Partial Persistence of Excitation in RKHS Embedded Adaptive Estimation
PPersistence of Excitation in Uniformly EmbeddedReproducing Kernel Hilbert (RKH) Spaces
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Jia Guo, Sai Tej Paruchuri, Andrew Kurdila
Department of Mechanical EngineeringVirginia TechBlacksburg, VA 24060 [email protected], [email protected], [email protected]
February 20, 2020 A BSTRACT
This paper introduces two new notions of persistence of excitation (PE) in reproducing kernel Hilbert(RKH) spaces that can be used to establish convergence of function estimates generated by the RKHspace embedding method. The equivalence of these two PE conditions is shown to hold if U ( ¯ S ) is uniformly equicontinuous, where U is the Koopman operator and ¯ S is the closed unit spherein the RKH space. The paper establishes sufficient conditions for the uniform asymptotic stability(UAS) of the error equations of RKH space embedding in term of these PE conditions. The proof isself-contained, and treats the general case, extending the analysis of special cases studied in [1, 2].Numerical examples are presented that illustrate qualitatively the convergence of the RKH spaceembedding method where function estimates converge over the positive limit set, which is assumedto be a smooth, regularly embedded submanifold of R d . K eywords Adaptive Estimation, Reproducing Kernel, Persistence of Excitation
Adaptive online estimation for uncertain systems governed by nonlinear ordinary differential equations (ODEs) isnow a classical topic in estimation and control theory. Systematic study of this topic has a long history, and manyof the first principles can be found in texts on adaptive estimation and control theory [3–5]. It is known that ingeneral, convergence of state estimates in such schemes is easier to establish than to guarantee parameter convergence.Parameter convergence refers here to estimates of the (real) constants that characterize an unknown function appearingin the uncertain governing ODEs. Beginning with analyses such as in [5–7], sufficient conditions for parameterconvergence in terms of various definitions of persistence of excitation in finite dimensional state spaces have beenstudied carefully. These initial investigations have been the source of numerous generalizations of PE conditions forevolution laws in R d , with notable examples including [8–13], to name a few. The sequence of papers [14–18] furtherextends the analysis of PE conditions for evolution equations defined in terms of a pivot space structure in Banach andHilbert spaces.This paper studies novel persistence of excitation (PE) conditions that play a role in recently introduced method ofreproducing kernel Hilbert (RKH) space embedding for adaptive estimation of uncertain, nonlinear ODE systems[1, 2, 19]. The RKH space embedding method analyzes the uncertain nonlinear system of ODEs by replacing themwith a distributed parameter system (DPS). While the usual approach such as in the texts above describe evolution ofstates and parameters in the finite dimensional space R d × R n , the RKH space embedding method considers evolutionof states and function estimates in R d × H with H an RKH space of functions.Some of the key theoretical questions regarding the RKH space embedding method have been studied in refer-ences [1, 2, 19–21]. References [1, 2] introduce the study of the well-posedness of the governing equations under a r X i v : . [ m a t h . O C ] F e b PREPRINT - F
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20, 2020some elementary hypotheses: existence and uniqueness of solutions as well as continuous dependence on initial con-ditions are discussed. References [1, 2] also describe some preliminary convergence results for finite dimensionalapproximations. Reference [20] studies the relationship between persistence of excitation results in an RKH space andpositive limit sets, including some cases when the positive limit set is a smooth manifold. Reference [19] describesthe extension of the basic theory in [1, 2] to consensus estimation. Reference [21] is an early example of applicationof the RKH space embedding strategy to an L adaptive control problem.One of the limitations of the analyses in [1, 2, 19, 21] is that the uniform asymptotic stability of the error equationsfor the RKH space embedding method is proven in detail only for a very specific case. In fact, one of the simplerproofs in [1] is unfortunately incorrect, which we correct in this paper. Here we study various alternative statementsof the persistence of excitation conditions in the class of uniformly embedded RKH spaces. When the RKH space isuniformly embedded in the space of continuous functions, we find that many of the well-known classical statementsabout the PE condition in finite dimensional spaces have analogous counterparts in the RKH space. For the classicalstatements, PE is understood as positivity of integrated regressors over subspaces of R n . In the RKH space embeddingmethod, the PE condition is cast in term of the evaluation functional E x : f (cid:55)→ f ( x ) for x ∈ X , with X := R d thestate space of a continuous flow. The conventional notion of persistence of excitation is defined to study the uniform asymptotic stability of errorequations that have the form (cid:20) ˙˜ x ( t )˙˜ α ( t ) (cid:21) = (cid:20) A B Φ T ( x ( t )) − µ Φ( x ( t )) B T P (cid:21) (cid:20) ˜ x ( t )˜ α ( t ) (cid:21) (1)where ˜ x ( t ) = x ( t ) − ˆ x ( t ) ∈ R d is the error in state estimates ˆ x ( t ) of the true trajectory x ( t ) , ˜ α ( t ) = α ∗ − ˆ α ( t ) ∈ R n is the error in the parameter estimates ˆ α ( t ) of the true parameters α ∗ , A ∈ R d × d is Hurwitz, B ∈ R d × , P ∈ R d × d isthe unique positive definite solution of the Lyapunov equation A T P + P A = − Q for some fixed symmetric positivedefinite Q ∈ R d × d , and Φ : R d → R m is the collection of regressors for the system. The associated PE condition forEq. 1 follows in Definition 1. This condition is a sufficient condition for the UAS of the error Equations 1. Details arediscussed in references [5–13]. Definition 1. (PE in R n ) A trajectory x : t (cid:55)→ x ( t ) ∈ R d persistently excites a family of regressor functions Φ : R d → R n if there exist constants T , ∆ , γ > such that (cid:90) t +∆ t v T Φ( x ( τ ))Φ T ( x ( τ )) vdτ ≥ γ > (2) for each t ≥ T and v ∈ R n with (cid:107) v (cid:107) = 1 . In this paper, we introduce the two definitions below of the PE condition for the RKH space embedding method.
