Nonlinear Balanced Realizations

Transcription

1 Nonlinear Balanced Realizations Erik I. Verriest School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia USA Joint work with W. Steven Gray, Old Dominion University Lorentz Center September 22, 2005

2 Overview 1. Introduction 2. Reachability and Observability - Gramians 3. Sliding Interval Balancing 4. Nonlinear Balancing 5. Mayer-Lie Integration 6. Mayer-Lie Interpolation 7. Global State Space Topology 8. Conclusions 2

3 1. Introduction 3

4 1. Introduction - The Problem Balanced Realizations for LTI systems successful in Model Reduction (by Projection of Dynamics) Parameterization Sensitivity Analysis System Identification What can be generalized to nonlinear systems and how? 4

5 1. Introduction - History LTI: Moore, Pernebo-Silverman, Antoulas,... LTV: Shokoohi-Silverman, V.-Kailath LQG: Jonckheere, Fuhrman, V. Singular: Gray Infinite Dimensions: Curtain, Ober, Young Periodic: Van Dooren, Varga, V.-Helmke Nonlinear: Scherpen-Fujimoto, Newman-Krishnaprasad, Lall, Gray, V. RKHS: V. 5

6 2. Reachability and Observability Maps Gramians 6

7 2. RO: Finite Time Gramians General time varying linear minimal system Single Input - Single Output ẋ(t) = A(t)x(t) + b(t)u(t) (1) y(t) = c(t)x(t) (2) Finite time reachability and observability Gramians (interval of length δ) R(t, δ) = O(t, δ) = t t δ t+δ t Φ(t, τ)b(τ)b (τ)φ (t, τ) dτ (3) Φ (τ, t)c(τ) c(τ)φ(τ, t) dτ, (4) Φ(t, τ) is the transition matrix, Φ(t, τ) = A(t)Φ(t, τ), Φ(τ, τ) = I. t 7

8 2. RO: Reachability Maps Reachability maps L t (u( )) = With metrics L t : L 2 ([t δ, t], R) R n t t δ Φ(t, τ)b(τ)u(τ) dτ. u, v L 2 : u( ), v( ) L2 = and t t δ u(τ)v(τ) dτ x, y R n : x, y = x y Adjoint maps: L t : R n L 2 ([t δ, t], R) L t x = b ( )Φ (t, ) x 8

9 2. RO: Reachability Maps: Energy Interpretation 1 Minimum norm solution to L t (u( )) = x is given by where z is any solution to u( ) = L t (z) L t L t (z) = x L t L t : R n R n : Reachability Gramian R(t, δ). Minimum energy u 2 : u 2 = x 2 (L t L t ) 1 = x (L t L t ) 1 x def = E i (x) Fact: Cost associated with transfer of event (0, t δ) R n R to the event (x, t) is the above quadratic form in x. It relates to the effort needed to reach x at time t from the origin. 9

10 2. RO: Reachability Maps: Energy Interpretation 2 Given ν S n 1, what is the maximal excursion that can be reached from (t δ, 0) at time t in the direction ν with a unit energy input? Solution max ν, L tu def = P i (ν) u =1 u ext = kl t ν, with k = L t ν, L t ν 1/2 P i (ν) = (ν R t ν) 1/2 Fact: Maximal state excursion at t form the origin at time t δ in the direction ν is the above quadratic form in ν. It relates to the influencability of the state space in the direction ν. 10

11 2. RO: Reachability Maps: Uncertainty Interpretation Let the input be zero mean, unit variance white noise (=standard white noise). The resulting state is a random vector L t u in the Hilbert space L 2 (Ω, R n ). The inner product in this space is x, y = E x y This random vector defines an additive measure on the subspaces of R n by µ i (A) def = P A L t u 2 = Tr (L t L t P A ) = Tr (R t P A ) where P A is the projector onto A. Fact: The uncertainty of the state reached form the origin by the white noise process from t δ to t in the direction ν is given by ν R t ν. It relates the influencability of the subspace spanned by ν. 11

