Formulario Model Identification

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Model Identification ↪→ Cheat Sheet ←↩ Stationary Stochastic Process (SSP). • m(t) =m∀t • γν(t1,t 2) depends on τ =t1−t2 only or: the transfer function of the process is asymptotically stable, i.e. all|poles|< 1 MA(n) process. Always SSP. my(t) = (c0 +c1 +c2 +... +cn)me γy(τ) =    ( c2 0 +c2 1 +c2 2 +... +c2 n ) λ2 ,τ = 0 (c0c1 +c1c2 +... +cn−1cn)λ2 ,τ =±1 (c0c2 +c1c3 +... +cn−2cn)λ2 ,τ =±2 ... (c0cn)λ2 ,τ =±n 0 ,τ >±n Yule-Walker equations for AR(1). y(t) =ay(t− 1) +e(t). my(t) = 0 γy(0) = λ2 1−a2 γy(τ) =aγy(τ− 1) Spectrum. • Fourier Transform Γy(ω) = +∞∑ τ =−∞ γy·e−jωτ = F{γy(τ)} • Inverse Fourier Transform γy(τ) = F(Γy(ω)) = 1 2π ∫ +π −π Γy(ω)·e+jωτdω • for SSP: y(t) =F (z)ν(t) Γy(ω) =|F (ejω)|2· Γν(ω) • if the transfer function F(z) has two complex conjugate zeros, the spectrum in the corresponding ω =z(zero)→ Γy(z) = 0 Canonical Form. y(t) = C(z) A(z)e(t) • max degree terms of C(z) and A(z) equal to 1 • C(z) and A(z) with same degree • C(z) and A(z) with no common factors • |poles|< 1, |zeros|≤ 1 in case of unstable zero (a)→ ALL-PASS FILTER: y(t) = z +a A(z)e(t) = z + 1 a A(z) · η(t)    z +a z + 1 a   all-pass e(t) k-Step Predictor. AAAAAAAAAA C(z) A(z) ... E(z) z−kF (z) C(z) A(z) =E(z) +z−kF (z) A(z) ˆy(t|t−k) = F (z)z−k A(z) ·e(t) predictor from noise ˆy(t|t−k) = F (z)z−k C(z) ·y(t) predictor from data 1-step Predictor (ARMA). AAAAAAAAAA ˆy(t|t− 1) = C(z)−A(z) A(z) ·e(t) predictor from noise ˆy(t|t− 1) = C(z)−A(z) C(z) ·y(t) predictor from data Non-zero Mean ARMA Predictor. AAAAAAAAAA ˆy(t|t−k) = F (z)·z−k C(z) ·y(t) + ( 1− F (1) C(1) ) ·my ARMAX Predictor. AAAAAAAAAA ˆy(t|t−k) = F (z)·z−k C(z) ·y(t) + B(z)E(z) C(z) ·u(t−d) Prediction Error. AAAAAAAAAA ϵ(t|t−k) =y(t)− ˆy(t|t−k) prediction error var = E [ ϵ(t|t−k)2] variance of prediction…