Definition 2. (PE.1) The trajectory x : t (cid:55)→ x ( t ) ∈ R d persistently excites the indexing set Ω and the RKH space H Ω provided there exist positive constants T , γ, δ, and ∆ , such that for each t ≥ T and any g ∈ H Ω with (cid:107) g (cid:107) H Ω = 1 ,there exists s ∈ [ t, t + ∆] such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s + δs E x ( τ ) gdτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ γ > . (3) Definition 3. (PE.2) The trajectory x : t (cid:55)→ x ( t ) ∈ R d persistently excites the indexing set Ω and the RKH space H Ω provided there exist positive constants T , γ, and ∆ , such that (cid:90) t +∆ t (cid:16) E ∗ x ( τ ) E x ( τ ) g, g (cid:17) H Ω dτ ≥ γ > (4) for all t ≥ T and any g ∈ H Ω with (cid:107) g (cid:107) H Ω = 1 . These two definitions are analogous to those studied in the classical scenario in [5, 6], but here they are expressed interms of the evaluation operator E x rather than the regressor functions Φ : R d → R n . Also, Definition 3 should becompared to that given in [14] for evolution equations cast in terms of Gelfand triples. We define the Koopman operator U associated with the motion x : t (cid:55)→ x ( t ) to be the mapping U : g (cid:55)→ g ◦ x . Let ¯ S := { g ∈ H Ω : (cid:107) g (cid:107) H Ω = 1 } denote the unit sphere in RKH space H Ω . The first primary result of this paper is stated in terms of the collection offunctions U ( ¯ S ) := { g ( x ( · )) : (cid:107) g (cid:107) H Ω = 1 , g ∈ H Ω } . We establish in Theorems 1 and 2 that2 PREPRINT - F
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20, 2020“PE.1 = ⇒ PE.2,” and“PE.2 and U ( ¯ S ) uniformly equicontinuous = ⇒ PE.1.”This theorem can be viewed as a type of generalization of the results in [5, 6] to the DPS that defines the RKH spaceembedding method. Essentially, the assumption that U ( ¯ S ) is uniformly equicontinuous eliminates in the infinitedimensional case the possibility of “pathological” rapid switching that has been commented in [5, 6] for the finitedimensional case.The role of these PE conditions above is studied for the following DPS that is associated with estimation errors in theRKH space embedding formulation: (cid:34) ˙˜ x ( t )˙˜ f ( t ) (cid:35) = (cid:20) A B E x ( t ) − µ ( B E x ( t ) ) ∗ P (cid:21) (cid:20) ˜ x ( t )˜ f ( t ) (cid:21) . (5)Here ˜ x ( t ) ∈ R d , A ∈ R d × d Hurwitz, and B ∈ R d × are defined as above, but now ˜ f ( t ) = f − ˆ f ( t ) ∈ H Ω is the errorof the function estimates ˆ f ( t ) of the true function f . The second fundamental result of this paper is a detailed proof inTheorem 3 of the fact that “PE.1 = ⇒ the error equations are UAS in R d × H Ω ”.We should note that references [1,2,19] establish this fact as a special case of [14] when P = I and A is in fact negativedefinite, but here we treat the general situation. The proof is self-contained, avoids much of the complexity of the pivotspace framework of [14], and essentially lifts the finite dimensional proof to the DPS setting. We emphasize that thesimplicity of the proof relies fundamentally on the uniform embedding of H Ω into C ( X ) , the space of continuousfunctions on X . In this paper, (cid:107) · (cid:107) denotes the Euclidean norm on R d , (cid:107) · (cid:107) op is the operator norm, and (cid:107) · (cid:107) H X is the norm on a Hilbertspace H X of functions that map the set X → R . The inner product on R d and H X are written as ( · , · ) and ( · , · ) H X respectively. The set Ω ⊆ X is