A sufficient condition for output feedback stabilization ofuncertain dynamical systems

A different proof than the one given by E. Zeheb (ibid., vol.AC-31, p.1055-7, Nov. 1986) is given for the existence of an output feedback controller which stabilizes an uncertain single-input/single-output dynamical system with a linear nominal part and matched uncertainties. However, there is no need to assume that the nominal system is stable. A simple expression is obtained for the feedback gain which is necessary for the closed-loop nominal system to become strictly positive real     

A necessary and sufficient condition for feedback stabilization in a factorial ring

The necessity for a coprime fractional representation of a plant, in order to completely parametrize all stabilizing compensators by the Youla theory, is examined. It is shown that the coprimeness is necessary for systems in a factorial ring, a class inclusive of multidimensional scalar systems. Some observations on partial-state transfer functions in the feedback configuration are found to be useful in showing the necessity.     

A sufficient condition of output feedback stabilization of uncertain systems

A sufficient condition is derived for the existence of an output constant gain feedback controller which stabilizes a single-input single-output dynamical system with a linear nominal part and matched uncertainties. The sufficient condition is significantly less restrictive than a recently derived one [1]. If such a controller exists, it may readily be computed.     

A necessary and sufficient stabilization condition for discrete time-varying stochastic systems with multiplicative noise

This paper investigates a fundamental problem of stabilization for time-varying multiplicative noise stochastic systems. A necessary and sufficient stabilization condition is presented based on the receding horizon approach. The explicit time-varying controller is designed if the condition is satisfied. The presented results are new to the best of our knowledge.     

Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions

In this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control. We offer sufficient conditions for the existence of ‘Control Lyapunov functions’ that according to [3,23] and [28–30] guarantee stabilization at a specified equilibrium by means of a feedback law, which is smooth except possibly at the equilibrium. We note that the results of the paper present a local nature.     

A sufficient condition for pancyclic graphs

In 2005, Rahman and Kaykobad proved that if G is a connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for each pair x, y of distinct nonadjacent vertices in G, where d(x,y) is the length of a shortest path between x and y in G, then G has a Hamiltonian path [Inform. Process. Lett. 94 (2005) 37–41]. In 2006 Li proved that if G is a 2-connected graph of order n3 such that d(x)+d(y)+d(x,y)n+2 for each pair x,y of nonadjacent vertices in G, then G is pancyclic or G=K_n/2,n/2 where n4 is an even integer [Inform. Process. Lett. 98 (2006) 159–161]. In this work we prove that if G is a 2-connected graph of order n such that d(x)+d(y)+d(x,y)n+1 for all pairs x, y of distinct nonadjacent vertices in G, then G is pancyclic or G belongs to one of four specified families of graphs.     

A sufficient condition for global constrained extrema†

Sufficient conditions for global constrained minima and for global minima in some restricted domain of the problem are given. Both conditions express the fact that the Lagrangian function is convex, which does not require any convexity property of the functional and the constraints.     

A sufficient condition for non-overestimation in interval arithmetic

The range of an arithmetic expression on an interval can be estimated by evaluating the expression using interval arithmetic. This gives always a superset of the range, which is usually proper. However, there are cases where no overestimation error occurs. The subject of this paper is to study such cases systematically and to give a sufficient condition when interval evaluation yields the exact range. Testing the condition is inexpensive: Given the intermediate values which are obtained during the evaluation of the expression, only some comparisons and bookkeeping but no additional arithmetic operations need to be performed. We show that for certain classes of expressions there is no overestimation if the input interval has a sufficiently large distance to zero. Formulae for the minimal distance are derived for polynomials in Horner form and in factorized form.     

A new sufficient condition for Hamiltonicity of graphs

Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u)+d(v)+δ(u,v)?n+1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u,v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.     

A clique-covering sufficient condition for hamiltonicity of graphs

A graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as NP-complete problem and no satisfactory characterization for hamiltonian graphs has been found. There are several necessary conditions for hamiltonicity and since the seminal work of Dirac in 1952, many sufficient conditions were found. These conditions are usually expressed in terms of node degree, connectivity, density, toughness, independent sets, regularity and forbidden subgraphs. In this article we give an extended clique decomposition condition ensuring the hamiltonicity of a large class of graphs. Then we discuss briefly the possibility of broader extensions as well as algorithmic issues.     

A necessary and sufficient minimality condition for uncertainsystems

A necessary and sufficient condition is given for the exact reduction of systems modeled by linear fractional transformations (LFTs) on structured operator sets. This condition is based on the existence of a rank-deficient solution to either of a pair of linear matrix inequalities which generalize Lyapunov equations; the notion of Gramians is thus also generalized to uncertain systems, as well as Kalman-like decomposition structures. A related minimality condition, the converse of the reducibility condition, may then be inferred from these results and the equivalence class of all minimal LFT realizations defined. These results comprise the first stage of a complete generalization of realization theory concepts to uncertain systems. Subsequent results, such as the definition of and rank tests on structured controllability and observability matrices are also given. The minimality results described are applicable to multidimensional system realizations as well as to uncertain systems; connections to formal powers series representations also exist     

基于强传递性的线性时不变系统同时稳定化的充分条件

本文考虑了离散线性时不变系统的同时稳定化问题.基于强传递性的方法,本文利用互素因子分解法建立了同时存在多个稳定控制器的充分条件.此外,在这个条件下,本文提出了同时稳定3个系统的控制器的设计方法.最后,给出了一个例子验证结果的有效性.     

A sufficient condition for the stability of interval matrixpolynomials

The root location of sets of scalar polynomials whose coefficients are confined to intervals and the associated extension to eigenvalues of sets of constant matrices whose coefficients are contained in intervals are reviewed. A central result for complex scalar interval polynomials is a theorem developed by V.L. Kharatonov (1978), which states that each member of a set of such polynomials is stable if and only if eight special polynomials from the set are stable. The case of interval matrix polynomials is examined, and a Kharitonov-like result for their strong stability is provided. This in turn yields a sufficient condition for stability of a set of interval matrix polynomials     

A sufficient condition for the stability of matrix polynomials

A sufficient condition is presented for the stability of the matrix polynomials based on algebraic properties of the matrix coefficients. The stability condition is derived from the Lyapunov theory by a multivariable feedback system that is associated to the matrix polynomial. Illustrative examples are given. A block-Schwarz form related to the matrix polynomial is obtained directly from the given realization algorithm     

A state-space interpretation of simultaneous stabilization

A state-space interpretation of some previously known results on simultaneous stabilization is given. The results reviewed were obtained using the Yuola lemma     

A sufficient condition for structural decouplability of linear nonsquare systems

This note presents a sufficient condition for structural decouplability of linear time-invariant nonsquare systems. Using graph representation for structured systems, the case of a square system (withm-inputs andm-outputs) is examined first. The condition for decouplability of this case is characterized by the existence of a "complete shortest matching" in the graph of that system. The above result plays an essential role in deriving a sufficient condition for the case of nonsquare systems. This is accomplished by reducing a nonsquare system into an appropriate square system, and trying to find a complete shortest matching after removing some arcs from the graph of the transformed system.     

A sufficient condition for optimal control problems with time delays

This paper considers a class of systems modeled by ordinary differential equations which contain a delay in both the state and control variables. A sufficient condition for the optimal control of such systems is presented. The condition is a modification of a sufficient condition for optimal control problems without time delays. The application of the result is illustrated by examples.