{2}-b^{2}}{2ac}})} y = s i n ( 25 x 2 ) {\displaystyle y=sin(25x^{2})} x = s i n ( z ) {\displaystyle x=sin(z)} y = c o s ( z ) {\displaystyle y=cos(z)} y = s i n ( x ) {\displaystyle y=sin(x)} z = c o s ( x ) {\displaystyle z=cos(x)} R = x 2 + y 2 {\displaystyle R={\sqrt {x^{2}+y^{2}}}} z = s i n ( R ) {\displaystyle z=sin(R)} z = ( x − 100 ) 2 y {\displaystyle z={\frac {(x-100)^{2}}{y}}} e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + . . . . . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+.....} c o s ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + x 8 8 ! − x 10 10 ! + . . . . . {\displaystyle cos(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n}}{(2n)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}-{\frac {x^{10}}{10!}}+.....} s i n ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + x 9 9 ! − x 11 11 ! + . . . . . {\displaystyle sin(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{(2n+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+{\frac {x^{9}}{9!}}-{\frac {x^{11}}{11!}}+.....} 10 x 1 + 5 x 2 + 3.5 x 3 = 2.5 {\displaystyle 10x_{1}+5x_{2}+3.5x_{3}=2.5} 5 x 1 + 0.5 x 3 = 0 {\displaystyle 5x_{1}+0.5x_{3}=0} 2 x 1 − x 2 + 2 x 3 = 10 {\displaystyle 2x_{1}-x_{2}+2x_{3}=10} d 2 x d t 2 = − x − 0.2 d x d t {\displaystyle {\frac {d^{2}x}{dt^{2}}}=-x-0.2{\frac {dx}{dt}}} [ x 1 , x 2 ] = [ x , d x d t ] {\displaystyle [x_{1},x_{2}]=[x,{\frac {dx}{dt}}]} d x 1 d t = x 2 {\displaystyle {\frac {dx_{1}}{dt}}=x_{2}} d x 2 d t = − x 1 − 0.2 x 2 {\displaystyle {\frac {dx_{2}}{dt}}=-x_{1}-0.2x_{2}} d x d t = − 2 x {\displaystyle {\frac {dx}{dt}}=-2x} d 2 x d t 2 = − k x − c d x d t {\displaystyle {\frac {d^{2}x}{dt^{2}}}=-kx-c{\frac {dx}{dt}}} ω d = π t p {\displaystyle {\omega }_{d}={\frac {\pi }{t_{p}}}} F C o r i o l i s = 2 m ( v × Ω ) {\displaystyle \mathbf {F_{Coriolis}} =2m\left(\mathbf {v} \times \mathbf {\Omega } \right)} ω d = π t p {\displaystyle {\omega }_{d}={\frac {\pi }{t_{p}}}} x k + 1 = x k + v k T s s i n θ {\displaystyle x_{k+1}=x_{k}+v_{k}T_{s}sin{\theta }} y k + 1 = y k + v k T s c o s θ {\displaystyle y_{k+1}=y_{k}+v_{k}T_{s}cos{\theta }} θ k + 1 = θ k + ω k T s {\displaystyle {\theta }_{k+1}={\theta }_{k}+{\omega }_{k}T_{s}} x = x o + x ′ c o s ( θ ) − y ′ s i n ( θ ) {\displaystyle x=x_{o}+x^{'}cos(\theta )-y^{'}sin(\theta )} y = y o + x ′ s i n ( θ ) + y ′ c o s ( θ ) {\displaystyle