12 2. RO: Observability maps Observability maps M t : R n L 2 ([t, t + δ], R) M t (x) = C( )Φ(, t) x. With metric the adjoint maps are u, v L 2 : u( ), v( ) L2 = t+δ t u(τ)v(τ) dτ M t : L 2 ([t, t + δ], R) R n M t u( ) = t+δ t Φ (τ, t)c (τ)u(τ) dτ 12

13 2. RO: Observability Maps: Energy Interpretation-1 Given x R n, the energy in the output x y( ) = M t x is E o (x) def = M t x, M t x = M t M t x, x M t M t = O(t, δ) : R n R n, Observability Gramian Fact: Cost associated with transfer of event (x, t) R n R to the event (x f, t + δ) is the above quadratic form in x. It relates to the signal energy available to detect x. (SNR if embedded in unit variance white noise) 13

14 2. RO: Observability Maps: Energy Interpretation-2 Given ν S n 1, and let an output signal, M t x 0, be embedded in a unit energy disturbance, what is the ambiguity of the initial state component in direction ν? max ν, (x 0 ˆx def = P o (ν), where, min M ˆx y, y = Mx 0 + v v =1 ˆx Solution: (Cauchy-Schwarz) x = x 0 ˆx = 1 ν O 1 t ν O 1 t ν P o (ν) = (ν O 1 t ν) 1/2 Fact: Induced ambiguity of initial state (x 0, t) in direction ν is the above quadratic form in ν. It relates to the uncertainty in determining x. 14

15 2. RO: Observability Maps: Uncertainty Interpretation Given y( ) L 2 ([t, t + δ], R) the error norm y M tˆx L2 is minimal for ˆx = (M t M t ) 1 M t y( ). If y = M t x + u (signal + standard white noise), the estimate ˆx is a random vector with covariance O t = M t M t. Hence the uncertainty in the subspace A is given by the additive measure µ o (A) def = P A (M t M t ) 1 M t u 2 = Tr ((M t M t ) 1 P A ) = Tr (O 1 t P A ) where P A is the projector onto A. Fact: Residual uncertainty of the state at t in the direction ν after observation of the output from t to t + δ is ν O 1 t ν. It relates the difficulty of observing the ν-component of the state x. 15

16 2. RO: Summary Energy maps - 1 Energy maps - 2 E i : R n R + : x x R 1 t x E o : R n R + : x x O t x P i : S n 1 R + : ν (ν R t ν) 1/2 P o : S n 1 R + : ν (ν O 1 t ν) 1/2 Uncertainty maps µ i : Proj R n R + : A Tr (R t P A ) or, for ν S n 1, respectively µ o : Proj R n R + : A Tr (O 1 t P A ) U i : S n 1 R + : ν ν R t ν U o : S n 1 R + : ν ν O 1 t ν 16

17 3. Sliding Interval Balancing 17

18 3. Sliding Interval Balancing Derive a uniform measure (input and output) for the relative importance of subspaces of the state space: Find a state space transformation, T ( ), so that the transformed input to state and state to output maps are symmetric, i.e., R t = O t and both diagonal. Rationale: In balanced coordinates, for each coordinate direction (more general, for each subspace) the degrees of reachability and observability are equal. Fact: The similarity transformation T ( ) has the following effects R t T (t)r t T (t) O t T (t) T O t T (t) 1 Hence T (t) (pointwise) and differentiable under suitable conditions, such that the new realization is SIB. [V. 1980]. 18

19 3. SIB Time-weighted Gramians SIB Sign Symmetric R(δ) = O(δ) = Λ(δ) A = SAS b = Sc M-balanced R M = RMR O M = O MO where R = [b, Ab,..., A n 1 b], O = [c, A c,..., (A ) n 1 c ]. Interpretation: time weighted Gramians E i = 0 γ 2 (t)u 2 (t) dt R γ = RM γ R 19

