Approach to simultaneous stabilization of linear dynamic plants with delay

For a finite family of continuous linear scalar stationary plants with a common constant delay in the control, we study the existence of a discrete controller simultaneously stabilizing the family with respect to the output and suggest an algorithm for constructing such a controller. The approach to the stabilization includes the construction of a continuousdiscrete (hybrid) closed system for each plant, passage to a discrete model of the system, and the subsequent finding of a common stabilizing discrete controller.     

Topological approach to the simultaneous stabilization of plants with delay

We consider the problem of constructing a common stabilizing controller for a family of k linear stationary nth-order plants with incommensurable delays in the control, output, and state variables. To solve it, we suggest to use a topological approach based on finding a common stabilizer in the form of a linear combination of stabilizers for the individual plants.     

Robust finite-time H∞ control of singular stochastic systems via static output feedback

This paper addresses the problem of robust finite-time stabilization of singular stochastic systems via static output feedback. Firstly, sufficient conditions of singular stochastic finite-time boundedness on static output feedback are obtained for the family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Then the results are extended to singular stochastic H_∞ finite-time boundedness for the class of singular stochastic systems. Designed algorithm for static output feedback controller is provided to guarantee that the underlying closed-loop singular stochastic system is singular stochastic H_∞ finite-time boundedness in terms of strict linear matrix equalities with a fixed parameter. Finally, an illustrative example is presented to show the validity of the developed methodology.     

Simultaneous stabilization of plants of various orders

We consider the problem of constructing a common controller stabilizing a finite family of linear dynamic plants of various orders with respect to state. For the solution, we use the method of extension of dynamic order, which permits one to reduce the original problem to a problem of simultaneous stabilization of plants of the same order. We state conditions for the existence of a universal stabilizer for a given finite family of dynamic plants.     

The dimension of intersections of convex sets II

The subject of this paper is a Helly type theorem which solves the following problem: Find the smallest number β= β(j,k,n) having the following property: IfG is any finite family ofβ + 1 convex sets in Euclideann-space and if the intersection of everyβ members ofG is at leastk-dimensional, then the intersection of all members ofG is at leastj-dimensional.     

On the complexity of a family of generalized matching problems

We consider a family of generalized matching problems called k-feasible matching (k-RM) problems, where k? {1,2,3,…} ∪ {∞}. We show each k-FM problem to be NP-complete even for very restricted cases. We develop a dynamic programming algorithm that solves in polynomial time the k-FM problem for graphs with width bounded by 2k. We also show that for any subset S of {1,2,…} ∪ {∞}, there is a set D of problem instances such that for k in S the k-FM problem is NP-complete on D, while for k not in S the k-FM problem is polynomially solvable on D.     

Stabilization of irregular systems

We consider the stabilization problem for the zero equilibrium of bilinear and affine systems in canonical form. We obtain necessary and sufficient conditions for the stabilizability of second-order bilinear and affine systems in canonical form and generalize these conditions to the case of simultaneous stabilization of a family of bilinear systems and to nth-order bilinear systems.     

Connected graphs with a minimal number of spanning trees

The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number t_μ(G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges qis ≥(₂^P)?p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case whereq<(₂^P)?p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t₀(G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.     

Scheduling an unbounded batching machine with family jobs and setup times

We consider the problem of scheduling n jobs on an unbounded batching machine that can process any number of jobs belonging to the same family simultaneously in the same batch. All jobs in the same batch complete at the same time. Jobs belonging to different families cannot be processed in the same batch, and setup times are required to switch between batches that process jobs from different families. For a fixed number of families m, we present a generic forward dynamic programming algorithm that solves the problem of minimizing an arbitrary regular cost function in pseudopolynomial time. We also present a polynomial-time backward dynamic programming algorithm with time complexity O (mn(n/m+1) ^m ) for minimizing any additive regular minsum objective function and any incremental regular minmax objective function. The effectiveness of our dynamic programming algorithm is shown by computational experiments based on the scheduling of the heat-treating process in a steel manufacturing plant.     

Digital Stabilizer Design for a Switched Linear Control Delay System

The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.     

A mechanical system containing weakly coupled subsystems

The concept of a mechanical system (model), containing coupled subsystems (MSCCS) is introduced. Examples of such a system are the Sun–planets–satellites system, a system of interacting moving objects, a system of translationally and rotationally moving celestial bodies, chains of coupled oscillators, Sommerfeld pendulums, spring systems, etc. The MSCCS subsystems and the entire system are analysed, and problems related to the investigation of the oscillations, bifurcation, stability, stabilization and resonance are stated. A solution of the oscillations problem is given for a class of MSCCS, described by reversible mechanical systems. It is proved that the autonomous MSCCS retains its family of symmetrical periodic motions (SPMs) under parametric perturbations, while in the periodic MSCCS a family of SPMs bifurcates by producing two families of SPMs. The two-body problem and the N-planet problem are investigated as applications. The generating properties of the two-body problem are established. For the N-planet problem it is proved that an (N + 1)-parametric family of orbits exists, close to elliptic orbits of arbitrary eccentricity, the family being parametrized by energy integral constant, and a syzygy of planets occurs.     