referred to as the indexing set contained in X . As we discuss more fully in Section2.2, Ω is termed the index set since it is used to define a subspace H Ω ⊆ H X that is defined in terms of basis functionsthat are indexed by the true trajectory. In this paper, when we refer to the classical problem of adaptive estimation for an unknown nonlinear set of ODEs, weassume that the state trajectory x : t (cid:55)→ x ( t ) ∈ R d satisfies for t ∈ R + the set of equations ˙ x ( t ) = Ax ( t ) + Bf ( x ( t )) , (6)where, as in the error equations, A ∈ R d × d is a known Hurwitz matrix, B ∈ R d × is known, and f : R d → R isunknown and to be identified. Under the assumption that f ( · ) = (cid:80) ni =1 α ∗ i φ i ( · ) for a set of given regressor functions φ i : R d → R , the choice of estimator and learning law ˙ˆ x ( t ) = A ˆ x ( t ) + B Φ T ( x ( t ))ˆ α ( t ) , ˙ˆ α ( t ) = µ (cid:2) B Φ T ( x ( t )) (cid:3) T P ( x ( t ) − ˆ x ( t )) (7)induces the error equations in Eq. 1. To define the RKH space embedding method, we briefly review RKH spaces. A real RKH space H X over a subset X is defined in terms of an admissible kernel K X : X × X → R . When X is a subset of R d , or even some types of(sub)manifolds, many choices of admissible kernels exist. See [22] for summaries of possible kernels over (subsetsof) R d and [23] for kernels over some choices of manifolds. We define a basis function K X,x centered at x ∈ X interms of the kernel K X by setting K X,x ( · ) := K X ( x, · ) . The associated RKH space H X is the closed linear space H X = span { K X,x : x ∈ X } . It is a defining property of RKH spaces that they satisfy the reproducing property f ( x ) = E x f = ( K X,x , f ) H X (8)3 PREPRINT - F
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20, 2020for each x ∈ X and f ∈ H X . In this paper, we only consider the RKH spaces for which we have the uniformlycontinuous embedding H X (cid:44) → C ( X ) , which holds provided that there is a constant c > such that (cid:107) f (cid:107) C ( X ) ≤ c (cid:107) f (cid:107) H X (9)for all f ∈ H X . In the view of Eq. 8 and Eq. 9 above, a sufficient condition for uniform embedding is that a constant ¯ k exists such that K X ( x, x ) ≤ ¯ k < ∞ for all x ∈ X . In this we can write | f ( x ) | = |E x f | = | ( K X,x , f ) H X | ≤ (cid:107) K X,x (cid:107) H X (cid:107) f (cid:107) H X ≤ (cid:112) K X ( x, x ) (cid:107) f (cid:107) H X ≤ ¯ k (cid:107) f (cid:107) H X . The condition that K X ( x, x ) ≤ ¯ k guarantees that (cid:107)E x (cid:107) op ≤ ¯ k , that is, the evaluation operator is uniformly boundedin x ∈ X .When H X is an RKH space of functions defined over a set X , we define the closed subspace H Ω = span { K Ω ,x : x ∈ Ω } when Ω ⊆ X . In practice, we can roughly view H Ω as the set of functions in H X indexed bytrajectories x ( t ) ∈ Ω . For this reason, we refer to Ω as an indexing set in this paper. We define (cid:107) · (cid:107) H Ω := (cid:107) P Ω ( · ) (cid:107) H X where P Ω is the H X -orthogonal projection onto H Ω ⊆ H X . Note that it is always possible that Ω = X in this setup.In the RKH space embedding formulation, Eq. 6 is interpreted as the functional equation ˙ x ( t ) = Ax ( t ) + B E x ( t ) f, (10)and the corresponding estimation equation and learning law are ˙ˆ x ( t ) = A ˆ x ( t ) + B E x ( t ) ˆ f ( t ) , ˙ˆ f ( t ) = − µ ( B E x ( t ) ) ∗ P ( x ( t ) − ˆ x ( t )) , (11)which induces the dynamics of the error in R d × H Ω in terms of Eq. 5. Theorem 1.
