Partial exponential stability of semi-state systems

The general class of non-linear non-autonomous semi-state systems is considered from the viewpoint of stability of system motions. A theorem is proved giving necessary and sufficient conditions for partial exponential stability of arbitrary motion of these systems over finite or infinite time intervals.

The results are illustrated by the analysis of a non-linear electrical circuit.     

Equations of perturbed motions and stability of state and semi-state systems

Problems in the construction of equations of perturbed motions for state and semi-state systems are discussed. It is shown that in most cases it is not possible to construct the equations of the perturbed motions, so that the classical approach to the stability analysis of a selected motion is not applicable. The necessary and sufficient stability conditions that provide the testing of stability of an arbitrary system motion, without the utilization of the equations of perturbed motions, are developed. The results obtained concern the original Lyapunov stability definition and contain as special cases some other existing stability results. Two examples are worked out: one of them concerns the analysis of a non-linear feedback system that does not possess a state model.     

离散奇异摄动系统稳定性分析

讨论了关于离散奇异摄动系统的摄动稳定性解析条件, 给出了离散奇异摄动系统两种模型下的摄动稳定性充要条件. 通过分析状态矩阵的结构特性, 得出了系统稳定所需要的状态矩阵形式和限制条件. 同时, 通过广义系统途径也得到了 2-D 离散奇异摄动 Roessor 模型稳定的充分条件.     

ER isomorphisms and uniqueness conditions

Given a collection of entity types (database tables) there is usually more than one way to model their associations. Consequently, two data models may appear different while essentially they are the same. To simplify the task of comparing data models, necessary and sufficient conditions are defined for a collection of entity types to have a unique Entity Relationship Diagram (ERD). The sufficient conditions for uniqueness are translated into modeling constraints that can be easily used to build an Entity-Relationship model. It is shown that the constraints do not prevent the representation of information requirements except for rare types of involuted relationships that seldom appear in the real world. Additionally, sufficient conditions are established for two ERDs to be isomorphic. All of this is done under the assumption that relationships are degree 2 or less. The results are extended to models containing relationships of higher degree.     

The regulator problem with indefinite quadratic cost for boundary control systems: the finite horizon case

In this paper we study the finite horizon non-standard LQ-problem for an abstract dynamics, which models a large class of hyperbolic-like partial differential equations. We provide necessary/sufficient conditions for finiteness of the value function corresponding to the control problem. Sharpness of sufficient conditions is shown by means of counterexamples. The specific features of the finite, in contrast to infinite, horizon case are illustrated.     

Generalized sampled data feedback control of distributed parameter systems

In the sampled-data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled-data control. For finite-dimensional systems the necessary conditions are also known to be sufficient. We show that this equivalence extends to the infinite-dimensional case if the underlying semigroup is analytic. However, for general systems, the necessary conditions are not sufficient, nor are the sufficient conditions necessary. We prove this by a single example with a free parameter – one choice of parameter shows that the necessary conditions are too weak, and another choice shows that the sufficient conditions are too strong.     

Generalized sampled data feedback control of distributed parameter systems

In the sampled-data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled-data control. For finite-dimensional systems the necessary conditions are also known to be sufficient. We show that this equivalence extends to the infinite-dimensional case if the underlying semigroup is analytic. However, for general systems, the necessary conditions are not sufficient, nor are the sufficient conditions necessary. We prove this by a single example with a free parameter – one choice of parameter shows that the necessary conditions are too weak, and another choice shows that the sufficient conditions are too strong.     

Ergodicity of hidden Markov models

In this paper we study ergodic properties of hidden Markov models with a generalized observation structure. In particular sufficient conditions for the existence of a unique invariant measure for the pair filter-observation are given. Furthermore, necessary and sufficient conditions for the existence of a unique invariant measure of the triple state-observation-filter are provided in terms of asymptotic stability in probability of incorrectly initialized filters. We also study the asymptotic properties of the filter and of the state estimator based on the observations as well as on the knowledge of the initial state. Their connection with minimal and maximal invariant measures is also studied. Work partially supported by grants MIUR-PRIN 2001, PBZ KBN 016/P03/99 and IMPAN-BC Centre of Excellence     

State feedback control of linear systems in the presence of devices with finite signal-to-noise ratio

This paper provides necessary and sufficient conditions for mean-square state-feedback stabilization of linear systems whose white noise sources have intensities affinely related to the variance of the signal they corrupt. Systems with such noise sources have been called FSN (finite signal-to-noise) models, and the stability results provided in prior work were only sufficient conditions. Upper bounded ?₂ performance is also guaranteed herein by solving a control problem which is non-convex only due to a certain scaling parameter. By fixing this parameter convex programming algorithms provide controllers.     

Uniqueness of rest points for optimal control models in economics

Optimal control solutions for dynamic multi-sector economic models in general may exhibit multiple rest points having different stability properties, and under such circumstances the qualitative behavior of the system depends crucially upon initial conditions. Both necessary and sufficient conditions for a unique rest point solution are presented, and a class of ‘regular economies’ is defined for which this uniqueness property prevails.     

