Four rendezvous problems for n non-identical simple-integral plants

Four rendezvous problems for n non-identical simple-integral plants (K ₁/s, K₂/s,[tdot],K_n/s), with a common input x(t) = A_mi^m u(t), m = 0,1,2,[tdot], and initial output conditions of (ξ₁,ξ₂,[tdot], ξ_n), are studied in this paper. These rendezvous problems, generally speaking, are concerned with transferring the n plants outputs to meet one another in a finite time (at a. fixed target or on a non-stationary one).     

Lumped model of a non-linear distributed system by finite element method

The finite element method has been used to find an approximate lumped parameter model of a non-linear distributed parameter system. A one dimensional non-linear dispersion system is considered. The space domain is divided into a finite set of k elements. Each element, has n nodes. Within each element the concentration is represented by C(x,t)^(e) = [N][C] ^T where [N] = [n₁(x),n₂(x), [tdot] n_n(x)] and [C] = [C₁(t),C₂(t), [tdot] C_n(t)]. By using Galerkin's criterion a set of (k × n ? n+ 1) first order differential equations are obtained for C_i(t). These equations are solved by an iterative method. The concepts are illustrated by an example taking five three-node elements in the space domain. The results are compared with those obtained by a finite difference method. It is shown that the finite element method can be used effectively in modelling of a distributed system by a lumped system.     

General eigenvalue placement in linear control systems by output feedback

Consider the time-invariant system E[xdot] = Ax + Bu, y = Cx where E is a square matrix that may be singular. The problem is to find constant matrices K and L, such that the feedback law u = Ky+L[ydot] yields x = exp (λt)v_i (where v_i is some constant vector) for some preassigned λ_i (i=l, 2, [tdot], r). This problem is equivalent to that of finding K and L which makes a preassigned λ _i an eigenvalue corresponding to the general eigenvalue problem {λ(E ? BLC) ? (A + BKC)}v=0. Using matrix generalized inverses, a method is developed for the construction of a linear system of equations from which the elements of K and L may be computed.     

An approximation method for high-order fractional derivatives of algebraically singular functions

The fractional derivative D^qf(s) (0≤s≤1) of a given function f(s) with a positive non-integer q is defined in terms of an indefinite integral. We propose a uniform approximation scheme to D^qf(s) for algebraically singular functions f(s)=s^αg(s) (α>−1) with smooth functions g(s). The present method consists of interpolating g(s) at sample points t_j in [0,1] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer m such that m<q<m+1, the use of high-order derivatives g⁽ⁱ⁾(0) and g⁽ⁱ⁾(1) (0≤i≤m) at both ends of [0,1] as well as g(t_j), t_j∈[0,1] in interpolating g(s), is essential to uniformly approximate D^q{s^αg(s)} for 0≤s≤1 when α≥q−m−1. Some numerical examples in the simplest case 1<q<2 are included.     

An enlarged family of packing polynomials on multidimensional lattices

HereN = {0, 1, 2, ...}, while a functionf onN ^m or a larger domain is apacking function if its restrictionf|N ^m is a bijection ontoN. (Packing functions generalize Cantor's [1]pairing polynomials, and yield multidimensional-array storage schemes.) We call two functionsequivalent if permuting arguments makes them equal. Alsos(x) =x ₁ + ... +x _m when x = (x ₁,...,x _m); and such anf is adiagonal mapping iff(x) <f(y) whenever x, y εN ^m ands(x) <s(y). Lew [7] composed Skolem's [14], [15] diagonal packing polynomials (essentially one for eachm) to constructc(m) inequivalent nondiagonal packing polynomials on eachN ^m. For eachm > 1 we now construct 2^m−2 inequivalent diagonal packing polynomials. Then, extending the tree arguments of the prior work, we obtaind(m) inequivalent nondiagonal packing polynomials, whered(m)/c(m) → ∞ asm → ∞. Among these we count the polynomials of extremal degree.     

Optimal testing policy for a computer system with fault margin

The periodic testing policy is considered, for a computer system which fails when the total number of hidden faults exceeds a threshold level N of tolerence. Periodic tests are scheduled at times kT(k = 1,2,[tdot]) to detect hidden faults. A fault occurs according to a non-homogeneous Poisson processes. When the ith fault occurs at age S_i < T, (i) it causes a system failure with probability p₁(S_i), (ii) it becomes a hidden fault with probability p₂(S_i) and is accumulated, or (iii) it is removed with probability p₃(S_i). The expected cost rate is derived. The aim of the paper is to find the optimal T which minimizes the expected cost rate per unit time of the policy. Various special cases are considered.     