20 3. SIB Projection of Dynamics A 11 A 12 A 21 A 22 [ C 1 C 2 ] B 1 B 2 POD A [ C 1 0 ] B 1 0 according to Λ 1 Λ 2 POD Λ

21 4. Nonlinear Balancing 21

22 4. NLB: Tenets of Nonlinear Balancing 1. Nominal Flow Balancing should be defined for a perturbation system with respect to some nominal flow, as opposed to a single equilibrium point. 2. Perturbation short time Only small perturbations should be allowed if the linear variational equation (the perturbation system) along the nominal trajectory is to remain sufficiently accurate. 3. Commutation: balancing/linearization (f, g, h) linearization global balancing ( ˆf, ĝ, ĥ) linearization (A(t), B(t), C(t)) local balancing (Â(t), ˆB(t), Ĉ(t)) 22

23 4. NLB: Local Balancing - 1 Smooth nonlinear system ẋ = f(x) + g(x)u y = h(x) 23

24 Infinitesimal Balancing For analytic system: 4. NLB: Local Balancing - 1b Φ(t, τ)b(τ) = Φ (τ, t)c (τ) = i=0 i=0 (t τ) i i! (τ t) i D: differentiation operator (acting to the right) i! [ ] (A D) i B t [ (A + D) i C ] t Local reachability and observability matrices: R (t) = [B, (A D)B, (A D) 2 B,... ] t O (t) = [C, (A + D)C, (A + D) 2 C,... ] t 24

25 4. NLB: Local Balancing - 1c Response to impulsive input u(t) = i=0 g iδ (i 1) (t τ) g1 x τ = R (τ)g, g = g2. For initial condition x at τ Y(τ) = y(τ) ẏ(τ). = O (τ)x. R(t δ, t) = R (t) m (δ)r (t) = R n (t) m (δ)r n(t) + O(δ n+1 ) O(t, t + δ) = O (t) p (δ)o (t) = O n(t) p (δ)o n (t) + O(δ n+1 ) [ m (δ)] ij = δ i+j 1 (i 1)!(j 1)!(i + j 1) I m. 25

26 Nominal flow (wolog u = 0) : 4. NLB: Local Balancing - 2 ẋ = f(x) y = h(x) Linear Variational Equation (Perturbation System) Local Reachability: Local Observability: A x = f x x = df, C x = h x x = dh, B x = g(x) (A x D)B x = f x g g f = [ f, g] x (A x + D)C x = dl f h 26

27 4. NLB: Local Balancing - 2 Lie-product: Gradient: Lie-derivative: ad f g = [f, g] = g x f f x g ad k f g = [f, ad k 1 f g] d L f h, directional derivative Nonlinear local reachability and observability matrices R n (f, g) = [g, ad f g,..., ad n 1 f g] O n (f, h) = dh d L f h. dl n 1 f h. 27

28 4. NLB: Local Balancing - 3 Local reachability and observability Gramian R n (f, g) = t0 t 0 δ Φ(t 0, τ)b(τ)b (τ)φ (t 0, τ) dτ O n (f, h) = t0 +δ t 0 Φ (τ, t 0 )C (τ)c(τ)φ(t, t 0 ) dτ where: x(t 0 ) = x 0, (A(t), B(t), C(t)) : linearization of (f, g, h) along the nominal flow. Φ(, ) : transition matrix of A( ). 28

29 4. NLB: Local Balancing - 4 For small δ: R n (f, g) = R n (f, g) n (δ)r n(f, h) O n (f, h) = O n(f, g) n (δ)o n (f, h) [ n (δ)] ij = δ i+j 1 (i 1)!(j 1)!(i + j 1) 29

30 4. NLB: Local Balancing - 5 Let x 0 be a point in the state space where the local reachability and observability matrices have full rank. A nonsingular matrix T 0 exists at x 0 such that the transformed Gramians, at x 0, T 0 R n (f, g)t0 T, and T T 0 O n (f, h)t 1 0 are equal and diagonal. In general this T 0 will depend on x 0. Thus, we arrive at a map (omit the subscript o ) T : IR n GL n (IR), Can the local transformation, T (x), be extended to a global one? 30