The PE condition in Definition 2 implies the one in Definition .Proof. If the condition in Definition 3 holds, there exist constants T , γ, δ, and ∆ , such that for each t ≥ T and any g ∈ H Ω with (cid:107) g (cid:107) H Ω = 1 , there exists s ∈ [ t, t + ∆] such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s + δs E x ( τ ) gdτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ γ > . By the definition of the adjoint operator, the integral in Eq. 4 equals (cid:90) t +∆ t (cid:16) E ∗ x ( τ ) E x ( τ ) g, g (cid:17) H Ω dτ, = (cid:90) t +∆ t (cid:0) E x ( τ ) g, E x ( τ ) g (cid:1) dτ ≥ (cid:90) s + δs (cid:0) E x ( τ ) g (cid:1) dτ. It is assumed that x : t (cid:55)→ x ( t ) is continuous, and g : y (cid:55)→ g ( y ) is continuous since H Ω (cid:44) → C ( X ) . Hence, g ◦ x is also a continuous mapping. Moreover, the interval [ t, t + ∆] is compact, so g ( x ( t )) is bounded on this interval.Therefore, the function E x ( · ) g = g ( x ( · )) ∈ L ([ t, t + ∆] , R ) , and the same applies to the integrand (cid:0) E x ( t ) g (cid:1) . By theCauchy-Schwarz inequality, we have (cid:90) s + δs dτ (cid:90) s + δs (cid:0) E x ( τ ) g (cid:1) dτ, ≥ (cid:32)(cid:90) s + δs (cid:12)(cid:12) E x ( τ ) g (cid:12)(cid:12) dτ (cid:33) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s + δs E x ( τ ) gdτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ γ , which then implies (cid:90) t +∆ t (cid:16) E ∗ x ( τ ) E x ( τ ) g, g (cid:17) H Ω dτ ≥ γ /δ > . PREPRINT - F
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20, 2020The implication above is straightforward, but the derivation of converse requires more conditions on the function g . Recall that a function g : Ω ⊆ R d → R is said to be uniformly continuous on Ω if for each (cid:15) > , thereis a δ = δ ( (cid:15) ) > such that (cid:107) x − y (cid:107) < δ implies | g ( x ) − g ( y ) | < (cid:15) for all x, y ∈ Ω . A family of uniformlycontinuous functions G is called uniformly equicontinuous if for each (cid:15) > , there is a δ depending solely on (cid:15) suchthat (cid:107) x − y (cid:107) < δ implies | g ( x ) − g ( y ) | < (cid:15) for all g ∈ G and x, y ∈ Ω . Theorem 2.
Suppose that the PE condition in Definition 3 holds and the family of functions U ( ¯ S ) is uniformlyequi-continuous. Then the PE condition in Definition 3 (PE.2) implies the one in Definition 2 (PE.1).Proof. Suppose the condition in Definition 3 holds. For each t ≥ T and g ∈ H Ω with (cid:107) g (cid:107) H Ω = 1 , we have (cid:90) t +∆ t (cid:16) E ∗ x ( τ ) E x ( τ ) g, g (cid:17) H Ω dτ = (cid:90) t +∆ t g ( x ( τ )) dτ ≥ γ. By the mean value theorem, for each g there exist ξ ∈ [ t, t + ∆] such that (cid:82) t +∆ t g ( x ( τ )) dτ = g ( x ( ξ )) ∆ . Thus wehave g ( x ( ξ )) ∆ ≥ γ ⇒ | g ( x ( ξ )) | ≥ (cid:112) γ/ ∆ . (12)For (cid:15) = (cid:112) γ/ ∆ , since g ∈ U ( ¯ S ) , which is a family of uniformly equicontinuous functions, there exist δ = δ ( (cid:15) ) > such that | s − ξ | < δ implies | g ( x ( s )) − g ( x ( ξ )) | < (cid:15) = (cid:112) γ/ ∆ for all g ( x ( · )) . In other words, for all s ∈ [ ξ − δ, ξ + δ ] ,we have | g ( x ( s )) − g ( x ( ξ )) | < (cid:112) γ/ ∆ , which implies that | g ( x ( s )) | > | g ( x ( ξ )) | − (cid:112) γ/ ∆ ≥ (cid:112) γ/ ∆ . This implies that g ( x ( s )) does not change its sign in the interval [ s, s + δ ] . Therefore, we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s + δs E x ( τ ) gdτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ (cid:112) γ/ ∆ > . The following lemma is one intuitive way that the uniform equicontinuity condition can be achieved. It relies on thefact that a uniformly bounded derivative can be used to show a function is Lipschitz continuous [24].
Lemma 1.
Let g ∈ ¯ S , that is, g ∈ H Ω with (cid:107) g (cid:107) H Ω = 1 . The family of functions U ( ¯ S ) is defined as stated above.Suppose there is a constant L > such that (cid:107) ∂g ( x ) /∂x (cid:107) ≤ L for all x ∈ Ω and g ∈ ¯ S . Then U ( ¯ S ) is uniformlyequicontinuous.Proof. The norm of evaluation operator (cid:107)E x (cid:107) op is uniformly bounded due to the uniform embedding, which thenguarantees from Eq. 6 that (cid:107) ˙ x (cid:107) ≤ c for all t ≥ where c is a positive constant. Then the lemma is an immediateconsequence of Lemma 3.1 in [24].It is clear that the condition PE.2 is stronger than condition PE.1. In the following theorem, we prove that the DefinitionPE.1 is a sufficient condition for UAS of the error system in Eq. 5. Although PE.2 is more specific, its statement seemsbetter suited for interpretation from the geometric perspective. One result established in [20] is that PE.2 relates theindexing set Ω , the RKH space H Ω , and the positive orbit Γ + ( x ) of system in Eq. 6 in an intuitive manner. That is,the positive orbit Γ + ( x ) persistently excites H Ω implies the indexing set Ω is a subset of the ω -limit set of Γ + ( x ) .Readers are referred to [20] for a detailed discussion. From the proof of the following theorem, one finds that theconvergence of function estimates ˜ f ( t ) and the state estimates ˜ x ( t ) occurs in terms of the same small constant (cid:15) → .This may provide inspiration for the design of practical adaptive laws. Theorem 3.