Factorization of m-D polynomials in linear m-D factors

This paper presents a solution to the factorization problem of high-order m-D polynomials into special m-D factors that are linear in all variables. By special it is meant that the factors have the form of a sum of one independent variable, say z ₁, and a general first-order polynomial not involving z ₁, i.e. z ₁ + a _j2 z ₂ + [tdot] + a _jm z _m (j = 1, 2,[tdot],N ₁). Two theorems providing the necessary and sufficient conditions for this factorization are given, which have an algorithmic form and are thus appropriate for direct computer use. The results are illustrated by means of three examples.     

Blending curves for landing problems by numerical differential equations II. Numerical methods

A landing curve of airplane is a blending curve that smoothly joins the two given boundary points, which is described by the parametric functions x(s), y(s), and z(s) governed by a system of ordinary differential equations (ODEs) with certain boundary conditions. In Part I, Mathematical Modelling [1], existence and uniqueness of the ODE system are explored to produce the optimal landing curves in minimum energy. In this paper, numerical techniques are provided by the finite element method (FEM) using piecewise cubic Hermite polynomials, to give the optimal solutions. An important issue is how to deal with infinite solutions occurring in the landing problems reported in [1]. Moreover, error analysis is made, and numerical examples are carried to verify the theoretical results made. This paper displays again the effectiveness and flexibility of the ODE approach to complicated blending curves. Besides, the numerical techniques in this paper can be applied directly to other landing and trajectory problems given in [1], as well as other kinds of blending curves and surfaces of airplane, ships, grand building, and astronautic shuttle-station.     

Time evolution of a system of two valued interacting elements : a microscopic interpretation of birth and death equations

We consider here one of the simplest possible systems with N interacting particles. It has the following features : (i) the state variable of each particle takes the values σ_i(= ?1 ; (ii) the interaction is chosen in such a way to preserve the symmetry of the distribution function p(σ₁, σ₂, [tdot], σ _N ; t) with respect to the σ _i and (iii) the evolution of the system is defined in a stochastic way by the transition probabilities of each particle as depending on the state of all other particles. The master equation of this Markov process is shown to be the equation of a general birth and death process in one dimension. More precisely, the birth and death process is : linear if the particles are independent ; quadratic if there is a binary interaction ; or cubic if there is a third-order interaction. We develop the reduced distribution equations hierarchy (which is the analogue of the BBGKY hierarchy) and we study under what conditions this hierarchy closes. Then we show that for specific systems there is a conserved quantity (in the mean) and we discuss for what kind of intercation there is respectively an H-theorem and a postulate of equal a priori probabilities at equilibrium. It appears in particular that this postulate should not be true in the strong form in which it is usually stated.     

On decoupling the H-optimal sensitivity problem for products of plants

In this note we show how to solve the H^∞-optimal sensitivity problem for a SISO plant P(s) = P₁(s)P₂(s), given the solutions for P₁(s), P₂(s). This allows us to solve the problem for systems of the form e^−hsP₀(s), where P₀(s) is the transfer function of a stable, LTI, finite dimensional system.     

On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫₀ ^t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p _i t and q _i t, 0<p _i ,q _i ≤1 and complex numbers a,b _i and c _i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u _it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M _m (t) of v(th) or M^ _m (t) of u _it (th), t∈[0,1] such that the rational approximant v(h) or u _it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)−y(h)|=O(h ^{2 m +1}). If they exist, then we actually give the conditions of M _m (t) and M^ _m (t), respectively. Received September 17, 1998; revised September 30, 1999     

A new subclass of cyclic Goppa codes

We propose a subclass of cyclic Goppa codes given by separable self-reciprocal Goppa polynomials of degree two. We prove that this subclass contains all reversible cyclic codes of length n, n | (q ^m ± 1), with a generator polynomial g(x), g(α ^±i ) = 0, i = 0, 1, α ⁿ = 1, α ∈ GF(q ^2m ).     

Synthesis of a control system for a single-input multiple-output plant

Given a constrained plant P with input x and outputs y ₁ y ₂,[tdot],y _n , the assigned specifications require that, at each instant, y i follows the corresponding reference value r j for one index i, while for the other indices, j≠i, y j should be smaller or equal to r j. A quantitative synthesis technique is presented to solve this problem, that reduces it, in quite a transparent way, to the solution of several single-input single-output problems.     