y=y_{o}+x^{'}sin(\theta )+y^{'}cos(\theta )} R = L s i n α + r {\displaystyle R={\frac {L}{sin\alpha }}+r} V = R 2 ( ω R + ω L ) {\displaystyle V={\frac {R}{2}}(\omega _{R}+\omega _{L})} ω = R D ( ω R − ω L ) {\displaystyle \omega ={\frac {R}{D}}(\omega _{R}-\omega _{L})} R c = D 2 ω R + ω L ω R − ω L {\displaystyle R_{c}={\frac {D}{2}}{\frac {\omega _{R}+\omega _{L}}{\omega _{R}-\omega _{L}}}} v R = R ω R {\displaystyle v_{R}=R\omega _{R}} v L = R ω L {\displaystyle v_{L}=R\omega _{L}} V = 1 2 ( v R + v L ) {\displaystyle V={\frac {1}{2}}(v_{R}+v_{L})} ω = 1 D ( v R − v L ) {\displaystyle \omega ={\frac {1}{D}}(v_{R}-v_{L})} τ {\displaystyle \tau } J {\displaystyle J} J = K e K t τ R {\displaystyle J={\frac {K_{e}K_{t}\tau }{R}}} J = τ R K 2 {\displaystyle J={\frac {\tau }{RK^{2}}}} K {\displaystyle K} K e {\displaystyle K_{e}} K t {\displaystyle K_{t}} K e = 1 K {\displaystyle K_{e}={\frac {1}{K}}} K t = 1 K {\displaystyle K_{t}={\frac {1}{K}}} K = ω n 2 τ K p {\displaystyle K={\frac {{{\omega }_{n}}^{2}\tau }{K_{p}}}} K = ( ω d 2 + σ 2 ) τ K p {\displaystyle K={\frac {({{\omega }_{d}}^{2}+{\sigma }^{2})\tau }{K_{p}}}} K = ω d 2 + σ 2 2 K p σ {\displaystyle K={\frac {{{\omega }_{d}}^{2}+{\sigma }^{2}}{2K_{p}\sigma }}} ω n 2 = ω d 2 + σ 2 {\displaystyle {{\omega }_{n}}^{2}={{\omega }_{d}}^{2}+{\sigma }^{2}} τ = − 1 2 t p log ( M p ) {\displaystyle \tau =-{\frac {1}{2t_{p}{\log }(M_{p})}}} τ = 1 2 ξ ω n {\displaystyle \tau ={\frac {1}{2\xi {\omega }_{n}}}} M p = e − σ ω d π {\displaystyle M_{p}=e^{-{\frac {\sigma }{{\omega }_{d}}}\pi }} log ( M p ) = − ( σ ω d ) π {\displaystyle {\log }(M_{p})=-({\frac {\sigma }{{\omega }_{d}}})\pi } σ = − ω d log ( M p ) / π {\displaystyle {\sigma }=-{\omega }_{d}{\log }(M_{p})/\pi } σ = − log ( M p ) t p {\displaystyle {\sigma }=-{\frac {{\log }(M_{p})}{t_{p}}}} σ = ξ ω n {\displaystyle {\sigma }={\xi }{\omega }_{n}} σ {\displaystyle {\sigma }} ω d {\displaystyle {\omega }_{d}} ω ( s ) T m ( s ) = 1 J s + b {\displaystyle {\frac {\omega (s)}{T_{m}(s)}}={\frac {1}{Js+b}}} K 1 s + K 2 s 2 {\displaystyle K_{1}s+K_{2}s^{2}} K 1 = 1 K {\displaystyle K_{1}={\frac {1}{K}}} K 2 = τ K {\displaystyle K_{2}={\frac {\tau }{K}}} U f f 0 ( s ) = K 0 D ( s ) {\displaystyle U_{ff0}(s)=K_{0}D(s)} K 0 = R K t {\displaystyle K_{0}={\frac {R}{K_{t}}}} 2 = n m {\displaystyle {\sqrt {2}}={\frac {n}{m}}} y = log ( 2 x − 2 ) {\displaystyle y=\log {(2x-2)}} lim