A software package for system identification in the behavioral setting

Transcription

1 1 / 20 A software package for system identification in the behavioral setting Ivan Markovsky Vrije Universiteit Brussel

2 2 / 20 Outline Introduction: system identification in the behavioral setting Solution approach: structured low-rank approximation Examples

3 Work plan 1. define a problem 2. develop methods/algorithms to solve the problem 3. implement the algorithm in software an identification problem involves: data collection set of time-series (trajectories) choice of model class bounded complexity LTI choice of identification criteria weighted 2-norm 3 / 20

4 4 / 20 Representation free problem formulation model B set of trajectories w : Z R q representation equations, which solution set = B parameters specify the representation behavioral setting problem definitions involve the model, not its representations solution methods do involve representations, however they are implementation details of the methods

5 LTI model representations kernel representation B = {w R 0 w + R 1 σw + + R l σ l w = 0} (KER) where σ is the shift operator (σw)(t) = w(t + 1) input/state/output representation B = {w = Π(u,y) x, such that σx = Ax + Bu, y = Cx + Du } (I/S/O) Π is a permutation matrix, defining the I/O partitioning 5 / 20

6 6 / 20 Model class L m,l the smallest l, for which (KER) exists, is the lag of B the smallest n = dim(x), for which (I/S/O) exists, is the order of B the number of inputs m is invariant of the repr. (I/S/O) (m,l) and (m,n) are measures of model s complexity L m,l LTI models with m inputs and lag l

7 7 / 20 Approximation criterion orthogonal distance between data and model M(w,B) := min w ŵ 2 ŵ B M(w,B) shows how much B fails to explain w called misfit (lack of fit) between w and B system identification problem minimize over B Lm,l M(w, B) (SYSID)

8 Generalizations multiple time-series w = {w 1,...,w N } M(w, B) := min N {ŵ 1,...,ŵ N i=1 w i ŵ i 2 2 } B fixed initial conditions w ini M(w, B) := fixed variables I {1,...,q } M(w, B) := min w ŵ 2 (w ini,ŵ) B min ŵ B, ŵ I =w I w ŵ 2 missing data: w i j (t) = NaN = w i j (t) is missing 8 / 20

9 9 / 20 Mosaic-Hankel matrix 1 N block matrix H l+1 (w) := [ H l+1 (w 1 ) H l+1 (w N ) ] with block-hankel blocks w i (1) w i (2) w i (T l) H l+1 (w i w i (2) w i (3) w i (T l + 1) ) :=... w i (l + 1) w i (l + 2) w i (T )

10 Mosaic-Hankel low-rank approximation ( w i (1),...,w i (T i l) ) B L m,l, for i = 1,...,N rank ( H l+1 (w) ) lq + m (SYSID) is mosaic-hankel low-rank approx. problem: minimize over B Lm,l M(w, B) minimize over ŵ (w ŵ) diag(v)(w ŵ) }{{} w ŵ v subject to rank ( H l+1 (w) ) lq + m v i = 0 if w i = NaN, v i = if w i is exact, w i = 1 otherwise 10 / 20

11 11 / 20 Solution method kernel representation of the rank constraint rank ( H l+1 (ŵ) ) r RH l+1(ŵ) = 0 RR = I (l+1)q r variable projection: elimination of the variable ŵ M(R) := min w ŵ v subject to RH l+1 (ŵ) = 0 ŵ is a least-norm problem with analytic solution M(R) = vec (w)γ 1 (R)vec(w) where Γ is a positive definite banded Toeplitz matrix

12 12 / 20 SLRA software package the identification problem is then minimize over R M(R) subject to RR = I nonconvex optimization problem on a manifold efficient evaluation of M(R) exploiting the structure software implementation is available

13 13 / 20 Usage and implementation [sysh,info,wh] = ident(w, m, ell, opt) sysh (I/S/O) repr. of the identified model opt.sys0 (I/S/O) repr. of initial approximation opt.wini initial conditions opt.exct exact variables info.rh parameter R of (KER) info.m misfit [M, wh, xini] = misfit(w, sysh, opt) implemented as a literate program the code lives in research papers the papers document the code the code reveals the full implementation details

14 14 / 20 The simulation setup simulation parameters ell = 2; m = 1; p = 1; T = 30; s = 0.02; define constants q = m + p; n = ell * p; the "true" system and "true" data sys0 = drss(n, p, m); u0 = rand(t, 1); y0 = lsim(sys0, u0);

15 15 / 20 The simulation setup add noise w0 = [u0 y0]; w = w0 + s * randn(t, q); generate noisy data for the examples clear all ell = 2; m = 1; p = 1; T = 30; s = 0.02; q = m + p; n = ell * p; sys0 = drss(n, p, m); u0 = rand(t, 1); y0 = lsim(sys0, u0); w0 = [u0 y0]; w = w0 + s * randn(t, q);

16 16 / 20 Invariance to variables permutation identify a model with permuted variables io = fliplr(1:q); [sysh, info] = ident(w(:, io), m, ell); disp(info.m) identify a model with the original variables order io = 1:q; [sysh, info] = ident(w(:, io), m, ell); disp(info.m)

17 Zero initial conditions identify the system with option opt.wini = 0 opt.wini = 0; [sysh0, info0, wh] = ident(w, m, ell, opt); disp(info0.m) verify that (0,...,0,ŵ) B whext = [zeros(ell, q); wh]; disp(misfit(whext, sysh0)) e-32 compare the misfit with free initial conditions zero initial conditions / 20

18 18 / 20 Identification from multiple trajectories split w into two parts and apply ident on them wm{1} = w(1:10, :); wm{2} = w(11:t, :); [sysh, infom, wmh] = ident(wm, m, ell); disp(infom.m) compare the misfit when fitting one trajectory the same trajectory split into two parts

19 19 / 20 System identification with missing data w 1 is the noisy trajectory of the system w 2 = (δ,nan) exact input, missing output, and zero initial conditions ŵ 2 = (δ,ĥ) estimate of the system s impulse resp. data-driven simulation problem ŷ 2 is compared with the true impulse response and the impulse response of the model identified from w 1

20 20 / 20 System identification with missing data wm{1} = w; wm{2} = [1 NaN; 0 NaN; 0 NaN; 0 NaN]; opt.wini = {[] 0}; opt.exct = {[] 1}; [~, info, wh] = ident(wm, m, ell, opt); disp(wh{2}(:, 2) ) disp(impulse(sys0, 3) )