Standard Canonical Form Research Articles

Data-Driven-Based Vector Space Decomposition Modeling of Multiphase Induction Machines

For contemporary variable-speed electric drives, the accuracy of the machine's mathematical model is critical for optimal control performance. Basically, phase variables of multiphase machines are preferably decomposed into multiple orthogonal subspaces based on vector space decomposition (VSD). In the available literature, identifying the correlation between states governed by the dynamic equations and the parameter estimate of different subspaces of multiphase IM remains scarce, especially under unbalanced conditions, where the effect of secondary subspaces sounds influential. Most available literature has relied on simple RL circuit representation to model these secondary subspaces. To this end, this paper presents an effective data-driven-based space harmonic model for n-phase IMs using sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover the IM governing equations. Moreover, the proposed approach is computationally efficient, and it precisely identifies both the electrical and mechanical dynamics of all subspaces of an IM using a single transient startup run. Additionally, the derived model can be reformulated into the standard canonical form of the induction machine model to easily extract the parameters of all subspaces based on online measurements. Eventually, the proposed modeling approach is experimentally validated using a 1.5 Hp asymmetrical six-phase induction machine.

О представлении решения краевой задачи Гурса для стохастических гиперболических уравнений с частными производными первого порядка

We study the standard canonical form of a stochastic analog of a system of linear partial differential equations of first order hyperbolic type with Goursat boundary conditions. The stochastic analogue of the Riemann matrix in block form is introduced, an integral representation of the solution of the boundary value problem under consideration is obtained in an explicit integral form in terms of boundary conditions.

Data-Driven Safety Verification of Discrete-Time Networks: A Compositional Approach

This letter proposes a compositional data-driven approach for safety verification of networks of discrete-time subsystems with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">formal guarantees</i> . Following a modular approach and for each subsystem, we search for a so-called sub-barrier candidate represented as a linear combination of a priori user-defined basis functions. We formulate the conditions on sub-barrier candidates as robust convex programs (RCPs) which are semi-infinite linear programs. Collecting sampled data from each subsystem, we approximate each RCP via a scenario convex program (SCP) which is a finite linear program. We provide an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">explicit</i> formula to compute the minimum number of sampled data guaranteeing a desired mismatch between the optimal value of each RCP and that of the corresponding SCP in a probabilistic sense. To ensure that the sum of the sub-barrier candidates is a barrier function for the whole network, we define a global dissipativity condition on top of the local SCPs. The local SCPs are thus related to each other via this global condition. This results in a large-scale optimization problem in a standard canonical form which is efficiently solved by the alternating direction method of multipliers (ADMM) algorithm. The effectiveness of our approach is illustrated by applying it to a room temperature control problem in a 100-room building.

Robust multi-input multi-output adaptive fuzzy terminal sliding mode control of deep brain stimulation in Parkinson\u2019s disease: a simulation study

Deep brain stimulation (DBS) has become an effective therapeutic solution for Parkinson’s disease (PD). Adaptive closed-loop DBS can be used to minimize stimulation-induced side effects by automatically determining the stimulation parameters based on the PD dynamics. In this paper, by modeling the interaction between the neurons in populations of the thalamic, the network-level modulation of thalamic is represented in a standard canonical form as a multi-input multi-output (MIMO) nonlinear first-order system with uncertainty and external disturbances. A class of fast and robust MIMO adaptive fuzzy terminal sliding mode control (AFTSMC) has been presented for control of membrane potential of thalamic neuron populations through continuous adaptive DBS current applied to the thalamus. A fuzzy logic system (FLS) is used to estimate the unknown nonlinear dynamics of the model, and the weights of FLS are adjusted online to guarantee the convergence of FLS parameters to optimal values. The simulation results show that the proposed AFTSMC not only significantly produces lower tracking errors in comparison with the classical adaptive fuzzy sliding mode control (AFSMC), but also makes more robust and reliable outputs. The results suggest that the proposed AFTSMC provides a more robust and smooth control input which is highly desirable for hardware design and implementation.

