Structural modification of systems using discretization and generalized sampled-data hold functions

This paper investigates the decentralized control of LTI continuous-time plants using generalized sampled-data hold functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the discrete-time equivalent model to form a hierarchical system model of the plant. This is a new application of discretization and has, as its motivation, the design of decentralized controllers using centralized methods.     

Decoupling of continuous-time linear time-invariant systems using generalized sampled-data hold functions

We investigate the possibility of input-output decoupling without stability a square continuous-time LTI plant (C, A, B) by designing discrete devices, the generalized sampled-data hold functions. By adopting sequential design procedures for the resulting diagonalizing problem, two cases are found to be solvable and yield nontrivial solutions: (1) CB nonsingular; (2) mn2m, rk(CB)=n-m where n, m are the number of states and inputs (outputs) respectively; C=I_n-C^T(CC^T)⁻¹C.     

Gain/phase margin improvement using static generalized sampled-data hold functions

This paper considers the robust control of a finite-dimensional linear time-invariant (FDLTI) continuous-time plant by a static generalized sampled-data hold function (GSHF) controller. It is shown that it is possible to design a static GSHF controller to provide a gain margin as large as desired, and a phase margin of up to 90°. The design is straight-forward, and we illustrate it with an example.     

Discrete-time loop transfer recovery via generalized sampled-data hold functions based compensator

Loop transfer recovery (LTR) techniques are known to enhance the input or output robustness properties of linear quadratic gaussian (LQG) designs. One restriction of the existing discrete-time LQG/LTR methods is that they can obtain arbitrarily good recovery only for minimum-phase plants. A number of researchers have attempted to devise new techniques to cope with non-minimum-phase plants and have achieved some degrees of success.^6-9 Nevertheless, their methods only work for a restricted class of non-minimum-phase systems. Here, we explore the zero placement capability of generalized sampled-data hold functions (GSHF) developed in Reference 14 and show that using the arbitrary zero placement capability of GSHF, the discretized plant can always be made minimum-phase. As a consequence, we are able to achieve discrete-time perfect recovery using a GSHF-based compensator irrespective of whether the underlying continuous-time plant is minimum-phase or not.     

A note on generalized rank aggregation

In the classic rank aggregation (RA) problem, we are given L input lists with potentially inconsistent orders of n elements; our goal is to find a single order of all elements that minimizes the total number of disagreements with the given orders. The problem is well known to be NP-hard, already for L=4. We consider a generalization of RA, where each list is associated with a set of orderings, and our goal is to choose one ordering per list and to find a permutation of the elements that minimizes the total disagreements with the chosen orderings. For the case in which the lists completely overlap, i.e. each list contains all n elements, we show that a simple Greedy algorithm yields a (2−2/L)-approximation for generalized RA. The case in which the lists only partially overlap, i.e. each list contains a subset of the n elements, is much harder to approximate. In fact, we show that RA with multiple orderings per list and partial overlaps cannot be approximated within any bounded ratio.     

A note on generalized Euler numbers and polynomials

In this paper we construct new generalized Euler polynomials and generalized Euler numbers attached to χ. We investigate some of the properties that are related to generalized Euler polynomials. We also derive the existence of a specific interpolation function that interpolates generalized Euler polynomials at negative integers. Finally, we investigate computationally the roots of the generalized Euler polynomials E _{n, χ}(x) for values of the index n.     

A note on ranking generalized fuzzy numbers

Ranking fuzzy numbers plays an important role in decision making under uncertain environment. Recently, Chen and Sanguansat (2011) [Chen, S. M. & Sanguansat, K. (2011). Analyzing fuzzy risk based on a new fuzzy ranking method between generalized fuzzy numbers. Expert Systems with Applications, 38(3), (pp. 2163-2171)] proposed a method for ranking generalized fuzzy numbers. It considers the areas on the positive side, the areas on the negative side and the heights of the generalized fuzzy numbers to evaluate ranking scores of the generalized fuzzy numbers. Chen and Sanguansat’s method (2011) can overcome the drawbacks of some existing methods for ranking generalized fuzzy numbers. However, in the situation when the score is zero, the results of the Chen and Sanguansat’s ranking method (2011) ranking method are unreasonable. The aim of this short note is to give a modification on Chen and Sanguansat’s method (2011) to make the method more reasonable.     

