c84a6723-96ad-4efb-93f8-01b2e5b2cf8d 9 4 5 decimal

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

2 f 2 g x0 NumIt dt dynSys ξ x0 : 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 x0 NumIt dynSysDiscr ξ x0 : x ξ : k 1 NumIt range ξ ξ ξ r g f eval : x x ξ augment : 21 line for x 41 line :

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

system parameters

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

M 1 :

η 0.7 :

shift ▲  ▼ K 1 :

x r Ctrl1 111 mat :

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

initial condition x0 0021 mat :

NumIt 150 :

Sol x u MasSprDmp x r Ctrl1 x0 NumIt 0.1 dynSys :

Sol 0 0.1 0.193 0.2785 0.3561 0.4254 0.4864 0.5388 0.5827 00 0.01 0.0293 0.0571 0.0928 0.1353 0.1839 0.2378 29 mat

vx Sol transpose 1 col : vy Sol transpose 2 col 0 + : 12 mat

v Sol 1 row transpose :

s Sol 2 row transpose :

ux 0 Sol transpose 2 col rows 1 - (range :

alternate dummy style any1 any2 MasSprDmp any1 any2 Ctrl1 x0 NumIt 0.1 dynSys

plot v x el s x el NumIt 1.5 NumIt 0.5 - 22 mat 31 sys :

plot

vx ux : vy Sol transpose 2 col : 12 mat x sp vx vy x ainterp : 21 sys

x λ 10 x ≤ (x NumIt ≤ (& cases :

x sp x λ *

The ROBUST root(s) Intersection Solve scanner f q x0 x1 TOL Solve ρ x f x q - x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : 12 mat xn b a + (2 / : n 0 : xn f xn q - abs TOL > xn f xn q - (a f a q - (* 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 41 line X i el xn : 21 line for X n 21 mat 31 line :

x f x sp x λ * :

TOL 1013 -^:

x q K :

Inclusive Parametric command line 
Bracket wisely, observing the plot ZACroots f q 0 NumIt TOL Solve :

ZACroots 19.9276 52.3921 84.8572 117.3216 41 mat 3621 mat

X ZACroots 1 el : 11 mat 19.9276 52.3921 84.8572 117.3216 41 mat 11 mat

vx vy Clear 1

vx Sol transpose 1 col : vy Sol transpose 2 col : 12 mat

x sp x λ * K X 1 el K . 14 black 1 5 mat X 2 el K . 14 black 1 5 mat X 3 el K . 14 black 1 5 mat X 4 el K . 14 black 1 5 mat 6 1 sys

x sp x λ * K X 1 el K . 14 black 15 mat X 2 el K . 14 black 15 mat X 3 el K . 14 black 15 mat X 4 el K . 14 black 15 mat 61 sys

Spiral style ... bonus

vx vy augment

NOT offered ... you can locate 1. max/min spiral X directions, 2. max Y spiral

Mass spring damped system

Damped system in continuous time ... η => damping coefficient.
f(2), g(2) go in pair with "init" 2 f 2 g init NumIt dt dynSys t init : u t : k 1 NumIt range t t dt t t r g f * + eval : u u t augment : 21 line for u 41 line :

X Clear 1

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

M 1 : η 0.875 : K 2 : 31 line

initial condition init 0021 mat :

n 150 : number of samples ≡ step 0.1 : increment size ≡ K step input ≡ 31 line

For "zoom detail" of the front end <= decrease step size. <= experiment various 'K'

The mass-spring differential system X u MasSprDmp M u 1 el η X 1 el * - K X 2 el * - (* X 1 el 21 mat :

n 150

t r Ctrl 111 mat :

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

Sol any1 any2 MasSprDmp any1 any2 Ctrl init n step dynSys :

v Sol 1 row transpose :

s Sol 2 row transpose :

Response sol v1 v2 MasSprDmp v1 v2 Ctrl init n step dynSys : U i 1 n 1 + range idx i el i : for idx sol transpose augment 21 line : U 1 col U 2 col augment U 1 col U 3 col augment 21 mat 31 line :

Auto-index from the canvas var 'x'

v x el s x el 1 K / 3 1 sys

Export discrete response

Response 1 el Response 2 el 1 K / 3 1 sys

sol v1 v2 MasSprDmp v1 v2 Ctrl init n step dynSys :

sol 0 0.1 0.1912 0.2725 0.3428 0.4016 0.4483 0.4829 0.5055 00 0.01 0.0291 0.0564 0.0907 0.1308 0.1756 0.2239 29 mat