x → 0 tan x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\tan {x}}{x}}=1} lim x → 0 tan 5 x sin 2 x {\displaystyle \lim _{x\to 0}{\frac {\tan {5x}}{\sin {2x}}}} lim n → ∞ ( 4 n 2 + 2 n − 2 n ) {\displaystyle \lim _{n\to \infty }({\sqrt {4n^{2}+2n}}-2n)} x = 1 2 e y + 1 {\displaystyle x={\frac {1}{2}}e^{y}+1} y = 1 2 e x + 1 {\displaystyle y={\frac {1}{2}}e^{x}+1} 2 x − 2 = e y {\displaystyle 2x-2=e^{y}} 2 {\displaystyle {\sqrt {2}}} n 2 = 2 m 2 {\displaystyle n^{2}=2m^{2}} n = 2 p {\displaystyle n=2p} m 2 = 2 p 2 {\displaystyle m^{2}=2p^{2}} 5 = n m {\displaystyle {\sqrt {5}}={\frac {n}{m}}} 5 {\displaystyle {\sqrt {5}}} n 2 = 5 m 2 {\displaystyle n^{2}=5m^{2}} n = 5 p {\displaystyle n=5p} m 2 = 5 p 2 {\displaystyle m^{2}=5p^{2}} X ( s ) R ( s ) = ( K p + K d s ) K / τ s 2 + ( 1 + K d K ) s / τ + K p K / τ {\displaystyle {\frac {X(s)}{R(s)}}={\frac {(K_{p}+K_{d}s)K/{\tau }}{s^{2}+(1+K_{d}K)s/{\tau }+K_{p}K/{\tau }}}} ξ = 1 + K d K 2 ω n τ = 1 + K d K 2 τ K p K {\displaystyle \xi ={\frac {1+K_{d}K}{2\omega _{n}\tau }}={\frac {1+K_{d}K}{2{\sqrt {{\tau }K_{p}K}}}}} ω n = K p K / τ {\displaystyle \omega _{n}={\sqrt {K_{p}K/\tau }}} K t L s + R {\displaystyle {\frac {K_{t}}{Ls+R}}} K p + K d s {\displaystyle K_{p}+K_{d}s} x ( ∞ ) = R K t K p {\displaystyle x(\infty )={\frac {R}{K_{t}K_{p}}}} lim t → ∞ x ( t ) = lim s → 0 s J s 3 + ( b + K t K e R ) s 2 + K t K p R s + K t K i R = 0 {\displaystyle \lim _{t\to \infty }x(t)=\lim _{s\to 0}{\frac {s}{Js^{3}+(b+{\frac {K_{t}K_{e}}{R}})s^{2}+{\frac {K_{t}K_{p}}{R}}s+{\frac {K_{t}K_{i}}{R}}}}=0} X ( s ) T L ( s ) = 1 J s 2 + ( b + K t K e R ) s + K t K p R {\displaystyle {\frac {X(s)}{T_{L}(s)}}={\frac {1}{Js^{2}+(b+{\frac {K_{t}K_{e}}{R}})s+{\frac {K_{t}K_{p}}{R}}}}} X ( s ) T L ( s ) = s J s 3 + ( b + K t K e R ) s 2 + K t K p R s + K t K i R {\displaystyle {\frac {X(s)}{T_{L}(s)}}={\frac {s}{Js^{3}+(b+{\frac {K_{t}K_{e}}{R}})s^{2}+{\frac {K_{t}K_{p}}{R}}s+{\frac {K_{t}K_{i}}{R}}}}} K p + K e s {\displaystyle K_{p}+K_{e}s} T L T m {\displaystyle T_{L}T_{m}} G ( s ) = X ( s ) E ( s ) = K ( K p s + K i ) s 2 ( τ s + 1 ) {\displaystyle G(s)={\frac {X(s)}{E(s)}}={\frac {K(K_{p}s+K_{i})}{s^{2}({\tau }s+1)}}} G ( s ) = X ( s ) E ( s ) = K p K s ( τ s + 1 ) {\displaystyle G(s)={\frac {X(s)}{E(s)}}={\frac {K_{p}K}{s({\tau }s+1)}}} G ( s ) = X ( s ) E ( s ) = K p K τ s + 1 {\displaystyle G(s)={\frac {X(s)}{E(s)}}={\frac {K_{p}K}{{\tau }s+1}}} G c ( s ) = K p + K i s + K d s = K d s 2 + K p s + K i s {\displaystyle G_{c}(s)=K_{p}+{\frac {K_{i}}{s}}+K_{d}s={\frac {K_{d}s^{2}+K_{p}s+K_{i}}{s}}} G c ( s ) = K p + K i s = K p s + K i s {\displaystyle G_{c}(s)=K_{p}+{\frac {K_{i}}{s}}={\frac {K_{p}s+K_{i}}{s}}} ω d = 1 − ξ 2 ω n {\displaystyle \omega _{d}={\sqrt {1-\xi ^{2}}}\omega _{n}} ξ = 1 2 ω n τ = 1 2 τ K p K {\displaystyle \xi ={\frac {1}{2\omega _{n}\tau }}={\frac {1}{2{\sqrt {{\tau }K_{p}K}}}}} ω n = K p K / τ {\displaystyle \omega _{n}={\sqrt {K_{p}K/\tau }}} t p = π ω d {\displaystyle t_{p}={\frac {\pi }{\omega _{d}}}} t s = log ( 0.02 ) − ξ ω n {\displaystyle t_{s}={\frac {\log(0.02)}{-\xi \omega _{n}}}} x m a x = 1 + e − ξ π 1 − ξ 2 {\displaystyle x_{max}=1+e^{\frac {-{\xi }\pi }{\sqrt {1-\xi ^{2}}}}} M p = e − ξ π 1 − ξ 2 × 100 ( % ) {\displaystyle M_{p}=e^{\frac {-{\xi }\pi }{\sqrt {1-\xi ^{2}}}}\times 100(\%)} x ( t ) = 1 − e − ξ ω n t 1 − ξ 2 ( 1 − ξ 2 cos ω d t + ξ sin ω d t ) {\displaystyle x(t)=1-{\frac {e^{-{\xi }\omega _{n}t}}{\sqrt {1-\xi ^{2}}}}({\sqrt {1-\xi ^{2}}}\cos \omega _{d}t+{\xi }\sin \omega _{d}t)} x ( t ) = 1 − e − ξ ω n t ( cos ω d t + ξ 1 − ξ 2 sin ω d t ) {\displaystyle x(t)=1-{e^{-{\xi }\omega _{n}t}}(\cos \omega _{d}t+{\frac {\xi }{\sqrt {1-\xi ^{2}}}}\sin \omega _{d}t)} x ( t ) = 1 − e − ξ ω n t 1 − ξ 2 sin ( ω d t + ϕ ) {\displaystyle x(t)=1-{\frac {e^{-{\xi }\omega _{n}t}}{\sqrt {1-\xi ^{2}}}}\sin(\omega _{d}t+\phi )}
1 s {\displaystyle {\frac {1}{s}}} K p {\displaystyle K_{p}} E ( s ) {\displaystyle E(s)} V ( s ) = K p E ( s ) {\displaystyle V(s)=K_{p}E(s)} E ( s ) = R ( s ) − X ( s ) {\displaystyle E(s)=R(s)-X(s)} R ( s ) = 1 s {\displaystyle R(s)={\frac {1}{s}}} R ( s ) = 1 s 2 {\displaystyle R(s)={\frac {1}{s^{2}}}} J = K e K t τ R {\displaystyle J={\frac {K_{e}K_{t}\tau }{R}}} R ( s ) {\displaystyle R(s)} X ( s ) {\displaystyle X(s)} X ( s ) R ( s ) = K p K s ( τ s + 1 ) 1 + K p K s ( τ s + 1 ) {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K_{p}{\frac {K}{s({\tau }s+1)}}}{1+K_{p}{\frac {K}{s({\tau }s+1)}}}}} X ( s ) R ( s ) = K p K s ( τ s + 1 ) 1 + K p K s ( τ s + 1 ) = K p K τ s 2 + s + K p K {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K_{p}{\frac {K}{s({\tau }s+1)}}}{1+K_{p}{\frac {K}{s({\tau }s+1)}}}}={\frac {K_{p}K}{{\tau }s^{2}+s+K_{p}K}}} X ( s ) R ( s ) = K p K / τ s 2 + s / τ + K p K / τ {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K_{p}K/{\tau }}{s^{2}+s/{\tau }+K_{p}K/{\tau }}}}
