© Hindawi Publishing Corp. STOCHASTIC LINEARIZATION OF NONLINEAR POINT DISSIPATIVE SYSTEMS

Stochastic linearization produces a linear system with the same 
covariance kernel as the original nonlinear system. The method 
passes from factorization of finite-dimensional covariance 
kernels through convergence results to the final input/output 
operator representation of the linear system.

by a Wiener process the linearization yields a process with covariance R, that is, one which is indistinguishable from the original process. A natural way to proceed, building on our experience with linear systems, is to seek a factorization of R in terms of limits of the Cholesky factors [11] of discrete approximations R t of R. The next section provides the background for a reasonable notion of convergence required to make this approach feasible.

Background.
The RKH space approach to linear system modeling [24] provides discrete nonparametric model representations in terms of factorizations of the discrete covariances of the input and output processes for the system. Thus the representations are in terms of data, avoiding the dimension or order problem associated with parametric approaches. The RKH space method eliminates decisions about the form of the model, such as the number of terms to be included, which require a high level logic.
Let R d denote the space of d-tuples of real numbers with the usual inner product ·, · and norm |·|. Let G denote the class of continuous functions f from [0, ∞) into R d such that f (0) = 0. We define a family of pseudonorms {N x , x ≥ 0} on G by N x (f ) = sup z≤x |f (z)| for each f in G and x ≥ 0. Two classes of linear operators defined initially on G are introduced. These operators are used to describe the systems of interest. Let Ꮾ denote the set of linear operators on G to which B belongs only when (1) [Bf ](0) = 0 for each f in G, (2) for each T > 0 there is a number b such that for each f in G and 0 ≤ s ≤ t ≤ T . Let Ꮽ denote the set of linear operators on G to which A belongs only in case A − I is in Ꮾ. If B is in Ꮾ, then I − B is an invertible operator from G onto G and (I − B) −1 is in Ꮽ. If A is in Ꮽ, then A is an invertible operator from G onto G and I − A −1 is in Ꮾ.
The classes of operators Ꮽ and Ꮾ can serve to describe linear systems, but what is their relation with nonlinear systems? In particular, how do we relate these operators to the available observation process {Y (t), 0 ≤ t}? The next three theorems provide answers to these questions.
In general, the covariance function R of a stochastic process {Y (t), 0 ≤ t} defined by where E denotes the expectation operator, is nonnegative, that is, n p,q=0 R t p ,t q x q ,x p ≥ 0 (2.3) for each sequence {t p } n 0 in [0, ∞) and each sequence {x p } n 0 in R d [18,21]. From this point we will reserve R to denote the covariance function of some observation process {Y (t), 0 ≤ t}. In order to see the structure of the problem, modeling the process {Y (t), 0 ≤ t} from partial information, we begin by assuming complete information, that is, R(s, t) is known exactly for 0 ≤ s, t.
Theorem 2.1 [1,18]. The theorem only asserts the existence of the RKH space {G R ,Q R } with kernel R. When our observation process is of the form Y = AW , where A is an invertible operator in Ꮽ∪Ꮾ, W is the standard d-dimensional Wiener process, and R(s, t) = EY (s)Y (t) T , we can obtain more, namely, an explicit representation of {G R ,Q R }. (See [21] for an alternative representation.) In order to accomplish this we will introduce another RKH space, this time associated with the input process. Let k denote an increasing scalar function with k(0) = 0. Let G K denote the subspace of functions in G which are Hellinger integrable with respect to k, . The least such number M is denoted by ∞ 0 |df | 2 /dk. Finally, let Q K denote the inner product for G K defined by Q K (f , g) = ∞ 0 df dg/dk, the limit through refinement of sums t df dg/dk. We will use the short notation dk(x, y) for the difference k(y) − k(x).
The space {G K ,Q K } is an RKH space with kernel given by K(s, t) = k(min(s, t))I, where I is the d × d identity matrix [18]. Elements of Ꮾ map G into G K and elements of Ꮽ map G K onto G K . From now on, we will be concerned primarily with the restrictions of elements of Ꮽ and Ꮾ to G K .
Let L denote the function from Ꮽ∪Ꮾ into the space of d×d matrix-valued functions In general, the variance of a scalar input process is an increasing function k. When the input process is the standard scalar Wiener process, we can use the special case k(t) = t. Note that, when d = 1, K is the covariance function of the Wiener process.
Theorem 2.2 [24]. Suppose R is the covariance of the process Let LA denote the matrix representation of the assumed operator A. We can write for 0 ≤ s, t, that is, we can use A to obtain a representation of R.
The following example illustrates this last observation. Note that a state space formulation of the model would have to be infinite dimensional; however, the input and output processes are scalar. Example 2.3. Suppose W is the standard scalar Wiener process and which agrees with the direct calculation.
Theorem 2.4 [24]. Given that A in Ꮽ ∪ Ꮾ is invertible and R is represented in terms of A (see (2.5)), the RKH space with kernel R is given by G R = G K and Q R (f , g) = Q K (f , (AA * ) −1 g), for each f and g in G R .
For our problem, that is, R associated with a general observation process {Y (t), 0 ≤ t}, the underlying system might be nonlinear and the linear operator A assumed in Theorems 2.2 and 2.4 unavailable. We seek a linearization in Ꮽ ∪ Ꮾ of the underlying system, which will play the role of A, through a factorization of the covariance function R. Since R can be factored in many different ways, we will have to justify our choice in the end. The method returns an element of Ꮽ ∪ Ꮾ, which we will denote by A, with matrix representation LA.
Finite-dimensional approximations. For calculations, the matrix representations of the operators have to be projected down to finite-dimensional spaces. A more detailed explication appears in [24].
A class of polygonal functions, the K-polygonal functions, arises naturally in RKH spaces and can be used along with projection methods to develop finite-dimensional approximations to system operators. Any function f on [0, ∞) of the form where t = {t p } n 0 is an increasing sequence in [0, ∞) and {x p } n 0 is a sequence in R d , is called a K-polygonal function. The subspace of all K-polygonal functions based on a fixed increasing finite sequence t in [0, ∞) is a closed linear subspace of G K . We let Π t denote the orthogonal projection of G K onto this subspace. Also, let P t denote the projection on G K defined by (2.11) for each f in G K and 0 ≤ s, t.
Theorem 2.5 [1,18]. For each positive number T , the union of the finite-dimensional For convenience and clarity we restrict our attention in the rest of the paper to the case d = 1 (observations and inputs are both scalar). This, of course, does not restrict the underlying dynamical system to be one-dimensional. (See Example 2.3.) Furthermore, we assume k is an increasing function on [0, ∞) with k(0) = 0. Recall that K(s, t) = k(min(s, t)).
is an increasing sequence in [0, ∞). We will use the same notation for various functions without comment. For instance, R t (p, q) = R(t p−1 ,t q−1 ) for 1 ≤ p, q ≤ n + 1. With this understanding, for s in [0, ∞), and, for p = 1, 2,...,n+ 1, Note that the finite-dimensional approximations converge [24] but covergence is not a sequential convergence but rather a net convergence, that is, through refinements of partitions.

