Rational approximation of a class of infinite-dimensional systems I: Singular values of hankel operators

In order to establish the optimal rates of convergence for the infinity-norm rational approximation problem, upper and lower bounds on the singular values of a class of Hankel operators are established. These asymptotically accurate estimates are derived from results on the singular values of Hankel operators with symbol equal to the product of a rational function and an exponential function, combined with results on Hankel integral operators (in continuous time) whose kernels have certain smoothness properties.     

Optimal H∞ general distance problem with degree constraint

In this paper the optimal H_∞, general distance problem, for continuous-time systems, with a prescribed degree on the solution is studied. The approach is based on designing the Hankel singular values using an imbedding idea. The problem is first imbedded into another problem with desirable characteristics on the Hankel singular values, then the solution to the original problem is retracted via a compression. The result is applicable to both the one-block and the four-block problems. A special case is given for illustration.     

Admissible Observation Operators for the Right-Shift Semigroup

We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L ²[0,∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space. Date received: January 15, 1999. Date revised: September 24, 1999.     

A simpler formula for the singular values of a certain Hankel operator

This paper deals with the problem of computing the singular values and vectors of a Hankel operator with symbol m^*W where m ε H^∞ is arbitrary inner and W ε H^∞ is rational. A simplified version of the formula given in [6] is obtained for computing the singular values of the Hankel operator. This result is applied to the (one-block) H^∞ optimal control of SISO stable infinite dimensional plants and rational weights. Using this new formula a simple expression is derived for the H^∞ optimal controller whose structure was observed in [9].     

System reduction via truncated Hankel matrices

The problem of approximating Hankel operators of finite or infinite rank by lower-rank Hankel operators is considered. For efficiency, truncated Hankel matrices are used as the intermediate step before other existing algorithms such as theCF algorithms are applied to yield the desirable approximants. If the Hankel operator to be approximated is of finite rank, the order of approximation by truncated Hankel operators is obtained. It is also shown that when themths-number is simple, then rational symbols of the best rank-m Hankel approximants of thenth truncated Hankel matrices converge uniformly to the corresponding rational symbol of the best rank-m Hankel approximant of the original Hankel operator asn tends to infinity. Supported by SDIO/IST managed by the U.S. Army under Contract No. DAAL03-87-K-0025 and also supported by the National Science Foundation under Grant No. DMS 8602337. Supported by SDIO/IST managed by the U.S. Army under Contract No. DAAL03-87-K-0025. Supported by the National Science Foundation under Grant No. DMS 8602337.     

Rate of convergence of schmidt pairs and rational functions corresponding to best approximants of truncated hankel operators

The problem of approximating Hankel operators of infinite rank by finite-rank Hankel operators is considered. For efficiency, truncated infinite Hankel matrices ⁿ of are utilized. In this paper for any compact Hankel operator of the Wiener class, we derive the rate of l₂-convergence of the Schmidt pairs of ⁿ to the corresponding Schmidt pairs of . For a certain subclass of Hankel operators of the Wiener class, we also obtain the rate of l₁-convergence. In addition, an upper bound for the rate of uniform convergence of the rational symbols of best rank-k Hankel approximants of ⁿ to the corresponding rational symbol of the best rank-k Hankel approximant to asn is derived.Supported by SDIO/IST managed by the U.S. Army under Contract No. DAAL03-87-K-0025 and also supported by the National Science Foundation under Grant No. DMS 89-01345.     

Singular Linear Behaviors and Their AR-Representations

A linear behavior is defined essentially as a family of finite but sufficiently long trajectories that is invariant with respect to restriction operators. This may be singular in the sense that it may contain trajectories that cannot be continued to the right. An AR-model is defined essentially as a pair (D,R), where D is a nonsingular rational matrix and R is a full row rank polynomial matrix such that D ⁻¹ R is proper. A canonical one-to-one correspondence is established between linear behaviors and equivalence classes of AR-models. The result is a natural generalization of a well-known result due to Willems. Date received: April 19, 1999. Date revised: April 18, 2000.     

Model reduction of systems with zeros interlacing the poles

The problem of reducing a system with zeros interlacing the poles (ZIP) on the real axis is considered. It is proved that many model reduction methods, such as the balanced truncation, balanced residualization, suboptimal and optimal Hankel approximations, inherit the ZIP property. Properties of the Hankel singular values of ZIP systems are also listed.     

What can routh table offer in addition to stability？

Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a litde beyond , the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc.     

On the Hankel singular values of linear continuous and sampled-data multivariable systems

This paper considers the Hankel singular values of the sample-data systems obtained from continuous-time models via standard sampling techniques. It is shown that the Hankel singular values of the sampled-data system recover the Hankel singular values of the continuous-time system as the sampling period approaches zero. A state space proof of this result is given.     

An extension of A-A-K Hankel approximation theory using state-space formulation

This paper presents a state-space extension of A-A-K Hankel approximation theory. Conventional A-A-K theory can only be applied to solve minimum-degree approximation problems. Here, we extend the A-A-K results and show that it can also be generalized to minimum-norm problems for multivariable systems. State-space formula for generating enough independent Hankel singular vectors corresponding to a prescribed singular value is derived and is used to construct all the optimal Hankel approximants for a given rational function matrix.     

