After input-output linearizaton of my model, I obtain this equation (x represents the tracking error) from wich I can extract the command.
d6X/dt6+ kn-1 x d5X/dt5+ ...+k1 dx/dt + k0 x = 0
the ki must be chosen to satisty hurwitz.
But how i can play with the coefficents ki if i want for example a faster response or stability?
Thxs
Hi Nicolas,
The ki coefficients can be determined based on the desired deadbeat response, which yields a fast response with minimum overshoot.
The error dynamics
x'''''' + k5x''''' + k4x'''' + k3x''' + k2x'' + k1x' + k0x = 0
is first converted to the Transfer Function
T(s) = k0 / (s6 + k5s5 + k4s4 + k3s3 + k2s2 + k1s + k0).
Next, express it in Standard Transfer Function
T(s) = ω6 / (s6 + αωs5 + βω2s4 + γω3s3 + δω4s2 + εω5s + ω6).
Then, normalize it by dividing both numerator and denominator by ω6 yields
T(s) = 1 / (s6 + αs5 + βs4 + γs3 + δs2 + εs + 1)
where s = s/ω. To achieve the deadbeat response, the coefficients of the characteristic equation {α, β, γ, δ, ε} are specifically chosen to be
α = 3.15, β = 6.50, γ = 8.70, δ = 7.55, ε = 4.05.
The value of ω can be chosen based on the desired settling time, Ts. Since the normalized settling time of a 6th-order deadbeat system is 6.04, then ω is computed by
ω = 6.04 / Ts.
Now, the ki coefficients can be determined as follows:
k5 = αω, k4 = βω2, k3 = γω3, k2 = δω4, k1 = εω5, k0 = ω6.
Source: Dorf & Bishop, Modern Control Systems, 13th Edition - Pearson
https://www.pearson.com/us/higher-education/program/Dorf-Modern-Control-Systems-13th-Edition/PGM328731.html
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