3505da33-766f-4e0a-8db9-35143970f8e9 19 4 5 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

q r q 3^r 2^+ 31 mat 0.1111 - 4.037 16.2963 31 mat

Cubic roots invariants

Z 1 el Z 2 el * Z 3 el * abs 40

q 3^r 2^+ 0 > q 3^r 2^+ 0 ≡ q 3^r 2^+ 0 < 31 mat 1 real pair complex conjugate all real roots, at least TwoEqual all real roots 31 mat ≡

[1]

Z 1 el Z 2 el * Z 1 el Z 3 el * + Z 2 el Z 3 el * + 40

Z 1 el Z 2 el + Z 3 el + (- 11

... polyroots -----------

404011141 mat polyroots 1.6051 - 4.6974 - 1.6894 i * + 4.6974 - 1.6894 i * - 31 mat

<= dotted plot complex x Lagrange x Z 1 el - (x Z 2 el - (* x Z 3 el - (* :

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 / + 1.6 - 2 1.6 - 2 - 2 2 mat 2 1 sys

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

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

x f x r n el - (n 13 product :

x f maple 10000 x * 16051 + (234875000 x * + 8447 i * - (* 234875000 x * + 8447 i * + (* 250000000000 /

x f 1 1 sys