Generation of robust root loci for linear systems with parametric uncertainties in an ellipsoid

Given a parametric polynomial family p(s; Q) := {ⁿ _k=0 a_k (q)s^k : q ] Q}, Q R ^m , the robust root locus of p(s; Q) is defined as the two-dimensional zero set _p,Q := {s ] C:p(s; q) = 0 for some q ] Q}. In this paper we are concerned with the problem of generating robust root loci for the parametric polynomial family p(s; E) whose polynomial coefficients depend polynomially on elements of the parameter vector q ] E which lies in an m-dimensional ellipsoid E. More precisely, we present a computational technique for testing the zero inclusion/exclusion of the value set p(z; E) for a fixed point z in C, and then apply an integer-labelled pivoting procedure to generate the boundary of each subregion of the robust root locus _p,E . The proposed zero inclusion/exclusion test algorithm is based on using some simple sufficient conditions for the zero inclusion and exclusion of the value set p(z,E) and subdividing the domain E iteratively. Furthermore, an interval method is incorporated in the algorithm to speed up the process of zero inclusion/exclusion test by reducing the number of zero inclusion test operations. To illustrate the effectiveness of the proposed algorithm for the generation of robust root locus, an example is provided.     

An interpolation problem associated with H-optimal design in systems with distributed input lags

A variety of H^∞ optimal design problems reduce to interpolation of compressed multiplication operators, f(s) → π_k(w(s)f(s)), where w(s) is a given rational function and the subspace K is of the form K=H² φ(s)H². Here we consider φ(s) = (1-e^α-5)/(s - α), which stands for a distributed delay in a system's input. The interpolation scheme we develop, adapts to a broader class of distributed lags, namely, those determined by transfer functions of the form B(e^−s)/b(s), where B(z) and b(s) are polynomials and b(s) = 0 implies B(e^−s) = 0.     

Modified Laguerre operational matrices for fractional calculus and applications

The modified Laguerre polynomial is defined with an additional parameter from the conventional one and is applied to approach the problems of fractional calculus. First, the operational matrices for the integration and the differentiation of the modified Laguerre polynomials are derived. The generalized operational matrices corresponding to s, 1/s, s/(s₂ + 1) ^1/2 exp [ ? s/(s + 1)] are derived as examples. Comparison of the modified Laguerre series approximate inversions of irrational Laplace transforms with exact solution shows that the present modification method is much better than the conventional one. In addition, the present proposed modified Laguerre polynomials can also be used to approximate the solution of fractional calculus which cannot be obtained from conventional Laguerre polynomials.     

Mapping Between 2-D Meshes of the Same Size

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Stochastic adaptive pole-zero assignment with convergence analysis

The adaptive control u_n is designed for the stochastic system A(z)y_n+1 = B(z)u_n+C(z)w_n+1 with unknown constant matrix coefficients in the polynomials A(z), B(z) and C(z) in the shift-back operator with the purposes that (1) the unknown matrices are strongly consistently estimated and (2) the poles and zeros are replaced in such a way that the system itself is transferred to A⁰(z)y_n+1 = B⁰(z)u_n⁰+_n+1 with given A⁰(z), B⁰(z) and u_n⁰ so that the pole-zero assignment error {_n+1} is minimized. The problem of adaptive pole-zero assignment combined with tracking is also considered in this paper. Conditions used are imposed only on A(z), B(z) and C(z).     

Numerical Computation of the Least Common Multiple of a Set of Polynomials

The paper presents a number of properties of the least common multiple (LCM) m(s) of a given set of polynomials P. These results lead to the formulation of a new procedure for computing the LCM that avoids the computation of roots. This procedure involves the computation of the greatest common divisor (GCD) z(s) of a set of polynomials T derived from P, and the factorisation of the product of the original set P, p(s) as p(s) = m(s)·z(s). The symbolic procedure leads to a numerical one, where robust methods for the computation of GCD are first used. In this numerical method the approximate factorisation of polynomials is an important part of the overall algorithm. The latter problem is handled by studying two associated problems: evaluation of order of approximation and the optimal completion problem. The new method provides a robust procedure for the computation of LCM and enables the computation of approximate values, when the original data are inaccurate.