Appunti completi del corso "Control and actuating devices for mechanical systems"

Control Strategy

FeedForward:

• It's fast
• But it can't detect deviations of the response caused by disturbances, because the response of the system is not measured.

It's not able to change the stiffness and the damping of the system. So it's not affecting the stability of the system.

FeedBack:

• The response is slower
• But we are able to recover the deviation of the response of the system thanks to the feedback of the sensor.

So we can change the stability of the system.

FeedForward + FeedBack:

It combines the advantages of the two.

Control Strategy

FeedForward:

• It's fast
• But it can't detect deviations of the response caused by disturbances, because the response of the system is not measured.

It's not able to change the stiffness and the damping of the system. So it's not affecting the stability of the system.

FeedBack:

• The response is slower
• But we are able to recover the deviation of the response of the system thanks to the feedback of the sensor.

So we can change the stability of the system.

FeedForward + FeedBack:

It combines the advantages of the two.

FeedForward Control (FF)

Equation of Motion of the system:

In this system we want to control the displacement x of the mass.

Control Target: X → X_ref

H_p: ideal actuatorTarget: x ≡ x_r

Model: _estm_ẍ + _estc_ẋ + _estk_x = f_c + f_d

we can't include the unknown force into the system

Substituting f_c in the equation of motion of the system:

m_ẍ + c_ẋ + k_x = _estm_ẍ + _estc_ẋ + _estk_x + f_u

Equation of Motion of the FF Control System

mẍ + cẋ + kx = m _t + c _t + k _t = f_u

2 inputs: f_u, x_r

Response of the system: x(t) = x_h(t) + x_ph(t) + x_pv(t)

transient steady response

• Homogeneous Solution: x_h (solution of the equation of free motion)

Free Motion: mẍ + cẋ + kx = 0

Solution: x_h = x_h0 e^λt λ ∈ ℂ

(mλ² + cλ + k) x_h0 e^λt = 0

mλ² + cλ + k = 0 characteristic equation of the system

λ₁,₂ = -^c/_2m ± √(^c/_2m)² - ^k/_m

= -α ± ω_o √(h²-1)

ω_o = √(^k/_m)

h = ^c/_2mωo, α = ^c/_2m = h ω_o

natural damping

frequency factor

h > 1 overdamped

h = 1 critical

h < 1 underdamped → oscillations

h > 1:

x_h(t) = x_h01 e^λ₁t + x_h02 e^λ₂t = e^-αt (Acos ωt + Bsin ωt)

The FeedForward control does not affect the homogeneous solution of the system,

the response of the system only depends on the parameters of the passive system (m, c, k),

so the FeedForward control doesn't affect the system stability.

• Particular Solutions: since the system is linear, we can apply the superposition principle:

It is convenient to solve this not in the time domain, but in the frequency domain.

To shift from one another we can use the Fourier Transform:

Let's make the assumption that the mean value is equal to 0,

and let's calculate the response of the system only to one of the armonic function:

Equation:              _w (t) = Re (F_wo e ^iΩt)

Solution:  x_pu = Re (x_puo e^iΩt)

(it's of the same type of the input)

     x_pu = x_pu, e^iΩt

 x_pu =

 x_pu = -mΩ² + icΩ + k)

 x_pu

  &emsp