Linearization of logical functions defined by a set of orthogonal terms. I. Theoretical aspects

Consideration was given to the linearization of logical functions defined by a set of pairwise orthogonal terms. The linearization is carried out by computing the autocorrelation functions. Proposed was a method consisting of (i) calculation of the autocorrelation function in the space of orthogonal terms, (ii) generation of the corresponding matrix of linear transformation, and (iii) the linear transformation proper of the variables in the space of orthogonal terms. Complexity of the proposed method and its effectiveness were estimated. Effectiveness was verified by a series of experiments with standard benchmarks. The distinctions of the proposed method from other existing methods of linearization were examined.     

Structure analysis via orthogonal functions

The use of orthogonal functions to analyse the structure of a system is investigated. Applying the definitions of observability and controllability to a system that is approximated with the help of orthogonal functions, it is shown that the concepts of the state space and the space of orthogonal functions are equivalent, provided that two weak conditions are met. This result ensures that the observability and controllability properties remain invariant under the transformation introduced by the approximation. Furthermore, new criteria to test observability and controllability are given in terms of the coefficient matrix of the orthogonal expansion. Because this test does not require the knowledge of the system matrices A, B and C, the results derived may be used for the identification of systems. It is demonstrated that all the results obtained remain true, even for an approximation with low accuracy. These properties allow the application of orthogonal functions for the analysis of systems     

Improved scaling functions for robust synthesis

An algorithm for finding improved D-scale transfer functions for robust control synthesis is presented. Generalized Popov multipliers and a linear matrix inequality (LMI) multiplier formulation are used to find D-scalings of polynomial order and arbitrary structure which provide advantages over the normal curve-fit scalings in a D–K iteration. The two primary benefits are enhanced convergence properties and the capability to handle repeated scalar block uncertainty. A flexible manipulator control example illustrates the advantages of the new D-scaling approach. © 1998 John Wiley & Sons, Ltd.     

State estimation using continuous orthogonal functions

A new approach is presented to facilitate research in the state estimation of linear systems using continuous orthogonal functions. The principle of the Luenberger observer is utilized in developing simple algebraic expressions for the estimates of states. This approach has the distinct advantage that the smoothing effect of integration reduces the effect of noise. Hence, this observer gives acceptable estimates of the states in the presence of zero-mean observation noise, even without a filter. Results of simulation indicate that the proposed method works quite well. In addition, the algorithms are recursive and suitable for on-line implementation.     

Analysis of singular systems using orthogonal functions

The use of orthogonal functions to analyze singular systems is investigated. It is shown that the differential-algebraic system equation may be converted to an algebraic generalized Lyapunov equation that can be solved for the coefficients ofx(t)in terms of the orthogonal basis functions. This generalized Lyapunov equation may be considered as a "discrete" equation on the slow subspace of the system, and as a "continuous" equation on its fast subspace. Necessary and sufficient conditions for the existence of a unique solution are given in terms of the relative spectrum of the system. A generalized Bartels/Stewart algorithm based on theQZalgorithm is presented for its efficient solution. Relations are drawn with the invariant subspaces of the system.     

Some sets of orthogonal polynomial kernel functions

Kernel methods provide high performance in a variety of machine learning tasks. However, the success of kernel methods is heavily dependent on the selection of the right kernel function and proper setting of its parameters. Several sets of kernel functions based on orthogonal polynomials have been proposed recently. Besides their good performance in the error rate, these kernel functions have only one parameter chosen from a small set of integers, and it facilitates kernel selection greatly. Two sets of orthogonal polynomial kernel functions, namely the triangularly modified Chebyshev kernels and the triangularly modified Legendre kernels, are proposed in this study. Furthermore, we compare the construction methods of some orthogonal polynomial kernels and highlight the similarities and differences among them. Experiments on 32 data sets are performed for better illustration and comparison of these kernel functions in classification and regression scenarios. In general, there is difference among these orthogonal polynomial kernels in terms of accuracy, and most orthogonal polynomial kernels can match the commonly used kernels, such as the polynomial kernel, the Gaussian kernel and the wavelet kernel. Compared with these universal kernels, the orthogonal polynomial kernels each have a unique easily optimized parameter, and they store statistically significantly less support vectors in support vector classification. New presented kernels can obtain better generalization performance both for classification tasks and regression tasks.     

Computing orthogonal rational functions with poles near the boundary

Computing orthogonal rational functions is a far from trivial problem, especially for poles close to the boundary of the support of the orthogonality measure. In this paper we analyze some of the difficulties involved and present two different approaches for solving this problem.     

Universal scaling functions of a dissipative Manna model

This study uses a losing probability f to measure the magnitude of the bulk dissipation of a dissipative Manna model where P(s;r,L,f), the probability distribution of toppling size s, on a L×(L/r) square lattice is calculated. This approach assumes that the bulk dissipation corresponds to a characteristic length ξ. The finding ξ∼f−ν leads to P(s;r,L,f)=s−τH(s/D[L/Ar],[L/Ar]fν) where H is a two-variable universal scaling function, τ, D, and ν are exponents, and Ar is a nonuniversal metric factor determined by the calculation of moment.     

Analytical properties of bivariate fractal interpolation functions with vertical scaling factor functions

Based on the construction of bivariate fractal interpolation functions (FIFs), a class of FIFs with vertical scaling factor functions are presented and the analytical properties of smoothness and stability are proved.     

Optimal control of singular systems via orthogonal functions

In this paper two new recursive algorithms are presented for computing optimal control law of linear time-invariant singular systems with quadratic performance index by using the elegant properties of block-pulse functions (BPFs) and shifted Legendre polynomials (SLPs). Also a unified approach is given to solve the optimal control problem of singular systems via BPFs or SLPs. Two numerical examples are included to demonstrate the validity of the proposed algorithms and approach.     

