Factorability of lossless time-varying filters and filter banks

We study the factorability of linear time-varying (LTV) lossless filters and filter banks. We give a complete characterization of all, degree-one lossless LTV systems and show that all degree-one lossless systems can be decomposed into a time-dependent unitary matrix followed by a lossless dyadic-based LTV system. The lossless dyadic-based system has several properties that make it useful in the factorization of lossless LTV systems. The traditional lapped orthogonal transform (LOT) is also generalized to the LTV case. We identify two classes of TVLOTs, namely, the invertible inverse lossless (IIL) and noninvertible inverse lossless (NIL) TVLOTs. The minimum number of delays required to implement a TVLOT is shown to be a nondecreasing function of time, and it is a constant if and only if the TVLOT is IIL. We also show that all IIL TVLOTs can be factorized uniquely into the proposed degree-one lossless building block. The factorization is minimal in terms of the delay elements. For NIL TVLOTs, there are factorable and unfactorable examples. Both necessary and sufficient conditions for the factorability of lossless LTV systems are given. We also introduce the concept of strong eternal reachability (SER) and strong eternal observability (SEO) of LTV systems. The SER and SEO of an implementation of LTV systems imply the minimality of the structure. Using these concepts, we are able to show that the cascade structure for a factorable IIL LTV system is minimal. That implies that if a IIL LTV system is factorable in terms of the lossless dyadic-based building blocks, the factorization is minimal in terms of delays as well as the number of building blocks. We also prove the BIBO stability of the LTV normalized IIR lattice

A generalization of Serre's conjecture and some related issues

AbstractSeveral topics concerned with multivariate polynomial matrices like unimodular matrix completion, matrix determinantal or primitive factorization, matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original Serre problem (conjecture) solved nonconstructively in 1976 and constructively, more recently. This generalized Serre conjecture is proved to be equivalent to several other unsettled conjectures and, therfore, all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

Optimal pre- and post-filtering in noisy sampled-data systems

Originally presented as author's thesis (Ph. D.-- Massachusetts Institute of Technology), 1986.Includes bibliographies."This work has been supported in part by the Brazillian Government through its Conselho Nacional de Desenvolvimento Cientifico e Tecnologico. It has also been supported in part by the Center for Advanced Television Studies, an industry group consisting of the Amercian Broadcasting Company, Ampex Corporatin, Columbia Broadcasting Systems, Harris Corporation, Home Box Office, Public Broadcasting Service, National Broadcasting Company, RCA Corporation, Tektronix, and the 3M Company."Henrique Sarmento Malvar

A System Approach to the Design of Multirate Filter Banks.

This dissertation studies the design of multirate filter banks by adopting a so-called system approach. The design issue of Johnston\u27s method is first investigated in which an explicit expression of the reconstruction error is derived using Lyapunov stability theory, and new convergent iterative algorithms are proposed through non-linear optimization. The results are extended to the two-dimensional filter banks. The design issue of more general multirate filter banks is also investigated through model matching method. Using standard results from modern control theory, new design algorithms are developed which minimize the reconstruction error while completely eliminating the aliasing error. State-space realizations, inner-outer factorizations, and optimal Hankel norm approximation are used to reduce the complexity of computation and improve the accuracy of the proposed design algorithms

Stability and Stabilization of the Wave Model.

The stability properties of 2-D systems are an important aspect of the design of acoustic, seismic, image and sonar signal processors. This research utilizes the Wave model format to transport 1-D stability techniques to the 2-D setting. The research studies stability through multistep growth bounds on the Wave state. The use of Lyapunov theory is also considered. The research considers also the problem of stabilizing a 2-D system using state and/or output information feedback to interior and/or boundary controls. Finally the problem of observer design for 2-D systems is considered, with the new stability criteria being used to assure observer/system convergence. New results based on symmetrizability are also discussed. The principal results are illustrated by a number of examples. The results are also interpreted in the context of other contemporary local state models

