Multiplier Matrices Research Articles

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

State and adaptive disturbance observer co‐design for incrementally quadratic nonlinear descriptor systems with nonlinear outputs

AbstractThe problem of designing state and adaptive disturbance observers for a class of nonlinear descriptor systems with the disturbance generated by an unknown exogenous system is addressed in this paper. Especially, the nonlinear terms satisfy incremental quadratic constraints parameterized by a bunch of multiplier matrices, which can represent various classes of nonlinearities. For three different cases in the output equation, the state and adaptive disturbance observers are proposed. Some sufficient conditions formulated in terms of linear matrix inequalities and algebraic constraints are derived to guarantee that state and disturbance error systems, simultaneously, are asymptotically stable with performance level. The availability of proposed observers is validated by two practical examples.

A complete genus-level phylogeny reveals the Cretaceous biogeographic diversification of the poppy family

Angiosperms, a trigger for the Cretaceous Terrestrial Revolution (KTR), underwent a rapid expansion and occupied all the environments during the Mid-Upper Cretaceous. Yet, Cretaceous biogeographic patterns and processes underlying the distribution of angiosperm diversity in the Northern Hemisphere are still poorly known. Here, we elucidated the biogeographic diversification of the angiosperm family Papaveraceae, an ancient Northern Hemisphere clade characterized by poor dispersal ability and high level of regional endemism. Based on both plastome and multi-locus datasets, we reconstructed a robust time-calibrated phylogeny that includes all currently recognized 45 genera of this family. Within the time-calibrated phylogenetic framework, we conducted 72 biogeographic analyses by testing the sensitivity of uncertainties of area delimitation, maxarea constraints, and the parameters of the model, i.e., j (describing jump-dispersal events) and w (modifying dispersal multiplier matrices), to ancestral range estimations. We also inferred ancestral habitat and ecological niches. Phylogenetic analyses strongly support Papaveraceae as monophyletic. Pteridophylloideae is strongly supported as sister to Hypecoideae-Fumarioideae. Our results indicate that the j parameter and number of predefined areas strongly affect ancestral range estimates, generating questionable ancestral ranges, whereas maxarea constraint and w parameter have no effect and improve model fit. After accounting for these uncertainties, our results indicate that Papaveraceae differentiated in Asian wet forests during the Lower Cretaceous and subsequently occupied the Asian and western North American arid and open areas. Three dispersals from Asia to western North America via the Bering land bridge occurred in the Mid-Upper Cretaceous, largely in agreement with the KTR. Habitat shift and ecological niche divergence resulted in the subsequent disjunctions between Asia and western North America. These findings suggest that the interplay of range expansion and niche divergence-driven vicariance might have shaped Cretaceous biogeographic patterns of angiosperms with Papaveraceae-like ecological requirements and dispersal abilities in the Northern Hemisphere, hence contributing to the knowledge on the geographic expansion of angiosperms during the KTR.

Observer-Based Synchronization of Chaotic Systems Satisfying Incremental Quadratic Constraints and Its Application in Secure Communication

This paper presents a secure chaotic communication scheme using observer-based synchronization of chaotic systems. A class of chaotic systems satisfying incremental quadratic constraints are considered as the transmitter. The original message is encrypted in the noise-like output of the system. A circle criterion-based observer is designed for the system as the receiver by using the linear matrix inequalities technique. The message is recovered at the receiver end from the estimation error of the observer. To design the observer, a general algorithm to calculate incremental multiplier matrices for nonlinearities in chaotic systems is presented. Moreover, the proposed observer-based chaotic secure communication scheme is feasible for many common chaotic systems with ellipsoidal bounds. The simulation results show the good performance of the proposed communication scheme through an example of image transmission.

Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

There is a natural conjugation action on the set of endomorphism of $¶^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $¶^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant functions determine the conjugacy class in the moduli space. Several different types of isospectral families are constructed and a generalization of McMullen's theorem on the multiplier mapping of dimension 1 is proposed. Finally, this generalization is shown to hold when restricted to several specific families in $\mathcal{M}_d^N$.