31 System: 4. NLB: Local Balancing - Discrete Case x k+1 = f(x k, u k ) y k = h(x k ) Nonlinear local reachability and observability matrices R n (f, g) = [ g (n 1), it 1 f g (n 2),..., it n 1 f g 0 ] O n (f, g) = where it k f g l = df l df l k+1 g l k, dh 0 dh 1 df 0.. dh n 2 df n 1 df 0 Nonlinear Gramians: as in continuous case with = I. 31

32 4. NLB: Global Balancing - 1 Jacobian Problem: Find a diffeomorphism ξ such that ξ x = T (x), x D This is a Mayer-Lie system of PDE s. It is generically not solvable. NASC for solvability: T ij (x) x k T ik(x) x j = 0. for all i, j, k = 1,..., n and x D. also known as the Frobenius conditions. 32

33 4. NLB: Global Balancing

34 5. Mayer-Lie Integration 34

35 5. Mayer-Lie Integration - 1 When the Frobenius conditions hold, a diffeomorphism can be found on some D 1 D by the method of characteristics. Transform the Mayer-Lie system to a system of nonlinear equations: G(x) = T 1 (x) G(x) : R n R n is an element of L(R n, R n ), thus: G : R n L(R n, R n ). The Frobenius conditions imply the existence of a vector field F x : R n R n, such that F x (ξ) ξ = G(F x (ξ)) F x (0) = x 0. 35

36 5. NLB: Global Balancing - 2 If F x is an invertible map, set h x : R n R n i.e., h x (z) = Fx 1 (z) h x (z) = ξ F x (ξ) = z then h x (z) z = T (z), i.e., h x (z) is a solution for ξ with h x (x 0 ) = 0. 36

37 5. MLI: Frobenius condition: alternative form DG(x) : R n L(R n, L(R n, R n )) i.e., (DG(x))[u] is a linear map in L(R n, R n ) ((DG(x))[u])[v] R n. Let u = G(x)w Frobenius condition: v, w R n. ((DG(x))[G(x)w])[v] = ((DG(x))[G(x)v])[w] 37

38 5. MLI: Frobenius Condition Example G(x) = 1 0 x G(x)w = [w 1, x 2 1w 1 + w 2 ] T (DG(x))[G(x)w] = 0 0 2x 1 w 1 0 thus ((DG(x))[G(x)w])[v] = [0, 2x 1 w 1 v 1 ] which is symmetric upon permutation of v and w, so that G is integrable. 38

39 5. MLI Example Local balancing transformation on R 2 : T (x, y) = cos y sin y x sin y x cos y Inversion G(x, y) = cos y sin y 1 x sin y 1 x cos y. Solve (in two steps) along the path (0, 0) (ξ, 0) (ξ, η): df 1 dξ = cos F 2 df 2 dξ = 1 F 1 sin F 2 F 1 (0, 0) = x 0 F 2 (0, 0) = y 0. 39

40 Noting that F 1 F 1 + (F 1) 2 = 1, The solution is readily found to be F 1 (ξ, 0) = (ξ + x 0 cos y 0 ) 2 + x 2 0 sin2 y 0. F 2 (ξ, 0) = arctan x 0 sin y 0 ξ+x 0 cos y 0 40

41 5. MLI Example df 1 = sin F dη 2 Next we solve, df 2 = 1 dη F 1 cos F 2 with the initial conditions F 1 (ξ, 0) and F 2 (ξ, 0) obtained above. F 1 (ξ, 0) = (ξ + x 0 cos y 0 ) 2 + (η + x 0 sin y 0 ) 2. F 2 (ξ, 0) = arctan η+x 0 sin y 0 ξ+x 0 cos y 0 Finally, inversion gives, upon resetting F 1 = x F 2 = y ξ = x cos y x 0 cos y 0 η = x sin y x 0 sin y 0, satisfying (ξ, η) = (0, 0) if (x, y) = (x 0, y 0 ). 41

