Operator-theoretic methods for prediction and control of nonlinear - PowerPoint PPT Presentation

Operator-theoretic methods for prediction and control of nonlinear dynamical systems Milan Milan Kor Korda da

Big picture Infinite-dimensional Lifting Approximation Linear object Difficult to handle Finite-dimensional Nonlinear object Linear object “Easy” to handle Difficult to handle Milan Korda 2

Big picture Lifting Linear operator Approximation Infinite-dimensional operator Approximation of x + = f ( x ) linear operator Nonlinear dynamics Matrix Milan Korda 3

Koopman operator Milan Korda 4

Koopman operator K : g �→ g ◦ f g : X → C Milan Korda 5

Koopman operator K : g �→ g ◦ f g : X → C K ( α g 1 + β g 2 ) = ( α g 1 + β g 2 ) � f Linearity = α g 1 � f + β g 2 � f = α K g 1 + β K g 2 Milan Korda 6

Koopman operator K : g �→ g ◦ f g : X → C (1900 − 1981) [B. O. Koopman, 1931] [Mezić, Banaszuk, 2004] Milan Korda 7

Koopman operator K : g �→ g ◦ f g : X → C Eigenfunctions K φ = λφ φ ◦ f = λφ ⇔ Milan Korda 8

Koopman operator K : g �→ g ◦ f g : X → C Eigenfunctions K φ = λφ φ ◦ f = λφ ⇔ φ ◦ f k = λ k φ Linear coordinate Milan Korda 9

Koopman operator Eigenfunctions φ ◦ f k = λ k φ invariant set λ = 1 { x : φ ( x ) = γ } ⇒ Chirikov standard map Milan Korda 10

Koopman operator Eigenfunctions φ ◦ f k = λ k φ invariant set | λ | ≤ 1 { x : | φ | ( x ) ≤ γ } ⇒ Chirikov standard map Milan Korda 11

Koopman operator Eigenfunctions φ ◦ f k = λ k φ periodic set λ = e i ω { x : φ ( x ) = γ } ⇒ ( ω rational) Chirikov standard map [ Budisic et al. 2012] Milan Korda 12

Koopman operator Eigenfunctions φ ◦ f k = λ k φ Isostables Isochrons Ergodic partition [ Mauroy et al. 2013] Model reduction . . . [ Rowley et al. 2009] Milan Korda 13

Prediction Milan Korda 14

Prediction Invariant subspace H N = span { ψ 1 , . . . , ψ N } N � c i ψ i ∈ H N g = i =1 Milan Korda 15

Prediction Invariant subspace H N = span { ψ 1 , . . . , ψ N } N � c i ψ i ∈ H N g = i =1 g ( x k ) = CA k z 0 Linear predictor  ψ 1 ( x 0 )  . . C = [ c 1 , . . . , c N ] z 0 = ψ ( x 0 ) = .     ψ N ( x 0 ) Milan Korda 16

Prediction Invariant subspace H N = span { ψ 1 , . . . , ψ N } N � c i ψ i ∈ H N g = i =1 g ( x k ) = CA k z 0 Linear predictor  ψ 1 ( x 0 )  . . C = [ c 1 , . . . , c N ] z 0 = ψ ( x 0 ) = .     ψ N ( x 0 ) ψ i ’s eigenfunctions A = diag( λ 1 , . . . , λ N ) ⇒ Milan Korda 17

Linear prediction Linear A z k +1 z k Z ψ f x k +1 Nonlinear X x k Milan Korda 18

Approximation of the Koopman operator Milan Korda 19

Approximation H N := span { ψ 1 , . . . , ψ N } Goal K N := P N K |H N K N : H N → H N • Construct • Analyze K |H N P N K N H N Milan Korda 20

Extended dynamic mode decomposition ( x i ) K i ) K ( x + x + Data = f ( x i ) i =1 i =1 i Basis functions ψ = [ ψ 1 , . . . , ψ N ] ⊤ K LS problem � ∥ ψ ( x + i ) − A ψ ( x i ) ∥ 2 min 2 A ∈ R N × N i =1 Koopman approximaton g = c ⊤ ψ K N,K g := c ⊤ A N,K ψ [Williams et al., 2015] Milan Korda 21