Observer-Based Controllers for Incrementally Quadratic Nonlinear Systems With Disturbances

Robust global stabilization of nonlinear systems by observer-based feedback controllers is a challenging task. This article investigates the problem of designing observer-based stabilizing controllers for incrementally quadratic nonlinear systems with external disturbances. The nonlinearities considered in the system model satisfy the incremental quadratic constraints, which are characterized by incremental multiplier matrices and encompass many common nonlinearities. The simultaneous search for the observer and the controller gain matrices is formulated as a feasibility problem of linear matrix inequalities, for two parameterizations (i.e., the block diagonal parameterization and the block anti-triangular parameterization) of the incremental multiplier matrices, respectively. The closed-loop system implementing the observer-based feedback controller is proven to be input-to-state stable with respect to external disturbances. Using the proposed continuous-time observer-based controllers, event-triggered controllers with time regularization are constructed for globally Lipschitz systems, such that the closed-loop system is Zeno-free and input-to-state practically stable.

Safe Approximate Dynamic Programming via Kernelized Lipschitz Estimation.

We develop a method for obtaining safe initial policies for reinforcement learning via approximate dynamic programming (ADP) techniques for uncertain systems evolving with discrete-time dynamics. We employ the kernelized Lipschitz estimation to learn multiplier matrices that are used in semidefinite programming frameworks for computing admissible initial control policies with provably high probability. Such admissible controllers enable safe initialization and constraint enforcement while providing exponential stability of the equilibrium of the closed-loop system.

Multiple model stochastic filtering for traffic density estimation on urban arterials

Traffic state estimation plays an important role in Intelligent Transportation Systems (ITS). It provides the latest traffic information to travelers and feedback to signal control systems. The Interactive Multiple Model (IMM) filtering provides a powerful estimation method to deal with the non-differentiable nonlinearity caused by the phase transitions between the under-critical and above-critical traffic density regimes. The IMM filtering also accounts for the uncertainty in the current ‘mode of operation’. In this paper, we develop an enhanced IMM filtering approach to traffic state estimation, with an underlying Cell Transmission Model (CTM) for traffic flow propagation. We improve the IMM filtering with CTM in two ways: (1) We apply two simplifying assumptions that are highly likely to hold in urban roads in incident-free conditions, which makes the computational complexity to grow with the number of cells only polynomially, rather than exponentially as reported in prior work. (2) We apply a novel approach to noise modeling wherein the process noise is explicitly obtained in terms of the randomness in more fundamental quantities (e.g., free-flow speed, maximum flow capacity, etc.), which not only makes noise calibration using real data convenient but also makes the computation of the cross-correlation between the process and measurement noises transparent. However, it leads to ‘process dynamic’ and ‘measurement’ equations that involve multiplier matrices whose elements are random variables rather than deterministic scalars, and hence, standard filtering equations cannot be applied. We derive the appropriate filtering equations from first principles. We calibrate the traffic parameters and the total inflow and outflow on the links using the SCATS loop detector data collected in Melbourne and report significant improvements in accuracy, which is due to the accurate computation of the cross-covariance of process and measurement noises.

Income distributions in multi-sector analysis; Miyazawa’s fundamental equation of income formation revisited

In standard type 1 input-output models, households’ activities are part of the exogenous final demand. This means that their scale and composition are exogenously determined. That is, if some other final demand categories change (say public investment or exports) this does not influence the behaviour of the household categories. In type 2 input-output models households’ activities are explained endogenously to capture the possibility of mutual interaction between household categories and productive sectors. In this area, Miyazawa (1976) proposed a novel way of modeling the endogenization of households’ activities. In modeling terms, Miyazawa’s proposition resulted in the so-called ‘fundamental equation of income formation’, core of which is an extended input coefficients matrix. This extended coefficients matrix produced several new types of multiplier matrices and explains industrial gross output and households’ income in terms of non-household final demand in great detail. The model is traditionally solved by inverting the new extended coefficients matrix, which often generates highly complex outcomes in terms of convoluted multiplier matrices. Consequently, the link between final demand impulses, gross outputs and income formation is not straightforward, working sometimes in different directions. Regarding this aspect, as we shall show, there is a second way to solve Miyazawa’s fundamental equation, which is much more transparent. This second way shows that Miyazawa type endogenization means that gross output and (remaining) final demand are directly linked via a new type of coefficients matrix. This matrix is the sum of the traditional matrix of intermediate input coefficients and a number of matrices of rank 1, each one corresponding to an endogenized households category. The existence of this matrix makes several new applications possible including the study of shifts over time in the distribution of income and (un)employment between the households categories involved. In an appendix we briefly focus on the link between the use of matrices of rank 1 in, respectively, Leontief, Sraffa and Miyazawa input-output economics.

