38606dcf-1b56-407a-945d-b6b7fccc0546 7 6 9 decimal

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

The discrete maths of a 2nd order system

Commonly, we are used to see a DE with visible 2nd member, i.e: = 0, UnitStep = 1, a forcing function f(t). Less commonly, the 2nd member is a DE ... implicit of UnitStep DE. Here are the maths of this demo, up to the plot.

9 y t 2 diff * y t diff + y + 8 x t 2 diff * 3 x t diff * + x + ≡

a1 9 :

b1 1 :

a2 8 :

b2 3 :

Δt 1 :

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r1 b1 - b1 2^4 a1 * - sqrt + 2 a1 * / :

r2 b1 - b1 2^4 a1 * - sqrt - 2 a1 * / :

r1 r2 21 mat 0.0556 - 0.3287 i * + 0.0556 - 0.3287 i * - 21 mat

α a1 a2 - (b1 b1 b2 - (* - Δt 2^/ :

β b1 b2 - (Δt / :

γ r1 r2 + (Δt * exp :

δ r1 Δt * exp r2 Δt * exp + :

γ 0.8948

δ 1.7906

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c1 γ - :

c2 δ :

α β γ δ 41 mat 32 - 0.894839 1.790649 41 mat

c3 γ α β 2 / - (1 γ + δ - (* - β 1 γ - (* - :

c4 δ - 2 α * 1 γ + δ - (* + 2 β * 1 γ - (* + :

c5 1 α β 2 / + (1 γ + δ - (* - β 1 γ - (* - :

c1 c2 c3 c4 c5 51 mat 0.894839 - 1.790649 0.688398 1.586147 - 1.00194 51 mat

Note that the recursive 3 coefficients collects the entire solution and that the final recursive settles to ε, either a setpoint [SP] change or perturbation from initial conditions s1,s2 = 0,0

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Fundamental properties

The recursive coefficients

c1 c2 + c3 + c4 + c5 + 1

a c1 :

b c2 :

c c3 c4 + c5 + :

c3 c4 + c5 + 0.104191

d c4 2 c5 * + :

a b c d 41 mat 0.894839 - 1.790649 0.104191 0.417733 41 mat

a b + c + 1

The system response

m 40 :

<= 'm' ramp length

sp 60 :

<= SetPoint final

X j 1 m range X j el j b2 / : for X 21 line :

a 0.894839 - : b 1.790649 : c 0.104191 : 31 line

U j 1 sp range U j el X X rows el : for U 21 line :

vx X U stack :

Modeling via initial 's2'. ramp s 1 el 0 : play with initial s2 s 2 el 1 : n 3 vx rows range s n el a s n 2 - el * b s n 1 - el * + c vx n 1 - el * + : for i 1 s rows range x i el i : for x s augment 61 line :

scale 1 vx max / : 0.075

X ramp 1 col : Y ramp 2 col scale * : rampOUT X Y augment : 31 line

rampOUT 1 m 2 m 2 - 2 2 mat 3 1 sys