The Behavioral Approach to Systems Theory Paolo Rapisarda, Un. of - PowerPoint PPT Presentation

The Behavioral Approach to Systems Theory Paolo Rapisarda, Un. of Southampton, U.K. & Jan C. Willems, K.U. Leuven, Belgium MTNS 2006 Kyoto, Japan, July 24–28, 2006

Lecture 6: System Identification Lecturer: Jan C. Willems

Issues to be discussed • Remarks on deterministic versus stochastic system identification.

Issues to be discussed • Remarks on deterministic versus stochastic system identification. • Deterministic SYSID via the notion of the most powerful unfalsified model (MPUM)

Issues to be discussed • Remarks on deterministic versus stochastic system identification. • Deterministic SYSID via the notion of the most powerful unfalsified model (MPUM) • What is subspace identification? • Algorithms for state construction • by past/future intersection • (by oblique projection) • by recursive annihilator computation

General Introduction

SYSID MODEL CLASS OBSERVED DATA MATHEMATICAL MODEL Basic difficulties: trade-off between overfitting and predictability learning essential features / rejecting non-essential ones

SYSID Data: an ‘observed’ vector time-series w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( T ) w ( t ) ∈ R w T finite, infinite, or T → ∞ ⇓ A dynamical model from a model class, e.g. a LTIDS R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = 0 or R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = M 0 ε ( t ) + · · · + M L ε ( t + L )

SYSID ‘deterministic’ ID observed variables MODEL w Model class: R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = 0 SYSID algorithm: R 0 , ˆ ˆ R 1 , . . . , ˆ w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( T ) �→ R ˆ L

SYSID ‘deterministic’ ID: I/O form observed observed variables variables MODEL u y Model class (with i/o partition): P 0 y ( t ) + · · · + P L y ( t + L ) = Q 0 u ( t ) + · · · + Q L u ( t + L ) , � u � , Π permutation , P ( ξ ) − 1 Q ( ξ ) proper w = Π y SYSID algorithm: Q 0 , ˆ ˆ Q 1 , · · · , ˆ P 0 , ˆ ˆ P 1 , · · · , ˆ w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( T ) �→ P ˆ L ; Q ˆ L

SYSID ID with unobserved latent inputs observed observed variables variables MODEL u y latent ! variables Model class: (unobserved) R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = M 0 ε ( t ) + M 1 ε ( t + 1 ) + · · · + M L ε ( t + L ) P 0 y ( t ) + · · · + P L y ( t + L ) = Q 0 u ( t ) + · · · + Q L u ( t + L ) + M 0 ε ( t ) + · · · + M L ε ( t + L ) SYSID algorithm (e.g. PEM): � ˆ R ( ξ ) , ˆ w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ � w ( T ) �→ M ( ξ ) Usual assumption: w , ε stochastic.

SYSID ID with unobserved latent inputs observed observed variables variables MODEL u y Why (unobserved) stochastic inputs? latent ! variables Model class: (unobserved) Why stochastics? R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = M 0 ε ( t ) + M 1 ε ( t + 1 ) + · · · + M L ε ( t + L ) P 0 y ( t ) + · · · + P L y ( t + L ) Is this physics? = Q 0 u ( t ) + · · · + Q L u ( t + L ) + M 0 ε ( t ) + · · · + M L ε ( t + L ) SYSID algorithm (e.g. PEM): � ˆ R ( ξ ) , ˆ w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ � w ( T ) �→ M ( ξ ) Usual assumption: w , ε stochastic.

SYSID Assumptions: • Data: w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( t ) ∈ R w w ( t ) , . . . T infinite

SYSID Assumptions: • Data: w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( t ) ∈ R w w ( t ) , . . . T infinite • Deterministic SYSID

SYSID Assumptions: • Data: w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( t ) ∈ R w w ( t ) , . . . T infinite • Deterministic SYSID • Exact modeling with an eye towards approximation

SYSID Assumptions: • Data: w ( 1 ) , ˜ ˜ w ( 2 ) , . . . , ˜ w ( t ) ∈ R w w ( t ) , . . . T infinite • Deterministic SYSID • Exact modeling with an eye towards approximation From the simple to the complex! Approximate Deterministic Exact Approximate Deterministic Stochastic Exact Stochastic

The MPUM The exact deterministic SYSID principle

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ A family of (vector) time series

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ • B is unfalsified by ˜ w : ⇔ ˜ w ∈ B � ˜ ˜ w ( 1 ) , ˜ w ( 2 ) , . . . , ˜ � w = w ( t ) , . . .

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ • B is unfalsified by ˜ w : ⇔ ˜ w ∈ B B 1 is more powerful than B 2 : ⇔ B 1 ⊂ B 2 • Every model is prohibition. The more a model forbids, the better it is. Karl Popper (1902-1994)

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ • B is unfalsified by ˜ w : ⇔ ˜ w ∈ B B 1 is more powerful than B 2 : ⇔ B 1 ⊂ B 2 • • A model class: a family, B , of models, e.g. L w .