Assume that the trajectory x : t (cid:55)→ x ( t ) persistently excites the RKH space H Ω in the sense of Definition2 (PE.1). Then the estimation error system in Eq. 5 is uniformly asymptotically stable at the origin. In particular, wehave lim t →∞ (cid:107) ˜ x ( t ) (cid:107) = 0 , and lim t →∞ (cid:107) ˜ f ( t ) (cid:107) H Ω = 0 . PREPRINT - F
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Proof.
Without loss of generality, we assume that µ = 1 in Eq. 5. First, we prove lim t →∞ (cid:107) ˜ x ( t ) (cid:107) = 0 . Consider thecandidate Lyapunov function V ( t ) = (˜ x ( t ) , P ˜ x ( t )) + ( ˜ f ( t ) , ˜ f ( t )) H Ω . Clearly, V ( x ) is bounded below by zero. We take the time derivative of V ( t ) along any trajectory of Eq. 5, then applythe Lyapunov equation: ˙ V ( t ) = (cid:0) ˜ x ( t ) , ( A T P + P A )˜ x ( t ) (cid:1) + 2 (cid:0) B E x ( t ) ˜ f ( t ) , P ˜ x ( t ) (cid:1) + 2 (cid:0) ˜ f ( t ) , − ( B E x ( t ) ) ∗ P x (cid:1) H Ω , = − (cid:0) ˜ x ( t ) , Q ˜ x ( t ) (cid:1) + 2 (cid:0) B E x ( t ) ˜ f ( t ) , P ˜ x ( t ) (cid:1) − (cid:0) B E x ( t ) ˜ f ( t ) , P x (cid:1) , = − (cid:0) ˜ x ( t ) , Q ˜ x ( t ) (cid:1) . Since Q is positive definite, ˙ V ( t ) is less than zero, which implies V ( t ) is nonincreasing. That is, for all t ≥ t , V ( t ) ≤ V ( t ) . We conclude that Eq. 5 is stable. By integrating ˙ V ( t ) , we have the following equation, (cid:90) tt (cid:0) ˜ x ( τ ) , Q ˜ x ( τ ) (cid:1) dτ = V ( t ) − V ( t ) , which implies lim t →∞ (cid:90) tt (cid:0) ˜ x ( τ ) , Q ˜ x ( τ ) (cid:1) dτ = (cid:90) ∞ t (cid:0) ˜ x ( τ ) , Q ˜ x ( τ ) (cid:1) dτ < ∞ . Finiteness of the limit is guaranteed since V ( t ) is nonincreasing and bounded below. The integrand (cid:0) ˜ x ( t ) , Q ˜ x ( t ) (cid:1) isuniformly continuous with respect to time t . By an extension of Barbalat’s lemma for Banach spaces [24, 25], it canbe deduced that lim t →∞ (cid:0) ˜ x ( t ) , Q ˜ x ( t ) (cid:1) = 0 , which then implies lim t →∞ (cid:107) ˜ x ( t ) (cid:107) = 0 . (13)To see why this is true, so far we have established that ˜ x ∈ L ([0 , ∞ ) , R d ) ∩ L ∞ ([0 , ∞ ) , R d ) , and ˜ f ∈L ([0 , ∞ ) , H Ω ) ∩ L ∞ ([0 , ∞ ) , H Ω ) [19]. From Eq. 5, (cid:107) ˙˜ x ( t ) (cid:107) can be bounded in terms of (cid:107) ˜ x ( t ) (cid:107) and (cid:107) ˜ f ( t ) (cid:107) H Ω . Thisstep relies on the fact that the Hilbert space is uniformly embedded in the continuous functions, so (cid:107)E x (cid:107) op ≤ ¯ k as statedin Section 2. Since ˜ x ( · ) and ˜ f ( · ) are both L ∞ mappings, (cid:107) ˜ x ( t ) (cid:107) and (cid:107) ˜ f ( t ) (cid:107) H Ω can be bounded by some constants.Thus (cid:107) ˙˜ x ( t ) (cid:107) is uniformly bounded, which further implies that ddt (cid:0) ˜ x ( t ) , Q ˜ x ( t ) (cid:1) is bounded uniformly. Therefore, theinner product is Lipschitz continuous with respect to t , which implies it is uniformly continuous. Hence, the resultsof [25] apply.According to Eq. 13, for all (cid:15) > , there exists T such that for all t ≥ T , (cid:107) ˜ x ( t ) (cid:107) < (cid:15) . Now we consider the PEcondition. Let g = ˜ f ( T ) / (cid:107) ˜ f ( T ) (cid:107) H Ω be the unit-norm function in Eq. 3. If PE condition in Definition 2 is satisfied,there exists s ∈ [ T, T + ∆] such that (cid:12)(cid:12)(cid:12)(cid:82) s + δs E x ( τ ) gdτ (cid:12)(cid:12)(cid:12) ≥ γ > . Consider the error in state ˜ x ( s + δ ) . It can be boundedby integrating the state equation in Eq. 5. (cid:107) ˜ x ( s + δ ) (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ x ( s ) + (cid:90) s + δs A ˜ x ( τ ) + B E x ( τ ) ˜ f ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) s + δs B E x ( τ ) ˜ f ( T ) dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:124) (cid:123)(cid:122) (cid:125) term 1 − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˜ x ( s ) + (cid:90) s + δs A ˜ x ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:124) (cid:123)(cid:122) (cid:125) term 2 − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) s + δs B E x ( τ ) ( ˜ f ( τ ) − ˜ f ( T )) dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:124) (cid:123)(cid:122) (cid:125) term 3 . (14)We will bound all the three terms in Eq. 14. As mentioned in Section 1, the kernel function K X ( · , · ) is assumed to beselected such that (cid:107)E x (cid:107) op is uniformly bounded by ¯ k for all x ∈ R d . In term 1, note that ˜ f ( T ) = g (cid:107) ˜ f ( T ) (cid:107) H Ω , and6 PREPRINT - F
