7e26afe5-9748-4331-bd12-109d2829bf4e 99 3 3 decimal

&[DATE] &[TIME] - &[FILENAME]

A general purpose, simple ODE solver to simulate (possibly non-linear) dynamic systems.

CONTINUOUS TIME

DISCRETE TIME

2 f 2 g x.0 NumIt dt dynSys ξ x.0 : x ξ : k 1 NumIt range ξ ξ dt ξ ξ r g f * + eval : x x ξ augment : 21 line for x 41 line :

2 f 2 g x.0 NumIt dynSysDiscr ξ x.0 : x ξ : k 1 NumIt range ξ ξ ξ r g f eval : x x ξ augment : 21 line for x 41 line :

EXAMPLE 1 - Mass-Spring-Damper

Linear system with 2 states and 1 input: unit step response.

x u MasSprDmp 1 M / u 1 el η x 1 el * - K x 2 el * - (* x 1 el 21 mat :

M 1 :

η 0.7 :

K 1 :

system parameters

x r Ctrl1 111 mat :

Constant input, i.e. no state feedback in this case

initial condition x0 0021 mat :

Sol1 any1 any2 MasSprDmp any1 any2 Ctrl1 x0 150 0.1 dynSys :

Sol1 0 0.1 0.193 0.2785 0.3561 0.4254 0.4864 0.5388 0.5827 0.6181 0.6452 0.6643 0.6755 0.6794 0.6762 0.6664 0.6506 0.6292 0.6028 0.5719 0.5372 0.4992 0.4585 0.4156 0.3712 0.3257 0.2797 0.2336 0.188 0.1432 0.0997 0.0578 0.0178 0.0199 - 0.0552 - 0.0878 - 0.1176 - 0.1444 - 0.1682 - 0.1888 - 0.2063 - 0.2207 - 0.2321 - 0.2404 - 0.2459 - 0.2485 - 0.2485 - 0.246 - 0.2412 - 0.2343 - 0.2255 - 0.2149 - 0.2028 - 0.1894 - 0.175 - 0.1596 - 0.1436 - 0.1271 - 0.1103 - 0.0934 - 0.0766 - 0.06 - 0.0439 - 0.0282 - 0.0132 - 0.001 0.0143 0.0267 0.0381 0.0485 0.0577 0.0658 0.0727 0.0785 0.0832 0.0868 0.0893 0.0907 0.0911 0.0906 0.0893 0.0871 0.0842 0.0806 0.0764 0.0717 0.0666 0.0611 0.0553 0.0494 0.0433 0.0371 0.0309 0.0248 0.0188 0.013 0.0074 0.002 0.003 - 0.0077 - 0.0121 - 0.016 - 0.0196 - 0.0228 - 0.0255 - 0.0278 - 0.0297 - 0.0312 - 0.0323 - 0.033 - 0.0333 - 0.0333 - 0.033 - 0.0323 - 0.0314 - 0.0302 - 0.0287 - 0.0271 - 0.0253 - 0.0233 - 0.0213 - 0.0191 - 0.0169 - 0.0146 - 0.0124 - 0.0101 - 0.0079 - 0.0057 - 0.0036 - 0.0016 - 0.0003 0.002 0.0037 0.0052 0.0066 0.0078 0.0089 0.0098 0.0106 0.0112 0.0117 0.012 0.0122 0.0122 0.0122 0.012 0.0117 0.0113 0.0108 0.0102 0.0096 00 0.01 0.0293 0.0571 0.0928 0.1353 0.1839 0.2378 0.2961 0.3579 0.4224 0.4888 0.5564 0.6243 0.692 0.7586 0.8237 0.8866 0.9468 1.004 1.0578 1.1077 1.1535 1.1951 1.2322 1.2648 1.2927 1.3161 1.3349 1.3492 1.3592 1.365 1.3667 1.3647 1.3592 1.3504 1.3387 1.3243 1.3074 1.2886 1.2679 1.2458 1.2226 1.1986 1.174 1.1492 1.1243 1.0997 1.0756 1.0521 1.0296 1.0081 0.9878 0.9689 0.9514 0.9354 0.9211 0.9084 0.8973 0.888 0.8803 0.8743 0.8699 0.8671 0.8658 0.8659 0.8673 0.87 0.8738 0.8787 0.8844 0.891 0.8983 0.9061 0.9145 0.9231 0.9321 0.9411 0.9502 0.9593 0.9682 0.9769 0.9854 0.9934 1.0011 1.0082 1.0149 1.021 1.0265 1.0315 1.0358 1.0395 1.0426 1.0451 1.047 1.0483 1.049 1.0492 1.0489 1.0481 1.0469 1.0453 1.0434 1.0411 1.0385 1.0357 1.0328 1.0296 1.0264 1.0231 1.0198 1.0164 1.0131 1.0099 1.0068 1.0038 1.0009 0.9982 0.9957 0.9933 0.9912 0.9893 0.9876 0.9861 0.9849 0.9839 0.9831 0.9825 0.9822 0.982 0.982 0.9822 0.9826 0.9831 0.9838 0.9846 0.9854 0.9864 0.9875 0.9886 0.9898 0.991 0.9922 0.9934 0.9946 0.9958 0.997 0.9981 0.9992 1.0002 2151 mat