X ( s ) = K s ( τ s + 1 ) {\displaystyle X(s)={\frac {K}{s({\tau }s+1)}}} X ( s ) R ( s ) = K τ s + 1 {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K}{{\tau }s+1}}} X ( s ) R ( s ) = K τ s 2 + s = n u m ( s ) d e n ( s ) {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K}{{\tau }s^{2}+s}}={\frac {num(s)}{den(s)}}} X ( s ) R ( s ) = K τ s + 1 = n u m ( s ) d e n ( s ) {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K}{{\tau }s+1}}={\frac {num(s)}{den(s)}}} X ( s ) R ( s ) = K τ s + 1 = n u m ( s ) d e n ( s ) {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K}{{\tau }s+1}}={\frac {num(s)}{den(s)}}} X ( s ) R ( s ) = K τ s + 1 = n u m ( s ) d e n ( s ) {\displaystyle {\frac {X(s)}{R(s)}}={\frac {K}{{\tau }s+1}}={\frac {num(s)}{den(s)}}}
X ( s ) R ( s ) = ω n 2 s 2 + 2 ξ ω n s + ω n 2 {\displaystyle {\frac {X(s)}{R(s)}}={\frac {\omega _{n}^{2}}{s^{2}+2\xi \omega _{n}s+\omega _{n}^{2}}}} X ( s ) = ω n 2 s ( s 2 + 2 ξ ω n s + ω n 2 ) {\displaystyle X(s)={\frac {\omega _{n}^{2}}{s(s^{2}+2\xi \omega _{n}s+\omega _{n}^{2})}}} s 2 + 2 ξ {\displaystyle {s^{2}+2\xi }}
L − 1 [ 1 s + a ] = e − a t {\displaystyle {\mathcal {L}}^{-1}[{\frac {1}{s+a}}]=e^{-at}} L − 1 [ 1 s ] = 1 {\displaystyle {\mathcal {L}}^{-1}[{\frac {1}{s}}]=1}
X ( s ) = K / τ s ( s + 1 / τ ) {\displaystyle X(s)={\frac {K/\tau }{s(s+1/\tau )}}} X ( s ) = K [ 1 s − 1 s + 1 / τ ] {\displaystyle X(s)=K[{\frac {1}{s}}-{\frac {1}{s+1/\tau }}]} X ( t ) = K ( 1 − e − t τ ) {\displaystyle X(t)=K(1-e^{-{\frac {t}{\tau }}})} x ( t ) = K ( 1 − e − t τ ) {\displaystyle x(t)=K(1-e^{-{\frac {t}{\tau }}})} x ( t ) = K ( t − T + T e − t T ) {\displaystyle x(t)=K(t-T+Te^{-{\frac {t}{T}}})}
ω ( t ) = R i ( t ) + L d i ( t ) d t + v e ( t ) {\displaystyle \omega (t)=Ri(t)+L{\frac {di(t)}{dt}}+v_{e}(t)} V ( s ) − V e ( s ) = R I ( s ) + L s I ( s ) {\displaystyle V(s)-V_{e}(s)=RI(s)+LsI(s)} I ( s ) = V ( s ) − V e ( s ) L s + R {\displaystyle I(s)={\frac {V(s)-V_{e}(s)}{Ls+R}}} τ m ( t ) = K t i ( t ) {\displaystyle \tau _{m}(t)=K_{t}i(t)} J d ω ( t ) d t + b ω ( t ) = τ m ( t ) {\displaystyle J{\frac {d\omega (t)}{dt}}+b\omega (t)=\tau _{m}(t)} v e ( t ) = K e ω ( t ) {\displaystyle v_{e}(t)=K_{e}\omega (t)} T m ( s ) = K t I ( s ) {\displaystyle T_{m}(s)=K_{t}I(s)} J s ω ( s ) + b ω ( s ) = T m ( s ) {\displaystyle Js\omega (s)+b\omega (s)=T_{m}(s)} ω ( s ) = T m ( s ) J s + b {\displaystyle \omega (s)={\frac {T_{m}(s)}{Js+b}}} V e ( s ) = K e ω ( s ) {\displaystyle V_{e}(s)=K_{e}\omega (s)} τ = R J R b + K e K t {\displaystyle \tau ={\frac {RJ}{Rb+K_{e}K_{t}}}} K = K t R b + K e K t {\displaystyle K={\frac {K_{t}}{Rb+K_{e}K_{t}}}} θ ( s ) V ( s ) = K s ( τ s + 1 ) {\displaystyle {\frac {\theta (s)}{V(s)}}={\frac {K}{s({\tau }s+1)}}}
M s V ( s ) = F ( s ) − B V ( s ) {\displaystyle MsV(s)=F(s)-BV(s)} V ( s ) = 1 M s ( F ( s ) − B V ( s ) ) {\displaystyle V(s)={\frac {1}{Ms}}(F(s)-BV(s))}
X ( s ) = 1 M s 2 ( F ( s ) − b s X ( s ) − k X ( s ) ) {\displaystyle X(s)={\frac {1}{Ms^{2}}}(F(s)-bsX(s)-kX(s))}
V ( s ) F ( s ) = 1 m s 1 + 1 m s b = 1 m s + b {\displaystyle {\frac {V(s)}{F(s)}}={\frac {\frac {1}{ms}}{1+{\frac {1}{ms}}b}}={\frac {1}{ms+b}}} X ( s ) F ( s ) = 1 m s + b 1 s 1 + 1 m s + b 1 s k = 1 m s 2 + b s + k {\displaystyle {\frac {X(s)}{F(s)}}={\frac {{\frac {1}{ms+b}}{\frac {1}{s}}}{1+{\frac {1}{ms+b}}{\frac {1}{s}}k}}={\frac {1}{ms^{2}+bs+k}}} E 0 ( s ) E i ( s ) = 1 R 1 C s 1 + 1 R 1 C s = 1 R C s + 1 {\displaystyle {\frac {E_{0}(s)}{E_{i}(s)}}={\frac {{\frac {1}{R}}{\frac {1}{Cs}}}{1+{\frac {1}{R}}{\frac {1}{Cs}}}}={\frac {1}{RCs+1}}} V ( s ) F ( s ) = 1 M s 1 + 1 M s B {\displaystyle {\frac {V(s)}{F(s)}}={\frac {\frac {1}{Ms}}{1+{\frac {1}{Ms}}B}}} n u m c ( s ) d e n c ( s ) {\displaystyle {\frac {numc(s)}{denc(s)}}} n u m p ( s ) d e n p ( s ) {\displaystyle {\frac {nump(s)}{denp(s)}}} K p J s 2 + K p = n u m ( s ) d e n ( s ) {\displaystyle {\frac {K_{p}}{Js^{2}+K_{p}}}={\frac {num(s)}{den(s)}}} Y ( s ) R ( s ) = 25 s 2 + 4 s + 25 = n u m ( s ) d e n ( s ) {\displaystyle {\frac {Y(s)}{R(s)}}={\frac {25}{s^{2}+4s+25}}={\frac {num(s)}{den(s)}}} Y ( s ) R ( s ) = 1 s 2 + 4 s = n u m ( s ) d e n ( s ) {\displaystyle {\frac {Y(s)}{R(s)}}={\frac {1}{s^{2}+4s}}={\frac {num(s)}{den(s)}}} V ( s ) F ( s ) = 1 M s + B {\displaystyle {\frac {V(s)}{F(s)}}={\frac {1}{Ms+B}}}
K = R 1 K p K a K c E s s v {\displaystyle K={\frac {R_{1}}{K_{p}K_{a}K_{c}E_{ssv}}}} E s s v = 0.01 {\displaystyle E_{ssv}=0.01} E s s v ′ = E s s v + λ ( M p p − M p ) {\displaystyle E_{ssv}^{'}=E_{ssv}+\lambda (M_{pp}-M_{p})} V ( s ) F ( s ) = 1 M s + B {\displaystyle {\frac {V(s)}{F(s)}}={\frac {1}{Ms+B}}} V ( s ) = 1 M s + B F ( s ) {\displaystyle V(s)={\frac {1}{Ms+B}}F(s)} X ( s ) F ( s ) = 1 M s 2 + B s = 1 s ( M s + B ) {\displaystyle {\frac {X(s)}{F(s)}}={\frac {1}{Ms^{2}+Bs}}={\frac {1}{s(Ms+B)}}} G ( s ) = X ( s ) F ( s ) = 1 m s 2 + c s + k {\displaystyle G(s)={\frac {X(s)}{F(s)}}={\frac {1}{ms^{2}+cs+k}}} i = e i − e 0 R {\displaystyle i={\frac {e_{i}-e_{0}}{R}}} e 0 = 1 C ∫ i d t {\displaystyle e_{0}={\frac {1}{C}}\int idt} I ( s ) = 1 R ( E i ( s ) − E 0 ( s ) ) {\displaystyle I(s)={\frac {1}{R}}(E_{i}(s)-E_{0}(s))}
E 0 ( s ) = 1 C s I ( s ) {\displaystyle E_{0}(s)={\frac {1}{Cs}}I(s)} C ( s ) = G ( s ) E ( s ) {\displaystyle C(s)=G(s)E(s)} E ( s ) = R ( s ) − H ( s ) C ( s ) {\displaystyle E(s)=R(s)-H(s)C(s)}
z ( t ) = 100 ∗ e x p − ( x ( t ) − 1.5 ) 2 ∗ e x p − ( y ( t ) − 1.5 ) 2 {\displaystyle z(t)=100*exp^{-(x(t)-1.5)^{2}}*exp^{-(y(t)-1.5)^{2}}}
C ( s ) R ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle {\frac {C(s)}{R(s)}}={\frac {G(s)}{1+G(s)H(s)}}} M s V ( s ) = F ( s ) − B V ( s ) {\displaystyle MsV(s)=F(s)-BV(s)} M d v ( t ) d t = f ( t ) − B v ( t ) {\displaystyle M{\frac {dv(t)}{dt}}=f(t)-Bv(t)} M d v ( t ) d t = f ( t ) − B v ( t ) {\displaystyle M{\frac {dv(t)}{dt}}=f(t)-Bv(t)} M d v ( t ) d t = f ( t ) − B v ( t ) {\displaystyle M{\frac {dv(t)}{dt}}=f(t)-Bv(t)}
m d 2 x d t 2 = f − b d x d t − k x {\displaystyle m{\frac {d^{2}x}{dt^{2}}}=f-b{\frac {dx}{dt}}-kx} m s 2 X ( s ) = F ( s ) − b s X ( s ) − k X ( s ) {\displaystyle ms^{2}X(s)=F(s)-bsX(s)-kX(s)} m d 2 x d t 2 + c d x d t + k x = f {\displaystyle m{\frac {d^{2}x}{dt^{2}}}+c{\frac {dx}{dt}}+kx=f}
= f ( t ) − B d x ( t ) d t {\displaystyle =f(t)-B{\frac {dx(t)}{dt}}} M s 2 X ( s ) = F ( s ) − B s X ( s ) {\displaystyle Ms^{2}X(s)=F(s)-BsX(s)}
F ( s ) = L [ f ( t ) ] = ∫ 0 − ∞ e − s t f ( t ) d t {\displaystyle F(s)={\mathcal {L}}[f(t)]=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt} L [ a f ( t ) + b g ( t ) ] = a L [ f ( t ) ] + b L [ g ( t ) ] = a F ( s ) + b G ( s ) {\displaystyle {\mathcal {L}}[af(t)+bg(t)]=a{\mathcal {L}}[f(t)]+b{\mathcal {L}}[g(t)]=aF(s)+bG(s)} L [ f ′ ( t ) ] = s F ( s ) − f ( 0 ) {\displaystyle {\mathcal {L}}[f'(t)]=sF(s)-f(0)} L [ f ″ ( t ) ] = s 2 F ( s ) − s f ( 0 ) − f ′ ( 0 ) {\displaystyle {\mathcal {L}}[f''(t)]=s^{2}F(s)-sf(0)-f'(0)} L [ ∫ 0 t f ( t ) d t ] = 1 s F ( s ) {\displaystyle {\mathcal {L}}[\int _{0}^{t}f(t)dt]={1 \over s}F(s)}
or 
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