NONLINEAR BEHAVIORAL BALANCING BY EXTENSION ... - CiteSeerX

3rd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nogoya 2006.

215

NONLINEAR BEHAVIORAL BALANCING BY EXTENSION OF LIE SEMIGROUPS Ricardo Lopezlena ∗,1 Jacquelien M. A. Scherpen ∗∗

∗

Instituto Mexicano del Petr´oleo (IMP), Eje Central L´azaro C´ardenas 152, Col. San Bartolo Atepehuacan CP 07730, M´exico D.F., M´exico, Apdo. Postal 14-805. Tel. +52-9175-7603, Fax +52-9175-7079. [email protected] ∗∗ Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands. Tel. +31-(0)15 278 6152, Fax +31-15-278 6679. [email protected]

Abstract: In a previous paper Lopezlena (2004) introduced a balancing condition for nonlinear systems. This paper provides an extended justification of such balancing condition in terms of semigroups of diffeomorphisms in a Hilbert submanifold framework. Moreover, it is argued that when such condition is satisfied the resulting group of diffeomorphisms describes the flow of the nonlinear system. Using the same framework the nonlinear behavioral operator is defined and a result regarding its spectral properties is c presented. Copyright !2006 IFAC Keywords: Nonlinear systems, model approximation, model reduction, differential geometric methods, geometric approaches.

1. INTRODUCTION In the classical linear balanced reduction method (Moore, 1981), an exponentially stable linear system realization is said to be balanced when the controllability Gramian equals the observability Gramian and it is called axis-balanced when additionally both Gramians are diagonal with an ordered spectrum, (Curtain and Zwart, 1995). Whenever such system is controllable and observable, there is an infinite number of transformations that provide us with a balanced realization and furthermore a family of transformations yield the axis-balanced condition. Balanced reduction is important for control purposes since the minimality properties of the original system are preserved in the reduced order system. Such preserved properties are clearly coordinate-invariant. 1

Enrolled as promovendus at the DCSC, TU Delft.

Nowadays nonlinear balanced reduction has been discussed in several publications. In (Scherpen, 1993) a nonlinear generalization of Moore’s approach has been presented using energy functions keeping certain similarity with the approach of behavioral balanced reduction for linear systems due to (Weiland, 1994). In (Fujimoto and Scherpen, 2005) it is shown that the nonlinear balancing problem can be solved with the solution of a nonlinear eigenvalue problem. Nonlinear adjoint operators for this purpose are presented in (Fujimoto et al., 2002). Schmidt pairs for the nonlinear balancing problem are proposed in (Gray and Scherpen, 2005). There is a rich geometric structure behind the nonlinear extension of Moore’s work. In (Lopezlena, 2004) some differential geometry-oriented research began towards an adequate geometric theory for the nonlinear extension of the behavioral balancing of (Weiland,

216 1994). In this paper such geometric approach continues in development with emphasis on the framework of dissipative systems initiated in (Lopezlena et al., 2003) and using Hilbert submanifold theory and semigroups of diffeomorphisms, firstly used in (Lopezlena, 2004) for nonlinear balanced reduction purposes. The exposition is planned as follows: In Sec. 2 essential geometric concepts are introduced along with semigroup theory and the Hilbert submanifold structures. Moreover, based on a theorem due to (Mirotin, 2001), it is shown that when the balancing condition alluded in (Lopezlena, 2004) is satisfied, then both semigroups define a group of diffeomorphisms, which in turn defines the flow of the system. In Sec. 3 is shown that the storage functions of the Dissipativity theory are generating functions of the action of past and future group maps and moreover they can be chosen as metrics of two dual Riemannian spaces. In Sec. 4 the behavioral operator is briefly discussed in terms of semigroups of diffeomorphisms. Finally, assuming the balancing condition is satisfied, in Sec. 5 some eigenvalue problems for such operator are discussed.

2. BEHAVIORAL-GEOMETRIC CONCEPTS Let us denote a dynamical system by Σ. Such system is perturbed by the environment through a set of variables called manifest or external signals defined on a space W where such signals are supported. As the system evolves in time, these external signals define (behavioral) trajectories on a space B ⊂ t × W called the behavioral space. The triad (t, W, B) is said to define a dynamical system in the behavioral approach (Weiland, 1994). Associated to such system is a map from past external signals into future external signals ˜ : Bp $→ Bf which will be referred hereafter as Γ the behavioral operator. This operator defines an exclusion law which discards trajectories outside the set of behavioral time-trajectories. Internally, such behavioral operator is built by summing up the effect of past external signals Bp on internal state-space trajectories on a manifold M and reconstructing from M the future external signals Bf .

