Positively invariant polyhedral sets of discrete-time linear systems

The problem of the existence of positively invariant polyhedral sets for linear discrete-time dynamical systems is studied. In the first part of the paper, necessary and sufficient conditions for a given polyhedral set to be a positively invariant set of a linear system are obtained. Then, the spectral properties of systems possessing this kind of invariant set are established. Finally the class of systems possessing positively invariant polyhedral cones is studied.     

On the positive invariance of polyhedral sets for discrete-time systems

In this note, necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a linear discrete-time system are established.     

线性控制系统的无界多面体不变集

本文利用非光滑分析方法,讨论了线性控制系统的无界多面体不变集问题.当无界多面体的极方向满足一定条件时,得到了该无界多面体为一类线性控制系统弱不变集的判别方法.然后在更一般的线性控制系统下给出了无界多面体为强不变集的充分条件.最后给出两个应用实例.     

Simple LP-type criteria for positively invariant polyhedral sets

Let S be a convex polyhedral set in the state space of a linear dynamical system. A new proof of an algebraic condition for the set S to be positively invariant is given by the dual set of S. The condition obtained in this correspondence can be examined by solving a set of standard LP-type formulas. It is also shown that the conventional algebraic conditions have redundancies in the LP-type formulas. Discrete-time systems as well as continuous-time systems are considered     

On the positive invariance of polyhedral sets in fractional-order linear systems

In this paper, we derive conditions for a given polyhedral set to be a positively invariant set with respect to fractional-order linear time-invariant (FO-LTI) systems with fractional order of 0<α<1$0<\alpha <1$. FO-LTI systems are described using Riemann–Liouville operator with initialization response. Then, the conditions are obtained from Farkas’ lemma and the definition of the Mittag-Leffler function. Furthermore, we apply these conditions to the constrained stabilization.     

Mesh watermarking based projection onto two convex sets

This paper proposes 3D-mesh watermarking based on projection onto convex sets (POCS). As such, a 3D-mesh model is iteratively projected onto two constraint convex sets until the convergence condition is satisfied. The sets consist of a robustness set and invisibility set designed to embed the watermark. The watermark can be extracted without the original mesh model using the decision values and index that the watermark was embedded with. Experimental results verify that the watermarked mesh model has both robustness against mesh simplification, cropping, scaling, rotation, translation, and vertex randomization and invisibility.     

多胞约束LPV系统多面体不变集RMPC算法

针对一类输入和输出受约束且具有多胞结构的离散LPV 系统, 提出一种基于多面体不变集的鲁棒模型预测控制(RMPC) 算法. 选取一系列收敛于原点的离散状态点, 计算每个状态的反馈控制率, 构建相应的多面体不变集. 在每一个采样时刻, 确定包含当前状态的最小多面体不变集, 通过计算与相邻两个多面体不变集的位置关系, 执行连续的状态反馈控制率. 仿真结果表明, 相比椭圆不变集离线RMPC算法, 所提出算法扩大了系统的稳定区域, 取得了保守性较小的结果.

     

On polyhedral estimates for reachable sets of discrete-time systems with bilinear uncertainty

We describe techniques for constructing external and internal polyhedral (parallelepiped-valued) estimates for reachability sets of discrete-time systems with bilinear uncertainty, i.e., systems with originally linear but incompletely specified dynamics, when system coefficients exhibit uncertainty of the interval type. Our primary attention is focused on internal estimates. The evolution of reachability set estimates is given by recursive relations. We give numerical modeling results.     

基于样本投影分布的平衡不平衡数据集分类*

提出一种平衡不平衡数据集统一分类方法,首先得到训练样本基于支持向量机（SVM）超平面法线方向上的投影;再借助支持向量数据描述（SVDD）对训练样本投影分布进行描述;测试样本在此基础上实现分类。平衡或不平衡数据集都可采用相同的方法进行分类。实验表明该方法能够同时对平衡或不平衡数据集进行有效的分类。     

A linear programming algorithm for invariant polyhedral sets of discrete-time linear systems

In this paper we formulate necessary and sufficient conditions for an arbitrary polyhedral set to be a positively invariant set of a linear discrete-time system. Polyhedral cones and linear subspaces are included in the analysis. A linear programming algorithm is presented that enables practical application of the results stated in this paper.     

Practical issues in the implementation of self-tuning control

Implementation aspects of self-tuning regulators are discussed in the paper. There is a large discrepancy between simulation or academic algorithms and practical algorithms. In the idealized environment of simulations it is easy to get different types of adaptive algorithms to perform well. In practice the situation is quite opposite. The adaptive or self-tuning controller must be able to handle nonlinearities, unmodelled dynamics and unmodelled disturbances over a wide range of operating conditions. Some aspects of how to implement self-tuning controllers are discussed in the paper. This includes robustness, signal conditioning, parameter tracking, estimator wind-up, reset action and start-up. Different ways to use the prior knowledge about the process are also discussed.     

Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones

This paper presents a new (geometrical) approach to the computation of polyhedral (robustly) positively invariant (PI) sets for general (possibly discontinuous) nonlinear discrete-time systems possibly affected by disturbances. Given a β-contractive ellipsoidal set , the key idea is to construct a polyhedral set that lies between the ellipsoidal sets and . A proof that the resulting polyhedral set is contractive and thus, PI, is given, and a new algorithm is developed to construct the desired polyhedral set. The problem of computing polyhedral invariant sets is formulated as a number of quadratic programming (QP) problems. The number of QP problems is guaranteed to be finite and therefore, the algorithm has finite termination. An important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs.     

