Showing papers in "Mathematics of Control, Signals, and Systems in 1991"
TL;DR: This paper proves general sampling theorems for functions belonging to a reproducing kernel Hilbert space (RKHS) which is also a closed subspace of a particular Sobolev space.
Abstract: In this paper we prove general sampling theorems for functions belonging to a reproducing kernel Hilbert space (RKHS) which is also a closed subspace of a particular Sobolev space. We present details of this approach as applied to the standard sampling theory and its extension to nonuniform sampling. The general theory for orthogonal sampling sequences and nonorthogonal sampling sequences is developed. Our approach includes as concrete cases many recent extensions, for example, those based on the Sturm-Liouville transforms, Jacobi transforms, Laguerre transforms, Hankel transforms, prolate spherical transforms, etc., finite-order sampling theorems, as well as new sampling theorems obtained by specific choices of the RKHS. In particular, our setting includes nonorthogonal sampling sequences based on the theory of frames. The setting and approach enable us to consider various types of errors (truncation, aliasing, jitter, and amplitude error) in the same general context.
155 citations
TL;DR: It is shown that the application of differential algebraic elimination theory leads to an effective method for deriving the equivalent representation of the state variables in nonlinear systems described by algebraic differential equations.
Abstract: For nonlinear systems described by algebraic differential equations (in terms of “state” or “latent” variables) we examine the converse to realization,elimination, which consists of deriving an externally equivalent representation not containing the state variables. The elimination in general yields not only differential equations but also differentialinequations. We show that the application of differential algebraic elimination theory (which goes back to J.F. Ritt and A. Seidenberg) leads to aneffective method for deriving the equivalent representation. Examples calculated by a computer algebra program are shown.
155 citations
TL;DR: It is shown that this rank equals the maximum number of vertex disjoint paths from the input vertices to the output vertices in the graph that can be associated to the structured system.
Abstract: In this paper structured systems are considered and the generic rank of the transfer matrix of such systems is introduced. It is shown that this rank equals the maximum number of vertex disjoint paths from the input vertices to the output vertices in the graph that can be associated to the structured system. This maximum number of disjoint paths can be calculated using techniques from combinatorics. As an application a structural version of the well-known almost disturbance decoupling problem is proposed.
85 citations
TL;DR: An elementary proof of the well-known fact that shift-invariant operators onL2[0, ∞) are represented by transfer functions which are bounded and analytic on the right open half-plane is given.
Abstract: We give an elementary proof of the well-known fact that shift-invariant operators onL
2[0, ∞) are represented by transfer functions which are bounded and analytic on the right open half-plane. We prove a generalization to Banach space-valuedL
p
-functions, where 1≤p<∞. We show that the result no longer holds forp=∞.
68 citations
TL;DR: A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study the robust stability of a linear time-invariant state space model subject to real parameter perturbations, and some explicit lower bounds on the distance are obtained.
Abstract: This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones.
49 citations
TL;DR: In this paper, the authors considered the achievable errors in infinity-norm approximation of an irrational transfer function by a rational one of given degree and showed that optimal convergence rates are achievable if the corresponding impulse response has certain smoothness properties.
Abstract: The achievable errors in infinity-norm approximation of an irrational transfer function by a rational one of given degree are considered. Error bounds are given which have particular application to delay systems, and it is shown that optimal convergence rates are achievable if the corresponding impulse response has certain smoothness properties.
43 citations
TL;DR: In this class of impulse responses calledpseudorational, it is shown that the difficulty is related to classical complex analysis, especially that of entire functions of exponential type, and the infinite-product representation makes it possible to prove that stability is indeed determined by the location of spectrum or by a modified H∞ condition.
Abstract: It is well known that for infinite-dimensional systems, exponential stability is not necessarily determined by the location of spectrum. Similarly, transfer functions in theH∞ space need not possess an exponentially stable realization. This paper addresses this problem for a class of impulse responses calledpseudorational. In this class, it is shown that the difficulty is related to classical complex analysis, especially that of entire functions of exponential type. The infinite-product representation for such entire functions makes it possible to prove that stability is indeed determined by the location of spectrum or by a modifiedH∞ condition. Examples are given to illustrate the theory.
35 citations
TL;DR: Different forms of Levinson-Durbin-type algorithms, which relate the coefficients of a continuous-time autoregressive model to the residual variances of certain regressions or their ratios, are derived.
Abstract: Different forms of Levinson-Durbin-type algorithms, which relate the coefficients of a continuous-time autoregressive model to the residual variances of certain regressions or their ratios, are derived. The algorithms provide parametrizations of the model by a finite set of positive numbers. They can be used for computing the covariance structure of the process, for testing the validity of such a structure, and for stability testing.
