Covariance and spectrum Repetition Covariance function: r w ( ) - - PowerPoint PPT Presentation

Covariance and spectrum

Repetition

◮ Covariance function:

rw(τ) Ew(t + τ)wT (t) ⇒ rw(τ) = rT

w(−τ) ◮ Spectral density:

◮ Discrete-time systems/processes: for −π ≤ ω < π

Φw(ω)

∞

• τ=−∞

rw(τ)e−iωτ ⇒ rw(0) = 1 2π π

−π

Φw(ω)dω

◮ Continuous-time systems/processes: for ω ∈ R

Φw(ω) ∞

−∞

rw(τ)e−iωτdτ ⇒ rw(0) = 1 2π ∞

−∞

Φw(ω)dω

◮ In both cases

Φw(ω) = Φ∗

w(ω) ≥ 0

∀ω (Φw(ω) = Φw(−ω) ≥ 0 in the scalar case)

1 / 7 hans.norlander@it.uu.se Disturbance models

Linear filtering and spectral factorization

Repetition

◮ White noise: w(t) white noise ⇔ Φw(ω) = Rw = constant

◮ Discrete-time white noise: rw(τ) = 0 for all τ = 0

◮ Linear filtering:

◮ Discrete-time systems/processes:

y(k) = G(q)u(k) ⇒ Φy(ω) = G(eiω)Φu(ω)G∗(eiω)

◮ Continuous-time systems/processes:

y(t) = G(p)u(t) ⇒ Φy(ω) = G(iω)Φu(ω)G∗(iω)

◮ In the scalar case: Φy(ω) = |G|2Φu(ω)

◮ Spectral factorization: If 0 ≤ Φw(ω) < ∞ is rational in

◮ (discrete-time:) cos ω there exists a rational G(z) ◮ (continuous-time:) ω2 there exists a rational G(s)

which is stable and minimum phase, such that Φw(ω) = |G|2 (with z = eiω/s = iω respectively).

2 / 7 hans.norlander@it.uu.se Disturbance models

State space models

Discrete-time systems/processes

◮ Let v(k) be white noise with Ev(k) = 0 and rv(0) = Rv. ◮ Assume that x(k) ∈ Rn is governed by the state equation

x(k + 1) = Fx(k) + Gv(k), with F stable.

◮ Then x(k) is a stationary stochastic process, and Ex(k) = 0. ◮ The covariance matrix Πx = Ex(k)xT (k) = rx(0) is the

solution of the discrete-time Lyapunov equation, Πx = FΠxF T + GRvGT .

◮ The covariance function of x is

rx(τ) = Ex(k + τ)xT (k) = F τΠx, for τ ≥ 0. (For τ < 0, use that rx(τ) = rx(−τ)T .)

3 / 7 hans.norlander@it.uu.se Disturbance models

State space models

Continuous-time systems/processes

◮ Let v(t) be continuous-time white noise with Ev(t) = 0 and

intensity Rv.

◮ Assume that x(t) ∈ Rn is governed by the state equation

˙ x(t) = Ax(t) + Bv(t), with A stable.

◮ Then x(t) is a stationary stochastic process, and Ex(t) = 0. ◮ The covariance matrix Πx = Ex(t)xT (t) is the solution of the

continuous-time Lyapunov equation, 0 = AΠx + ΠxAT + BRvBT .

4 / 7 hans.norlander@it.uu.se Disturbance models

Cross-covariance and cross-spectrum

Some more relevant properties

◮ Consider two stationary stochastic processes, x(t) and y(t),

with Ex(t) = Ey(t) = 0.

◮ Cross-covariance: rxy(τ) Ex(t + τ)yT (t) ◮ Cross-spectrum:

Discrete-time processes: Φxy(ω)

∞

• τ=−∞

rxy(τ)e−iωτ Continuous-time processes: Φxy(ω) ∞

−∞

rxy(τ)e−iωτdτ

◮ x(t) and y(t) independent ⇒ rxy(τ) ≡ 0 and Φxy(ω) ≡ 0. ◮ Discrete-time: y(k) = G(q)u(k) ⇒ Φyu(ω) = G(eiω)Φu(ω) ◮ Continuous-time: y(t) = G(p)u(t) ⇒ Φyu(ω) = G(iω)Φu(ω)

5 / 7 hans.norlander@it.uu.se Disturbance models

Aggregation of noise

Strategy: Lump several noise sources together into one single source

Let v and e be white noise of zero mean and Φv(ω) = Rv, Φe(ω) = Re and Φve(ω) = 0. w y y v e ε ⇔

Gw Gε

Σ

◮ Then Φy(ω) = |Gw|2Rv + Re = rational function. ◮ By spectral factorization we get Φy(ω) = |Gε|2 for som

stable, minimum phase, rational Gε

◮ ⇒ we can compute ε = G−1 ε y (since G−1 ε

is stable).

6 / 7 hans.norlander@it.uu.se Disturbance models

The total model

Incorporating the noise in the system model

w y v1 n u z v2

G Gw Gn

Σ

◮ u - input, ◮ z - controlled/performance variable, ◮ y - measured output, ◮ w - system/process noise, ◮ n - measurement noise, ◮ v1 - white process noise, ◮ v2 - white measurement noise.

7 / 7 hans.norlander@it.uu.se Disturbance models