2.1 Semigroups of diffeomorphisms The behavioral operator, as an evolutionary operator, is properly defined in terms of semigroups of diffeomorphisms, briefly recalled here: A family {Φ(x, t), t ∈ tf , x ∈ D ⊂ M} in a class of bounded operators in M is called a 1-parameter semigroup if it is such that the mapping Φ : R1 × D → D, Φ(t, x) = φt (x) depends smoothly on t ∈ R+ ; φ0(x) = x and φt2 ◦ φt1 (x) = φt2 +t1 (x), being called a strongly continuous semigroup (or C0 semigroup) if t $→ φt (x) is continuous on [0, ∞) for every x ∈ M. The procedure of internally building the

behavioral operator takes us to consider some class of model structure, e.g. the nonlinear system Σ written as x(t) ˙ = f(x(t), u(t)), y(t) = h(x(t)), where x ∈ Rn are local coordinates for a C ∞ state space manifold M, f and h are C ∞ . The set of external variables W ≈ Rω , p + q ≤ ω, includes u ∈ U ⊂ Rp and y ∈ Y ⊂ Rq as subsets. For piecewise constant control inputs u(t), u : t $→ U, the (smooth) time varying vector field x $→ f(x, u(t)) has an associated family of vector fields denoted by Fu = {fu : u ∈ U}. The semi-trajectories of this system are continuous curves -(t) on M on an interval [0, T ] that define integral curves of the family Fu if there exists a partition 0 = t0 ≤ t1 ≤ · · · ≤ tm = T and associated vector fields ξ1 , . . . , ξm ∈ Fu such that the restriction of -(t) to each open interval (ti , ti+1), i = 0, . . ., m, is differentiable and such that d-(t)/dt = ξi (-(t)), -(0) = -0 . The formal details can be reviewed in (Lopezlena, 2004) and references therein. Since usually we are concerned with a forward-time evolution of the system, the solution of Σ is provided in terms of 1-parameter C0-semigroups of diffeomorphisms x(t) = Φ(t; t0 , x0, u(t)), using appropriately defined piecewise-constant control inputs u(t) ∈ U that guarantee boundedness of such semi-group actions. Therefore troughout this paper we assume that such inputs u(t) are uniquely characterized using storage functions as in (Lopezlena, 2004). Recall that a vectorfield ξ(x) is called a generator of a 1-parameter group of diffeomorphisms if it is such that ξ(x) = [∂Φ(t, x)/∂t]|t=0. The exponential map expx : Tx M $→ M is a suggestive notation in Lie groups to express that gt (x) = exp(tξ)x is a 1parameter group of diffeomorphisms with generator ξ, see e.g. (Olver, 1993). A (closed) Lie semi-group SG of a Lie Group G is generated by the images of all the 1-parameter semigroups φt(x) : R1 $→ SG , t $→ exp(tξ), being called differentiable whenever each operation ◦ : SG × SG $→ SG yields a differentiable map. We will adopt as notation to describe the conventional forward-time evolution by the interval t = {t|t ∈ R+ } and a backward-time evolution by τ = {τ |τ = −t, t ∈ t}. Moreover we consider two half-spaces M×t and M∗ ×τ , where M and M∗ are dual spaces joined at t = 0 by duality relations at their boundary or edge M0 and M∗0 respectively. Under this notation, there can be defined a semi-trajectory x(t) ∈ M × t, t ∈ t generated by a positive semigroup and a negative semi-trajectory x ˆ(τ ) ∈ M∗ × τ , t ∈ τ generated by a negative semigroup. There is no reason to believe a priori that the integral trajectories of a tangent vectorfield ξ are defined for both positive t ∈ R+ and negative τ ∈ R− evolving + times. Denote by SF the positive semigroup of all the u t elements of {Φ x(t)} evolving in forward-time, i.e. Φt (x0 ) = exp tk ξk · · · exp t2 ξ2 · exp t1ξ1 x0, (1) such that ti ≥ 0 for i = 1, . . ., k. A vectorfield ξ(x) is a generator of the Lie semigroup of diffeo-