Convergence of EDMD g = c ⊤ ψ K N,K g := c ⊤ A N,K ψ µ K = empirical measure ˆ on x 1 , . . . , x K Fact : K N,K = P ˆ µ K N K |H N Milan Korda 22

Convergence of EDMD g = c ⊤ ψ K N,K g := c ⊤ A N,K ψ µ K = empirical measure ˆ on x 1 , . . . , x K Fact : K N,K = P ˆ µ K N K |H N Fact : lim K →∞ K N,K → K N = P µ N K |H N (iid or ergodic sampling from µ ) Milan Korda 23

Convergence of EDMD g = c ⊤ ψ K N,K g := c ⊤ A N,K ψ K |H N P µ N Fact : lim K →∞ K N,K → K N = P µ N K |H N (iid or ergodic sampling from µ ) K N H N Milan Korda 24

Convergence of EDMD H = L 2 ( µ ) Theorem • span { ψ i } ∞ i =1 = H • K : H → H bounded lim N →∞ � K N g � K g � = 0 for all g � H (Converge in strong operator topology) Milan Korda

Convergence of EDMD H = L 2 ( µ ) Theorem • span { ψ i } ∞ i =1 = H • K : H → H bounded lim N →∞ � K N g � K g � = 0 for all g � H (Converge in strong operator topology) As a result, any finite-horizon predictions converge! Corollary Under the same assumptions For any N p � N : lim N →∞ sup i ∈ { 1 ,...,N p } � K i N g � K i g � = 0 Milan Korda

Convergence of EDMD H = L 2 ( µ ) Theorem • span { ψ i } ∞ i =1 = H • K : H → H bounded lim N →∞ � K N g � K g � = 0 for all g � H (Converge in strong operator topology) As a result, any finite-horizon predictions converge! Corollary Under the same assumptions For any N p � N : lim N →∞ sup i ∈ { 1 ,...,N p } � K i N g � K i g � = 0 N ψ N � g � f i | 2 dµ � 0 X | CA i � lim N →∞ Milan Korda

Convergence of EDMD H = L 2 ( µ ) Theorem • span { ψ i } ∞ i =1 = H • K : H → H bounded lim N →∞ � K N g � K g � = 0 for all g � H (Converge in strong operator topology) As a result, any finite-horizon predictions converge! Corollary Under the same assumptions For any N p � N : lim N →∞ sup i ∈ { 1 ,...,N p } � K i N g � K i g � = 0 Spectral convergence more delicate. See Korda M. and Mezić I. On Convergence of Extended Dynamic Mode Decompo- sition to the Koopman Operator , Journal of Nonlinear Science, 2017 Milan Korda

Control (Joint work with Igor Mezić) Milan Korda 29

Predictor x 1 , x 2 , . . . x + = f ( x, u ) u 1 , u 2 , . . . x 0 z + = g ( z, u ) x 1 , ˆ ˆ x 2 , . . . z 0 = ψ ( x 0 ) x = h ( z ) ˆ ψ : R n � R N , N � n Milan Korda 30

Predictor x 1 , x 2 , . . . x + = f ( x, u ) u 1 , u 2 , . . . x 0 z + = Az + Bu x 1 , ˆ ˆ x 2 , . . . z 0 = ψ ( x 0 ) x = Cz ˆ ψ : R n � R N , N � n Milan Korda 31

Why linear predictors? Can design controllers using linear methods u = κ lift ( z ) = ⇒ u = κ ( x ) := κ lift ( ψ ( x )) Milan Korda 32

Why linear predictors? Can design controllers using linear methods u = κ lift ( z ) = ⇒ u = κ ( x ) := κ lift ( ψ ( x )) Especially suited for Model predictive control (MPC) • Optimization-based controller • Fast and effective for constrained linear systems • Computation speed independent of the size of the lift N Milan Korda 33

