Appunti Control Systems, Prof. Lanari

LTI System

^.x = Ax + Bu y = Cx + Du

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^Atx₀ + ∫₀^t e^A(t-τ)B u(τ) dτ y(t) = Ce^Atx₀ + ∫₀^t Ce^A(t-τ)B u(τ) dτ + D u(t)

ZIR = zero input response (free)

ZSR = zero state response (forced)

e^At = state-transition matrix

Ce^At = output-transition matrix

Impulse Response u(t) = δ_Dir(t)

x(t) = e^Atx₀ + H(t) y(t) = Ce^Atx₀ + W(t)

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

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

Change of Coordinates

z = Tx => S → S_̅

(det T ≠ 0)

N.Bo: Ẇ(t) = W(t) invariant!!

If A is diagonalisable then we can use T = U^-1 and

Z_t(t) = e^λ_it z₀

General Response u(t)=f(t)

x(t) = Φ(t)x₀ + ∫₀^t H(t-τ)u(τ)dτ

y(t) = Ψ(t)x₀ + ∫₀^t W(t-τ)u(τ)dτ

Φ(t) = e^At

H(t) = Ce^At

Ψ(t) = Ce^At

W(t) = Ce^AtB + D δ(t)

LTI System

ẋ = Ax + Buy = Cx + Du

• 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 + ∫ e^A(t-τ) Bu(τ) dτy(t) = Ce^At x₀ + ∫ e^A(t-τ) B μ(t) dτ + D u(t)

e^At = state-transition matrixCe^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)

Property: ∫ f(t-σ) δ(σ) dσ = f(t)

H(t) = ∫ e^A(t-τ) B δ_Dir(t) dτ = e^At BW(t) = ∫ Ce^A(t-τ) B δ_Dir(t) dτ + D δ_Dir(t) = Ce^At B + D δ(t)

Change of Coordinates

ż = Ãz + B̃μy = Ćz + D̃μ

z = Tx, S → ṠĂż = Ṡ˙

New matrices:Ã = TAṠ^-1B̃ = TBĆ = CṠ^-1D̃ = D

N.B.: Ṽ(t) = W(t) invariant!!

If A is diagonalisable then we can use T = U^-1 and

Z_i(t) = e^λ_it z₀

General Response u(t) = f(t)

x(t) = Φ(t) x₀ + ∫ H(t-τ) u(τ) dτy(t) = Ψ(t) x₀ + ∫ W(t-τ) u(τ) dτ

Φ(t) = e^AtΨ(t) = Ce^AtH(t) = Ce^AtW(t) = Ce^At B + D δ(t)

LINEAR ALGEBRA

CHARACTERISTIC POLYNOMIAL

EIGENVALUES :

Are all the solutions of P_A(λ) = 0

EIGENVECTORS :

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

so u_i belongs to the nullspace of (A - λ_iI) {ker(A - λ_iI)}

ALGEBRAIC MULTIPLICITY

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

GEOMETRIC MULTIPLICITY

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

DIAGONALIZABLE MATRIX

An (mxn) matrix is diagonalizable if and only if mg(λ_i) = m_a(λ_i) ∀ i

if T = U^-1

CASE OF COMPLEX EIGENVALUES

• DIAGONALIZATION → [ λ_i 0 ] complex elements [ 0 λ_i^* ] ( T^-1 = [u_i u_i^*] )
• REAL BLOCKS → [ α_i ω_i ] real elements [ -ω_i α_i ]
( T^-1 = [u_a, u_b] )

λ_i ∈ ℝ → APERIODIC