Appunti Control Systems, Prof. Lanari

LTI System

_ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t)

x = state y = output u = input

A = dynamic matrix (mxm) B = input matrix (mxm) C = output matrix (pxm) D = feedthrough matrix (pxm)

Response u(t) = f(t)

x(t) = e^At x₀ + ∫₀^t e^A(t-τ) B u(τ) dτ

ZIR: Zero Input Response (FREE) ZSR: Zero State Response (forced)

y(t) = Ce^At x₀ + ∫₀^t Ce^A(t-τ) B u(τ) dτ + D u(t)

e^At = state-transition matrix Ce^At = output-transition matrix

Impulse Response u(t) = δ_Dir(t)

x(t) = e^At x₀ + H(t)

y(t) = Ce^At x₀ + W(t)

H(t) = ∫₀^t e^A(t-τ) B δ_Dir(t) dτ = e^At B

W(t) = ∫₀^t Ce^A(t-τ) B δ_smn(t) dτ + D δ_smn(t) = Ce^At B + D δ(t)

Change of Coordinates

z = T x ⇒ S → Ṡ : (detT ≠ 0)

_ż = Ã z + B̃ u y = C̃ z + D̃ u

Ŵ(t) = W(t) invariant!!

If A is diagonalisable then we can use T = U^-1 and Z_t(t) = e^Λt z₀

à = T A T^-1 B̃ = T B C̃ = C T^-1 D̃ = D

General Response u(t) = f(t)

x(t) = Φ(t) x₀ + ∫₀^t Φ(t-τ) u(τ) dτ

y(t) = Ψ(t) x₀ + ∫₀^t W(t-τ) u(τ) dτ

Φ(t) = e^At Ψ(t) = Ce^At

H(t) = Ce^At

W(t) = Ce^At B + D δ(t)

Linear Algebra

Characteristic Polynomial

P_A(λ) = det[A - λI]

Eigenvalues

λ_i : Are all the solutions of P_A(λ) = 0

Eigenvectors

u_i : All the vectors such that (A - λ_iI)u_i = 0

so u_i belongs to the nullspace of (A - λ_iI) i.e ker(A - λI)

Algebraic Multiplicity

m_A(λ_i) : Is the multiplicity of the solution of λ_i in P_A(λ) = 0

Geometric Multiplicity

m_G(λ_i) = dim[ker(A - λ_iI)] = n - rk[A - λ_iI]

(number of parameters in (A - λ_iI)u_i = 0)

Diagonalizable Matrix

An (m x m) matrix is diagonalizable if and only if m_G(λ_i) = m_A(λ_i) ∀ i

Λ = _A [λ₁ 0 ⋯ 0 0 λ₂ ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ -λ_m]

_U = [u₁, ..., u_m]

eigenvectors matrix

if T = U^-1

Λ = TAT^-1 = U^-1AU

Spectral Form: A = ∑_i=1^m λ_i u_i u_i^T

Case of Complex Eigenvalues

If : λ_i = α_i + jω_i and u_i = u_a + ju_b :

(λ_i^* = α_i - jω_i ; u_i^* = u_a; - ju_b;)

• Diagonalisation → [λ_i 00 λ_i^*]

complex elements

(T^-1 = [u_i u_i*])

• Real Blocks (2x2) → [α_i ω_i -ω_i α_i]

real elements

(T^-1 = [u_a; u_b;])

λ_i ∈ ℝ → Aperiodic natural modes

λ_i ∈ ℂ → Pseudo periodic natural modes

Foglio delle Trasformate

• Linearità Addizione:

  L [α f(t) + β g(t)] = α L [f(t)] + β L [g(t)]

• Linearità Prodotto:

  L [f(t) · g(t)] = L [f(t)] · L [g(t)]

• Teorema della Derivazione:

  L [d/dt f(t)] = s L [f(t)] - f(0)

• L [δ(t)] = 1
• L [u(t)] = k / s
• L [et] = 1 / s-a
• L [t^k / k!] = 1 / s^k+1
• L [t^k et] / k!] = 1 / (s-a)^k+1

• L [sin(ωt)] = ω / (s² + ω²) = ω/2 j / s-jω + -ω/2 j / s+jω
• L [cos(ωt)] = s / (s² + ω²) = ω/2 / s-jω + ω/2 / s+jω
• L [sin(ωt+φ)] = s sin φ + ω cos φ / s² + ω²

• et sin(ωt) = (s-a) / (s-a)² + ω²
• et cos(ωt) = s-a / (s-a)² + ω²

Trasformata di funzione con il ritardo:

L [f(t-τ) · g(t-τ)] = L [f(t) · g(t)] · e^-sτ

FATTORE BINOMIO (1+jωτ)/(1+jt*ω)

Si fa un'approssimazione intorno alla PULSAVILE DI ROTTURA ω* = 1/|τ| (CUTOFF FREQUENCY)

-1 1 per ωτ → P(s) is STRICTLY PROPER

      s^τ + a_τ-1s^τ-1 + ... + a₁s + a₀

C(s) = d_nsⁿ + d_n-1s^n-1 + ... + d₁s + d₀     → C(s) is PROPER at maximum

    s^χ + c_χs^χ-1 + ... + c₁s + c₀

D_wr = D_cDp + N_cN_p   → poles of S_cl are solutions of D_wr(s) = 0 (order max = n+τ)

I want to assign n+τ poles, P_1^z, P_2^z,...,P_n+τ^z → P_cl(s) = (s-P_1^z)(s-P_2^z)...(s-P_n+τ^z)

Pole Assignment: I choose χ = n + τ in order to obtain a solution (at least χ = n+τ, it's possible χ = n-1)

So: C(s) = ^{d_n+τsn+τ + ... + d₁s + d₀}⁄_{s^n+τ + cn+τs^n+τ-1 + ... + c1s + c0}

And (s-P_3^z)(s-P_2^z)...(s-P_n+τ^z) = D_p(s)·(s^n+χ + c_n+χs^n+χ-1 + ... + c₁s + c₀) + N_p(s)·(d_nχs^n+τ + ... + d₁s + d₀)