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.

FeedForward Control (FF)

Equation of Motion of the system:

m ẍ + c ẋ + k x = f_c + f_d + f_u

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

Control Target:

x → x_ref

Hp: ideal actuator

Target: x ≡ x_v

Model: ^̂m ẍ + ^̂c ẋ + ^̂k x_v = f_c + f_d

f_c = ^̂m ẍ_v + ^̂c ẋ_v + ^̂k x_v - 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 = ^̂m ẍ_v + ^̂c ẋ_v + ^̂k x_v + f_u

Equation of Motion of the FF Control System

Quasistatic zone:no amplification or filtering

Resonance zone:they are amplified

Seismic zone:they are filtered out

Once again we have found that using a FF control system we can't modify the filtering capabilities of the system, because G depends only on m, c, k.We can only modify the value of the damping coefficient c.

Let's now see the capability of the system to follow the reference:

mẍ + cẋ + kx = f_c = m̈x_r + cẋ_r + kx_r

Also here is convenient to apply the Fourier Transform:

x_r = ∑_j=1^∞|x_r0,j| cos(Ω_jt + ψ_j)

f_c = mẍ_r + cẋ_r + kx_r = ∑_j=1^∞(-mΩ_j² + icΩ_j + k) x_r0,je^iΩjt

f_c(t) = ∑_j=1^∞F_c0,je^iΩjt

Response:

x_pr = ∑_j=1^∞Re  (x_pr0,j e^iΩjt) = ∑_j=1^∞ Re (G(iΩ_j)F_c0,j e^iΩjt)

• Particular Solutions:

Once again we move to the frequency domain using the Fourier Transform:

Let's consider for simplicity only one single armonic component:

Input:

Solution:

This function L represent the the ratio between the reference motion and the response of the system.

To achieve the target we must have that L = 1 in all the frequency range:

Let's see the behavior of L:

It's clear that is not possible to achieve L = 1 in all the frequency range.

However:

• increasing k_p we approach 1 in the quasistatic zone
• increasing k_d we are better approximating the value of 1 in the resonance zone

So with a FB controller we are only able to have an approximation of the target.

So the steps to analyze the stability of a system are:

1. write the equation of motion of the non-linear system
2. calculate the static equilibrium position
3. linearize the system in the neighborhood of the static equilibrium position
4. study the perturbed motion of the linearized system
5. apply the Lyapunov theorem

Example:

1) Equation of motion:

mẍ + cẋ + kx = f(x,ẋ)

2) Static equilibrium position:

{ x = x_o ẋ = ẍ = 0 wx_o + cẋ + kx = f(x,ẋ) ⇒ kx_o = f(x_o,0) → x_o

Example:

3) Linearize the system in the neighborhood of one of the static equilibrium position:

To linearize the system we use Taylor series:

f(x,ẋ) = f(x_o,0) + ∂f/∂x|_o(x-x_o) + ∂f/∂ẋ|_o(ẋ-ẋ_o)

-K_F -C_F

Summary:

λ_1/2 complex conjugate

λ_1/2 both real

λ_1/2 c.c.:

• Re(λ_1/2) = 0 → λ_1/2 = ± iω₀ stability
• Re(λ_1/2) < 0 → λ_1/2 = -α ± iω asymptotic stability
• Re(λ_1/2) > 0 → λ_1/2 = |α| ± iω dynamic instability

λ_1/2 real:

• λ_1/2 negative asymptotic stability
• at least one of λ_1/2 positive static instability

What we have found is that:

if Re(λ_1/2) > 0 → UNSTABLE SYSTEM

So the system will be unstable when:

• c_ξ < 0
• c_τ > 0 & k_ξ < 0 instability

stability asymptotic stability

static instability (k_τ < 0) dynamic instability (c_ξ < 0)

Conservative Force Field

A force field is conservative if exits a function so that:

F = - grad V

The work done by the force field F to move the system is only dependent on the potential energy of the system at the initial and final position, it doesn't depend on the path.

Let's verify it:

F = f_x i + f_y j

grad V = (∂V/∂x) i + (∂V/∂y) j

F = - grad V   ⇒   f_x = - ∂V/∂x   f_y = - ∂V/∂y

W = ∫_r F ⋅ ds

F ⋅ ds = ( f_x i + f_y j ) ⋅ ( dx i + dy j ) =

  = f_x dx + f_y dy = -( ∂V/∂x dx + ∂V/∂y dy )

W = - ∫₁² ( ∂V/∂x dx + ∂V/∂y dy ) = - ∫₁² dV = V₁ - V₂

Along a closed path, the work done by the conservative force field is zero.

∮ F ⋅ ds = 0

So if β > 0 we have the same cases of conservative force fields, where we have stability only if Δ and γ are both > 0

• β < 0

β = _{(kxx + kyy)} / 2

λ_{1, 2, π, π̅} = - γ ± √β . . . . .

λ_{3, 4} = ± √M e^{-i φ̅ / 2}

In this condition the energy introduced by the non conservative force field makes the frequencies of the two modes of vibration to become equal.

So the energy introduced by the non conservative force field couples the two modes of vibration of the system and synchronize them at the same frequency ω.

Re (λ₁), Re (λ₃) > 0 => flutter instability (type of dynamic instability)

Flutter instability synchronizes the two modes of vibration of the system at the same frequency ω.

Let's analyze the motion of the system when flutter instability occurs.

Classical Control

• Classical Control Method: defined in frequency domain
• Modern Control Method: defined in time domain

The classical control it's easier to apply, since it's based on the FRF. It considers the relationships between the input and the output. It can only be used for single input - single output or single input - multi output systems, not for multi input - multi output.

While the modern control uses differential equations.

The classical control is based on the transfer function, and the transfer function is based on the Laplace Transform.

Laplace Transform

It's a way to solve differential equations.

Given: \( f(t) \), \( s = \sigma + i \omega \) Laplace variable

If \( \int_0^{\infty} f(t) e^{-st} dt \) exist, \( f(t) = 0 \; \text{for} \; t < 0 \)

this integral is called Laplace Transform of \( f \):

\( \mathcal{L} [f(t)] = F(s) = \int_0^{\infty} f(t) e^{-st} dt \)

If this integral exist for \( \sigma > \bar{\sigma} \), we can define also the Inverse Transform:

abscissa of absolute convergence

\( f(t) = \mathcal{L}^{-1}[F(s)] = \frac{1}{2\pi i} \int_{\sigma - i\infty}^{\sigma + i\infty} f(t) e^{-st} dt \; \text{for} \; \sigma > \bar{\sigma} \)