217 + morphisms SG = {Φt (x)}t∈t if the limit ξ(x) = t limt→0+ (Φ (x) − x)/t exists in a domain D(ξ). When the solution of system Σ is solved in negative time, it defines an evolution operator map (C0semigroup of diffeomorphisms) defined by x ˆ(τ ) = Θ(τ ; τ0, x ˆ(0), u ˆ(τ )), τ ∈ τ , u ˆ(τ ) ∈ U ∗ . Using apropriately defined u ˆ(τ ) ∈ U ∗ , the evolution of such backward-time semigroup can be expressed as

Θτ (ˆ x0) = exp τk ξk · · · exp τ2ξ2 · exp τ1ξ1 x ˆ0, (2) − . with τi ≤ 0 for i = 0, . . . , k − 1, k, denoted by SF u − τ A vectorfield ξ(x) generates SFu = {Θ (ˆ x)}τ ∈τ whenever ξ(x) = limτ →0− (Θτ (ˆ x) − x ˆ)/τ exists in ˆ a domain D(ξ). Φ

Θ−1∗

Mf ←−−−− M0 ←−−−− Mp ! ! ! ! ! ! ∗! ∗! ∗! Φ−1∗

Θ

M∗f −−−−→ M∗0 −−−−→ M∗p

Consider now the set of points at the edge M0. Since each edge-point x0 ∈ M defines uniquely a positive semi-trajectory x(t) in forward-time and analogously for x ˆ0 ∈ M∗ in backward-time, then the duality pairing between such dual spaces defines the required condition to define a complete integral trajectory, namely the semi-groups Φt (x), t ∈ t and the dualized Θτ (ˆ x), τ ∈ τ must be such that " −1 x(τ ))] = Φt (x(t)), s.t. [Θτ (ˆ # −1 τ t # [∇xΘ (ˆ x(τ ))]τ =0 = ∇xΦ (x(t)) t=0 , is regular (3) is satisfied. Let G(Fu) denote the (connected) Lie group of diffeomorphisms in M generated by the union of {exp tξ | t ∈ t, ξ ∈ Fu}. When such ξ generates integral trajectories for all times (positive and negative), then ξ(x) is a complete vector field and the family Fu is complete. Such complete vectorfield generates the group G(Fu) = {Φt (x)}t∈t and furthermore it is the unique group extension, see e.g. (Hofmann and Lawson, 1983), whenever G(Fu) is connected and the tangent wedge of SFu , defined as £(SFu ) := {ξ ∈ £(G(Fu))| exp(tξ) ⊆ SFu } is a Lie-semialgebra, (Mirotin, 2001). Furthermore, such unique group extension G(Fu) = Φt (x(t)), + $ − G(Fu) = SFu SFu , defines the flow of Σ, being the + smallest local group containing SF , (Hofmann and u Lawson, 1983). Remark 2.1. Eq. (3) is valid for lumped-parameter systems. The infinite-dimensional semigroup extension for distributed parameter systems is equivalent to Eq. (3) and can be obtained based on (Mirotin, 2002).

space (Palais, 1963; Lang, 1999). Being M Riemannian, it has as inner product gM for Tx M equivalent to the inner product 9·, ·: in (dF, 9·, ·:) for all x ∈ M, where F consists of (the equivalence class of) Lebesgue-measurable, (square) integrable functions mapping the interval [a, b] into Rn , denoted by Ln 2 [a, b]. Four Hilbert manifold structures are needed throughout the paper. The first one characterizes the natural duality of the state and costate spaces of the compact, differentiable, manifold (M, 9·, ·:T M) where the internal system trajectories and associated functions are supported for inner products defined by % b 9ξ, ζ:T M = (ρ0 , -0)0 + ξ i ζ i dt, (4) a

9α, β:T ∗ M = (f(x0 ), g(x0))0 +

A Riemannian Hilbert manifold M is a differentiable manifold locally modelled on a separable Hilbert

x(b)

αiβi dx,(5)

x(a)

where ξ, ζ ∈ T M and α = df, β = dg ∈ T ∗ M for a duality pairing % b ∗ 9α, ξ:T M×T M = (x0, F (x0))0 + iξ α dt a

= F (x0) + F (x(t))|ba.