State and unknown input observers for nonlinear systems with delayed measurements

The use of connected devices or humans-in-the-loop for measuring outputs of dynamical systems inevitably produces time-varying measurement delays. These delays can lead to instability or severe degradation of system performance. In this paper, linear matrix inequality-based sufficient conditions are proposed for the design of state and unknown input observers based on delayed measurements for a class of nonlinear systems, where the nonlinearities are characterized by incremental multiplier matrices. The proposed observer is guaranteed to perform at specified operational levels in the presence of unknown exogenous inputs acting on the states and measurement outputs. Sufficient conditions are also provided for the estimation of these unknown inputs to a specified degree of accuracy. The potential of the proposed approach is illustrated via estimation of enzyme kinetics.

State and Unknown Input Observers for Nonlinear Systems With Bounded Exogenous Inputs

A systematic design methodology for state observers for a large class of nonlinear systems with bounded exogenous inputs (disturbance inputs and sensor noise) is proposed. The nonlinearities under consideration are characterized by an incremental quadratic constraint parameterized by a set of multiplier matrices. Linear matrix inequalities are developed to construct observer gains, which ensure that a performance output based on the state estimation error satisfies a prescribed degree of accuracy. Furthermore, conditions guaranteeing estimation of the unknown inputs to arbitrary degrees of accuracy are provided. The proposed scheme is illustrated with a numerical example, which does not satisfy the so-called “matching conditions.”

Stability of nonlinear time-delay systems satisfying a quadratic constraint

The stability problem of nonlinear time-delay systems is addressed. A quadratic constraint is employed to exploit the structure of nonlinearity in dynamical systems via a set of multiplier matrices. This yields less conservative results concerning stability analysis. By employing a Wirtinger-based inequality, a delay-dependent stability criterion is derived in terms of linear matrix inequalities for the nominal and uncertain systems. A numerical example is used to demonstrate the effectiveness of the proposed stability conditions in dealing with some larger class of nonlinearities.

Evolutionary and biogeographical history of the land snail family Urocoptidae (Gastropoda: Pulmonata) across the Caribbean region

AbstractAimLand snails, because of their low vagility, are ideal organisms for probing biogeographical hypotheses. This study is the first to examine the evolutionary and biogeographical history of a group of land snails from the Caribbean region using molecular phylogenetics. It focuses on the family Urocoptidae that displays extraordinary diversity, particularly in the Greater Antilles.LocationCaribbean region.MethodsWe reconstructed the phylogeny of Urocoptidae based on multi‐locus (partial 28S, H3 andCOIsequences) analyses (MrBayes,BEAST,GARLI) of 65 species, representing 44 recognized genera. Biogeographical analyses of a subset of the time‐calibratedBEASTtrees were made both with (DECandDEC+J analysis inBioGeoBEARS) and without (S‐DIVAinRASP) palaeogeographical assumptions. In theDECandDEC+J analyses we examined the effect of different settings for dispersal between directly connected areas relative to that between areas without direct land connection.ResultsUrocoptidae has been present on the Greater Antilles Arc from at least Middle Eocene onwards. Morphologically diverse and previously unrecognized clades evolved on most Caribbean (palaeo)islands. Jamaica was colonized at least twice. Dispersal multiplier matrices with moderately constrained dispersal between areas without direct land connections describe the phylogeographical history of the family with higherDECandDEC+J lnL scores than uniform matrices.Main conclusionsUrocoptids constitute an old element of the Greater Antillean biota, predating a proposedGAARlandia landspan connection to South America. The biogeographical history and evolution of Urocoptidae were shaped primarily by the geographical distribution of Caribbean landmasses, in combination with occasional oversea dispersal. Oversea dispersal allowed colonization of palaeogeographically isolated areas, such as Jamaica and present‐day western Cuba, where presumably the absence of ecological competitors led to independent radiations into similar shell types.

Conservatism of Analysis and Controller Synthesis of Decomposable Systems

Recently, distributed controller synthesis approaches for decomposable systems, a subclass of distributed systems with identical subsystems, where the interconnection can be described as LFT interconnection, have been proposed. In order to make these approaches tractable for systems containing a very large number of subsystems, constraints on the Lyapunov matrix and the multiplier matrices are introduced that render the complexity of analysis and controller synthesis smaller and in best case independent on the number of subsystems. Those assumptions have to be paid for with conservatism, which is investigated in this work. It is proven that the conservatism is not reduced if either only the Lyapunov or only the multiplier matrices are constrained, when compared with having constraints on both simultaneously.