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Global optimization algorithms for semi-infinite and generalized semi-infinite programs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.Includes bibliographical references (p. 235-249).The goals of this thesis are the development of global optimization algorithms for semi-infinite and generalized semi-infinite programs and the application of these algorithms to kinetic model reduction. The outstanding issue with semi-infinite programming (SIP) was a methodology that could provide a certificate of global optimality on finite termination for SIP with nonconvex functions participating. We have developed the first methodology that can generate guaranteed feasible points for SIP and provide e-global optimality on finite termination. The algorithm has been implemented in a branch-and-bound (B&B) framework and uses discretization coupled with convexification for the lower bounding problem and the interval constrained reformulation for the upper bounding problem. Within the framework of SIP we have also proposed a number of feasible-point methods that all rely on the same basic principle; the relaxation of the lower-level problem causes a restriction of the outer problem and vice versa. All these methodologies were tested using the Watson test set. It was concluded that the concave overestimation of the SIP constraint using McCormcick relaxations and a KKT treatment of the resulting expression is the most computationally expensive method but provides tighter bounds than the interval constrained reformulation or a concave overestimator of the SIP constraint followed by linearization. All methods can work very efficiently for small problems (1-3 parameters) but suffer from the drawback that in order to converge to the global solution value the parameter set needs to subdivided. Therefore, for problems with more than 4 parameters, intractable subproblems arise very high in the B&B tree and render global solution of the whole problem infeasible.(cont.) The second contribution of the thesis was the development of the first finite procedure that generates guaranteed feasible points and a certificate of e-global optimality for generalized semi-infinite programs (GSIP) with nonconvex functions participating. The algorithm employs interval extensions on the lower-level inequality constraints and then uses discretization and the interval constrained reformulation for the lower and upper bounding subproblems, respectively. We have demonstrated that our method can handle the irregular behavior of GSIP, such as the non-closedness of the feasible set, the existence of re-entrant corner points, the infimum not being attained and above all, problems with nonconvex functions participating. Finally, we have proposed an extensive test set consisting of both literature an original examples. Similar to the case of SIP, to guarantee e-convergence the parameter set needs to be subdivided and therefore, only small examples (1-3 parameters) can be handled in this framework in reasonable computational times (at present). The final contribution of the thesis was the development of techniques to provide optimal ranges of valid reduction between full and reduced kinetic models. First of all, we demonstrated that kinetic model reduction is a design centering problem and explored alternative optimization formulations such as SIP, GSIP and bilevel programming. Secondly, we showed that our SIP and GSIP techniques are probably not capable of handling large-scale systems, even if kinetic model reduction has a very special structure, because of the need for subdivision which leads to an explosion in the number of constraints. Finally, we propose alternative ways of estimating feasible regions of valid reduction using interval theory, critical points and line minimization.by Panayiotis Lemonidis.Ph.D

Guidance, Flight Mechanics and Trajectory Optimization. Volume 5 - State Determination And/or Estimation

Guidance, flight mechanics, and trajectory optimizatio

Dynamical Systems in Spiking Neuromorphic Hardware

Dynamical systems are universal computers. They can perceive stimuli, remember, learn from feedback, plan sequences of actions, and coordinate complex behavioural responses. The Neural Engineering Framework (NEF) provides a general recipe to formulate models of such systems as coupled sets of nonlinear differential equations and compile them onto recurrently connected spiking neural networks – akin to a programming language for spiking models of computation. The Nengo software ecosystem supports the NEF and compiles such models onto neuromorphic hardware. In this thesis, we analyze the theory driving the success of the NEF, and expose several core principles underpinning its correctness, scalability, completeness, robustness, and extensibility. We also derive novel theoretical extensions to the framework that enable it to far more effectively leverage a wide variety of dynamics in digital hardware, and to exploit the device-level physics in analog hardware. At the same time, we propose a novel set of spiking algorithms that recruit an optimal nonlinear encoding of time, which we call the Delay Network (DN). Backpropagation across stacked layers of DNs dramatically outperforms stacked Long Short-Term Memory (LSTM) networks—a state-of-the-art deep recurrent architecture—in accuracy and training time, on a continuous-time memory task, and a chaotic time-series prediction benchmark. The basic component of this network is shown to function on state-of-the-art spiking neuromorphic hardware including Braindrop and Loihi. This implementation approaches the energy-efficiency of the human brain in the former case, and the precision of conventional computation in the latter case