3505da33-766f-4e0a-8db9-35143970f8e9 10 4 3 decimal

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

The synthetic cubic solvers

a0 11 :

b0 20 :

<= 2nd oder system dynamics

P 20 :

I 40 :

<= P [Proportional] I [integral] control parameters

a I b0 P + a0 31 mat :

a 40401131 mat

These two cubic solvers are equivalent for modeling PI & PID control Laplace algorithms. Naturally, the preference is type 2

The cubic roots solver: 1

S1 36 a 2 el * a 3 el * 108 a 1 el * - 8 a 3 el (3^* - 1212 a 2 el (3^* 3 a 2 el (2^* a 3 el (2^* - 54 a 1 el * a 2 el * a 3 el * - 81 a 1 el (2^* + 12 a 1 el * a 3 el (3^* + (12 /^* + (13 /^:

q 13 / a 2 el * 19 / a 3 el (2^* - :

q 0.1111 -

S1 12.0368

Z 16 / S1 * 6 q S1 / * - 13 / a 3 el * - 1 - 12 / S1 * 3 q S1 / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S1 * 6 q S1 / * + (* + 1 - 12 / S1 * 3 q S1 / * + 13 / a 3 el * - i 3 sqrt 2 / * 16 / S1 * 6 q S1 / * + (* - 31 mat :

a Z 12 mat 40401131 mat 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat 12 mat

The cubic roots solver: 2

q 13 / a 2 el * 19 / a 3 el (2^* - :

r 16 / a 2 el a 3 el * 3 a 1 el * - (* 127 / a 3 el (3^* - :

u1 r q 3^r 2^+ sqrt + (13 /^:

u2 r q 3^r 2^+ sqrt - (13 /^:

Z u1 u2 + (a 3 el 3 / - u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* + u1 u2 + 2 / (- a 3 el 3 / - i 3 sqrt 2 / u1 u2 - (* (* - 31 mat :

Z 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat

... polyroots fails ----------------------

a polyroots 1.8182 - 0.575 i * + 1.8182 - 0.575 i * - 21 mat

x Lagrange x Z 1 el - (x Z 2 el - (* x Z 3 el - (* :

<= Does NOT plot complex

40 40 x * + 11 x 2 ^ * + x 3 ^ + 1 1 x * + 11 40 / x 2 ^ * + x 3 ^ 40 / + 2 1 sys 1 1 x * + 11 40 / x 2 ^ * + x 3 ^ 40 / +

b 11 0.275 0.025 14 mat transpose :

r b polyroots :

r 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat

Z 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat :

x Lagrange x Z 1 el - (x Z 2 el - (* x Z 3 el - (* :

Force Lagrange plot complex: isolate 'Z'

x Lagrange 0 2 1 sys

xy 5 - 76.1428 - 547.92 i * + 4.9 - 43.9359 - 546.6588 i * + 4.8 - 11.729 - 545.3977 i * + 4.7 - 20.4779 544.1365 i * + 4.6 - 52.6848 542.8754 i * + 4.5 - 84.8917 541.6143 i * + 4.4 - 117.0986 540.3531 i * + 4.3 - 149.3055 539.092 i * + 4.2 - 181.5123 537.8308 i * + 4.1 - 213.7192 536.5697 i * + 4 - 245.9261 535.3085 i * + 3.9 - 278.133 534.0474 i * + 3.8 - 310.3399 532.7862 i * + 3.7 - 342.5468 531.5251 i * + 3.6 - 374.7537 530.2639 i * + 3.5 - 406.9606 529.0028 i * + 3.4 - 439.1675 527.7417 i * + 3.3 - 471.3744 526.4805 i * + 3.2 - 503.5813 525.2194 i * + 3.1 - 535.7882 523.9582 i * + 3 - 567.995 522.6971 i * + 2.9 - 600.2019 521.4359 i * + 2.8 - 632.4088 520.1748 i * + 2.7 - 664.6157 518.9136 i * + 2.6 - 696.8226 517.6525 i * + 2.5 - 729.0295 516.3914 i * + 2.4 - 761.2364 515.1302 i * + 2.3 - 793.4433 513.8691 i * + 2.2 - 825.6502 512.6079 i * + 2.1 - 857.8571 511.3468 i * + 2 - 890.064 510.0856 i * + 1.9 - 922.2709 508.8245 i * + 1.8 - 954.4778 507.5633 i * + 1.7 - 986.6846 506.3022 i * + 1.6 - 1.0189 103^* 505.0411 i * + 1.5 - 1.0511 103^* 503.7799 i * + 1.4 - 1.0833 103^* 502.5188 i * + 1.3 - 1.1155 103^* 501.2576 i * + 1.2 - 1.1477 103^* 499.9965 i * + 1.1 - 1.1799 103^* 498.7353 i * + 1 - 1.2121 103^* 497.4742 i * + 0.9 - 1.2443 103^* 496.213 i * + 0.8 - 1.2765 103^* 494.9519 i * + 0.7 - 1.3088 103^* 493.6907 i * + 0.6 - 1.341 103^* 492.4296 i * + 0.5 - 1.3732 103^* 491.1685 i * + 0.4 - 1.4054 103^* 489.9073 i * + 0.3 - 1.4376 103^* 488.6462 i * + 0.2 - 1.4698 103^* 487.385 i * + 0.1 - 1.502 103^* 486.1239 i * + 1.8 1014 -^* - 1.5342 103^* 484.8627 i * + 0.1 1.5664 103^* 483.6016 i * + 0.2 1.5986 103^* 482.3404 i * + 0.3 1.6308 103^* 481.0793 i * + 0.4 1.663 103^* 479.8182 i * + 0.5 1.6952 103^* 478.557 i * + 0.6 1.7274 103^* 477.2959 i * + 0.7 1.7597 103^* 476.0347 i * + 0.8 1.7919 103^* 474.7736 i * + 0.9 1.8241 103^* 473.5124 i * + 1 1.8563 103^* 472.2513 i * + 1.1 1.8885 103^* 470.9901 i * + 1.2 1.9207 103^* 469.729 i * + 1.3 1.9529 103^* 468.4678 i * + 1.4 1.9851 103^* 467.2067 i * + 1.5 2.0173 103^* 465.9456 i * + 1.6 2.0495 103^* 464.6844 i * + 1.7 2.0817 103^* 463.4233 i * + 1.8 2.1139 103^* 462.1621 i * + 1.9 2.1461 103^* 460.901 i * + 2 2.1783 103^* 459.6398 i * + 712 mat :