MFD models and time delays; some consequences for identification

A general relationship is established between the coefficients of a general MFD model and its associated Markov parameters. This result is used to derive necessary and sufficient conditions for the appearance of time delays in general MFD models. These conditions involve dependencies between some coefficients of the MFD models. Finally, it is indicated how these dependencies can possibly be incorporated in equation-error identification techniques retaining the favourable property of linearity in the parameters.     

Near optimality conditions in stochastic control of jump diffusion processes

This paper is concerned with necessary as well as sufficient conditions for near-optimality of controlled jump diffusion processes. Necessary conditions for a control to be near-optimal are derived, using Ekeland’s variational principle and some stability results on the state and adjoint processes, with respect to the control variable. In a second step, we show that the necessary conditions for near-optimality, are in fact sufficient for near-optimality provided some concavity conditions are fulfilled. Finally, as an illustration some examples are solved explicitly.     

Petri网共享T型子网合成结构性质分析及其应用

为了解决系统设计中的子系统共享问题,提出了经由Petri网共享T-型子网构成共享T-型子网合成网的解决方案;研究了共享T-型子网合成网的结构性质,提出了共享T-型子网合成网保持结构有界性、守恒性、可重复性、相容性、P-不变量、T-不变量、公平性和结构活性的充分条件或充要条件;特别在证明结构活性保持性的过程中,体现了Petri网层次化的描述方法。本文的结果可为Petri网系统合成性质的考察提供有效途径,为复杂大系统的分析提供重要手段,并特别适合于一类系统的设计和分析,具有一定的实用价值。     

k轨道任务分配问题的可解性条件: 图论方法

将图论及一种新的数学分析工具–—矩阵的半张量积(semi-tensor product of matrices, STP), 作为研究工具,通过研究图的k内稳定集的充分必要条件, 研究了k轨道任务分配问题的可解性条件. 定义了图的顶点子集的特征向量, 利用STP方法得到图的k内稳定集新的若干充分必要条件. 基于这些新的充分必要条件, 建立了能够搜索出图的所有k内稳定集的两种算法. 进而将上述结果应用到k轨道任务分配问题, 得到了该问题可解性的两个充分必要条件. 此外, 通过这些充分必要条件, 也发现了一些有趣的现象. 例如, 完全最优方案(completely optimal schedules)的存在.     

The positive solutions of difference equations relevant to an economic problem

We establish some sufficient and necessary conditions, as well as sufficient conditions for the existence and nonexistence of positive solutions of certain difference equations relevant to economic theory.     

An interpolated controller for stabilization of a plant withvariable operating conditions

This paper considers stabilization of a linear time-invariant single-input/single-output plant with variable operating conditions. It is assumed that the plant is described by a linear interpolation of proper stable coprime factorizations of the transfer functions of two representative models which are defined at two representative operating points. A linear interpolation of stabilizing controllers for the representative models is applied to the interpolated plant. The necessary and sufficient condition for the interpolated plant to be stabilizable by the interpolated controller is presented using the Nevanlinna-Pick theory. It is also shown that simultaneous stabilizability of the two representative models is necessary for such stabilization. An example is given, which illustrates that the class of stabilizable plants via interpolated controllers is larger than that of plants stabilizable by fixed controllers     

Structured system models Part 2. Directed graphs and boolean matrices

Parameterized system models and their augmented controllability indices are studied. Analytically parameterized system models arc shown to have generic index lists, i.e. lists which arc actual lists for all parameter values except those in a proper analytic variety. For structured system models, as a subset of analytically parameterized system models, there exists a known graph-theoretic method for determining the Hermite indices. A new method for examining both Hermite and Kronecker indices is introduced. The method is based on a necessary graph-theoretic condition for the examined columns of the controllability matrix to be linearly independent. Additional sufficient conditions are given to complete the method. The given graph-theoretic results are also expressed using boolean matrices associated with graphs.     

Multidimensional state-space models: A comparative overview

This paper provides a comparative survey of the existing multidimensional state-space models. The 2-Dimensional state-space models are first presented and compared to each other from both the mathematical and the applications point of view. Then a canonical 3-Dimensional state space realization is considered and its applications are briefly discussed. Finally the special cases of 2-D and 3-D first-order transfer functions are investigated and necessary and sufficient conditions for existence of respective minimal state-space realizations are introduced. The models considered in the paper have been used for treating various design problems of multidimensional systems such as model mathing, transfer function separation, etc., as can be seen in the corresponding references.     

Fractional discrete-time consensus models for single- and double-summator dynamics

The leader–following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the systems, interactions between opinions are defined like in Krause and Cucker–Smale models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear systems. In this paper, we investigate fractional-order models of opinions for the single- and double-summator dynamics of discrete-time by analytical methods as well as by computer simulations. The necessary and sufficient conditions for the leader–following consensus are formulated by proposing a consensus control law for tracking the virtual leader.