The minimal dimension of stable faces required to guaranteestability of a matrix polytope

Considers the problem of determining whether each point in a polytope n×n matrices is stable. The approach is to check stability of certain faces of the polytope. For n⩾3, the authors show that stability of each point in every (2n-4)-dimensional face guarantees stability of the entire polytope. Furthermore, they prove that, for any k⩽n², there exists a k-dimensional polytope containing a strictly unstable point and such that all its subpolytopes of dimension min {k-1,2n-5} are stable     

The minimal dimension of stable faces required to guaranteestability of a matrix polytope: D-stability

The problem of determining whether a polytope P of n ×n matrices is D-stable-i.e. whether each point in P has all its eigenvalues in a given nonempty, open, convex, conjugate-symmetric subset D of the complex plane-is discussed. An approach which checks the D-stability of certain faces of P is used. In particular, for each D and n the smallest integer m such that D-stability of every m-dimensional face guarantees D-stability of P is determined. It is shown that, without further information describing the particular structure of a polytope, either (2n-4)-dimensional or (2n-2)-dimensional faces need to be checked for D-stability, depending on the structure of D. Thus more work needs to be done before a computationally tractable algorithm for checking D-stability can be devised     

Complete instability of a box of polynomials

The authors study the converse of V.L. Kharitonov's polynomial problem (1978) by asking whether the complete instability of a box of polynomials can be determined from extreme sets. They show that it is not enough to check the (n-4)-dimensional boundary, but prove that the complete instability of the (n-1)-dimensional boundary is sufficient     

n-D polynomial matrix equations

Linear matrix equations in the ring of polynomials in n indeterminates (n-D) are studied. General- and minimum-degree solutions are discussed. Simple and constructive, necessary and sufficient solvability conditions are derived. An algorithm to solve the equations with general n-D polynomial matrices is presented. It is based on elementary reductions in a greater ring of polynomials in one indeterminate, having as coefficients polynomial fractions in the other n-1 indeterminates, which makes the use of Euclidean division possible     

Fault-tolerant embedding of complete binary trees in hypercubes

The focus is on the following graph-theoretic question associated with the simulation of complete binary trees by faulty hypercubes: if a certain number of nodes or links are removed from an n-cube, will an (n-1)-tree still exists as a subgraph? While the general problem of determining whether a k-tree, k< n, still exists when an arbitrary number of nodes/links are removed from the n-cube is found to be NP-complete, an upper bound is found on how many nodes/links can be removed and an (n-1)-tree still be guaranteed to exist. In fact, as a corollary of this, it is found that if no more than n-3 nodes/links are removed from an (n-1)-subcube of the n-cube, an (n-1)-tree is also guaranteed to exist     

On time mapping of uniform dependence algorithms into lowerdimensional processor arrays

Most existing methods of mapping algorithms into processor arrays are restricted to the case where n-dimensional algorithms, or algorithms with n nested loops, are mapped into (n-1)-dimensional arrays. However, in practice, it is interesting to map n-dimensional algorithms into (k-1)-dimensional arrays where k<n. A computational conflict occurs if two or more computations of an algorithm are mapped into the same execution time. Based on the Hermite normal form of the mapping matrix, necessary and sufficient conditions are derived to identify mapping without computational conflicts. These conditions are used to find time mappings of n-dimensional algorithms into (k-1)-dimensional arrays, k<n , without computational conflicts. For some applications, the mapping is time-optimal     

Stabilization by reduced-order controllers

A recent result by A. Linnemann (Syst. Contr. Lett., vol.11, p.27-32, 1988) gives conditions under which a continuous-time single-loop plant of order n can be stabilized by a reduced-order controller. Specifically, if the Euclidean algorithm is applied to the numerator and denominator polynomials of the transfer function and one of the remainders is a kth-order Hurwitz polynomial, then a stabilizing controller of order n-k-1 exists. The author provides an alternative proof of this result     