(6)

A second Hilbert structure characterizes the space of external signals (W, 9·, ·:T W ) and its dual. Since it is completely analog to the previous structure, it is omitted. The third and fourth Hilbert structures are defined by duality of the halfspaces of past M∗ × τ and future M × t system trajectories and the past W ∗ × τ and future W × t behavioral trajectories. Each pair of spaces can be interpreted as two intertwined dual manifolds with forward-time evolution in M × t and backward-time evolution in M∗ × τ . The edge can always be defined by dualization via the metric. The system trajectories m = (x, x ˆ) ∈ M ⊕ M∗ with tangent vectorfields µ = (ξ, α), ∈ T M⊕T ∗ M are on Hilbert manifolds (M∗ ×τ , 9·, ·:p) and (M×t, 9·, ·:f ) with inner products defined by % 1 −T 9µp1 , µp2:p = iξ1 α2 + iξ2 α1 dτ, (7) 2 0 % 1 T iξ1 α2 + iξ2 α1 dt, (8) 9µ1f , µ2f :f = 2 0 ˜ M) ˜ in a duality pairing defined as with Zd ∈ L(M, % 1 T f 9µf , µp :f ×p = µ Zd µp dt. (9) 2 −T The halfspaces of behavioral trajectories (Bp , 9·, ·:T Bp ) and (Bf , 9·, ·:T Bf ) have inner products defined by 9w1p , w2p :T Bp =

2.2 Hilbert manifold structures

%

9wf1 , wf2 :T Bf

=

% %

−T 0

0

w1pT Zp w2p dτ,

(10)

T

wf1T Zf wf2 dt,

(11)

where w1p , w2p ∈ T Bp and wf1 , wf2 ∈ T Bf for a duality pairing defined by

218 p

9wf , w :T Bf×p =

%

T

−T

wfT Zd (w(t))wp dt,

(12)

where throughout the paper we assume Zp , Zf , Zd ∈ L(T W, T W) are self-adjoint linear operators with associated quadratic function r : T W × T W $→ R1 satisfying r(w(t)) = wT (t)Zi w(t) ≥ 0, ∀t ∈ t, w ∈ T W. Influenced by the past Bp ⊂ W ∗ × τ , the system trajectories are used to define the future behavior Bf ⊂ W ×t. The Behavior and the system trajectories are related by two storage functions associated to system Σ, the backward-time required supply, Sr : M∗ → R+ , x0 , rr ) Sr∗ (ˆ

= − sup v(·)∈V

%

−T p

rr (w (τ ))dτ , (13)

0

x0 =x, T ≥0

and the available storage, Sa : M → R+ , defined by Sa (x0 , ra) =

sup − u(·)∈U

%

T

ra (wf (t))dt, (14) 0

x0 =x, T ≥0

where the function of external signals defined by r : W → R1, r(w(t)), is called supply rate relative to Sr∗ or Sa respectively. The functionals Sr∗ (ˆ x0, rr ), Sa (x0, ra) ∗

Assumption 2.1. associated to state-trajectories on M and M are induced metrics of past gpB and future gfB behavioral trajectories preserving the following relations Sr∗ (ˆ x0 , rr ) = 9µp , µp :p = 9wp , wp :T Bp ,

(15)

0

Sa (x , ra) = 9µf , µf :f = 9wf , wf :T Bf . (16)

3. PROPERTIES OF THE STORAGE FUNCTIONS Eqs. (13) and (14) are generating functions of statecostate group actions: Proposition 3.1. Assume that Sa : Rn $→ R1 exists and is smooth on (the compact) M. The (maximal) flow {Qt(x)}t≥0 generated by the vectorfield ξq = −∇Sa is a positive semigroup; meaning that for all t ∈ tf , Qt(x) is defined on all M. Moreover, for any x ∈ M, {Qt(x)}t≥0, has at least one critical point of Sa as a limit point as t → ∞. Proof. (Sketch). The smooth vectorfield ∇Sa (dual to dSa ) points to the direction of fastest increase of Sa . Let {Qt(x)} denote the maximal flow generated d by ξq = −∇Sa , then we may write dt Qt(x) = d t t ξq (Q (x)) = −∇Sa (Q (x)), thus dt Sa (Qt (x)) = ∇Sa (−∇Sa (Qt (x))) = −>∇Sa (Qt(x))>2 . The proof is completed by showing that ξq is of bounded length and thus is a generator, see (Palais and Terng, 1988). !