Designing the predictors Milan Korda 34

Koopman operator for controlled systems x + = f ( x, u ) , x ∈ R n , u ∈ R m Milan Korda 35

Koopman operator for controlled systems x + = f ( x, u ) , x ∈ R n , u ∈ R m = ⇒ � f ( x, u (0)) � χ + = F ( χ ) := S u • Extended state � := ( x, u ) ∈ X := R n × � ( R m ) • Shift operator ( S u )( i ) = u ( i + 1) Space of all control sequences ( u i ) ∞ i =0 =: u Milan Korda 36

Koopman operator for controlled systems x + = f ( x, u ) , x ∈ R n , u ∈ R m = ⇒ � f ( x, u (0)) � χ + = F ( χ ) := S u • Extended state � := ( x, u ) ∈ X := R n × � ( R m ) • Shift operator ( S u )( i ) = u ( i + 1) Space of all control sequences ( u i ) ∞ i =0 =: u Koopman operator K φ = φ � F φ : X → R Milan Korda 37

Koopman operator for controlled systems x + = f ( x, u ) , x ∈ R n , u ∈ R m = ⇒ � f ( x, u (0)) � χ + = F ( χ ) := S u • Extended state � := ( x, u ) ∈ X := R n × � ( R m ) • Shift operator ( S u )( i ) = u ( i + 1) Space of all control sequences ( u i ) ∞ i =0 =: u [Proctor et al, 2016] [Proctor et al, 2016] Other work [Williams et al, 2016] [Brunton et al, 2016] Milan Korda 38

Linear predictors from Koopman - EDMD ( χ i ) K ( χ + i ) K Data χ + i = F ( χ i ) i =1 i =1 K � ∥ φ ( χ + i ) − A φ ( χ i ) ∥ 2 min LS problem 2 A ∈ R N φ × N φ i =1 φ ( χ ) = [ φ 1 ( χ ) , . . . , φ N φ ( χ )] � Milan Korda 39

Linear predictors from Koopman - EDMD ( χ i ) K ( χ + i ) K Data χ + i = F ( χ i ) i =1 i =1 K � ∥ φ ( χ + i ) − A φ ( χ i ) ∥ 2 min LS problem 2 A ∈ R N φ × N φ i =1 linear operator φ ( χ ) = [ φ 1 ( χ ) , . . . , φ N φ ( χ )] � Predictor linear in u φ i ( x, u ) = ψ i ( x ) + L i ( u ) ⇒ Milan Korda 40

Linear predictors from Koopman - EDMD ( χ i ) K ( χ + i ) K Data χ + i = F ( χ i ) i =1 i =1 K � ∥ φ ( χ + i ) − A φ ( χ i ) ∥ 2 min LS problem 2 A ∈ R N φ × N φ i =1 linear operator φ ( χ ) = [ φ 1 ( χ ) , . . . , φ N φ ( χ )] � Predictor linear in u φ i ( x, u ) = ψ i ( x ) + L i ( u ) ⇒ Without loss of generality φ ( x, u ) = [ ψ 1 ( x ) , . . . , ψ N ( x ) , u (0) 1 , . . . , u (0) m ] ⊤ Milan Korda 41

Linear predictors from Koopman - EDMD ( χ i ) K ( χ + i ) K Data χ + i = F ( χ i ) i =1 i =1 K � ∥ φ ( χ + i ) − A φ ( χ i ) ∥ 2 min LS problem 2 A ∈ R N φ × N φ i =1 linear operator φ ( χ ) = [ φ 1 ( χ ) , . . . , φ N φ ( χ )] � Predictor linear in u φ i ( x, u ) = ψ i ( x ) + L i ( u ) ⇒ Without loss of generality φ ( x, u ) = [ ψ 1 ( x ) , . . . , ψ N ( x ) , u (0) 1 , . . . , u (0) m ] ⊤ K � ∥ ψ ( x + i ) − A ψ ( x i ) − B u i (0) ∥ 2 min 2 A ∈ R N × N , B ∈ R N × m i =1 Milan Korda 42