v Sol1 1 row transpose :

s Sol1 2 row transpose :

v x el s x el 2 1 sys

Note: subscript "x" forces vectors elements to be plotted.

EXAMPLE 2 - Linear Feedback Stabilization

Linear system with 3 states and 2 inputs

x u LinSys1 0.5 x 1 el * x 2 el + u 2 el + 0.5 x 1 el * x 3 el + 2 x 1 el * - u 1 el + u 2 el + 31 mat :

x r Ctrl2 1 0.5 21 mat :

Constant input

x0 00031 mat :

Sol2 any1 any2 LinSys1 any1 any2 Ctrl2 x0 50 0.1 dynSys :

Sol2 0 0.05 0.1025 0.1594 0.2225 0.2938 0.3751 0.4684 0.5756 0.6986 0.8392 0.9994 1.1809 1.3855 1.6145 1.8695 2.1517 2.4618 2.8006 3.1681 3.5642 3.9878 4.4377 4.9115 5.4062 5.9179 6.4416 6.9711 7.4991 8.0166 8.5133 8.9774 9.3951 9.751 10.0274 10.205 10.2623 10.1754 9.9186 9.464 8.7814 7.839 6.6028 5.0375 3.1061 0.7709 2.0067 - 5.2649 - 9.042 - 13.3745 - 18.297 - 00 0.0175 0.0516 0.1015 0.1664 0.2454 0.3376 0.442 0.5573 0.6823 0.8154 0.9546 1.098 1.2429 1.3866 1.5258 1.6567 1.7751 1.8761 1.9545 2.0044 2.0192 1.9917 1.9141 1.778 1.5744 1.2937 0.9255 0.4594 0.1159 - 0.8116 - 1.6394 - 2.6109 - 3.7375 - 5.0303 - 6.4997 - 8.1554 - 10.0057 - 12.0573 - 14.3151 - 16.7812 - 19.4551 - 22.3326 - 25.4054 - 28.6605 - 32.0795 - 35.6378 - 39.3039 - 43.0385 - 46.7939 - 0 0.15 0.29 0.4195 0.5376 0.6431 0.7344 0.8093 0.8657 0.9005 0.9108 0.893 0.8431 0.7569 0.6298 0.4569 0.233 0.0473 - 0.3897 - 0.7998 - 1.2835 - 1.8463 - 2.4939 - 3.2314 - 4.0637 - 4.9949 - 6.0285 - 7.1668 - 8.4111 - 9.7609 - 11.2142 - 12.7669 - 14.4123 - 16.1414 - 17.9416 - 19.797 - 21.6881 - 23.5905 - 25.4756 - 27.3093 - 29.0521 - 30.6584 - 32.0762 - 33.2468 - 34.1043 - 34.5755 - 34.5796 - 34.0283 - 32.8253 - 30.8669 - 28.042 - 351 mat

x1 Sol2 1 row transpose :

x2 Sol2 2 row transpose :

x3 Sol2 3 row transpose :

x1 x el x2 x el x3 x el 3 1 sys

The state grows indefinitely, i.e. the system is unstable.