34 citations
TL;DR: Pontryagin’s maximum principle is derived for a class of control systems described by semilinear equations in Hilbert space and convergence properties of sequences of near-optimal controls for these systems are studied.
Abstract: We consider infinite-dimensional nonlinear programming problems which consist of minimizing a functionf
0(u) under a target set constraint. We obtain necessary conditions for minima that reduce to the Kuhn-Tucker conditions in the finite-dimensional case. Among other applications of these necessary conditions and related results, we derive Pontryagin’s maximum principle for a class of control systems described by semilinear equations in Hilbert space and study convergence properties of sequences of near-optimal controls for these systems.
32 citations
TL;DR: An algorithm to compute a minimal realization and study the feedback classification problem is described and feedback invariants related to the singularities of the input-output mapping and canonical forms are exhibited.
Abstract: We outline a geometric theory for a class of homogeneous polynomial control systems called quadratic systems. We describe an algorithm to compute a minimal realization and study the feedback classification problem. Feedback invariants are related to the singularities of the input-output mapping and canonical forms are exhibited.
25 citations
TL;DR: In the particular case of time-varying linear systems, it is shown how the structure of their solutions in the case of regular and irregular singularities affects solutions to the tracking problem.
Abstract: In this paper we continue the analysis of the problem of output tracking in the presence of singularities, whose study was begun by R. Hirschorn and J. Davis. We introduce further structure which is important in quantifying the multiplicity and smoothness of solutions to the problem. The paper is motivated by the analysis of those singular ordinary differential equations whose structure ultimately governs solutions to the singular tracking problem. In the particular case of time-varying linear systems, it is shown how the structure of their solutions in the case of regular and irregular singularities affects solutions to the tracking problem. Less specific results are also obtained in the full nonlinear case.
TL;DR: In this article, the tracking error bounds for the unknown randomly varying parameters, and some results on sample path deviations of the estimates, were established, and the asymptotic properties of the algorithm were developed.
Abstract: In linear stochastic system identification, when the unknown parameters are randomly time varying and can be represented by a Markov model, a natural estimation algorithm to use is the Kalman filter. In seeking an understanding of the properties of this algorithm, existing Kalman-filter theory yields useful results only for the case where the noises are gaussian with covariances precisely known. In other cases, the stochastic and unbounded nature of the regression vector (which is regarded as the output gain matrix in state-space terminology) precludes application of standard theory. Here we develop asymptotic properties of the algorithm. In particular, we establish the tracking error bounds for the unknown randomly varying parameters, and some results on sample path deviations of the estimates.
TL;DR: This paper introduces the concept of linear translation-invariant systems and uses this notion in studying impulse response, z-transforms, and transfer functions for multidimensional systems.
Abstract: A significant part of the theory of one-dimensional linear shift-invariant systems is based on the concept of weighting function (or impulse response): the output is the convolution of the weighting function with the input. This paper introduces the concept of linear translation-invariant systems and uses this notion in studying impulse response, z-transforms, and transfer functions for multidimensional systems.
TL;DR: It turns out that the zero-sum, infinite-horizon, linear quadratic differential game has a strong relation with the singularH∞ problem with state feedback, i.e., the H∞problem where the direct feedthrough matrix from control input to output is not necessarily injective.
Abstract: In this paper we consider the zero-sum, infinite-horizon, linear quadratic differential game. We derive sufficient conditions for the existence of (almost) equilibria as well as necessary conditions. Contrary to all classical references we allow for singular weighting on the minimizing player in the cost criterion. It turns out that this problem has a strong relation with the singularH∞ problem with state feedback, i.e., theH∞ problem where the direct feedthrough matrix from control input to output is not necessarily injective.
TL;DR: Under generic conditions a local feedback synthesis for the problem of time-optimally stabilizing an equilibrium point in dimension three is constructed and gives the precise structure of the small-time reachable set for an associated system to which time has been added as an extra coordinate.
Abstract: Under generic conditions a local feedback synthesis for the problem of time-optimally stabilizing an equilibrium point in dimension three is constructed. There exist two surfaces which are glued together along a singular are on which the optimal control is singular. Away from these surfaces the optimal controls are piecewise constant with at most two switchings. Bang-bang trajectories with two switchings but different switching orders intersect in a nontrivial cut-locus and optimality of trajectories ceases at this cut-locus. The construction is based on an earlier result by Krener and Schattler which gives the precise structure of the small-time reachable set for an associated system to which time has been added as an extra coordinate.