Hankel singular value functions from Schmidt pairs for nonlinear input–output systems

This paper presents three results in singular value analysis of Hankel operators for nonlinear input–output systems. First, the notion of a Schmidt pair is defined for a nonlinear Hankel operator. This makes it possible to define a Hankel singular value function from a purely input–output point of view and without introducing a state space setting. However, if a state space realization is known to exist then a set of sufficient conditions is given for the existence of a Schmidt pair, and the state space provides a convenient representation of the corresponding singular value function. Finally, it is shown that in a specific coordinate frame it is possible to relate this new singular value function definition to the original state space notion used to describe nonlinear balanced realizations.     

On the decay rate of Hankel singular values and related issues

This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues, like the Hankel singular values, are rapidly decaying. This fact has motivated the development of several algorithms for computing low-rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low-rank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions.     

A note on a system theoretic approach to a conjecture by Peller-Khrushchev

Based on the construction of infinite dimensional balanced realizations an alternative solution to the following inverse spectral problem is presented: Given a monotonically decreasing sequence of positive numbers (σ_n)_{n 1}, does there exist a Hankel operator whose sequence of singular values is (σ_n)_{n 1}?     

Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions

Given a stable rational transfer function G(s) and weighting function W(s), the problem of finding of MacMillan degree k so as to minimise is considered. This problem is solved for W(s) =(s-β)/(s-α) with no assumptions on the signs of α and β. This gives rise to approximations where can be accurately bounded from above and below in terms of the Hankel singular values of WG (when α, β > 0).     

Admissible factorizations of Hankel operators induce well-posed linear systems

One of the basic axioms of a well-posed linear system says that the Hankel operator of the input–output map of the system factors into the product of the input map and the output map. Here we prove the converse: every factorization of the Hankel operator of a bounded causal time-invariant map from L² to L² which satisfies a certain admissibility condition induces a stable well-posed linear system. In particular, there is a one-to-one correspondence between the set of all minimal stable well-posed realizations of a given stable causal time-invariant input–output map (or equivalently, of a given H^∞ transfer function) and all minimal stable admissible factorizations of the Hankel operator of this input–output map.     

Linear Operator Inequalities for Strongly Stable Weakly Regular Linear Systems

We consider the question of the existence of solutions to certain linear operator inequalities (Lur'e equations) for strongly stable, weakly regular linear systems with generating operators A, B, C, 0. These operator inequalities are related to the spectral factorization of an associated Popov function and to singular optimal control problems with a nonnegative definite quadratic cost functional. We split our problem into two subproblems: the existence of spectral factors of the nonnegative Popov function and the existence of a certain extended output map. Sufficient conditions for the solvability of the first problem are known and for the case that A has compact resolvent and its eigenvectors form a Riesz basis for the state space, we give an explicit solution to the second problem in terms of A, B, C and the spectral factor. The applicability of these results is demonstrated by various heat equation examples satisfying a positive-real condition. If (A, B) is approximately controllable, we obtain an alternative criterion for the existence of an extended output operator which is applicable to retarded systems. The above results have been used to design adaptive observers for positive-real infinite-dimensional systems. Date received: July 25, 1997. Date revised: February 10, 2001.     

State-space algorithm for the computation of superoptimal matrix interpolating functions

A state-space algorithm is studied which generates the (unique) superoptimal Nehari extension of a general rational matrix $. The procedure is to use a set of all-pass transformations to sequentially minimize each frequency-dependent singular value (of the interpolating function) in a dimension peeling algorithm. These all-pass transformations are determined by the maximal Schmidt pairs of a sequence of Hankel operators. The process terminates when the original problem is reduced to one of rank one; at this stage all the available degrees of freedom have been exhausted. The work is an extension of that by Young (1986) and gives a ‘concrete’ state-space implementation of his operator-theoretic arguments. In addition, bounds are given on the minimum achievable values for s₁ ^∞ (E) = sup_ωεRSi (E(jω)), i = 1, 2,..., rank (G₀), and also the McMillan degree of the final superoptimal extension. Here S_i.(.) denotes the ith singular value of a (frequency-dependent) matrix, and the numbering is taken to be in decreasing order of magnitude. The algorithm has the property that it may be stopped after minimizing s_i ^∞ (.),i = 1. 2,...,l< rank (G₀) if it continues further it is deemed ‘not worth it’ in some sense. A premature termination of the algorithm carries with it an expected saving in computational effort and a predictable reduction in the degree of the extension. A shortened version of the present work has already appeared in work by Limebeeref al, (1987).     

The Realization Problem for Hidden Markov Models

If {X _t} is a finite-state Markov process, and {Y _t} is a finite-valued output process with Y _t+1 depending (possibly probabilistically) on X _t, then the process pair is said to constitute a hidden Markov model. This paper considers the realization question: given the probabilities of all finite-length output strings, under what circumstances and how can one construct a finite-state Markov process and a state-to-output mapping which generates an output process whose finite-length strings have the given probabilities? After reviewing known results dealing with this problem involving Hankel matrices and polyhedral cones, we develop new theory on the existence and construction of the cones in question, which effectively provides a solution to the realization problem. This theory is an extension of recent theoretical developments on the positive realization problem of linear system theory. Date received: December 13, 1996. Date revised: October 9, 1998.