RECENT ADVANCES IN . SUBSPACE IDENTIFICATION GIORGIO PICCI Dept. of Information Engineering, Universit` a di Padova, Italy MTNS 2006 KYOTO July 2006
OUTLINE OF THE TALK • BASIC IDEA OF SUBSPACE IDENTIFICATION: STOCHASTIC REAL- IZATION + REGRESSION • UNIFIED ANALYSIS OF SUBSPACE METHODS WITH INPUTS (CCA, N4SID, MOESP ,...): ILL-CONDITIONING, “WORST-CASE” INPUTS, CON- SISTENCY CONDITIONS.. • TRANSPARENT ASYMPTOTIC VARIANCE EXPRESSIONS, RELATION TO ILL-CONDITIONING • SAME IDEA SOLVES THE PROBLEM OF FEEDBACK: STATE SPACE CONSTRUCTION WORKS WITH CLOSED LOOP DATA 1
DATA GENERATING MECHANISM I/O data: sample paths of second order stationary processes y , u , zero mean, with rational spectrum ⇒ described by � � � � � � � � x ( t + 1 ) x ( t ) A B K = + e ( t ) y ( t ) u ( t ) C D I y d ( t ) : = C ( zI − A ) − 1 B u ( t )+ D u ( t ) y s ( t ) : = C ( zI − A ) − 1 K e ( t )+ e ( t ) deterministic + stochastic components 2
e ( t ) ❄ C ( zI − A ) − 1 K + I y s ( t ) u ( t ) y d ( t ) + y ( t ) ✛✘ ❄ + C ( zI − A ) − 1 B + D ✲ ✲ ✲ ✚✙ • No feedback from y to u : e ( t ) ⊥ u ( τ ) ∀ t τ 3
(UNIQUE) INNOVATION MODEL e ( t ) Innovation white noise: one step ahead prediction error of y ( t ) . Stationarity and no feedback ⇒ A stable: ( | λ ( A ) | < 1 ) With feedback A may be unstable x ( t ) : steady-state Kalman predictor based on joint infinite past of y , u . 4
PARAMETRIZATION OF LINEAR STOCHASTIC SYSTEMS y , u second order stationary processes described by � � � � � � � � x ( t + 1 ) x ( t ) A B K = + e ( t ) (1) y ( t ) u ( t ) C D I • No feedback from y to u : e ( t ) ⊥ u ( τ ) ∀ t , τ � − 1 � ⊤ � � ⊤ � � � � � � � � x ( t + 1 ) x ( t ) x ( t ) x ( t ) A B = E { } E { } y ( t ) u ( t ) u ( t ) u ( t ) C D Parameters are uniquely determined by choosing basis x ( t ) in the state space ! 5
HILBERT SPACES OF RANDOM VARIABLES Inner product � ξ , η � : = E { ξ η } , E mathematical expectation. For − ∞ ≤ t 0 ≤ t ≤ T ≤ + ∞ define the Hilbert space of scalar scalar zero- mean random variables : = span { u k ( s ) ; k = 1 ,..., p , t 0 ≤ s < t } U [ t 0 , t ) : = span { y k ( s ) ; k = 1 ,..., m , t 0 ≤ s < t } Y [ t 0 , t ) future spaces up to time T : = span { u k ( s ) ; k = 1 ,..., p , t ≤ s ≤ T } U [ t , T ] Y [ t , T ] : = span { y k ( s ) ; k = 1 ,..., m , t ≤ s ≤ T } U − t , Y − When t 0 = − ∞ use t for U [ − ∞ , t ) , Y [ − ∞ , t ) . 6
ELEMENTARY HILBERT SPACE GEOMETRY E [ z | X ] (vector of) orthogonal projections (conditional expectations in the Gaussian case) of the components of z onto the subspace X . E [ Y | X ] Hilbert subspace spanned by the r. v’s { E [ η | X ] | η ∈ Y } . Let A ∩ B = { 0 } i.e. A + B direct sum, E { z | A + B } = E || A { z | B } + E || B { z | A } E || A { z | B } is the oblique projection of z onto B along A . 7
IDENTIFICATION From observed input-output time series y t ∈ R m u t ∈ R p { y 0 , y 1 , y 2 ,..., y N } , { u 0 , u 1 , u 2 ,..., u N } , ˆ � � A B find estimates (in a certain basis) C D N such that ( consistency ) ˆ � � � � A B A B = lim N → ∞ C D C D N 8
BASIC IDEA OF SUBSPACE IDENTIFICATION Assume we can observe also the state trajectory { x 0 , x 1 , x 2 ,..., x N } , cor- rresponding to the I/O data y t ∈ R m u t ∈ R p { y 0 , y 1 , y 2 ,..., y N } , { u 0 , u 1 , u 2 ,..., u N } , Form “tail” matrices Y t , X t , U t Y t : = [ y t , y t + 1 , y t + 2 ,... ] X t : = [ x t , x t + 1 , x t + 2 ,... ] U t : = [ u t , u t + 1 , u t + 2 ,... ] Every sample trajectory { y t } , { x t } , { u t } of the system must satisfy the ”true” model equations, so � � � �� � � � X t + 1 A B X t K = + E t Y t C D U t I 9
BASIC IDEA OF SUBSPACE IDENTIFICATION (cont’d) � � � �� � � � X t + 1 A B X t K = + E t Y t C D U t I Linear Regression ! Solve by Least Squares : � � � �� � X t + 1 A B X t A , C , B , D � − � min C D Y t U t getting � ⊤ � − 1 � ⊤ � ˆ � � � �� � �� : = 1 1 A B X t + 1 X t X t X t C D Y t U t U t U t N N N 10
BASIC IDEA OF SUBSPACE IDENTIFICATION (cont’d) � ⊤ � − 1 � ⊤ � ˆ � � � �� � �� : = 1 1 A B X t + 1 X t X t X t C D Y t U t U t U t N N N If the data are second order ergodic and the inverse exists: ˆ � � � � A B A B = lim N → ∞ C D C D N consistent estimate of A , B , C , D . 11