Balanced parallel sort on hypercube multiprocessors

A parallel sorting algorithm for sorting n elements evenly distributed over 2^d p nodes of a d-dimensional hypercube is presented. The average running time of the algorithm is O((n log n)/p+p log 2n). The algorithm maintains a perfect load balance in the nodes by determining the (kn/p)th elements (k1,. . ., (p-1)) of the final sorted list in advance. These p-1 keys are used to partition the sorted sublists in each node to redistribute data to the nodes to be merged in parallel. The nodes finish the sort with an equal number of elements (n/ p) regardless of the data distribution. A parallel selection algorithm for determining the balanced partition keys in O(p log2n) time is presented. The speed of the sorting algorithm is further enhanced by the distance-d communication capability of the iPSC/2 hypercube computer and a novel conflict-free routing algorithm. Experimental results on a 16-node hypercube computer show that the sorting algorithm is competitive with the previous algorithms and faster for skewed data distributions     

Rotator graphs: an efficient topology for point-to-pointmultiprocessor networks

Rotator graphs, a set of directed permutation graphs, are proposed as an alternative to star and pancake graphs. Rotator graphs are defined in a way similar to the recently proposed Faber-Moore graphs. They have smaller diameter, n-1 in a graph with n factorial vertices, than either the star or pancake graphs or the k-ary n-cubes. A simple optimal routing algorithm is presented for rotator graphs. The n-rotator graphs are defined as a subset of all rotator graphs. The distribution of distances of vertices in the n-rotator graphs is presented, and the average distance between vertices is found. The n-rotator graphs are shown to be optimally fault tolerant and maximally one-step fault diagnosable. The n-rotator graphs are shown to be Hamiltonian, and an algorithm for finding a Hamiltonian circuit in the graphs is given     

Minimal parameter solution of the orthogonal matrix differentialequation

The straightforward solution of the first-order differential equation satisfied by all nth-order orthogonal matrices requires n² integrations to obtain the matrix elements. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation, and expressing the orthogonal matrix in terms of these parameters are considered in the present work. Several possibilities which are based on attitude determination in three dimensions are examined. It is concluded that not all 3-D methods have useful extensions to other dimensions, and that the 3-D Gibbs vector (or Cayley parameters) provide the most useful extension. An algorithm is developed using the resulting parameters, which are termed extended Rodrigues parameters, and numerical results are presented of the application of the algorithm to a fourth-order matrix     

A processor-time-minimal systolic array for transitive closure

Using a directed acyclic graph (DAG) model of algorithms, the authors focus on processor-time-minimal multiprocessor schedules: time-minimal multiprocessor schedules that use as few processors as possible. The Kung, Lo, and Lewis (KLL) algorithm for computing the transitive closure of a relation over a set of n elements requires at least 5n-4 parallel steps. As originally reported. their systolic array comprises n² processing elements. It is shown that any time-minimal multiprocessor schedule of the KLL algorithm's dag needs at least n²/3 processing elements. Then a processor-time-minimal systolic array realizing the KLL dag is constructed. Its processing elements are organized as a cylindrically connected 2-D mesh, when n=0 mod 3. When n≠0 mod 3, the 2-D mesh is connected as a torus     

A class of weak Kharitonov regions for robust stability of linearuncertain systems

Kharitonov's theorems are generalized to the problem of so-called weak Kharitonov regions for robust stability of linear uncertain systems. Given a polytope of (characteristic) polynomials P and a stability region D in the complex plane, P is called D-stable if the zeros of every polynomial in P are contained in D. It is of interest to know whether the D-stability of the vertices of P implies the D-stability of P. A simple approach is developed which unifies and generalizes many known results on this problem     

Stability of a family of polynomials with coefficients bounded in adiamond

The robust stability property is examined for family of nth-order real polynomials where the coefficients are bounded within a diamond in the (n+1)-dimensional space. It is shown that such a family of polynomials is Hurwitz if and only if four specially selected edge polynomials are Hurwitz     