Introduction to a special issue on concept mapping

Concept mapping was developed in the 1980s as a unique integration of qualitative (group process, brainstorming, unstructured sorting, interpretation) and quantitative (multidimensional scaling, hierarchical cluster analysis) methods designed to enable a group of people to articulate and depict graphically a coherent conceptual framework or model of any topic or issue of interest. This introduction provides the basic definition and description of the methodology for the newcomer and describes the steps typically followed in its most standard canonical form (preparation, generation, structuring, representation, interpretation and utilization). It also introduces this special issue which reviews the history of the methodology, describes its use in a variety of contexts, shows the latest ways it can be integrated with other methodologies, considers methodological advances and developments, and sketches a vision of the future of the method’s evolution.

Rhotacism in Spanish and its decline in Ciudad Real (Spain). A case study of a change in progress nearing completion

This paper begins with an examination of rhotacism as a phonetic process and the conditioning factors that govern its occurrence. Its importance among Spanish varieties and its place within the syllable-final s weakening continuum is also considered. Next we turn to the gradual disappearance of this phenomenon in Ciudad Real, using both apparent time inferences and real time evidence. The results from our investigation show a somewhat atypical change from below and an interesting development of the syllable-final s weakening process; namely, rhotacism is not replaced by the standard canonical form but rather gives way to a more advanced form of weakening. This implies that, although the standard variant has shown complete stability during the last century, the weakening process is progressing among the non-standard allophones. From a methodological point of view, this study has revealed the importance of analyzing different individual contexts in which the syllable-final s can appear, since global data can conceal some important patterns.

Energy and Uncertainty: Models and Algorithms for Complex Energy Systems

The problem of controlling energy systems (generation, transmission, storage, investment) introduces a number of optimization problems which need to be solved in the presence of different types of uncertainty. We highlight several of these applications, using a simple energy storage problem as a case application. Using this setting, we describe a modeling framework based around five fundamental dimensions which is more natural than the standard canonical form widely used in the reinforcement learning community. The framework focuses on finding the best policy, where we identify four fundamental classes of policies consisting of policy function approximations (PFAs), cost function approximations (CFAs), policies based on value function approximations (VFAs), and lookahead policies. This organization unifies a number of competing strategies under a common umbrella.

On stability of time-varying linear differential-algebraic equations

We develop a stability theory for time-varying linear differential algebraic equations (DAEs). Well-known stability concepts of ordinary differential equations are generalised to DAEs and characterised. Lyapunov’s direct method is derived as well as the converse of the stability theorems. Stronger results are achieved for DAEs, which are transferable into standard canonical form; in this case the existence of the generalised transition matrix is exploited.

On the standard canonical form of time-varying linear DAEs

We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) E ( t ) x ˙ = A ( t ) x E(t)\dot x=A(t)x which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE z ˙ 1 = J ( t ) z 1 \dot z_1 = J(t)z_1 and a pure DAE N ( t ) z ˙ 2 = z 2 N(t) \dot z_2 = z_2 , where N N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformation matrices which put a DAE into SCF.

Sortals and criteria of identity

In a recent article, Harold Noonan argues that application conditions and criteria of identity are not distinct from one another. This seems to threaten the standard approach to distinguishing sortals from adjectival terms. I propose that his observation, while correct, does not have this consequence. I present a simple scheme for distinguishing sortals from adjectival terms. I also propose an amended version of the standard canonical form of criteria of identity.

Tractable forms of the bond pricing equation

So far, a small number of analytically tractable single-factor models have been devised for the well-known bond pricing equation (BPE). In this paper, new tractable models are formulated in a systematic manner. First, the BPE is transformed to a standard canonical form in which only one coefficient function appears. In some interesting cases, this single coefficient function is identically zero, leaving nothing more to solve than the classical heat equation. In many cases, the canonical form allows a general solution by standard mathematical techniques such as separation of variables and Laplace transforms. In other cases, the general solution of the BPE is reduced to a single inverse Laplace transform.

Impulse observability and impulse controllability of linear time-varying singular systems

Impulse controllability and impulse observability of linear time-varying singular systems in a standard canonical form are examined. We focus on the fast subsystems with constant-rank N matrices. Sufficient and necessary conditions are derived and presented directly in terms of system parameters. Conditions obtained for these two properties are algebraically dual to each other.

