ELEC system identification workshop. Behavioral approach

Transcription

1 1 / 33 ELEC system identification workshop Behavioral approach Ivan Markovsky

2 2 / 33 The course consists of lectures and exercises Session 1: behavioral approach to data modeling Session 2: subspace identification methods Session 3: optimization-based identification methods "I hear, I forget; I see, I remember; I do, I understand." session = lecture (you hear and see) + exercises (you do)

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

4 4 / 33 Outline From Ax = B to low-rank approximation Linear static model representations Linear time-invariant model representations

5 5 / 33 Outline From Ax = B to low-rank approximation Linear static model representations Linear time-invariant model representations

6 6 / 33 A classic line fitting method is solving Ax B problem: fit points d 1,...,d N R 2 by line going through 0 approach: find approximate solution x R of a 1 b 1. x =., where d i = a N b N the fitting line is B := {[ b a] R2 ax = b } (x is model parameter) [ ai b i ]

7 7 / 33 The choice of a and b is arbitrary another approach: find approximate solution x R of b 1 a 1. x =. b N a N the fitting line is B := {[ b a] R2 a = bx } (x is model parameter) exceptions: vertical line x = x = 0 horizontal line x = 0 x =

8 8 / 33 In general, the two solutions differ: B B solving Ax = B and Bx = A leads to different solutions the fitting criterion depends on how we choose a and b the mode representation affects the fitting criterion

9 9 / 33 Ax = B imposes input/output model structure functional relations Ax = B defined a function a b Bx = A defined a function b a in the model B := {[ a b ] R2 ax = b } a is input, b is output (a causes b) in the model B := {[ a b ] R2 bx = a} b is input, a is output (b causes a)

10 10 / 33 Model class set of all candidate models in the example, the model class is M := {lines through 0} separately, ax = b and bx = a don t represent all B M any B M is representable as B = {Π[ a b ] ax = b } with Π a permutation matrix

11 11 / 33 Definition of least-squares line fitting problem given points D = {d 1,...,d N } R 2 and model class M minimize over B M error(d, B) where notes: error(d, B) := min D B N d i d i 2 2 i=1 D B means that B fits { d1,..., d N } exactly di is the projection of d i on the line B di d i 2 is the orthogonal distance from d i to B

12 12 / 33 Any B M can be represented as kernel any B M can be represented as B = ker(r) := {d R 2 Rd = 0} (R R 1 2, R 0 is a model parameter) Rd = 0 defines a relation (implicit fucntion) between a and b exact modeling condition {d 1,...,d N } ker(r) ] R [d 1 d N = 0 } {{ } D

13 13 / 33 Any B M can be represented as image any B M can be represented as B = image(p) := {d = Pl l R} (P R 2 1 is a model parameter) d = Pl also defines a relation between a and b exact modeling condition D image(p) [d 1 d N ] = PL (L R 1 N is a latent variable)

14 For exact data, rank ([ d 1 d N ] ) 1 14 / 33 common feature of the representations considered [ ] x R,Π permut. x 1 ΠD = 0 R R 1 2,R 0 RD = 0 P R 2 1,L R 1 N D = PL rank(d) 1 representation free characterization of exact data D B M rank(d) = 1

15 Approximate modeling of data is equivalent to low-rank approximation minimize over D error(d, D) subject to exact model for D exists minimize over D error(d, D) subject to D is rank deficient 15 / 33

16 16 / 33 Low-rank approximation is a general concept 1. multivariable data fitting U = R q linear static model subspace model complexity subspace dimension rank(d) upper bound on the model complexity 2. nonlinear static modeling D D nonlinear function nonlinearly structured low-rank approximation 3. linear time-invariant dynamical models D Hankel matrix D Hankel structured low-rank approximation

17 The matrix structure corresponds to the model class 17 / 33 structure S unstructured Hankel q 1 Hankel q N Hankel mosaic Hankel [ ] Hankel unstructured model class M linear static scalar LTI q-variate LTI N equal length traj. N general trajectory finite impulse response block-hankel Hankel-block 2D linear shift-invariant

18 18 / 33 EIV, PCA, and factor analysis are related errors-in-variables modeling all variables are perturbed by noise maximum likelihood estimation LRA principal component analysis another statistical setting for LRA factor analysis factors latent variables in an image representation

19 19 / 33 Outline From Ax = B to low-rank approximation Linear static model representations Linear time-invariant model representations