Proposition 3.2. Assume that Sr∗ : M∗ $→ R1 exists and is smooth on (the compact) M∗ . The (maximal) flow {P τ (ˆ x)}τ ≤0 generated by the vectorfield ξp = −∇Sr∗ is a negative semigroup on M∗ × τ ; meaning that for all τ ∈ τ , P τ (ˆ x) is defined on all M∗ . ∗ Moreover, for any x ˆ ∈ M , {P τ (ˆ x)} has at least one ∗ critical point of Sr as a limit point as τ → −∞. Proof. Omitted. Similar to the proof of Prop. 3.1.

!

Group extension of these semigroup actions brings about interesting consequences: Proposition 3.3. If condition (3) for Qt : t × M $→ M∗ , Qt (x) = −∇Sa (x) and P τ : τ × M∗ $→ M, P τ (ˆ x) = −∇Sr∗ (ˆ x), expressed by " −1 x(τ ))] = Qt(x(t)), s.t. [P τ (ˆ & −1 2 ∗ 2 x, rr ) = ∇x Sa (x, ra), is regular ∇xˆSr (ˆ (17) is satisfied, then: x) have a com(1) Both semigroups Qt(x) and P τ (ˆ mon complete generator −ξp = ξq = ξ. (2) Generated by ξ, G(Fu) = Qt (x) is the unique Lie group extension with associated singular value problem defined by Υt(x(t)) − σ2 x(t) = 0,

(18)

for σ2 ∈ R1 and x(t) ∈ M × t with the homeomorphism Υt : t × M $→ M given by Υt (x(t)) = P τ ◦ Qt (x(t)).

(19)

Proof. (1) By Props. 3.1 and 3.2 there are generators ξp , ξq (possibly other than −ξp of Θ and ξf of Φ). Preservation of the group extension condition (17) implies −ξp = ξq . Moreover item (2) follows straightforwardly from Eq. (17) since P τ ◦ Qt (x(t)) = [Qt]−1 ◦ Qt(x(t)), t ∈ t being thus a singular value problem.! In view of the previous results, there remains the question of whether there exist an invariant relationship preserved throughout the evolution of the system. The following result provides an answer: Proposition 3.4. (Duality Legendre transform). The following statements are equivalent: (1) The functions Sa (x(t), r), x(t) ∈ M, t ∈ t and Sr∗ (ˆ x(τ ), r), x ˆ(τ ) ∈ M∗ , τ ∈ τ are such that: def

L(x(t), x ˆ(τ )) = Sa (x(t), r) + Sr∗ (ˆ x(τ ), r) + 9ξ(x), ζ(ˆ x):T ∗ M×T M = 0.

(20)

is preserved, where x(t) ˙ = ξ(x(t)) and x ˆ˙ (τ ) = ζ(ˆ x(τ )) and 9ξ(x), ζ(ˆ x):T ∗ M×T M is defined in Eq. (6). (2) Invariance of the Legendre transform (20) is equivalent to

219 x ˆ(τ ) = −∇T Sa (x(t), r),

(21)

x(τ ), r). x(t) = −∇T Sr∗ (ˆ

(22)

(3) Assuming that ∇T Sa (x(t), r) has a regular inverse, (21) and (22) are related by [∇T Sa (x(t), r)]−1 = ∇T Sr∗ (ˆ x(τ ), r). (23) Proof. (Outline). (1) Coordinate independence is inherited by the dual pairing of Eq. (6) and structural arguments. (2) Since L(x, x ˆ) is invariant then ∇L(x, x ˆ) = 0, i.e. # # ∂L(x, x ˆ) ∂L(x, x ˆ) ## ∂L(x, x ˆ) ## = + = 0, ∂[x, x ˆ] ∂x # ∂x ˆ # 0 0 x ˆ=ˆ x

− + where Θ ∈ SF and Φ ∈ SF . The composition of u u Eq. (24)-(25) defines the operator Γu(t) = Ψf ◦ Ψp ◦ u(t). Moreover, using Def. 4.1, the adjoint operators Ψ†p : Rn $→ L2 [−T, 0], Ψ†f : L2 [0, T ] $→ Rn are defined by

Ψ†p (x0 ) := g−1 [Θ(−T, 0, x ˆ0, u ˆ(τ ))],

ˆ(τ ); h−1(ˆ y )), (27) Ψ†f yˆ(τ ) := Φ−1∗ (0, T, 0, u + − where Φ ∈ SF , Θ ∈ SF . The composition of the u u Eq. (26) and (27) defines the operator Γ† yˆ(τ ) = Ψ†p ◦ Ψ†f ◦ yˆ(τ ).