SECOND ORDER ERGODICITY For N → ∞ sample covariances converge to true covariances, say t + N 1 k } = 1 { y k u ⊤ N Y t U ⊤ s → E { y ( t ) u ( s ) ⊤ } � N → ∞ N k = t For N → ∞ the average Euclidean inner product of tail sequences con- verges to the inner product of the corresp. random variables (asymp- totic isometry). As N → ∞ : Hilbert space geometry of (semi-infinite) tail sequences is the same as Hilbert space geometry of random variables y ( t ) ≡ Y t , u ( t ) ≡ U t , isometry 12
(SEMI-INFINITE) TAIL SUBSPACES Sample fluctuations (i.e. finite data length) play no role in the analysis, can assume that N → ∞ and work as with the stochastic setting. u ( t ) U t u ( t + 1 ) U t + 1 u + U [ t , T ] : = → t : = . . . . . . u ( T ) U T u ( t 0 ) U t 0 u ( t 0 + 1 ) U t 0 + 1 u − U [ t 0 , t ) : = → t : = . . . . . . u ( t − 1 ) U t − 1 same notation for Y [ t 0 , t ) etc.. 13
BACK TO THE (IDEAL) SUBSPACE ID PROCEDURE STATE SEQUENCE IS NOT AVAILABLE: NEED TO CONSTRUCT THE STATE FROM INPUT-OUTPUT DATA (STOCHASTIC REALIZATION) Fundamental step: Stochastic realization to construct the state from I/O data. Easy if infinite past data were available at time t : U − Y − t : = span { U s | s < t } , t : = span { Y s | s < t } H-spaces generated by all past inputs/outputs from ( − ∞ , t ] Generalize procedure of Akaike, and L.P .: Construct the oblique predictor space , X + / − Y + t | Y − t ∨ U − � � : = E || U + t t t Pick basis vector in X + / − .... ⇒ innovation model ! t 14
NUISANCE: ONLY FINITE DATA ARE AVAILABLE ! In practice can regress only on finite past data at time t In practice can work with U [ t 0 , t ) , Y [ t 0 , t ) from (small) finite past/future in- tervals [ t 0 , t ) , [ t , T ] . L.S. Estimates depend on sample covariances.... Finite-interval approximation of infinite-past regression leads to errors (bias) in the estimate which do not → 0 as N → ∞ . If zeros of the system arbitrarily close to the unit circle, bias can be made arbitrarily large. Want consistency with finite regression data: NEED FINITE-INTERVAL (NON–STATIONARY) STOCHASTIC REALIZATION 15
CONSTRUCTING THE STATE FROM FINITE INPUT-OUTPUT DATA PROBLEM: Construct the state space of a stochastic realization of y using ONLY the r.v.’s of input and output processes from a finite interval [ t 0 , T ] . Try to mimic the infinite past construction: Future output predictor + oblique projection 16
THE OUTPUT PREDICTOR (TAIL MATRICES) Output Predictor based on joint input-output data (wedge denotes vector sum): � � = Γ ˆ ˆ Y [ t , T ] : = E Y [ t , T ] | Y [ t 0 , t ) ∨ U [ t 0 , T ] X t + H U [ t , T ] set ν : = T − t (future horizon) ... C D 0 0 0 ... CA CB D 0 0 Γ : = H : = . . . ... . . . . . . CA ν − 1 CA ν − 1 B CA ν − 2 B ... CB D ˆ X t : Transient conditional Kalman filter on [ t 0 , T ] : � ˆ A ˆ X t + B U t + K ( t ) ˆ = X t + 1 E t C ˆ X t + D U t + ˆ = Y t E t 17
A TECHNICAL DIFFICULTY WITH FINITE DATA With finite data need to “factor out” the dynamics of u � � x ( t ) = E x ( t ) | y [ t 0 , t ) ∨ u [ t 0 , T ] ˆ initial condition depends on all input history � � x ( t 0 ) = E x ( t 0 ) | u [ t 0 T ] ˆ cannot recover ˆ x ( t ) by oblique projection along future u ’s since part of the state is in u [ t , T ] ! Leads to complications (a plethora of algorithms: MOESP , N4SID, CCA, etc...). Some people don’t care and use infinite-past approximation. 18
THE N4SID ALGORITHM [vanOverschee-DeMoor94] 1. Predictor matrix based on joint input-output data � � ˆ Y [ t , T ] : = E Y [ t , T ] | Y [ t 0 , t ) ∨ U [ t 0 , T ] (projection onto the joint rowspace). 2. From this compute the observability matrix Γ by an oblique projection + SVD factorization. X t : = Γ † ˆ 3. “Pseudostate” ¯ Y [ t , T ] obeys the recursion � ¯ � � � � � X t + 1 A K 1 U [ t , T ] + W ⊥ ¯ = X t + K 2 Y t C K 1 K 2 known linear functions of ( B , D ) . 4. Solve by LS for the unknown parameters ( A , C ) and ( K 1 , K 2 ) . 19
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