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20, 2020 (cid:107) ˜ f ( T ) (cid:107) H Ω is a constant. The coefficient matrix B ∈ R d × is in fact a d -dimensional vector, so (cid:107) B (cid:107) op = (cid:107) B ∗ (cid:107) op = (cid:107) B (cid:107) . In term 2, note that (cid:107) ˜ x ( t ) (cid:107) < (cid:15) for all t ≥ T . Then we haveterm 1 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s + δs E x ( τ ) gdτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) B (cid:107)(cid:107) ˜ f ( T ) (cid:107) H Ω , ≥ γ (cid:107) ˜ f ( T ) (cid:107) H Ω . (15)term 2 ≤ (cid:107) ˜ x ( s ) (cid:107) + (cid:90) s + δs (cid:107) A (cid:107) op (cid:107) ˜ x ( τ ) (cid:107) dτ, ≤ (cid:15) + (cid:107) A (cid:107) op δ(cid:15). (16)For term 3, we first derive a bound on ˜ f ( τ ) − ˜ f ( T ) , which can be obtained by integrating Eq. 5. (cid:107) ˜ f ( τ ) − ˜ f ( T ) (cid:107) H Ω = (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) τT ( B E x ( ξ ) ) ∗ P ˜ x ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) H Ω , ≤ (cid:90) τT (cid:107) B ∗ (cid:107) op (cid:107)E ∗ x ( ξ ) (cid:107) op (cid:107) P (cid:107) op (cid:107) ˜ x ( ξ ) (cid:107) dξ, ≤ ¯ k(cid:15) ( τ − T ) (cid:107) B (cid:107)(cid:107) P (cid:107) op . If we let c = (cid:107) B (cid:107)(cid:107) P (cid:107) op , then (cid:107) ˜ f ( τ ) − ˜ f ( T ) (cid:107) H Ω ≤ c ¯ k(cid:15) ( τ − T ) . In term 3, note that T ≤ s ≤ T + ∆ . This meansterm 3 ≤ (cid:90) s + δs (cid:107) B (cid:107)(cid:107)E x ( τ ) (cid:107) op (cid:107) ˜ f ( τ ) − ˜ f ( T )) (cid:107) H Ω dτ, ≤ ¯ k (cid:107) B (cid:107) (cid:90) s + δs c ¯ k(cid:15) ( τ − T ) dτ, ≤ c ¯ k (cid:15) (cid:107) B (cid:107) (cid:0) δ + ( s − T ) δ (cid:1) , ≤ c ¯ k (cid:15) (cid:107) B (cid:107) (cid:0) δ + ∆ δ (cid:1) . (17)Let c = (cid:107) B (cid:107) ¯ k (cid:0) δ + ∆ δ (cid:1) . Then term 3 ≤ c c (cid:15) . Substituting Eq. 15-17 into Eq. 14 gives a lower bound of ˜ x ( s + δ ) , (cid:107) ˜ x ( s + δ ) (cid:107) ≥ γ (cid:107) ˜ f ( T ) (cid:107) H Ω − (1 + (cid:107) A (cid:107) op δ ) (cid:15) − c c (cid:15). (18)On the other hand, since s + δ ≥ T , by assumption (cid:107) ˜ x ( s + δ ) (cid:107) < (cid:15). (19)Combining Eq. 18 and Eq. 19, and Eq. 20 gives an upper bound on (cid:107) ˜ f ( T ) (cid:107) H Ω . γ (cid:107) ˜ f ( T ) (cid:107) H Ω − (1 + (cid:107) A (cid:107) op δ ) (cid:15) − c c (cid:15) < (cid:15) ⇒ (cid:107) ˜ f ( T ) (cid:107) H Ω < (cid:15)γ (cid:0) (2 + (cid:107) A (cid:107) op δ ) + c c (cid:1) . (20)Now we have shown that ˜ f ( T ) is O ( (cid:15) ) for some T that depends on (cid:15) . However, it follows from this that ˜ f ( T (cid:48) ) is O ( (cid:15) ) for all T (cid:48) ≥ T . To see why this is so, choose any T (cid:48) > T . It is still true that (cid:107) ˜ x ( t ) (cid:107) < (cid:15) for all τ ≥ T (cid:48) > T . We canrepeat all of the steps above for τ ≥ T (cid:48) to conclude that (cid:107) ˜ f ( T (cid:48) ) (cid:107) H Ω is O ( (cid:15) ) . From this we eventually conclude that lim t →∞ (cid:107) ˜ f ( t ) (cid:107) H Ω = 0 . Therefore, the system in Eq. 5 is uniformly asymptotically stable. This paper studies the DPS defined by the error Equations 5, which are infinite-dimensional. Practical implementationrequires finite-dimensional approximations, a careful treatment of which exceeds the limits of this paper. See [1, 2]for some preliminary discussions of the theory of approximations. In this section, we study the qualitative behaviorof finite-dimensional approximations, since these are suggestive of the limiting guarantees of this paper. In particular,the analysis of the RKH space embedding method gives additional insights that have no counterpart in the usualfinite-dimensional framework. 7