42 5. MLI: Mayer-Lie Integration: Obstructions For scalar systems, there is no obstruction to global balancing. For second order systems, generically the Mayer-Lie conditions do not hold. However, one can always define integrating factors S = diag {S 1 (x), S 2 (x)} such that S(x)T (x) is integrable. Such non-uniform scaling retains the diagonality of the Gramians, as well as the fact that their product specifies the canonical Gramian and therefore the relative importance of the balanced local state components. Since this is the requisite information for model reduction such a scaled global balanced realization is still useful. Balanced Realization Uncorrelated Realization 42

43 6. Mayer-Lie Interpolation 43

44 6. Mayer-Lie Interpolation Caveat: Generically, the Mayer-Lie conditions form an obstruction to the application of the third principle. But this is not the end of our story. Propose an approximate solution via a special interpolation. 44

45 6. Mayer-Lie Interpolation Thus motivated, suppose one is given a distinct set of points {x 1,..., x N } D, with the corresponding matrices T (x i ). We want to find a diffeomorphism, η defined on D such that η x = T (x i ), for i = 1,..., N. xi 45

46 6. Mayer-Lie Interpolation For the special case of a discrete periodic or chain recurrent orbit, the proposed interpolation approach is easily justified: Only the neighborhoods of the successive points in the nominal orbit are important, hence the true form in between is rather immaterial [PSYCO-01]. 46

47 6. Mayer-Lie Interpolation One potential solution is to start form a parameterized set of diffeomorphisms, and identify the necessary parameters in order to satisfy the N point constraints [MTNS-02]. Example: Planar System Let ξ(x, y) = η(x, y) = 2N 1 i=0 2N 1 i=0 c (ξ) i x 2N 1 i y i c (η) i x 2N 1 i y i homogeneous polynomials in x and y of degree 2N 1, with c {ξ} j yet to be determined. c {η} j and 47

48 6. Mayer-Lie Interpolation - 2 Interpolation constraints imply T T (x 0, y 0 ). T T (x N 1, y N 1 ) = Z(x 0, y 0 ). Z(x N 1, y N 1 ) C. where C = [c (ξ), c (η) ] IR (2N 1) 2. Z(x, y) is (2N 1)x2N 2 (2N 2)x 2N 3 y y 2N x 2N 2 (2N 2)xy 2N 3 (2N 1)y 2N 2 More compact form: T T = Z(x 0, y 0,... x N 1, y N 1 ) C 48

49 6. Mayer-Lie Interpolation - 3 If Z(x 0, y 0,... x N 1, y N 1 ) is invertible, then C = Z(x 0, y 0,... x N 1, y N 1 ) 1 T T A candidate diffeomorphism is ξ(x, y) = T Z T (x 0, y 0,..., x N 1, y N 1 ) η(x, y) x 2N 1. y 2N 1 49

50 6. Mayer-Lie Interpolation - 4 Since Z is homogeneous of degree 2N 2, the Jacobian determinant has degree 4(N 1). Consequently, there are at most 4(N 1) lines through the origin where full rankness will fail. These lines define wedges in the original state space coordinates. Hence, if all interpolation states x 0,..., x N 1 fall inside a wedge, D, a globally defined balanced realization can be defined in D. 50

51 51

52 6. Mayer-Lie Interpolation - 5 Main Theorem. If no two states are collinear with the origin, then the matrix Z(x 0, y 0,..., x N 1, y N 1 ) is invertible. Proof: The matrix Z is a row permutation of the matrix D N (ρ) considered on the next slide for ρ i = x i /y i, for all i. The latter has a very special structure, consisting of the derivative N ρ 1 ρ N Vandermonde type matrices, but going in opposite directions. Since collinearity is equivalent to the disjointness of the ρ i for two i = 1... N, the theorem is a consequence of the following lemma. 52