Indication of proof. Let {r
Let s be a partition of [0,T ] refining r such that if {t p } n 0 refines s, then Cholesky factorizations. The upper Cholesky factor of a nonnegative symmetric matrix S is an upper triangular matrix S u with nonnegative diagonals such that We can tie R to {G K ,Q K } without supposing the existence of a continuous linear transformation A by assuming in the rest of the paper that for each positive number With this assumption R is the matrix representation of a nonnegative Hermitian member of Ꮽ ∪ Ꮾ [24] which we will denote by H in the rest of the paper. What happens when H = AA * and A is time-invariant?
Theorem 2.7 [24]. Suppose A is a time-invariant operator in Ꮽ ∪ Ꮾ. There is a continuous function M on [0, ∞) such that for each positive number T and partition t of Therefore, assuming equally spaced partition points, the diagonal elements of (K u t ) −T LA t all have the same sign. Hence ±(K u t ) −T LA t is an upper Cholesky factor. Further, In our problem the underlying system is nonlinear and we do not start with a factorization of H. Furthermore, the factorization we seek is not necessarily in terms of time-invariant linear operators. Even so, this result suggests that we seek a factorization of R as a limit in some sense of the upper Cholesky factors R u t of R t . We want convergence in terms of finite-dimensional linear operators associated with the upper Cholesky factors R u t in the following way. For each positive number T and partition t of [0,T ], let A t denote the linear trans- Indication of proof.
Hence the result.
Summary of standing assumptions and notation for the rest of the pa- (2) Let R denote the covariance function for a scalar observation process {Y (t), 0 ≤ t}. R is nonnegative (see (2.3)). Assume that R satisfies inequalities (2.21) and is the matrix representation of nonnegative Hermitian operator H in Ꮽ ∪ Ꮾ.
(3) P t and Π t are projections given by (2.11) and (2.12), respectively. (4) K t and R t are discretizations of K and R, respectively. For instance, 3. Main results. The objective is linearization of an unknown underlying nonlinear system generating the observation process {Y (t), 0 ≤ t} from data which we interpret as {R t }. The quality of the linearization should be measurable in terms of the sampling rates and statistics of the observations. The first part of this objective can be achieved by establishing convergence in some reasonable sense of the finite-dimensional operators {A t }. Conditions which imply convergence should be restricted to conditions on the data {R t } as opposed to conditions on the underlying system.
Given that A is in Ꮽ ∪ Ꮾ, T is a positive number, and 0 ≤ x ≤ T , we will say that the is Cauchy.
(2) If, for each positive number T and 0 ≤ x ≤ T , the net (3.2) is Cauchy, then there is a linear operator A in Ꮽ ∪ Ꮾ such that for each positive number T and 0 ≤ x ≤ T , the net (3.1) has limit 0.

