Non-parametric identification

Transcription

1 Non-parametric Non-parametric Transient Step-response using Spectral Transient Correlation Frequency function estimate Spectral System Identification, SSY230 Non-parametric 1

2 Non-parametric Transient Step-response using Spectral Consider the system described by or, equivalently, y(t) = y(t) =G 0 (q)u(t)+v(t) g 0 (k)u(t k)+v(t) k=1 Question: Can we determine G 0 (q) or {g 0 (k)} without parameterizing in θ? System Identification, SSY230 Non-parametric 2

3 Transient Non-parametric Transient Step-response using Spectral Applying the input gives the output u(t) = { k, t =0 0, t 0 y(t) =k g 0 (t)+v(t) which motivates the impulse response estimate ĝ(t) = y(t) k System Identification, SSY230 Non-parametric 3

4 Step-response Non-parametric Transient Step-response using Spectral Step input: u(t) = gives the output y(t) =k so that ĝ(t) = { k, t 0 0, t < 0 t g 0 (k)+v(t) k=1 y(t) y(t 1) k Problems with transient techniques: excitation disturbances nonlinearities System Identification, SSY230 Non-parametric 4

5 Some times one is not allowed to do anything else than small step changes. : For a tank with ideal mixing: V dc dt = Q(c i c) V :tankvolume,q: flow,c, c i : tank and flow concentration. Apply an impulsive concentration input: Stewart-Hamilton s equation: c i (t) =δ(t) c(t) = Q V e Q V t = h(t) 0 t h(t)dt = V Q Application: Blood volume measurement System Identification, SSY230 Non-parametric 5

6 using Non-parametric Transient Step-response using Spectral Assume that u is quasi-stationary and u and v are uncorrelated: Φ yu (ω) =G 0 (e iω )Φ u (ω) or R yu (τ) = g 0 (k)r u (τ k) k=1 Truncate the sum and solve the resulting system of equations for ĝ(k): M ˆR yu(τ) N = ĝ(k) ˆR u N (τ k) k=1 where estimates of the covariance function are formed as ˆR N u (τ) = 1 N N u(t)u(t τ) t=1 System Identification, SSY230 Non-parametric 6

7 Non-parametric Transient Step-response using Spectral Note: when u( ) is white noise the estimate becomes ĝ(k) = ˆR N yu(k) σ 2 u System Identification, SSY230 Non-parametric 7

8 Non-parametric Transient Step-response using Spectral So far we have estimated g 0 (k) what about estimating G 0 (e iω )? α cos ωt Measure amplitude and phase using : I c (N) = 1 N α y(t)cosωt N 2 G 0(e iω ) cos φ I s (N) = 1 N α y(t)sinωt N 2 G 0(e iω ) sin φ G α G cos(ωt + φ)+v(t) System Identification, SSY230 Non-parametric 8

9 Non-parametric Transient Step-response using Spectral implying I c (N) i I s (N) α 2 G 0(e iω ) (cos φ + i sin φ) so that Ĝ N = 2 α (I c(n) i I s (N)) System Identification, SSY230 Non-parametric 9

10 Interpretation of frequency response by the method: Recall that Y N (ω) = 1 N N t=1 y(t)e iωt = N(I c (N) i I s (N)) so that Ĝ N = 2 α (I c(n) i I s (N)) = 2 α N Y N(ω) = Y N(ω) U N (ω) where the last equality holds for ω = 2π N k. Hence, the estimate obtained using frequency by the method is simply the ratio between the DFT of the output and the input. This can be generalized... System Identification, SSY230 Non-parametric 10

11 Non-parametric Transient Step-response using Spectral Consider the open-loop system y(t) = G o (q)u(t)+v(t) Y N (ω) = G o (e iω )U N (ω)+v N (ω)+r N (ω) where R N (ω) 1/ N (or =0if u is periodic). function estimate () is: Ĝ(e iω )= Y N(ω) U N (ω), ω = k 2π N System Identification, SSY230 Non-parametric 11