53 6. Mayer-Lie Interpolation - 6 Lemma. Let ρ i 0; i = 1,... N. The 2N 2N matrix (2N 1)ρ 2N 2 1 (2N 2)ρ 2N (2N 1)ρ 2N 2 (2N 2)ρ 2N N N 0 1 (2N 2)/ρ 2N 3 1 (2N 1)/ρ 2N (2N 2)/ρ 2N 3 N (2N 1)/ρ 2N 2 N has determinant D N (ρ 1,..., ρ N ) = D N (ρ) = 1 if N = 1, and for N > 1: ( 1) N(N 1)/2 (2N 1) N i<j (ρ i ρ j ) 4 (ρ 1 ρ N ). 2N 2 53

54 6. Mayer-Lie Interpolation - 7 Example: Delayed Logistic Equation (MTNS-02) Planar system, observation of x, (Spencer: prediction of the influenza outbreak in England and Wales). x k+1 = µx k (1 y k ) y k+1 = x k. For µ = 2.1, the system exhibits an attracting limit cycle enclosed in [0, 1] 2. (Starting at a point on the limit cycle, the seventh iterate overtakes it). 54

55 0.8 iterates Limit cycle for delayed logistic equation 55

56 6. Mayer-Lie Interpolation - 8 The local reachability Gramian and observability Gramian are respectively R(x, y) = O(x, y) = 2 x2 xy µ 2 µ 2 xy y 2 µ 2 u µ2 (1 y) 2 µ 2 (1 y)x µ 2 (1 y)x µ 2 x 2 Interpolation points: {(0.2, 0.2), (0.5, 0.2)(0.2, 0.5)},. 56

57 Pseudo-balancing transformation ξ(x, y) = 2115x x 4 y x 3 y x 2 y xy y 5 η(x, y) = 1048x x 4 y x 3 y x 2 y xy y 5. 57

58 6. Mayer-Lie Interpolation - 9 The dominant value λ 1 (x, y) of the canonical Gramian is displayed. The height of the plot indicates the value, the shading is modulated by the angle of the dominant direction (mapped back to the original (x, y) coordinates). If T loc is the local balancing transformation, this is the direction of the first column of T 1 loc, i.e., the jointly most observable and reachable direction. As expected, along the limit cycle, the dynamics is almost one dimensional. Theorem: The dominant direction of (f, g, h) at x maximizes the form ξ (HH ) 1/2 ξ ξ (HH ) 1/2 ξ. where H = O 1/2 R 1/2 (symmetric pos. def square root matrices). 58

59 y x Dominant λ and corresponding direction. 59

60 Conclusions: Reasonable set of axioms for general balancing Obstruction: Mayer-Lie Conditions Integration of Jacobian (balanced or uncorrelated) Approximation via Mayer-Lie Interpolation Advantages: i) computationally feasible. ii) consistent with the linear theory 60

61 References: 1. Gray, W.S. and E.I. Verriest, Balanced Realizations near Stable Invariant Manifolds, MTNS-04, Leuven, Belgium [Submitted to Automatica]. 2. Verriest, E.I., Nonlinear Balancing and Mayer-Lie Interpolation, Proc. SSST-04, Atlanta, GA March 2004, pp Verriest, E.I. and W.S. Gray, Discrete Time Nonlinear Balancing, Proceedings IFAC NOLCOS 2001, St Petersburg, Russia, July Verriest, E.I., Balancing for Discrete Periodic Systems, IFAC Workshop Preprints: Periodic Control Systems, Como (Cernobbio), Italy, pp , September Verriest, E.I., Pseudo Balancing for Discrete Nonlinear Systems, Proceedings MTNS 2002, Notre Dame, August Verriest, E.I. and Gray, W.S., Flow Balancing Nonlinear Systems, Proc. MTNS 2000, Perpignan, France, June Verriest, E.I. and Gray, W.S., Nonlinear Balanced Realizations, Proc. 40-th IEEE Conference on Decision and Control, Orlando, FL: December Verriest, E.I. and T. Kailath, On Generalized Balanced Realizations, IEEE Transactions on Automatic Control, 28, pp ,