Indication of proof.
Assume that the hypothesis of (1) holds, c and T are positive numbers, and 0 ≤ x ≤ T . There is a partition r of [0,T ] refining {0,x,T } such that if s refines r and t refines s, then N K (Π t A * K(·,x) − Π s A * K(·, x)) < c/3. In addition, we may assume that N K ((A s ) * K(·,x)− Π s A * K(·, x)) < c/3. Hence that is, the net (3.2) is Cauchy. Assume the hypothesis of (2) holds. For each x ≥ 0 choose T > x and let LA(·,x) denote the limit of (3.2). Note that LA(·,x) is in G K and if x = t q−1 < u, then LA(·,x)) for each f in G K and x ≥ 0. Further, note that that is, the linear operator defined in terms of LA is in Ꮽ ∪ Ꮾ [24]. Further, R is the matrix representation of AA * . For each pair of positive numbers c and T and 0 ≤ x)) < c. Therefore (3.1) has limit 0.
Thus showing that {(A t ) * K(·,x)} is Cauchy is more basic since we do not need to assume a factorization AA * of the operator with matrix representation R. Our search then is for conditions on the finite-dimensional covariances {R t } which allow us to conclude that {(A t ) * K(·,x)} is Cauchy. (3.8) And hence (2) follows.

Assume that (2) holds, T is a positive number, and 0 ≤ x ≤ T . If s is a partition of [0,T ] refining {0,x,T } and t refines s, then
from which (1) follows.

Indication of proof. The net
is nonincreasing but bounded below. Let L(x) denote the limit. Then (3.20) Hence for each pair of positive numbers c and T and 0 ≤ x ≤ T , there is a partition r of [0,T ] refining {0,x,T } such that if s refines r and t refines s, then The conclusion follows from Theorem 3.2.  R(a, s)R(b, t) = R(b, s)R(a, t) for 0 ≤ a, b ≤ s, t and for for i = 2, 3,...,n + 1, then for each positive number T and 0 ≤ x ≤ T the net (3.2) is Cauchy.
for k > q.
Proof of Lemma 3.8. We will proceed by induction. Assume for convenience that (2,2).

