VNVe 2017/ Final project

Transcription

1 VNVe 2017/ Final project Athor s name Febrary 21, 2018 Solve all examples and send yor final soltion (pdf file) and all sorce codes (LaTex, MATLAB,, ++, etc.) to address satek@fit.vtbr.cz ntil May 9, Everything will be packed in one zip file srname.zip. 1 Problem - Derivation redction Solve linear differential eqation of the 4 th order y + a 3 y + a 2 y + a 1 y + a 0 y = b 4 z + b 3 z + b 2 z + b 1 z + b 0 z (1) with zero initial conditions, constant forcing fnction z = 1 and constants a 3 = 12, a 2 = 24, a 1 = 2, a 0 = 3, b 4 = 1, b 3 = 13, b 2 = 8, b 1 = 13, b 0 = 1. Use Method of Derivative Order edction with Axiliary Variable (MDO- wav) and the Method of Sccessive Integration (MSI). 1.1 Tasks: 1. Write the system of first order differential eqations for the both methods (MDOwAV and MSI), se symbolic operator p and 1/p for operation of derivation and integration, respectively. 2. Draw (e.g. in Dia software) the scheme with inverting blocks (inverting - integrators and smmators). 3. Show the stability of the system (se oth-hrwitz stability criterion). 4. Implement both schemes (MDOwAV and MSI) in MATLAB-Simlink. Plot the graphs of the soltions y MDOwAV and y MSI. Plot the absolte error y MDOwAV y MSI in the separate figre. 1

2 2 Problem - Discharging of apacitance Let s find a soltion of the voltage on capacitance (t) and the crrent i(t) in circit Fig. 1. Parameters of the circit are = 10Ω, = 10 2 F, (0) = 10V. Figre 1: ircit 2.1 Tasks: 1. Set ordinary differential eqation (initial vale problem) for voltage on capacitance (t) in the circit Fig Find analytic soltion of (t). 3. Find nmerical soltion of (t): se the Eler method, the nge- Ktta of 2nd and 4th order. Implement it sing yor own code (e.g. MATLAB,, ++, Java, etc. it is on yor choice, what yo prefer). ompare yor nmerical soltions with the Taylor series based nmerical soltion (in TKSL software). 4. ompare analytic soltion with nmerical soltions - plot figres with absolte error between nmerical soltions and analytical soltion. 5. Prepare a block scheme of this problem and solve it nmerically in MATLAB-Simlink. ompare the nmerical soltion from Simlink with the previos nmerical soltions. 6. hange the parameter of capacitance in the circit Fig. 1 to the new vale = 10 6 F and compare the total nmber of integration 2

3 steps sed in different nmerical methods (Eler method, nge-ktta methods and Taylor series based method - TKSL). Which nmerical method allow yo to make a larger integration step size? NOTE: Use smaller time of simlation. 3 Problem - harging of apacitance Let s find a soltion of the voltage on capacitance (t) and the crrent i(t) in circit in Fig. 2. Parameters of the circit are = 10Ω, = 10 2 F, U = 10V, (0) = 1V. + - U Figre 2: ircit with Spply Voltage U 3.1 Tasks: 1. Set ordinary differential eqation (initial vale problem) for voltage on capacitance (t) in the circit Fig Find analytic soltion of (t). 3. Find nmerical soltion of (t): se the Eler method, the nge- Ktta of 2nd and 4th order. Implement it sing yor own code (e.g. MATLAB,, ++, Java, etc. it is on yor choice, what yo prefer). ompare yor nmerical soltions with the Taylor series based nmerical soltion (in TKSL software). 4. ompare analytic soltion with nmerical soltions - plot figres with absolte error between nmerical soltions and analytical soltion. 3

4 5. Prepare a block scheme of this problem and solve it nmerically in MATLAB-Simlink. ompare the nmerical soltion from Simlink with the previos nmerical soltions. 6. hange the parameter of capacitance in the circit Fig. 2 to the new vale = 10 6 F and compare the total nmber of integration steps sed in different nmerical methods (Eler method, nge-ktta methods and Taylor series based method - TKSL). Which nmerical method allow yo to make a larger integration step size? NOTE: Use smaller time of simlation. 4 Problem - L circit Let s find a soltion of the voltage on capacitance (t) and the crrent i(t) in L circit Fig. 3. Parameters of the L circit are = 200Ω, L = 10 2 H, = 10 6 F, (0) = 0V, i(0) = 0A and inpt voltage (t) = U sin(ωt)v, where U = 1V and ω = 1000rad/s. L L U Figre 3: L ircit with A Spply Voltage U 4.1 Tasks: 1. Set ordinary differential eqation (initial vale problem) for voltage on capacitance (t) in the L circit Fig Find analytic soltion of (t) and i(t). 3. Find nmerical soltion of (t): se the Eler method, the nge- Ktta of 2nd and 4th order. Implement it sing yor own code (e.g. MATLAB,, ++, Java, etc. it is on yor choice, what yo prefer). ompare yor nmerical soltions with the Taylor series based nmerical soltion (in TKSL software). 4

5 4. ompare analytic soltion with nmerical soltions - plot figres with absolte error between nmerical soltions and analytical soltion. 5. Prepare a block scheme of this problem and solve it nmerically in MATLAB-Simlink. ompare the nmerical soltion from Simlink with the previos nmerical soltions. NOTE: Use axiliary differential eqations to generate the inpt voltage - the fnction (t) = sin(1000t). 5