K 1.4 0.5 - 1.4 - 3.3 - 1.9 - 023 mat :

Try to stabilize with proportional state feedback matrix:

x r Ctrl3 K x 1 el x 2 el x 3 el 31 mat * 1 0.5 21 mat + :

Sol3 any1 any2 LinSys1 any1 any2 Ctrl3 x0 50 0.1 dynSys :

x1 Sol3 1 row transpose :

x2 Sol3 2 row transpose :

x3 Sol3 3 row transpose :

x1 x el x2 x el x3 x el 3 1 sys

Sol3 0 0.05 0.086 0.1103 0.1252 0.1323 0.1334 0.1299 0.1231 0.1138 0.1029 0.0911 0.079 0.0669 0.0551 0.0439 0.0334 0.0237 0.0149 0.007 0.0001 - 0.0064 - 0.0118 - 0.0165 - 0.0206 - 0.0241 - 0.027 - 0.0294 - 0.0315 - 0.0331 - 0.0345 - 0.0356 - 0.0365 - 0.0372 - 0.0377 - 0.0381 - 0.0384 - 0.0386 - 0.0388 - 0.0389 - 0.039 - 0.039 - 0.0391 - 0.0391 - 0.0391 - 0.0391 - 0.039 - 0.039 - 0.039 - 0.039 - 0.039 - 00 0.0175 0.0478 0.0868 0.1315 0.1791 0.2278 0.2759 0.3223 0.3662 0.407 0.4445 0.4784 0.5087 0.5356 0.5592 0.5797 0.5974 0.6125 0.6252 0.6359 0.6448 0.6521 0.6581 0.6629 0.6667 0.6697 0.672 0.6738 0.6751 0.676 0.6767 0.6771 0.6774 0.6775 0.6776 0.6776 0.6775 0.6774 0.6773 0.6772 0.6772 0.6771 0.677 0.6769 0.6769 0.6768 0.6768 0.6768 0.6768 0 0.15 0.2595 0.3354 0.384 0.4106 0.4199 0.4161 0.4025 0.382 0.3568 0.3288 0.2995 0.2701 0.2414 0.214 0.1884 0.1648 0.1434 0.1241 0.107 0.092 0.079 0.0678 0.0582 0.0502 0.0434 0.0379 0.0333 0.0297 0.0267 0.0244 0.0226 0.0213 0.0203 0.0196 0.0191 0.0188 0.0186 0.0185 0.0185 0.0186 0.0187 0.0188 0.0189 0.019 0.0191 0.0192 0.0193 0.0194 0.0195 351 mat

EXAMPLE 3 - Nonlinear Equations (Lorentz Model)

x u Lorentz 5 x 2 el x 1 el - (* x 1 el x 3 el * - 15 x 1 el * + x 2 el - x 1 el x 2 el * 1 x 3 el * - 31 mat :

x r Ctrl4 0 :

x0 0.1 0.2 0.3 13 mat transpose :

Sol4 any1 any2 Lorentz any1 any2 Ctrl4 x0 500 0.02 dynSys :