is a partition of [0,T ] refining {0,x,T }, and t is a refinement of s (s = t[u]), then
Proof of Theorem 3.7. The theorem follows from Lemma 3.10 and Theorem 3.5. (3.40) We will show that R t is positive definite (see (2.3)) by developing a Cholesky factorization of R t . We accomplish this by first defining R u t inductively, a row at a time. Assuming R u t (1, 1) = R t (1, 1) 1/2 = 1, R u t (1,k) = R t (1, k)/R u t (1, 1) = e t(k) for k > 0, and for q = 2, 3,..., for k > q. We conclude that (R u t ) T R u t = R t by reading (

Simulations.
Starting with such a nice model for the covariance kernel in Theorem 3.7, one might ask if stochastic linearization contributes anything. That is, if the only information available for the discrete process Y t is the mean m t and the covariance kernel R t , then everything we can know of the approximating normal process determined by the first two moments is captured by the discrete distribution function obtained as follows. The finite-dimensional density function is given by The finite-dimensional distribution function is then given by However, a problem remains with the slow convergence of numerical evaluation of the iterated integral, especially when n is large, bigger than eight. Many methods rely on some kind of simulation to speed convergence [20].  Basic discussion. We propose another simulation methodology based on stochastic linearization. Let X t be an n-dimensional normally distributed random row vector. Note that EX t = 0 and EX T t X t = I n . If K u t is the upper Cholesky factor of K t , then W t (p) = [X t K u t ](p). To see this, note that EX t K u t = 0 and For a given finite-dimensional stochastic linearization A t with matrix representation Operator norm. In the system monitoring problem, there is a need to measure the distance between two nonlinear systems. One possiblity is to measure the distance between the systems' discrete linearizations rather than between the systems directly. If the systems have covariance kernels R 1 and R 2 , respectively, with upper Cholesky factorizations R u 1 and R u 2 , then an approximate operator norm for the difference of The distances between the systems represented by the covariance kernels given in Figure 3.1 are given in Table 4.1.
In the absence of an absolute scale, we can only conclude that the third system is further from the first than it is from the second. This certainly fits our intuition.

Significance of work.
We need a more robust condition on the finite-dimensional covariances {R t } implying convergence of the finite-dimensional operators {A t }. Theorems 3.5 and 3.7 are too delicate for application to estimates of R t . That is, the theorems assume we know R t exactly or, in other words, we have an infinite amount of data at our disposal.
One can easily move from statistics of observations of inputs and outputs to confidence intervals and other measures of the accuracy of the estimates of R t . We need to extend these possibilities to results on the quality of the estimates of A t .
Much of this material can be extended immediately to vector processes. Examples of vector processes have been explored; for instance, a Lorenz system [6,17,26] with a onedimensional noise input and a two-dimensional observation. The notion of convergence introduced for scalar inputs and outputs extends to the vector case. However, until the scalar case is settled, the condition which implies convergence for the vector case is hard to visualize.
Again, the dimension of the state space for the underlying system does not enter. The first example has an infinite-dimensional state space. So the method of linearization under investigation, if we can carry out our program, will apply to some systems governed by nonlinear partial differential equations as well.
The statistical linearization as presented in [25] is based on an assumption of the form of the underlying nonlinear system. Data enters the problem from simulations of a known nonlinear system. Applications are made to marine structures such as drilling platforms.
Application of Hilbert space ideas to system problems requires an additional time structure which can be used to guarantee the operators are realizable, that is, causal. This requirement as discussed in [8] can be satisfied in several different settings [2,10,29]. The framework of Hellinger integrable functions, associated with the covariance function of the Wiener process, has a built-in time structure. The elements of Ꮽ∪Ꮾ are immediately causal.
The starting point for the work in this investigation differs from that of [8,10,27] in that the covariance R, known only partially as R t from data, is the matrix representation of several positive definite Hermitian operators depending on choices made for the Hilbert space. Further, no assumption is made concerning the factorization of this operator. We are searching for conditions on R which yield the existence of a factorization.

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