Sensitivity analysis of linear systems using orthogonal functions

Some useful properties of coefficients for determining the sensitivity functions of linear systems are given. These properties simplify the numerical computations of sensitivity functions with respect to the system parameters.     

Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions

Two numerical techniques are presented for solving the solution of Riccati differential equation. These methods use the cubic B-spline scaling functions and Chebyshev cardinal functions. The methods consist of expanding the required approximate solution as the elements of cubic B-spline scaling function or Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the new techniques. The methods are easy to implement and produce very accurate results.     

An application of the interpolating scaling functions to wave packet propagation

The wave packet propagation in the basis of interpolating scaling functions (ISF) is studied. The ISF are well known in the multiresolution analysis based on spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding to an equidistantly spaced grid. They have compact support of the size determined by the order of the underlying interpolating polynomial. In this basis the potential energy matrix is diagonal. The kinetic energy matrix is sparse, and in the 1D case, has a band-diagonal structure. An important future of the basis is that matrix elements of a Hamiltonian are exactly computed by means of simple algebraic transformations efficiently implemented numerically. Therefore, the number of grid points and the order of the underlying interpolating polynomial can easily be varied allowing one to approach the accuracy of pseudospectral methods in a regular manner, similar to the high order finite difference methods. The results for the calculation of the H+H₂ collinear collision shows that the ISF provide one with an accurate and efficient representation for use in wave packet propagation method.     

Reducing the number of parameters of a fuzzy system using scaling functions

Learning techniques are tailored for fuzzy systems in order to tune them or even for deriving fuzzy rules from data. However, a compromise between accuracy and interpretability has to be found. Flexible fuzzy systems with a large number of parameters and high degrees of freedom tend to function as black boxes. In this paper, we introduce an interpretation of fuzzy systems that enables us to work with a small number of parameters without loosing flexibility or interpretability. In this way, we can provide a learning algorithm that is efficient and yields accuracy as well as interpretability. Our fuzzy system is based on extremely simple fuzzy sets and transformations using interpretable scaling functions of the input variables.     

A relativistic extension of the complex scaling method using oscillator basis functions

We develop the complex scaling method within the relativistic framework by expanding the Dirac spinors in the complete set of eigensolutions of a harmonic oscillator potential, and present the theoretical formalism of describing the discrete bound and resonant states on the same footing. Based on a well established and frequently used model, we demonstrate the utility and applicability of the extended method and examine the stability of the results with respect to the variations of the parameters of the model. Satisfactory agreements are found for all the calculated results in comparison with some other calculations in references. Especially, the present calculation in the nonrelativistic limit gives a consistent result with that in the nonrelativistic calculation.     

Exploiting submodular value functions for scaling up active perception

In active perception tasks, an agent aims to select sensory actions that reduce its uncertainty about one or more hidden variables. For example, a mobile robot takes sensory actions to efficiently navigate in a new environment. While partially observable Markov decision processes (POMDPs) provide a natural model for such problems, reward functions that directly penalize uncertainty in the agent’s belief can remove the piecewise-linear and convex (PWLC) property of the value function required by most POMDP planners. Furthermore, as the number of sensors available to the agent grows, the computational cost of POMDP planning grows exponentially with it, making POMDP planning infeasible with traditional methods. In this article, we address a twofold challenge of modeling and planning for active perception tasks. We analyze \(\rho \)POMDP and POMDP-IR, two frameworks for modeling active perception tasks, that restore the PWLC property of the value function. We show the mathematical equivalence of these two frameworks by showing that given a \(\rho \)POMDP along with a policy, they can be reduced to a POMDP-IR and an equivalent policy (and vice-versa). We prove that the value function for the given \(\rho \)POMDP (and the given policy) and the reduced POMDP-IR (and the reduced policy) is the same. To efficiently plan for active perception tasks, we identify and exploit the independence properties of POMDP-IR to reduce the computational cost of solving POMDP-IR (and \(\rho \)POMDP). We propose greedy point-based value iteration (PBVI), a new POMDP planning method that uses greedy maximization to greatly improve scalability in the action space of an active perception POMDP. Furthermore, we show that, under certain conditions, including submodularity, the value function computed using greedy PBVI is guaranteed to have bounded error with respect to the optimal value function. We establish the conditions under which the value function of an active perception POMDP is guaranteed to be submodular. Finally, we present a detailed empirical analysis on a dataset collected from a multi-camera tracking system employed in a shopping mall. Our method achieves similar performance to existing methods but at a fraction of the computational cost leading to better scalability for solving active perception tasks.     

Characterizations of detectability notions in terms of discontinuous dissipation functions

We consider a new Lyapunov-type characterization of detectability for non-linear systems without controls, in terms of lower-semicontinuous (not necessarily smooth, or even continuous) dissipation functions, and prove its equivalence to the GASMO (global asymptotic stability modulo outputs) and UOSS (uniform output-to-state stability) properties studied in previous work. The result is then extended to provide a construction of a discontinuous dissipation function characterization of the IOSS (input-to-state stability) property for systems with controls. This paper complements a recent result on smooth Lyapunov characterizations of IOSS. The utility of non-smooth Lyapunov characterizations is illustrated by application to a well-known transistor network example.     

Integration in finite terms with elementary functions and dilogarithms

In this paper, we report on a new theorem that generalizes Liouville’s theorem on integration in finite terms. The new theorem allows dilogarithms to occur in the integral in addition to transcendental elementary functions. The proof is based on two identities for the dilogarithm, that characterize all the possible algebraic relations among dilogarithms of functions that are built up from the rational functions by taking transcendental exponentials, dilogarithms, and logarithms. This means that we assume the integral lies in a transcendental tower.