A processor-time-minimal systolic array for cubical mesh algorithms

Using a directed acyclic graph (DAG) model of algorithms, the paper focuses on time-minimal multiprocessor schedules that use as few processors as possible. Such a processor-time-minimal scheduling of an algorithm's DAG first is illustrated using a triangular shaped 2-D directed mesh (representing, for example, an algorithm for solving a triangular system of linear equations). Then, algorithms represented by an n×n×n directed mesh are investigated. This cubical directed mesh is fundamental; it represents the standard algorithm for computing matrix product as well as many other algorithms. Completion of the cubical mesh required 3n-2 steps. It is shown that the number of processing elements needed to achieve this time bound is at least [3n^2/4]. A systolic array for the cubical directed mesh is then presented. It completes the mesh using the minimum number of steps and exactly [3n ^2/4] processing elements it is processor-time-minimal. The systolic array's topology is that of a hexagonally shaped, cylindrically connected, 2-D directed mesh     

Constructing submodule specifications and network protocols

Applications of an automated tool for module specification (ATMS) that finds the specification for a submodule of a system are presented. Given the specification of a system, together with the specification for n-1 submodules, the ATMS constructs the specification for the nth addition submodule such that the interaction among the n submodules is equivalent to the specification of the system. The implementation of the technique is based on an approach proposed by P. Merlin and G.B. Bochmann (1983). The specification of a system and its submodules consists of all possible execution sequences of their individual operations. The ATMS uses finite-state machine concepts to represent the specifications and interactions of the system and its submodules. The specification found by the ATMS for a missing module of a system is the most general one, if one exists. Application of the ATMS in the area of communication protocols is discussed. A manual process to find the specification for a missing module using the Merlin-Bochmann technique is time-consuming and prone to errors. The automated tool presented proves a reliable method for constructing such a module     

One more counterexample in n-D systems-unimodular versuselementary operations

By a simple counterexample, it is shown that there are unimodular matrices which cannot be written as products of elementary matrices in the ring of n-D polynomials. As a consequence, some methods for the solution of n-D matrix polynomial equations published recently, which are based on the elementary operations in the ring of n-D polynomials, do not work     

Cube connected Mobius ladders: an inherently deadlock-free fixeddegree network

The authors introduce a multiprocessor interconnection network, known as cube-connected Mobius ladders, which has an inherently deadlock-free routing strategy and hence has none of the buffering and computational overhead required by deadlock-avoidance message passing algorithms. The basic network has a diameter φ of 4n-1 for n2ⁿ⁺² nodes and has a fixed node degree of 4. The network can be interval routed in two stages and can be represented as a Cayley graph. This is the only practical fixed degree topology of size O(2^φ) which has an inherently deadlock-free routing strategy, making it ideally suited for medium and large sized transputer networks     

An efficient distributed knot detection algorithm

A distributed knot detection algorithm for general graphs is presented. The knot detection algorithm uses at most O(n log n+m) messages and O(m+n log n) bits of memory to detect all knots' nodes in the network (where n is the number of nodes and m is the number of links). This is compared to O(n²) messages needed in the best algorithm previously published. The knot detection algorithm makes use of efficient cycle detection and clustering techniques. Various applications for the knot detection algorithms are presented. In particular, its importance to deadlock detection in store and forward communication networks and in transaction systems is demonstrated     

A smoothly parameterized family of stabilizable, observable linearsystems containing realizations of all transfer functions of McMillandegree not exceeding n

It is shown that there is a continuously parameterized family F of n-dimensional single-input single-output (SISO) stabilizable detectable linear system Σ(p) which contains at least one realization of each reduced, strictly proper transfer function of McMillan degree not exceeding n. The parameterization map p→Σ(p) is a polynomial function in 2n indeterminates from an open convex polyhedron in R²ⁿ to the linear space of all SISO n-dimensional linear systems     

Using feedthrough for near one-step-ahead control of inverselyunstable plants

The one-step-ahead (OSA) control law is modified by the use of feedthrough to accommodate unstable plant zeros. An nth-order discrete-time system is guaranteed to track the desired input in one timestep. This is accomplished by subtracting the denominator dynamics and algebraically canceling the numerator dynamics. A problem occurs with OSA control when the plant has unstable zeros, that is, zeros that are outside the z-plane unit circle. In this case, an unstable zero is canceled and the resulting controller will be unstable. The authors address this problem. The approach taken is specific to OSA control