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20, 2020In particular, the results of this paper can be combined with those in [20]. Reference [20] shows that, with a judiciouschoice of the kernel K X , a persistently excited index set is contained in the positive limit set of the trajectory starting at x of original system. This suggests that that one logical choice of a reasonable finite-dimensional approximation canbe constructed using the bases { K X,z j } nj =1 located at the centers Ω n := { z j } nj =1 , which constitute a good sampling ofthe positive limit set ω + ( x ) of the orbit Γ + ( x ) := (cid:83) τ ≥ x ( τ ) . In this way, we seek estimates that converge in H Ω n ,that is, they converge over the indexing set Ω n ⊆ Ω ≡ ω + ( x ) . We specifically employ the extrinsic methodologysummarized in Corollary 11 of [20] to a system that has the positive limit set ω + ( x ) for the initial condition x ∈ R d = X . The positive limit set ω + ( x ) is unknown, but we assume that the trajectory x : t (cid:55)→ x ( t ) evolves on anunknown smooth, compact, connected m -dimensional regularly embedded submanifold M ≡ ω + ( x ) ⊆ R d . Wechoose the r -th order Sobolev-Matern kernel K r on R d for some r > d/ to define the RKH space H X over X ≡ R d .This is a classical choice of kernel over R d , and it has a closed form expression K rX ( x, y ) = K ( (cid:107) x − y (cid:107) )= K ( ξ ) := 2 − ( r − d/ Γ( r − d/ ξ r − d/ B r − d/ ( ξ ) , (21)with B r − d/ the Bessel function of order r − d/ . Even though we do not know the form of the unknown manifold M ⊆ R d , it is possible to define the subspace H M ⊆ H X : the kernel on H M is defined in terms of the kernel thatdefines H X . Calculations are carried out using the kernel K rX that is defined on X × X , with arguments contained in M × M ⊆ X × X . We assume that the trajectory x : t (cid:55)→ x ( t ) ∈ M is persistently exciting in H M . By Corollary 11of [20], we conclude that the PE condition implies M ⊆ ω + ( x ) . In other words, PE implies that the neighborhoodof points on the entire manifold M are visited infinitely often (and the time of occupation in such neighborhoods arebounded below in a certain sense).Suppose that by using the family { K X,z i } ni =1 with the choice of centers Ω n = { z j } nj =1 and the subspace H Ω n ⊆ H M ⊆ H X , finite dimensional approximations denoted by (ˆ x n ( t ) , ˆ f n ( t )) ∈ R d × H Ω n are generated from thefollowing equations ˙ˆ x n ( t ) = A ˆ x n ( t ) + B E x ( t ) Π ∗ n ˆ f n ( t ) , ˙ˆ f n ( t ) = µ Π n (cid:0) B E x ( t ) (cid:1) ∗ P ( x ( t ) − ˆ x n ( t )) . (22)In this equation, Π n is the H X -orthogonal projection operator that maps H X onto H Ω n . For a representation, we write ˆ f n ( t ) := n (cid:88) j =1 α n,j ( t ) K X,z j = a Tn ( t ) K X, Z , (23)where a n = [ α n, , ..., α n,n ] T is the vector of time-varying coefficients and K X, Z = [ K n,z , ..., K n,z n ] T is the vectorof basis functions. Then the equations of finite-dimensional approximation in Eq. 22 take the form ˙ˆ x n = A ˆ x n + B a Tn K X, Z ( x ) , ˙ a n = µ K − K X, Z ( x ) B T P ( x − ˆ x n ) , with K the Grammian operator associated with the samples { z j } nj =1 .At this point, carefully note that the finite dimensional equations can be understood as approximations of the infinitedimensional estimator in Eq. 11. The assumption of PE on the manifold M guarantees that as t → ∞ ˆ x ( t ) → x ( t ) in R d , and ˆ f ( t ) → f in H M , (24)which describes the convergence of infinite dimensional solutions of the DPS in Eq. 11. One elementary analysis thatshows ˆ x n ( t ) → ˆ x ( t ) in R d , and ˆ f n ( t ) → ˆ f ( t ) in H M , (25)as n → ∞ uniformly on each compact set of the form [0 , T ] is given in [2].Let P Ω be the projection operator from the RKH space H X to the RKH subspace H Ω . Carefully note that since P Ω f ∈ H Ω , and if { K z j } ∞ j =1 is assumed to be a basis for H Ω , we have P Ω f = ∞ (cid:88) j =1 α ∗ j K z j , PREPRINT - F
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Figure 1: Phase portrait of the original system -1.5 -1 -0.5 0 0.5 1 1.5-1-0.500.51
Positive limit setSamples
Figure 2: Samples along the positive limit setwhere { α ∗ j } ∞ j =1 are the unique true coefficients for P Ω f . Neither of the convergence results in Eq. 24 nor 25 directlyaddresses the issue of whether the solutions of Eq. 22 for some fixed n converge in the sense that ˆ α n,i ( t ) → α ∗ i , as t → ∞ . In fact, this is generally not the case, a fact that has been studied at length for adaptive estimation in finitedimensional settings. In fact, a conventional analysis of the finite dimensional estimator in Eq. 22 for a fixed n yieldultimate bounds on such error as expected.To illustrate the convergence of RKH space embedding method and the behavior of finite-dimensional approximation,an example of an