Sol4 0.1 0.11 0.1215 0.1347 0.1496 0.1665 0.1854 0.2066 0.2303 0.2569 0.2867 0.3199 0.357 0.3984 0.4447 0.4964 0.5541 0.6185 0.6904 0.7707 0.8603 0.9602 1.0717 1.196 1.3346 1.4891 1.6611 1.8526 2.0655 2.302 2.5644 2.8551 3.1763 3.5304 3.9194 4.3451 4.8084 5.3094 5.8466 6.4166 7.0131 7.6263 8.2422 8.8416 9.3999 9.8869 10.2685 10.5082 10.5716 10.4305 10.0686 9.4859 8.7 7.7454 6.6685 5.5208 4.352 3.2047 2.1118 1.0957 0.1697 0.66 - 1.3923 - 2.0291 - 2.5743 - 3.0326 - 3.4092 - 3.7092 - 3.9377 - 4.0997 - 4.2004 - 4.2449 - 4.2387 - 4.1873 - 4.0965 - 3.9723 - 3.8204 - 3.6467 - 3.4567 - 3.2557 - 3.0485 - 2.8393 - 2.6318 - 2.4291 - 2.2337 - 2.0476 - 1.8722 - 1.7085 - 1.557 - 1.418 - 1.2914 - 1.1771 - 1.0745 - 0.9831 - 0.9023 - 0.8314 - 0.7696 - 0.7164 - 0.6709 - 0.6326 - 0.6007 - 0.5748 - 0.5543 - 0.5387 - 0.5276 - 0.5207 - 0.5175 - 0.518 - 0.5217 - 0.5286 - 0.5385 - 0.5513 - 0.567 - 0.5855 - 0.6068 - 0.631 - 0.6581 - 0.6882 - 0.7215 - 0.7582 - 0.7983 - 0.8422 - 0.89 - 0.9421 - 0.9987 - 1.0603 - 1.127 - 1.1995 - 1.278 - 1.3631 - 1.4553 - 1.5551 - 1.6632 - 1.78 - 1.9064 - 2.0429 - 2.1903 - 2.3492 - 2.5203 - 2.7043 - 2.9019 - 3.1135 - 3.3397 - 3.5805 - 3.8361 - 4.1061 - 4.3898 - 4.6861 - 4.9931 - 5.3082 - 5.6279 - 5.9479 - 6.2626 - 6.5653 - 6.8482 - 7.1026 - 7.319 - 7.4874 - 7.5983 - 7.6429 - 7.6142 - 7.5075 - 7.3213 - 7.0574 - 6.7212 - 6.3216 - 5.8697 - 5.3786 - 4.862 - 4.3335 - 3.8053 - 3.2882 - 2.7909 - 2.32 - 1.8801 - 1.4741 - 1.1032 - 0.7673 - 0.4657 - 0.1965 - 0.0422 0.253 0.4384 0.6009 0.7433 0.868 0.9774 1.0737 1.159 1.2351 1.3038 1.3666 1.4249 1.48 1.533 1.5848 1.6364 1.6885 1.7419 1.7973 1.8551 1.9159 1.9802 2.0484 2.1209 2.1982 2.2804 2.368 2.4612 2.5603 2.6655 2.777 2.8949 3.0193 3.1502 3.2875 3.4312 3.5809 3.7361 3.8965 4.0612 4.2293 4.3996 4.5709 4.7413 4.9091 5.072 5.2277 5.3735 5.5065 5.6239 5.7226 5.7998 5.8527 5.879 5.8769 5.8449 5.7826 5.6899 5.568 5.4185 5.2439 5.0474 4.8325 4.6033 4.3637 4.1179 3.8698 3.6229 3.3805 3.1454 2.9198 2.7057 2.5043 2.3166 2.1431 1.984 1.8393 1.7088 1.5919 1.4881 1.3968 1.3172 1.2486 1.1902 1.1414 1.1015 1.0697 1.0456 1.0286 1.0182 1.014 1.0156 1.0227 1.0351 1.0526 1.0751 1.1025 1.1347 1.1717 1.2137 1.2607 1.3129 1.3704 1.4334 1.5022 1.5771 1.6585 1.7465 1.8417 1.9444 2.055 2.174 2.3018 2.4388 2.5854 2.7421 2.9091 3.0867 3.2751 3.4743 3.6842 3.9044 4.1343 4.3729 4.6188 4.8703 5.1248 5.3796 5.6309 5.8746 6.1057 6.3187 6.5076 6.6663 6.7883 6.8677 6.8992 6.8786 6.8031 6.6718 6.4858 6.248 5.9636 5.6391 5.2824 4.9022 4.5071 4.1056 3.7056 3.3137 2.9357 2.5759 2.2375 1.9227 1.6326 1.3675 1.1271 0.9106 0.7168 0.5441 0.391 0.2557 0.1365 0.0316 0.0605 - 0.1416 - 0.2131 - 0.2763 - 0.3325 - 0.3829 - 0.4286 - 0.4704 - 0.5092 - 0.5458 - 0.581 - 0.6152 - 0.6491 - 0.6831 - 0.7179 - 0.7537 - 0.791 - 0.8302 - 0.8717 - 0.9157 - 0.9628 - 1.0132 - 1.0673 - 1.1254 - 1.188 - 1.2555 - 1.3282 - 1.4066 - 1.4912 - 1.5824 - 1.6808 - 1.7868 - 1.9011 - 2.0241 - 2.1565 - 2.2988 - 2.4516 - 2.6154 - 2.7908 - 2.9782 - 3.1779 - 3.3903 - 3.6153 - 3.8527 - 4.1022 - 4.3629 - 4.6335 - 4.9122 - 5.1966 - 5.4835 - 5.769 - 6.0482 - 6.3154 - 6.5641 - 6.787 - 6.9764 - 7.1244 - 7.2232 - 7.266 - 7.2469 - 7.1624 - 7.0109 - 6.7936 - 6.5144 - 6.1798 - 5.7983 - 5.3801 - 4.936 - 4.477 - 4.0136 - 3.5551 - 3.1095 - 2.683 - 2.2805 - 1.9051 - 1.5587 - 1.2421 - 0.9551 - 0.6968 - 0.4658 - 0.2603 - 0.0783 - 0.0824 0.2239 0.3483 0.4578 0.5543 0.6397 0.7157 0.7838 0.8455 0.9021 0.9548 1.0046 1.0526 1.0995 1.1461 1.1932 1.2414 1.2912 1.3433 1.3981 1.456 1.5177 1.5833 1.6535 1.7287 1.8091 1.8953 1.9876 2.0864 2.1922 2.3053 2.4261 2.5549 2.692 2.8377 2.9923 3.1558 3.3283 3.5097 3.6998 3.8979 4.1036 4.3158 4.5331 4.754 4.9763 5.1975 5.4146 5.624 5.8218 6.0036 6.1648 6.3006 6.4061 6.4767 6.5085 6.4983 6.4437 6.344 6.1997 6.0127 5.7864 5.5256 5.2359 4.9237 4.5956 4.2583 3.9182 3.581 3.2517 2.9345 2.6325 2.3483 2.0834 0.2 0.2254 0.2532 0.2839 0.3179 0.3556 0.3975 0.4442 0.4962 0.5541 0.6188 0.691 0.7715 0.8613 0.9616 1.0735 1.1983 1.3376 1.4931 1.6664 1.8597 2.075 2.315 2.5821 2.8792 3.2093 3.5758 3.9818 4.4308 4.9261 5.4708 6.0672 6.7172 7.4208 8.1762 8.9784 9.8184 10.6816 11.5462 12.3812 13.1453 13.7853 14.2363 14.4241 14.2705 13.7024 12.6659 11.1417 9.1606 6.8123 4.2411 1.6271 0.8457 - 3.0237 - 4.8089 - 6.1675 - 7.1206 - 7.7243 - 8.0493 - 8.1644 - 8.1275 - 7.9825 - 7.7603 - 7.4811 - 7.1579 - 6.7987 - 6.4091 - 5.9939 - 5.5578 - 5.1063 - 4.6456 - 4.1827 - 3.7249 - 3.2797 - 2.8538 - 2.4534 - 2.0834 - 1.7472 - 1.447 - 1.1836 - 0.9565 - 0.7644 - 0.6049 - 0.4753 - 0.3725 - 0.2934 - 0.2347 - 0.1935 - 0.167 - 0.1525 - 0.1479 - 0.1511 - 0.1607 - 0.175 - 0.1931 - 0.214 - 0.2371 - 0.2617 - 0.2874 - 0.3141 - 0.3415 - 0.3696 - 0.3984 - 0.4279 - 0.4582 - 0.4894 - 0.5218 - 0.5555 - 0.5907 - 0.6276 - 0.6666 - 0.7079 - 0.7518 - 0.7986 - 0.8486 - 0.9021 - 0.9596 - 1.0214 - 1.0879 - 1.1596 - 1.237 - 1.3206 - 1.4108 - 1.5084 - 1.6139 - 1.7279 - 1.8513 - 1.9847 - 2.129 - 2.285 - 2.4536 - 2.6356 - 2.8319 - 3.0436 - 3.2715 - 3.5165 - 3.7793 - 4.0605 - 4.3607 - 4.68 - 5.0183 - 5.3747 - 5.7481 - 6.1363 - 6.5362 - 6.9435 - 7.3525 - 7.7558 - 8.144 - 8.5058 - 8.8278 - 9.0947 - 9.2896 - 9.3947 - 9.3922 - 9.2661 - 9.0034 - 8.5964 - 8.0445 - 7.3557 - 6.5472 - 5.645 - 4.6822 - 3.6961 - 2.7247 - 1.8028 - 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x1 Sol4 1 row transpose :

x2 Sol4 2 row transpose :

x3 Sol4 3 row transpose :

xyPlot x3 x2 augment :

xyPlot Sol4 transpose