ELEC system identification workshop. Subspace methods

Transcription

1 1 / 33 ELEC system identification workshop Subspace methods Ivan Markovsky

2 2 / 33 Plan 1. Behavioral approach 2. Subspace methods 3. Optimization methods

3 3 / 33 Outline Exact modeling Algorithms

4 4 / 33 Plan Exact modeling Algorithms

5 5 / 33 The goal is to obtain a model B from data D data D U identification model B M U data space (R q ) N : functions from N to R q D data: set of finite vector-valued time series D = {wd 1,...,w d N }, w d i = ( wd i (1),...,w d i (T i) ) B model: subset of the data space U M model class: set of models

6 6 / 33 Work plan 1. define a modeling problem (What is D B?) 2. find an algorithm that solves the problem 3. implement the algorithm (How to compute B?)

7 7 / 33 State the aim without hidden assumptions all user choices should enter in the problem formulation hyper-parameters should not appear in the solutions the resulting methods should be automatic

8 User choices reflect prior knowledge; they determine the model class and fitting criterion 8 / 33 the "true model" assumption D = }{{} D + D }{{} true data noise where D }{{} B M true model assuming, in addition, that D is a stochastic process noise distribution maximum likelihood principle fitting criterion we can specify M and as deterministic approximation

9 9 / 33 Examples of user choices for M and Model class linear static time-invariant nonlinear dynamic time-varying Fitting criterion exact deterministic approximate stochastic

10 10 / 33 Why exact identification? from simple to complex: exact approx. stoch. approx. stoch. exact identification is ingredient of the other problems exact methods lead to effective approximation heuristics

11 11 / 33 Exact identification in L given data D find B L, such that D B nonunique solution always exists Exact identification in L m,l given (m,l) and data D find B L m,l, such that D B solution may not exist

12 12 / 33 Most powerful unfalsified model B mpum (D) given data D find the smallest (m,l), such that B L q m,l, D B Why complexity minimization? makes the solution unique Occam s razor: "simpler = better"

13 13 / 33 Identifiability question Recover the data generating system B from exact data D D B L q Under what conditions B mpum (D) = B? the answer is given by the "fundamental lemma"

14 14 / 33 Hankel matrix consider the case D = w d (single trajectory) main tool w(1) w(2) w(3) w(t L + 1) w(2) w(3) w(4) w(t L + 2) H L (w) := w(3) w(4) w(5) w(t L + 3).... w(l) w(l + 1) w(l + 2) w(t ) if w d B L q, then image ( H L (w d ) ) B L extra conditions on w d and B are needed for image ( H L (w d ) ) = B L

15 15 / 33 Persistency of excitation (PE) u is PE of order L if H L (u) is full row rank system theoretic interpretation: u (R m ) T is PE of order L there is no B L m 1,L, such that u B Lemma 1. B L q m,l controllable and 2. w d = (u d,y d ) B with u d PE of order L + pl = image ( H L (w d ) ) = B L

16 16 / 33 Plan Exact modeling Algorithms

17 The main idea is that a desired trajectory w can be constructed directly from the data w d 17 / 33 any w B L can be obtained from w d B w = H L (w d )g, for some g g input and initial conditions, cf., image representation

18 18 / 33 Algorithms w d kernel parameter R w d impulse response H w d state/space parameters (A,B,C,D) wd R (A,B,C,D) or w d H (A,B,C,D) wd observability matrix (A,B,C,D) wd state sequence (A,B,C,D)

19 w d R under the assumptions of the lemma image ( H l+1 (w d ) ) = B l+1 leftker ( H l+1 (w d ) ) defines a kernel repr. of B [R 0 R 1 R l ] H l+1 (w d ) = 0, R i R g q kernel representation B = ker ( R(σ) ), with R(z) = l R i z i i=0 recursive computation (exploiting Hankel structure) 19 / 33

20 20 / 33 w d H impulse response (matrix values trajectory) ( [ ] [ ] W = 0,...,0, I }{{} H(0), 0 H(1),..., l by the lemma, W = H l+t (w d )G define H l+t (u d ) =: we have U p Y p U f G = [ 0 H(t) [ ] [ ] Up Yp and H U l+t (y f d ) =: Y f } 0 0 ] [ Im 0 ]) zero ini. conditions impulse input (1) Y f G = H (2)