x=x

then using Eq. (6), ∇Tx L(x, x ˆ) = ∇T Sa (x, r) + T x ˆ = 0 and ∇xˆ L(x, x ˆ) = ∇T Sr5 (ˆ x, r) + x = 0, which are precisely Eqs. (21) and (22). (3) Eq. (23) is another statement (17). In this context, since 'n of Eq. 5 dSr5 (ˆ x, r) = x, r)dˆ xi and on the other i ∂i Sr (ˆ 5 side by (20), dS x, r) = d[−(x, x ˆ) − Sa (x, r)] and r (ˆ ' n i 5 dS' (ˆ x, r) = a (x, r) = i ∂i Sa (x, r)dx then dSr' n i n i i i i − i x dˆ x +x ˆ dx + ∂i Sa (x, r)dx = − i xidˆ xi (due to Eq. (21)) and the last equality is precisely Eq. (22). Conclude that one transformation is the inverse of the other and thus Eq. (23) is obtained. !

Ψf

Ψp

Ψ†f

Ψ†p

Y ←−−−− M0 ←−−−− U ! ! ! ! ! ! ∗! ∗! ∗!

Y ∗ −−−−→ M∗0 −−−−→ U ∗ The Behavioral operator maps all past exogenous variables into all future exogenous variables: Definition 4.3. (Behavioral operator). The behavioral ˜ : L2 [−T, 0] $→ L2[0, T ], Γ ˜ : Bp $→ Bf operator Γ is defined by (

4. THE BEHAVIORAL OPERATOR

(26)

u ˆ(τ ) y(t)

)

= f

(

Γ† ◦ yˆ(τ ) Γ ◦ u(t)

)

p

,

τ ∈τ t∈t

(28)

In this section, the geometric structure of the nonlinear ˜ : Bp $→ Bf is presented. behavioral operator Γ Consider the set of input trajectories {u(t)|u(t) ∈ U, t ∈ t} on the set of admisible inputs U such that a point x0 ∈ M can be reached from the origin following a trajectory x(t). We assert that ui is equivalent to uj , ui ≡ uj mod x(t), ui , uj ∈ U, if both produce the same trajectory x(t) = G(ui ) = G(uj ) where G : U $→ M defines a (regular by assumption) equivalence relation ui Guj . Furthermore, consider the set of state trajectories {x(t)|x(t) ∈ M0, x(0) = x0, t ∈ t} that produce an output y(t) ∈ Y. We assert that xi is equivalent to xj , xi ≡ xj mod y(t), xi, xj ∈ M, if both produce the same output trajectory y(t) = h(xi) = h(xj ) where h : M $→ Y defines a (regular by assumption) equivalence relation xi h xj . The following definitions are used throughout:

˜ ∗ : T Bp $→ Remark 4.1. Denote tangent maps by Γ T Bf . The following facts are easily verified:

Definition 4.1. Denote by h−1 : Y $→ M, x(t) = h−1(y(t)) the inverse map of h and denote by g−1 : M $→ U, u(t) = g−1 (x(t)) the inverse map of G.

˜ : Bp $→ Bf is by construction an isometry Since Γ for the past and future metrics we may write

(1) Γ and Γ† satisfy 9Γ†∗ y, u:T W|u = 9y, Γ∗ u:T W|y , y ∈ T W|y , u ∈ T W|u and thus are adjoint. (2) Each homeomorphism map Γ† ◦ Γ : U $→ U and Γ ◦ Γ† : Y $→ Y is selfadjoint. ˜ is an isometric isomorphism (3) By construction Γ ˜ ∗ ξ, Γ ˜ ∗ ζ:T Bf . satisfying 9ξ, ζ:T Bp = 9Γ p ˜ ˜ ˜ ∗ ω p , ωf : T B , (4) Γ satisfies 9ω , Γ∗ ωf :T Bp = 9Γ f p ω ∈ T Bp , ωf ∈ T Bf and thus it is selfadjoint on B.