THE BIAS OF THE MULTIPLIER MATRIX WHEN SUPPLY AND USE TABLES ARE STOCHASTIC

The literature on stochastic input–output (I–O) analysis has paid considerable attention to the bias in the Leontief inverse. This paper extends previous studies by assuming supply and use tables (SUTs rather than I–O tables or input coefficients matrices) to be stochastic. This is a natural starting point because SUTs have become the basic data sources for I–O applications. In a Monte Carlo simulation experiment, a given SUT is randomized in two different ways and the effects are determined for eight different multiplier matrices. The analysis is carried out for Spain, Italy, the Netherlands, Germany and Finland, using their SUTs for 2006. The findings indicate that, in general, biases are statistically significant but negligibly small. This corroborates earlier findings obtained for stochastic I–O tables.

Incremental quadratic stability

The concept of incremental quadratic stability ($\delta$QS) is very useful in treating systems with persistently acting inputs. To illustrate, if a time-invariant $\delta$QS system is subject to a constant input or $T$-periodic input then, all its trajectories exponentially converge to a unique constant or $T$-periodic trajectory, respectively. By considering the relationship of $\delta$QS to the usual concept of quadratic stability, we obtain a useful necessary and sufficient condition for $\delta$QS. A main contribution of the paper is to consider nonlinear/uncertain systems whose state dependent nonlinear/uncertain terms satisfy an incremental quadratic constraint which is characterized by a bunch of symmetric matrices we call incremental multiplier matrices . We obtain linear matrix inequalities whose feasibility guarantee $\delta$QS of these systems. Frequency domain characterizations of $\delta$QS are then obtained from these conditions. By characterizing incremental multiplier matrices for many common classes of nonlinearities, we demonstrate the usefulness of our results.

THE ROLE OF SUPPLY CONSTRAINTS IN MULTIPLIER ANALYSIS

Multiplier analysis based upon the information contained in Leontief's inverse is undoubtedly part of the core of the input–output methodology and numerous applications and extensions have been developed that exploit its informational content, both at the national and regional levels. Nonetheless there are some implicit theoretical assumptions whose policy implications need to be assessed. This is the case for the ‘excess capacity’ assumption, which implies that resources are available as needed to adjust production to new equilibrium states. In an actual economy, however, new resources are often scarce and always costly. When supply constraints intervene, the assessment of the effects of government demand policies may be substantially different from that of the standard Leontief multiplier matrix. Using a closed general equilibrium model that incorporates supply constraints, we perform some simple numerical exercises and proceed to derive two ‘constrained’ multiplier matrices, based upon the implicit Jacobian matrix, that can be compared with the standard ‘unconstrained’ Leontief matrix.

Observers for systems with nonlinearities satisfying incremental quadratic constraints

We consider the problem of designing observers to asymptotically estimate the state of a system whose nonlinear time-varying terms satisfy an incremental quadratic inequality that is parameterized by a set of multiplier matrices. Observer design is reduced to solving linear matrix inequalities for the observer gain matrices. The proposed observers guarantee exponential convergence of the state estimation error to zero. In addition to considering a larger class of nonlinearities than previously considered, this paper unifies earlier related results in the literature. The results are illustrated by application to several examples.

Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions

In this paper, we present several conditions which are both necessary and sufficient for quadratic stability of an uncertain/nonlinear system. These conditions involve multiplier matrices which characterize the uncertain/nonlinear terms in the system description. It is known that some of these conditions are sufficient for quadratic stability. One of the main contributions of this paper is to demonstrate that these conditions are also necessary conditions, hence, they are not conservative conditions for quadratic stability. By presenting multiplier matrices for many common types of uncertain/nonlinear terms, the paper also demonstrates the usefulness of the multiplier matrix approach in the analysis and control of nonlinear/time-varying/uncertain systems.

INTERRELATIONAL INCOME DISTRIBUTION IN BRAZIL

We extend Miyazawa and Masegi's original idea of disaggregated interrelational income multipliers with regard to three aspects. First, we examine the effect of system closure on the inter‐income‐group matrix. Second, we formally include inter‐household transactions, or the informal economy, into Miyazawa's calculus. Third, we extend the interrelational multiplier to environmental factors. These extensions are applied to the Brazilian economy of 1995. Multiplier matrices excluding and including the informal economy are presented and interpreted, and the redistributive process of transfer payments is traced through consecutive spending rounds for various scenarios. Finally, these redistributive processes are enumerated in terms of transport fuel and electricity use. Our results indicate that, because of the distribution of ownership of productive capital, the income formation process is heavily skewed toward the highest incomes. Whether the existing process and potential redistributive policies alleviate environmental pressure depends on the factors as well as the population segment appraised.