undamped, nonlinear, piezoelectric oscillator is studied [26, 27]. The governing equations of theoscillator, after employing a single bending mode approximation, have the form (cid:20) ˙ x ˙ x (cid:21) = (cid:20) x − km x − k n, m x − k n, m x (cid:21) , (26)where k is the electromechanical stiffness, m is the mass, and k n, , k n, are the higher order electromechanical stiffnesscoefficients. In the governing equations above, we assume all the linear terms are known, and the nonlinear term f ( x ) = − k n, m x − k n, m x is to be identified. In this case, the Sobolev-Matern kernel and the associated RKH spaceis applied, which is uniformly embedded in the space of continuous functions [22]. From the conclusion of [20], apersistently excited set Ω must be contained in the positive limit set of the system ω + ( x ) , which we approximate withcarefully chosen samples Ω n = { z j } nj =1 .Fig. 1 shows the typical positive limit sets of this system. The limit sets form limit cycles around the equilibriumat the origin, which is prototypical for such conservative electromechanical oscillators. For this example, the initialcondition is selected such that the orbit Γ + ( x ) forms a limit cycle as shown in Fig.2. The samples { z j } serving asthe centers of basis functions { K X,z j } are chosen along the positive limit set.When the approximation of infinite dimensional adaptive estimator based on the RKH space embedding tech-nique is implemented for this problem, estimates of the unknown nonlinear function f ( x ) are obtained in H Ω n = span { K X,z j } nj =1 . Figures 2-5 depict the results of simulation for the specific case when the number of centers n = 120 and order of Sobolev-Matern kernel r = 3 . The parameters in the state equation 26 are m = 0 . k = 6 . , k n, = − . , and k n, = 3 . . Fig. 3 shows the time history of four typical coefficients α k,n ( t ) comparedto the corresponding true value calculated from projecting the true function f onto the RKH space H Ω n . There exista slight difference between the convergent coefficient with the true value as we would expect for the case when n isfixed: the coefficient estimates for fixed n satisfy ultimate boundedness conditions. Fig. 4 shows the error betweenthe actual function and function estimate over the state space. Qualitatively, convergence of the function estimate isexpected from the theory in this paper as n → ∞ over the positive limit set of the particular trajectory. Fig. 5 showsthe contour of the function error along with the positive limit set. Both the figures suggest that the function estimate in H Ω converges to the actual function over the indexing set, which when persistently excited is a subset of the positivelimit set. 9 PREPRINT - F
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Figure 3: Convergence of parametersFigure 4: Error in function estimate Figure 5: Error contour
In this paper, new PE conditions are defined for the adaptive estimator based on RKH space embedding. The twodefinitions of PE are both applied to the family of functions U ( ¯ S ) generated by the Koopman operator U acting onthe unit sphere ¯ S of RKH space. The paper establishes sufficient conditions of the equivalence of the two definitions.Condition PE.1 naturally implies PE.2, and PE.2 implies PE.1 when the family of functions U ( ¯ S ) is uniformlyequicontinuous. The paper then proves that PE.1 is a sufficient condition for the UAS of the error equations thatarise in the RKH space embedding framework. This constitutes a sufficient condition for the convergence of functionestimates. The proof relies in a fundamental way on the assumption that the RKH space is uniformly embedded inthe space of continuous functions. A numerical example is given to show qualitatively the convergence behavior ofthe RKH space embedding method. The numerical example shows that, roughly speaking, convergence is establishedover sets in which trajectories are eventually concentrated. Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could haveappeared to influence the work reported in this paper.
Acknowledgement
Andrew J. Kurdila would like to acknowledge the support of the Army Research Office under the award
DistributedConsensus Learning for Geometric and Abstract Surfaces , ARO Grant W911NF-13-1-0407.10
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