642feb45-6cc7-47c1-8658-ed40a3235eb2 5 4 5 decimal

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

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

Extensible image region to insert image objects [multiple images]. Image region does not blank the work sheet grid

Consultant: Geoff

Project: Edu-101 [page: 1/2]

Date: 20018_08_06

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 :

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

M 1 : η 0.75 : K 0.5 : 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 1 M / u 1 el η X 1 el * - 1 K / X 2 el * - (* X 1 el 21 mat :

Constant input, i.e. no state feedback in this case t r Ctrl 111 mat :

Observe: Ctrl(x,r):=[1]... step input [1] is a unit vector from matrix panel

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 K 3 1 sys

Export discrete response

Response 1 el Response 2 el K 3 1 sys 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

Extensible image region to insert image objects [multiple images]. Image region does not blank the work sheet grid

Consultant: Geoff

Project: Edu-101 [page: 2/2]

Date: 20018_08_06

2 f 2 g xo NumIt dynSysDiscr initialise 't & x' with xo t xo : x xo : 12 mat k 1 NumIt range t t t r g f eval : x x t augment : 21 line for x 41 line :

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 :

xo 00031 mat :

Constant input x r Ctrl2 1 0.5 21 mat :

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

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 xo 100 0.05 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