Set existence property for intuitionistic theories with dependent choice

Let TC be intuitionistic higher-order arithmetic or intuitionistic ZF (with Replacement), both with Relativized Dependent Choice, or just Countable Choice. We show that TC[boxvr]?x. A(x) (closed) gives TC[boxvr]A(t) for some comprehension term t. This solves a problem left open by Myhill in [4].     

Algorithm of robust stability region for interval plant with time delay using fractional order PID controller

By using PI^λD^μ controller, we investigate the problem of computing the robust stability region for interval plant with time delay. The fractional order interval quasi-polynomial is decomposed into several vertex characteristic quasi-polynomials by the lower and upper bounds, in which the value set of the characteristic quasi-polynomial for vertex quasi-polynomials in the complex plane is a polygon. The D-decomposition technique is used to characterize the stability boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters. We investigate how the fractional integrator order λ and the derivative order μ in the range (0, 2) affect the stabilizability of each vertex characteristic quasi-polynomial. The stability region of interval characteristic quasi-polynomial is determined by intersecting the stability region of each quasi-polynomial. The parameters of PI^λD^μ controller are obtained by selecting the control parameters from the stability region. Using the value set together with zero exclusion principle, the robust stability is tested and the algorithm of robust stability region is also proposed. The algorithm proposed here is useful in analyzing and designing the robust PI^λD^μ controller for interval plant. An example is given to show how the presented algorithm can be used to compute all the parameters of a PI^λD^μ controller which stabilize a interval plant family.     

A remark on the simultaneous <Emphasis Type="Italic">α</Emphasis>-stabilization of linear plants

In the present paper, we consider the problem of simultaneous α-stabilization (i.e., stabilization with degree of stability exceeding a given number α > 0) for linear scalar stationary plants of arbitrary order, for which we obtain numerically verifiable necessary conditions for simultaneous α-stabilization and a sufficient condition together with a constructive algorithm for the stabilizing controller design. To solve these problems, we use an approach based on a one-to-one transformation of the α-stability domains in the spaces of coefficients of polynomials.     

Parallel-machine scheduling with simple linear deterioration to minimize total completion time

We consider the parallel-machine scheduling problem in which the processing time of a job is a simple linear increasing function of its starting time. The objective is to minimize the total completion time. We give a fully polynomial-time approximation scheme (FPTAS) for the case with m identical machines, where m is fixed. This study solves an open problem that has been posed in the literature for ten years.     

Lagrangean relaxation for a lower bound to a set partitioning problem with side constraints: properties and algorithms

The problem (P) addressed here is a special set partitioning problem with two additional non-trivial constraints. A Lagrangean Relaxation (LR_u) is proposed to provide a lower bound to the optimal solution to this problem. This Lagrangean relaxation is accomplished by a subgradient optimization procedure which solves at each iteration a special 0–1 knapsack problem (KP-k). We give two procedures to solve (KP-k), namely an implicity enumeration algorithm and a column generation method. The approach is promising for it provides feasible integer solutions to the side constraints that will hopefully be optimal to most of the instances of the problem (P). Properties of the feasible solutions to (KP-k) are highlighted and it is shown that the linear programming relaxation to this problem has a worst case time bound of order O(n³).     

The distance-domination numbers of trees

The P-center problem is to locate P centers in a graph G so that the maximum distance between centers and non-centers is minimized. A related problem is to determine the maximum number of vertices that can be “covered” (within a distance of α) by a vertex set of cardinality P in G. In this paper we describe an O(n³P) algorithm which solves the maximum coverage problem on trees. We also apply the same idea to solve the P-median problem on trees.     

Clique-critical graphs: Maximum size and recognition

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K^-1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.     

Towards an Operator-Algebraic Construction of Integrable Global Gauge Theories

The recent construction of integrable quantum field theories on two-dimensional Minkowski space by operator-algebraic methods is extended to models with a richer particle spectrum, including finitely many massive particle species transforming under a global gauge group. Starting from a two-particle S-matrix satisfying the usual requirements (unitarity, Yang–Baxter equation, Poincaré and gauge invariance, crossing symmetry, . . .), a pair of relatively wedge-local quantum fields is constructed which determines the field net of the model. Although the verification of the modular nuclearity condition as a criterion for the existence of local fields is not carried out in this paper, arguments are presented that suggest it holds in typical examples such as non-linear O(N)   σ-models. It is also shown that for all models complying with this condition, the presented construction solves the inverse scattering problem by recovering the S-matrix from the model via Haag–Ruelle scattering theory, and a proof of asymptotic completeness is given.     

Inverse integer optimization with an imperfect observation

This study develops and evaluates methods for inverse integer optimization problems with an imperfect observation where the unknown parameters are the cost coefficients. We propose a cutting plane algorithm for this problem and compare it to a heuristic which solves the inverse of the linear relaxation of the forward problem. We then propose a hybrid approach that initializes the cutting plane algorithm from the solution of the heuristic.