Linear response theory for nonequilibrium stationary systems

A nonequilibrium statistical mechanics is developed by extending the Kubo-Nakano theory of linear response to spatially inhomogeneous systems under the action of external fields. It is argued that the standard linear response theory bases upon the fluctuation-dissipation theorem. The stationary density matrix of the nonequilibrium system is given by applying Zubarev's method. It is found that Onsager's reciprocity relations require a redefinition of the responses to external fields, reflecting the fact that the stationary density matrix is not of the standard canonical form with the unperturbed Hamiltonian. Following Nakano, a variational principle is developed for nonequilibrium stationary systems. A link of the variational principle with Machlup-Onsager theory is explored.

Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

This paper takes a fresh look at the application of quadratic penalty functions to linear programming. Recently, Madsen et al. (Ref. 1) described a continuation algorithm for linear programming based on smoothing a dual l 1-formulation of a linear program with unit bounds. The present paper is prompted by the observation that this is equivalent to applying a quadratic penalty function to the dual of a linear program in standard canonical form, in the sense that both approaches generate continuous, piecewise-linear paths leading to the optimal solution set. These paths lead to new characterizations of optimal solutions in linear programming. An important product of this analysis is a finite penalty algorithm for linear programming closely related to the least-norm algorithm of Mangasarian (Ref. 2) and to the continuation algorithm of Madsen et al. (Ref. 1). The algorithm is implemented, and promising numerical results are given.

System Simulation Using OCCAM

Methods are given for the mapping of a system representation into some standard form into an OCCAM program. Firstly, two different transform methods are converged into a standard canonical form. The different program elements, which extensively use the concept of OCCAM channels are demonstrated on a typical control problem. Secondly, the node-channel concepts used are extended to simulate MRAC scenarios which include state space notation.

Constrained least squares methods for linear timevarying DAE systems

This paper presents a family of methods for accurate solution of higher index linear variable DAE systems, $$A(t)\dot y(t) + B(t)y(t) = f(t)$$ . These methods use the DAE system and some of its first derivatives as constraints to a least squares problem that corresponds to a Taylor series ofy, or an approximative equality derived from a Pade' approximation of the exponential function. Accuracy results for systems transformable to standard canonical form are given. Advantages, disadvantages, stability properties and implementation of these methods are discussed and two numerical examples are given, where we compare our results with results from more traditional methods.

Analytical and numerical solutions to higher index linear variable coefficient DAE systems

Analytical and numerical solutions to the singular linear system of differential equations A( t) x′( t) + B( t) x( t) = f( t) are investigated. Closed-form analytical solutions of systems transformable to standard canonical form, SCF (see Campbell (1983)) are derived. Equivalent ways to compute the analytic solution are presented which are of interest in a symbolic/numeric computing environment. Many numerical methods for solving DAE systems are based on Gear's backward difference formula (BDF) method. It is proved that the BDF method and the modified BDF method, MBDF (see Clark (1986)) are stable and O( h k ) accurate after a maximum of ( s − 1) k + 1 steps for linear variable coefficient systems in SCF, where s is the dimension of the algebraic part. These new results are valid for arbitrarily higher index systems and generalize well-known results for linear constant coefficient singular systems to the variable coefficient case. Numerical experiments confirm the order and stability results.

Document formatting for interchange between word processors

There are many different types of word processor on the market today, each storing documents in a different way. Hence, a document created using one particular word processor cannot be edited using another word processor. Documents are often exchanged in an office environment, and should be in a form acceptable by other vendors' word processing packages. To make the exchange of documents created using word processors general and flexible, a standard canonical form has been adopted, and is outlined here. A methodology for converting the word processor documents into canonical form is described, detailing document and canonical structure, look-up tables, the intermediate file structure, and logical/layout object information. The author concludes by working through the methodology using an example.

The numerical solution of some higher-index time-varying semistate systems by difference methods

Necessary and sufficient conditions are given for the solvability of fully implicit index-one and semiexplicit index-two linear time-varying semistate systems of the formA(t)x′+B(t)x=f(t). Using these solvability criteria, we establish the convergence and stability of backward difference formulas (BDF) for solvable semiexplicit index-two systems. In the algebraic variables, these methods exhibit a boundary layer of instability of length 2k, wherek is the order of the method. Finally, we show that the systems considered in this paper belong to a class of problems, solvable by BDF, which strictly contains the systems safely transformable to standard canonical form.