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ • B is unfalsified by ˜ w : ⇔ ˜ w ∈ B B 1 is more powerful than B 2 : ⇔ B 1 ⊂ B 2 • • A model class: a family, B , of models, e.g. L w . • The MPUM ‘most powerful unfalsified model’ in B for w , denoted B ∗ ˜ w : ˜ 1. B ∗ w ∈ B ˜ 2. ˜ w ∈ B ∗ ˜ w 3. B ∈ B and ˜ w ∈ B ⇒ B ∗ w ⊆ B ˜

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ • B is unfalsified by ˜ w : ⇔ ˜ w ∈ B B 1 is more powerful than B 2 : ⇔ B 1 ⊂ B 2 • • A model class: a family, B , of models, e.g. L w . • The MPUM ‘most powerful unfalsified model’ in B for w , denoted B ∗ ˜ w : ˜ 1. B ∗ w ∈ B MPUM ˜ 2. ˜ w ∈ B ∗ Unfalsified ˜ w 3. B ∈ B and ˜ w ∈ B Falsified ⇒ B ∗ w ⊆ B ˜ OBSERVED DATA

Most Powerful & Unfalsified • A model:= a subset B ⊆ ( R w ) N , the ‘behavior’ • B is unfalsified by ˜ w : ⇔ ˜ w ∈ B B 1 is more powerful than B 2 : ⇔ B 1 ⊂ B 2 • • A model class: a family, B , of models, e.g. L w . • The MPUM ‘most powerful unfalsified model’ in B for w , denoted B ∗ ˜ w : ˜ 1. B ∗ w ∈ B ˜ 2. ˜ w ∈ B ∗ ˜ w 3. B ∈ B and ˜ w ∈ B ⇒ B ∗ w ⊆ B ˜ w and B , does B ∗ Given ˜ w exist? • ˜ • ‘Exact’ SYSID: Construct algorithms ˜ w �→ B ∗ w ˜

The Model Class Exceedingly familiar: The model B ⊆ ( R w ) N belongs to L w : ⇔ • B is linear, shift-invariant, and closed • B is linear, time-invariant, and complete : ⇔ ‘prefix determined’

The Model Class Exceedingly familiar: The model B ⊆ ( R w ) N belongs to L w : ⇔ • B is linear, shift-invariant, and closed • B is linear, time-invariant, and complete : ⇔ ‘prefix determined’ • ∃ matrices R 0 , R 1 , . . . , R L such that B : all w that satisfy R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = 0 ∀ t ∈ N In the obvious polynomial matrix notation R ( σ ) w = 0 • Including input/output partition � u w ∼ � P ( σ ) y = Q ( σ ) u , = det ( P ) � = 0 y

The Model Class Exceedingly familiar: The model B ⊆ ( R w ) N belongs to L w : ⇔ • B is linear, shift-invariant, and closed • B is linear, time-invariant, and complete : ⇔ ‘prefix determined’ R ( σ ) w = 0 • � u w ∼ � P ( σ ) y = Q ( σ ) u , = • y • ∃ matrices A , B , C , D such that B consists of all w ′ s generated by � u w ∼ � x ( t + 1 ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) + Du ( t ) , = y

The Model Class Exceedingly familiar: The model B ⊆ ( R w ) N belongs to L w : ⇔ • B is linear, shift-invariant, and closed • B is linear, time-invariant, and complete : ⇔ ‘prefix determined’ R ( σ ) w = 0 • � u w ∼ � P ( σ ) y = Q ( σ ) u , = • y � u w ∼ � σ x = Ax + Bu , y = Cx + Du , = • y • ∃ a matrix of rational functions G such that B = sol’ns of G ( σ ) w = 0 without LOG strictly proper with LOG (stabilizability) proper stable rational.

The lag L : L w → Z + , L ( B ) = smallest L such that there is a kernel repr.: R 0 w ( t ) + R 1 w ( t + 1 ) + · · · + R L w ( t + L ) = 0 . Polynomial matrix in R ( σ ) w = 0 has degree ( R ) ≤ L . One the important ‘integer invariants’: maps : L w → Z + , . Others: m , p , n : number of inputs, outputs, states, ν 1 , · · · , ν p : (kernel) lag indices, observability indices, κ 1 , · · · , κ m : (image) lag indices, controllability indices.

The MPUM in L w Theorem: For infinite obs. interval, T = ∞ (our case), w in L w exists. the MPUM for ˜ In fact, w , σ 2 ˜ B ∗ w , . . . } ) closure w = span ( { ˜ w , σ ˜ ˜ Same is true for model class L w with lag ≤ ℓ . We are looking for effective computational algorithms to go from ˜ w to (a representation of) B ∗ w , ˜ e.g., a kernel representation ❀ the corresponding R ; e.g. a generating set of annihilators � A B � of an i/s/o representation of B ∗ e.g., the matrices w . C D ˜