Practical issues in knowledge management

It is easy to get caught up in the potential surrounding KM. Yes, it is promising and, if companies can realize that promise, KM will become important. In point of fact, though, KM is a difficult, time-consuming, and expensive undertaking even under the best of circumstances. Unfortunately, today's world doesn't present the best of circumstances for most enterprises. Budgetary and resource constraints combine to put IT under the microscope of definable metrics, such as return on investment. This mindset challenges KM to deliver tangible and predictable results, when most KM benefits are difficult to realistically articulate. I examine two areas that present practical problems to any KM initiative. Rather than making a statement about knowledge or how to manage it, I simply want to identify the inherent challenges that any significant KM initiative faces. Coordinating the distribution of information in large companies is a challenge. It pays to think about fitting the information you have to an appropriate communication channel     

A polyhedral approach to solving multicriterion combinatorial optimization problems over sets of polyarrangements

Multicriterion discrete optimization problems over feasible combinatorial sets of polyarrangements are considered. Structural properties of feasible domains and different types of efficient solutions are investigated. Based on the ideas of Euclidean combinatorial optimization and the major criterion method, a polyhedral approach to the solution of the problems is developed and substantiated. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 118-126, May-June 2009.     

Computational and approximational complexity of combinatorial problems related to the committee polyhedral separability of finite sets

In [1, 2], results on the computational and approximational complexity of a minimum affine separating committee (MASC) problem were obtained for finite sets A, B ⊂ ℚ _n . In particular, it was shown that this problem is NP-hard and does not belong to the class Apx (under the assumption that P ≠ NP). Nevertheless, questions concerning the bounds for its effective approximability threshold and for the computational complexity of a number of practically important particular cases of the problem obtained by imposing additional constraints, for example, by fixing the dimension of the space, remained open. In this paper, a lower bound is presented for the polynomial approximability threshold of the problem in the general case, and the intractability of the problem in spaces of fixed dimension greater than unity is proved. In particular, it is shown that the problem of committee separability remains hard even when it is formulated on the plane (i.e., in the simplest non-trivial case). This result follows from the fact that the well-known PC problem on covering a finite planar set by straight lines, whose hardness was proved in [3], is polynomially reducible to the problem under consideration. The method of reduction represents a modification of the method that was described in [4] and was used there for proving the hardness of problems on piecewise linear separability of finite sets on the plane. Mikhail Yur’evich Khachai. Born 1970 in Krasnotur’insk, Sverdlovsk region. Graduated from the Faculty of Mathematics and Mechanics, Ural State University, Yekaterinburg, in 1993. Received candidate’s degree in mathematical cybernetics in 1996 and doctoral degree in 2005. Since 1994 he has been with the Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences. Since 1997 he has headed the Department of Pattern Recognition at the same institute. Scientific interests: theory and methods of patter recognition learning, committee (perceptron) decision rules, and theory and methods of improper optimization problems and combinatorial optimization. Author of more than 60 publications, including 10 papers in Pattern Recognition and Image Processing.     

Research issues in model-based visualization of complex data sets

At the most abstract level, data visualization maps discrete values computed over an n-dimensional domain onto pixel colors. It is largely a dimension-reducing process justified by its leverage on human perceptual capacities for extracting information from visual stimuli. The difficulty is to implement a mapping that reveals the data characteristics relevant to the application at hand. Effective visualization solutions let the user control the process parameters interactively and enhance the automatically extracted features. We argue for an intelligent, model-based approach to visualization, which extracts the intrinsic data characteristics and constructs multiresolution graphics models suitable for interactive rendering on commercially available hardware adapters. The model-based approach has four parts, which we summarize     

An off-line robust MPC algorithm for uncertain polytopic discrete-time systems using polyhedral invariant sets

In this paper, an off-line synthesis approach to robust model predictive control (MPC) using polyhedral invariant sets is presented. Most of the computational burdens are moved off-line by computing a sequence of state feedback control laws corresponding to a sequence of polyhedral invariant sets. At each sampling time, the smallest polyhedral invariant set that the currently measured state can be embedded is determined. The corresponding state feedback control law is then implemented to the process. The controller design is illustrated with two examples. Comparisons between the proposed algorithm and an ellipsoidal off-line robust MPC algorithm have been undertaken. The proposed algorithm yields a substantial expansion of the stabilizable region. Therefore, it can achieve less conservative result as compared to an ellipsoidal off-line robust MPC algorithm.     

Practical issues in designing knowledge-based expert systems

This paper examines problems, practical issues, and considerations in the design of knowledge-based expert systems. The state-of-the-art as represented by numerous systems in financial planning, accounting, and capital budgeting domains is summarized.     

Practical issues in multivariable feedback control performance assessment

This paper is concerned with practical issues in control loop performance assessment of MIMO processes. Performance assessment of feedback controllers is treated in the H₂ framework. A more practical benchmark, which is specified in terms of desired closed-loop dynamics, is proposed for performance assessment.     

On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams

This paper presents a three-dimensional extension of graphic statics using polyhedral form and force diagrams for the design of compression-only and tension-only spatial structures with externally applied loads. It explains the concept of 3D structural reciprocity based on Rankine’s original proposition for the equilibrium of spatial frames. It provides a definition for polyhedral reciprocal form and force diagrams that allows including external forces and discusses their geometrical and topological characteristics. This paper furthermore provides a geometrical procedure for constructing a pair of reciprocal polyhedral diagrams from a given polyhedron representing either the form or force diagram of a structural system. Using this method, this paper furthermore suggests a design strategy for finding complex funicular spatial forms in pure compression (or tension), based on the construction of force diagrams through the aggregation of convex polyhedral cells. Finally, it discusses the effect of changes in the geometry of the force diagram on the geometry of the form diagram and the distribution of forces in it.