5. EIGENVALUE PROBLEM FOR THE BEHAVIORAL OPERATOR

K(ξ) = 4.1 Structure of the nonlinear behavioral operator Definition 4.2. Associated to system Σ define the following operators Ψp : L2[−T, 0] $→ Rn, Ψf : Rn $→ L2[0, T ] by Ψp u(t) := Θ−1 (0, −T, 0, u(t)), 0

0

Ψf (x ) := h[Φ(T, 0, x , u(t))],

(24) (25)

˜ † ◦ Γ) ˜ ∗ ξ, ξ:T Bp 9(Γ IIB (ξ, ξ) , (29) := 9ξ, ξ:T Bp IB (ξ, ξ)

where K(ξ), ξ ∈ T B is the normal curvature of Bp , IB = 9ξ, ξ:T B is the first fundamental form of B and IIB = 9AB η (ξ), ξ:T B is the second fundamental form of B with Shape operator AB η : T B $→ T B, η ∈ (T B)⊥ . The eigenvalue problem of the quotient (29) consist in finding the principal directions ξ along T B where K(ξ) attains stationary values κ called

220 principal normal curvatures. Using classical Curvature Theory the following results are obtained:

by Ψ†p ◦Qt : t×M $→ U, u := Ψ†p ◦Qt ◦-(t), yielding Γ† ◦ Γ ◦ u(t) − λu(t) = 0. !

Proposition 5.1. Let Σ = (t, W, B) on a Hilbert submanifold (V, 9·, ·:T V ) of the Hilbert manifold of external signals (W, 9·, ·:T W ) s.t. suppB = V ⊂ W, dimV = υ, satisfying Assumption 2.1, IB = Sr (x0, rr ) and IIB = Sa (x0, ra) with AB η (ξ) = T ⊥ −∇ξ η, ξ ∈ (T B), η ∈ (T B) . The following can be asserted:

REFERENCES

(1) A vectorfield ζ ∈ T B, 9ζ, ζ:T B = 1 is solution to the eigenvalue problem associated to K(ζ) in (29) iff ζ is an eigenvector of AB η . (2) The set of eigenvectors of AB η , {ζi | i = 1, . . . υ; ζi ∈ Tp B}, defines an orthonormal basis of T B. (3) Denote by G = [gij ], Q = [qij ] the metric tensors of IB , IIB respectively. Then AB η = QG−1 ,∀ ζ ∈ Tp B and K(ζ) = det Q/ det G. Proof. (1), (2), (3) All these results can be proved using classical theory of Gaussian curvature. Due to space restrictions, these proofs are omitted. ! Definition 5.1. (Past and Future Gramians). The past and future map homeomorphisms P τ : τ × M∗ $→ M, Qt : t × M $→ M∗ defined as P τ (ˆ x0) = Ψp ◦ Ψ†p (ˆ x0 )

(30)

Ψ†f

(31)

t

0

Q (x ) =

0

◦ Ψf (x )

are called the nonlinear past and future Gramians. Their composition is denoted by the map Υ : t × M $→ M, Υt (x) = P τ ◦ Qt (x), Eq. (19). The following result provides an eigenvalue problem associated to Eq. (30)-(31): Theorem 5.1. Consider the nonlinear maps (30)- (31) with associated eigenvalue problem defined by Eq. (18), -(t) ∈ M. Then the resulting eigenvalues λ = σ2 are the same eigenvalues of the operator Γ† ◦ Γ ◦ u(t). Proof. The eigenvalue problem of the behavioral operator can be stated as finding the eigenvalue λ A= 0 and the eigenvector 0 A= u(t) ∈ U such that Γ† ◦ Γ ◦ u(t) = λu(t). Express such eigenvalue problem by Γ† ◦ Γ ◦ u(t) = Ψ†p ◦ Ψ†f ◦ Ψf ◦ Ψp ◦ u(t) = λu(t) which after being mapped by the nonlinear map Ψp : U $→ M, -(t) = Ψp (u) for some state trajectory -(t) ∈ M, yields Γ† ◦ Γ ◦ u(t) = Ψp ◦ Ψ†p ◦ Ψ†f ◦ Ψf ◦ Ψp ◦ u(t) = P τ ◦ Qt ◦ -(t) = λ-(t) where P τ ◦ -(t) and Qt ◦ -(t) are defined from (30) and (31) yielding P τ ◦ Qt ◦ -(t) − λ-(t) = 0. Assume now the eigenvalue λ A= 0 and the state trajectory (eigenvector) 0 A= -(t) ∈ M are solution of P τ ◦ Qt ◦ -(t) − λ-(t) = 0. Map this latter equation

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