TL;DR: An efficient algorithm for checking the robust stability of a polytope of polynomials is proposed, which can be solved by the Simplex procedure, with additional computations between steps consisting ofPolynomial evaluations and calculation of positive polynomial roots.
Abstract: An efficient algorithm for checking the robust stability of a polytope of polynomials is proposed. This problem is equivalent to a zero exclusion condition at each frequency. It is shown that such a condition has to be checked at only afinite number of frequencies. We formulate this problem as aparametric linear program which can be solved by the Simplex procedure, with additional computations between steps consisting of polynomial evaluations and calculation of positive polynomial roots. Our algorithm requires a finite number of steps (corresponding to frequency checks) and in the important case when the polytope of parameters is a hypercube, this number is at most of orderO(m
3
n
2), wheren is the degree of the polynomials in the family andm is the number of parameters.
TL;DR: The self-tuning approach to adaptive control is applied to a class of Markov chains called nearest-neighbor motions that have a countable state space and move from any state to at most finitely many neighboring states.
Abstract: The self-tuning approach to adaptive control is applied to a class of Markov chains called nearest-neighbor motions. These have a countable state space and move from any state to at most finitely many neighboring states. For compact parameter and control spaces, the almost-sure optimality of the self-tuner for an ergodic cost criterion is established under two sets of assumptions.
TL;DR: A sequence of approximating problems with the following properties is constructed: an optimal solution for each approximating problem can be obtained via dynamic programming; and given ε>0, there exists an approximation problem whose optimal solution is ε-optimal for the original problem.
Abstract: We study a discrete-time, finite-horizon adaptive stochastic control problem. The unknown parameter enters the state equation linearly, but otherwise the problem is nonlinear. We construct a sequence of approximating problems with the following properties: (i) an optimal solution for each approximating problem can be obtained via dynamic programming; (ii) given e>0, there exists an approximating problem whose optimal solution is e-optimal for the original problem.
TL;DR: If the Hankel operator to be approximated is of finite rank, the order of approximation by truncated Hankel operators is obtained and it is shown that when themths-number is simple, then rational symbols of the best rank-m Hankel approximants of thenth truncatedHankel matrices converge uniformly to the corresponding rational symbol of thebest rank- m Hankel approximation of the original Hankeloperator asn tends to infinity.
Abstract: 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.
TL;DR: In the application of the parallel algorithm to narrowband array processing in colored noise, the direction-of-arrival containing eigenspace information is provided recursively in order, facilitating early estimation of the number of sources as well as verification of results obtained at previous orders.
Abstract: We present a parallel algorithm which computes recursively, in increasing order, the complete generalized eigendecompositions of the successive subpencils contained in a maximum size Hermitian Toeplitz generalized eigenproblem. At each order a number of independent, structurally identical, nonlinear problems are solved in parallel, facilitating fast implementation. The multiple and clustered minimum eigenvalue cases are treated in detail. In the application of our algorithm to narrowband array processing in colored noise, the direction-of-arrival containing eigenspace information is provided recursively in order. This permits estimation of the angles of arrival for subsequent orders, facilitating early estimation of the number of sources as well as verification of results obtained at previous orders.
TL;DR: The concept of markovian basis is introduced, and its equivalence with the Malmquist basis in the spectral domain is shown, and an algorithm with polynomial complexity to compute an approximate model is given.
Abstract: The problem considered here is to represent a stationary stochastic processy with a low-dimensional stochastic model. This problem occurs when the state space of an exact realization ofy has a very large dimension. The reduction is obtained in this large state space, exploiting its markovian structure to characterize all markovian subspaces, among which a reducedk-dimensional model is sought. The concept of markovian basis is introduced, and its equivalence with the Malmquist basis in the spectral domain is shown. An algorithm with polynomial complexity to compute an approximate model is given.
TL;DR: The voltage-current regime of an infinite resistive network, whose current sources are distributed throughout the network and also connected to the network at infinity, is shown to be the limit in a certain Hilbert space of the regimes of an expanding sequence of subnetworks.
Abstract: The voltage-current regime of an infinite resistive network, whose current sources are distributed throughout the network and also connected to the network at infinity, is shown to be the limit in a certain Hilbert space of the regimes of an expanding sequence of subnetworks. This result is then applied to an infinite gridlike structure. A decomposition of that structure into ∞-ports yields an equivalent ladder network of operators. A procedure is established for determining the exact solution of the infinite gridlike structure by solving the finite truncations of the operator ladder and then passing to the limit appropriately. This closes a lacuna appearing in a number of prior finite-difference analyses of several exterior problems based upon infinite network theory.