12 Non-parametric Transient Step-response using Spectral Properties: u periodic number of ω fixed unbiased variance 1/N EĜ N (e iω ) = G 0 (e iω )+ R N (ω) U N (ω) VarĜ N (e iω Φ v (ω) ) U N (ω) 2 u stochastic process number of ω increases asymptotically unbiased variance (SNR) 1 as. uncorrelated System Identification, SSY230 Non-parametric 12

13 Spectral The is comprised of uncorrelated estimates at different ω but we know that the frequency response is smooth in reality! Idea: (1) Compute Ĝ(eiω ) as a weighted sum of the Ĝ around ω = ω 0 : Ĝ(e iω 0 )= k2 k=k 1 α k Ĝ(e iω k) k2 k=k 1 α k (1) where 2πk 1 /N = ω 0 Δ and 2πk 2 /N = ω 0 +Δ (2) Now choose {α k } inversely proportional to the variance of Ĝ(e iω k): Ĝ(e iω 0 )= UN (ω k ) 2 Φ v (ω k ) Ĝ(e iω k) UN (ω k ) 2 Φ v (ω k ) ω0 +Δ U N (ω) 2 ω 0 Δ Φ v (ω) ω0 +Δ U N (ω) 2 ω 0 Δ Φ v (ω) Ĝ(e iω )dω dω System Identification, SSY230 Non-parametric 13

14 (3) Put weight according to distance from ω 0 : Ĝ(e iω 0 )= π π W γ(ω ω 0 ) U N (ω) 2 Φ v (ω) Ĝ(e iω )dω π π π π W γ(ω ω 0 ) U W γ(ω ω 0 ) U N (ω) 2 Ĝ(e iω )dω N (ω) 2 π Φ v (ω) dω π W γ(ω ω 0 ) U N (ω) 2 dω π π = W γ(ω ω 0 )Y N (ω)u N (ω)dω π π W γ(ω ω 0 ) U N (ω) 2 = ˆΦ yu (ω 0 ) (2) dω ˆΦ u (ω 0 ) This is the estimate obtained by spectral according to Blackman-Tukey. The estimate is obtained by a natural replacement of the spectra in Φ yu (ω) =G 0 (e iω )Φ u (ω) (3) by the corresponding smoothed periodograms. System Identification, SSY230 Non-parametric 14

15 Non-parametric Transient Step-response using Spectral Consider the spectral estimate ˆΦ u (ω) = π π W γ (ω ω 0 ) U N (ω) 2 dω where the frequency window, W γ, is narrow for large γ. System Identification, SSY230 Non-parametric 15

16 Non-parametric Transient Step-response using Spectral The convolution above is transformed into multiplication in the time domain, so that where w γ (τ) = 1 2π ˆR u (τ) = 1 2π ˆΦ u (ω) = π π π π w γ (τ) ˆR u (τ)e iωτ W γ (ω)e iωτ dω U N (ω) 2 e iωτ dω = 1 N N u(t)u(t τ) Note: The time (lag) window w γ (τ) is wide when the frequency window W γ (ω) is narrow, and vice versa. System Identification, SSY230 Non-parametric 16

17 Non-parametric Transient Step-response using Spectral When γ increases (narrow frequency window), the bias decreases but the variance increases. This can be expressed in terms of the functions M(γ) = W (γ) =2π π π π π when γ.also, W (γ) γ. ω 2 W γ (ω)dω 0 (4) W 2 γ (ω)dω (5) System Identification, SSY230 Non-parametric 17

18 Non-parametric Transient Step-response using Spectral Theorem: Assume that y(t) =G 0 (q)u(t)+v(t) (6) where v( ) is a stochastic process, independent of u( ), whichis quasi-stationary. Then the following asymptotic results hold when γ,n,γ/n 0: [ ] 1 EĜN (e iω ) G 0 (e iω ) M(γ) 2 G 0 + G 0 Φ u Φ u E ĜN EĜN 2 1 N W γ Φv Φ u ReĜN, ImĜN are as. uncorrelated Estimates at different ω are as. uncorrelated System Identification, SSY230 Non-parametric 18