20 A linear static model is a subspace linear static model with q variables = subspace of R q model complexity subspace dimension Lm,0 linear static models with complexity at most m B L m,0 admits kernel, image, and I/O representations 20 / 33

21 A linear static model admits kernel, image, and input/output representations 21 / 33 kernel representation with parameter R R p q ker(r) := {d Rd = 0} image representation with parameter P R q m image(p) := {d = Pl l R m } input/output representation with parameters X R m p, Π B i/o (X,Π) := {d = Π [ ] u y u R m, y = X u }

22 22 / 33 The parameters R and P are not unique addition of linearly dependent rows of R columns of P minimal representations the smallest number of generators is m := dim(b) the max. number of annihilators is p := q dim(b) change of basis transformation ker(r) = ker(ur), U R p p,det(u) 0 image(p) = image(pv ), V R m m,det(v ) 0

23 23 / 33 Inputs and outputs can be deduced from B definition input is a "free" variable Π [ ] u y B and u input u R m output is bound by input and model fact: m := dim(b) number of inputs p := q m number of outputs choosing an I/O partition amounts to choosing full rank p p submatrix of R full rank m m submatrix of P

24 It is possible to convert a given representation into an equivalent one 24 / 33 B = ker(r) RP=0 B = image(p) X= (Ro 1 R i ) X=(P o P 1 i ) R=[X I]Π P =[I X]Π B = B i/o (X,Π) Π P =: [ P i P o ] m p m p ] and RΠ =: [R i R o

25 25 / 33 Outline From Ax = B to low-rank approximation Linear static model representations Linear time-invariant model representations

26 26 / 33 Dynamical models are sets of functions observations are trajectories w (R q ) N set of functions from N to R q shift operator: (σ τ w)(t) := w(t + τ), for all t N discrete-time dynamic model B is a subset of (R q ) N properties linearity: w,v B = αw + βv B, for all α,β time-invariance: σ τ B = B, for all τ N

27 Controllability can be defined in a representation free manner 27 / 33 w w c w f w p l T 1 T 2 t for all w p, w f B, there is w c, such that w p w c w f B (" " denotes "concatenation" of trajectories)

28 An LTI model admits kernel and input/state/output representations kernel representation with parameter R(z) R g q [z] ker(r) = {w R(σ)w = R 0 w + R 1 σw + + R l σ l w = 0} image representation with parameter P(z) R q g [z] image(p) = {w = P(σ)v for some v } input/state/output representation B(A,B,C,D,Π) := {w = Π [ u y ] exists x, such that σx = Ax + Bu and y = Cx + Du } 28 / 33

29 Minimal kernel and image representations have full rank R and P parameters 29 / 33 minimal row dim(r) = number of outputs minimal col dim(p) = number of inputs lag of B minimal l, for which kernel repr. exists

30 The I/S/O representation is not unique choice of an input/output partition redundant states (nonminimality of the representation) minimal representation n = order of B change of state space basis B(A,B,C,D) = B(T 1 AT,T 1 B,CT,D), for any nonsingular matrix T R n n 30 / 33

31 31 / 33 The complexity of an LTI model is determined by the number of inputs and the order restriction of B on an interval [1,T ] B T = {w = ( w(1),...,w(t ) ) there are w p,w f, for sufficiently large T such that w p w w f B } dim(b T ) = (# of inputs) T + (order) [ ] m # of inputs complexity(b) = l order or lag L q m,l LTI models with q variables and complexity bounded by (m, l)

32 Transition among different representations is a powerful problem solving tool 32 / 33 a problem is easier, when suitable representation is used examples: decoupling of a MIMO system diagonalization in linear algebra pole placement using canonical forms the problem becomes to transform the representation

33 33 / 33 data identification model B i/s/o (A,B,C,D) w = (u,y) B ( ) B i/o H(z) B i/o (H) 7 8 realization 1. H(z) = C(Iz A) 1 B + D 2. realization of a transfer function 3. Z or Laplace transform of H(t) 4. inverse transform of H(z) 5. convolution y d = H u d 6. exact identification 7. H(0) = D, H(t) = CA t 1 B (discrete-time), H(t) = Ce At B (continuous-time), for t > 0 8. realization of an impulse response 9. simulation with input u d and x(0) = exact identification 11. simulation with input u d and x(0) = x ini 12. exact identification