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.     

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.     

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.     

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.     

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.     

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.     

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     

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.     

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.     

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.     

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     

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.     

Computing minimal distances on polyhedral surfaces

The authors implement an algorithm that finds minimal (geodesic) distances on a three-dimensional polyhedral surface. The algorithm is intrinsically parallel, in as much as it deals with all nodes simultaneously, and is simple to implement. Although exponential in complexity, it can be used with a companion gradient-descent surface-flattening algorithm that produces an optimal flattening of a polyhedral surface. Together, these two algorithms have made it possible to obtain accurate flattening of biological surfaces consisting of several thousand triangular faces (monkey visual cortex) by providing a characterization of the distance geometry of these surfaces. The authors propose this approach as a pragmatic solution to characterizing the surface geometry of the complex polyhedral surfaces which are encountered in the cortex of vertebrates     

Robust set operations on polyhedral solids

An algorithm for performing regularized Boolean operation on polyhedral solids is described. Robustness is achieved by adding symbolic reasoning as a supplemental step to resolve possible numerical uncertainty. Additionally, numerical redundancy and numerical computation based on derived quantities are reduced as much as possible. Experience with an implementation of the algorithm, using a unit-cube example as a simple test object for robustness, is discussed     

Generalized shape operators on polyhedral surfaces

This work concerns the approximation of the shape operator of smooth surfaces in R³ from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions. The estimates are confirmed by numerical experiments.     

Real-time video watermarking system on the compressed domain for high-definition video contents: Practical issues

Everyday, we encounter high-quality multimedia contents from HDTV broadcasting, DVD, and high-speed Internet services. These contents are, unhappily, processed and distributed without protection. This paper proposes a practical video watermarking technique on the compressed domain that is real-time and robust against video processing attacks. In particular, we focus on video processing that is commonly used in practice such as downscaling resolution, framerate changing, and transcoding. Most previous watermarking algorithms are unable to survive when these processings are strong or composite. We extract low frequency coefficients of frames in fast by partly decoding videos and apply a quantization index modulation scheme to embed and detect the watermark. On an Intel architecture computer, we implement a prototype system and measure performance against video processing attacks frequently occur in the real world. Simulation results show that our video watermarking system satisfies real-time requirements and is robust to protect the copyright of HD video contents.     

Practical issues in the application of occlusion to measure visual demands imposed on drivers by in-vehicle tasks

Occlusion is a practical technique to measure the visual demand imposed by in-vehicle tasks and to assess whether a task can be resumed having been interrupted. This study describes a number of important factors and variables that need to be controlled to ensure reliability of results. Training of participants on in-vehicle tasks is found to help consistency and five training sessions are required for complex tasks. No significant differences in training with and without occlusion goggles are reported. The required sample size is dependent on the variability of the task; for those investigated an appropriate sample size is found to be 14. For in-vehicle systems that exhibit a delay in response to the user, consistency is improved when these delays are excluded from timing measurements. In terms of calculating the occlusion parameter R, the within-participant basis is most consistent by taking the ratio of the respective median total shutter open time and total task times across trial repetitions completed by one participant on each task under evaluation and, for the purposes of identifying interface designs that exhibit poor resumability, the 85th percentile value is identified as most suitable. Findings from the study are discussed in terms of future application of the occlusion technique to assess in-vehicle information systems (IVIS).