21 21 / 33 Block algorithm input: u d, y d, l, and t solve (2) and let G p be a solution compute H = Y f G p output: the first t samples of the impulse response H Exercise: implement and test the algorithm

22 22 / 33 Refinements solve (2) efficiently exploiting the Hankel structure do the computations iteratively for pieces of H automatically choose t, for a sufficient decay of H Exercise: try the improvements application for noisy data

23 23 / 33 w d (A,B,C,D) w d H(0 : 2l) or R(ξ ) realization (A,B,C,D) w d obs. matrix O l+1 (A,C) (3) (A,B,C,D) O l+1 (A,C) (A,C), (u d,y d,a,c) (B,C,x ini ) (3) (4) w d state sequence x d (A,B,C,D) [ ] [ ][ ] σx d A B x = d y d C D u d (4)

24 24 / 33 O l+1 (A,C) (A,B,C,D) C is the first block entry of O l+1 (A,C) A is given by the shift equation ( σ O l+1 (A,C) ) A = ( σo l+1 (A,C) ) (σ / σ removes first / last block entry) Once C and A are known, the system of equations y d (t) = CA t t 1 x d (1) + is linear in D, B, x d (1) τ=1 CA t 1 τ Bu d (τ) + Dδ(t + 1)

25 25 / 33 w d observability matrix columns of O t (A,C) are n indep. free resp. of B under the conditions of the lemma, [ ] [ ] H t (u d ) 0 zero inputs G = H t (y d ) free responses Y 0 lin. indep. free responses = G maximal rank rank revealing factorization ] Y 0 = O t (A,C) [x ini,1 x ini,j }{{} X ini

26 26 / 33 w d state sequence sequential free responses = Y 0 block-hankel then X ini is a state sequence of B computation of sequential free responses } U p U p sequential ini. conditions Y p G = Y p 0 zero inputs U f (5) Y f G = Y 0 rank revealing factorization [ Y 0 = O t (A,C) x d (1) ] x d (n + m + 1)

27 27 / 33 Refinements solve (5) efficiently exploiting the Hankel structure iteratively compute pieces of Y 0 iterative algorithm requires smaller persistency of excitation of u d could be more efficient (solve a few smaller systems of eqns than one big)

28 28 / 33 MOESP-type algorithms project the rows of H n (y d ) on rowspan ( H n (u) ) where Π u := Y 0 := H n (y d )Π u ( I Hn (u) ( H n (u)hn (u) ) ) 1 Hn (u) Observe that Π u is maximal rank and [ ] [ ] H n (u) Π 0 u = H n (y d ) = the orthogonal projection computes free responses Y 0

29 29 / 33 Comments H n (y d )Π u are T n + 1 free responses (n such responses suffice for exact identification) a geometric operation has system theoretic meaning condition for rank(y 0 ) = n given in the literature ([ ]) X rank ini = n + nm H n (u) is not verifiable from the data (u d,y d )

30 30 / 33 N4SID-type algorithms splitting of the data into "past" and "future" [ Up H 2n (u d ) =: ], H U f 2n (y d ) =: and define W p := oblique projection [ ] Up Y p [ ] Yp Y f ] + [ W p Y 0 := Y f / Uf W p := Y f [W ][ p W p Wp W p U U f f U f Wp U f U f 0 }{{} Π obl of the rows of Y f along rowspan(u f ) onto rowspan(w p ) ]

31 31 / 33 N4SID-type algorithms Observe that W p U f Y f Π obl = W p 0 (Π obl gives the least-norm, least-squares solution) Y 0 = oblique proj. computes sequential free responses

32 32 / 33 Comments Y 0 := Y f / Uf W p are T 2n + 1 sequential free responses (n + m + 1 such responses suffice for exact identification) geometric operation has system theoretic meaning conditions for rank(y 0 ) = n given in the literature 1. u d persistently exciting of order 2n and 2. rowspan(x ini ) rowspan(u f ) = {0} are not verifiable from the data (u d,y d )

33 33 / 33 Summary transitions among representations system theory exact identification aims at B mpum (w d ) H t (w d ) plays key role in both analysis and computation under controllability and u d persistently exciting image ( H t (w d ) ) = B t subspace methods construct special responses from data