A note on neta's family of sixth-order methods for solving equations

It has been shown that some three-step methods exist which, as well as Neta's methods, require one derivative and three function evaluations per iteration, but have an asymptotic convergence rate 7 which is better than the 6 of Neta.     

A note on identifying generalized diagonally dominant matrices

A successive method of descending degree has been presented by Guo, Liu and Jia in 2002 to judge whether or not a matrix is a generalized diagonally dominant matrix. In this paper, an alternative approach and comments on one of their examples are provided. Numerical examples are given to illustrate the algorithm processing.     

Simultaneous LQ control of a set of LTI systems using constrained generalized sampled-data hold functions

In this paper, sampled-data control of a set of continuous-time LTI systems is considered. It is assumed that a predefined guaranteed continuous-time quadratic cost function, which is, in fact, the sum of the performance indices for all systems, is given. The main objective here is to design a decentralized periodic output feedback controller with a prespecified form, e.g., polynomial, piecewise constant, exponential, etc., which minimizes the above mentioned guaranteed cost function. This problem is first formulated as a set of matrix inequalities, and then by using a well-known technique, it is reformulated as a LMI problem. The set of linear matrix inequalities obtained provides necessary and sufficient conditions for the existence of a decentralized optimal simultaneous stabilizing controller with the prespecified form (rather than a general form). Moreover, an algorithm is presented to solve the resultant LMI problem. Finally, the efficiency of the proposed method is demonstrated in two numerical examples.     

Advances on the theory of generalized Bessel functions and applications to multiphoton processes

The generalized Bessel functions (GBF) are presented within the context of a more comprehensive formalism. We also discuss the partial differential equations defining GBF and their modified versions of integer order (MGBF). Numerical results are given for the first-kind MGBF as well as for the GBF with the imaginary parameter, whose importance in multiphoton processes is extensively discussed.     

Bounds on generalized structured singular values via the Perron root of matrix majorants

The purpose of this paper is to present several bounds upon the structured singular value. We first adopt a generalized notion of the structured singular value which is useful for problems where uncertainties are assumed to be bounded in an l_p-induced matrix norm. Two different type of bounds, in terms of Perron root and interaction parameters respectively, are given for the new structured singular value and their relations are discussed. These bounds are useful in that they are easy to compute and may be further analyzed to provide insights useful in design.     

A note on kernel uncorrelated discriminant analysis

In this paper, we give a theoretical analysis on kernel uncorrelated discriminant analysis (KUDA) and point out the drawbacks underlying the current KUDA algorithm which was recently introduced by Liang and Shi [Pattern Recognition 38(2) (2005) 307-310]. Then we propose an effective algorithm to overcome these drawbacks. The effectiveness of the proposed method was confirmed by experiments.     

A note on observables on MV-algebras

 A criterion when a subset of an MV-algebra is contained in the range of a (finitely additive) observable is given. It is used to prove that every at most countable subset of any MV-algebra is contained in the range of an observable. A much stronger result holds for any bold fuzzy algebra [0,1] ^S , which is whole contained in the range of a (σ-additive) observable.     

基于最小二乘支持向量机的非线性广义预测控制

通过中值定理将一类非线性系统近似为时变线性系统,然后将提出的在线最小二乘支持向量机回归(OLSSVMR)与广义预测控制相结合,提出了一种基于OLS-SVMR的自适应直接广义预测控制.利用OLS-SVMR直接设计预测控制器,并基于广义误差估计对控制器参数和广义误差估计中的未知向量进行自适应调整.理论证明了该方法可使广义误差估计值收敛到原点的一个小邻域内.仿真算例也验证了该方法的有效性.     

A note on robust stabilization for systems with parameters

We consider the problem of stabilizing a linear time invariant multivariable system whose transfer function admits a factorization which contains parametric type uncertainties. The parameters are known to take values in some region Ω_a. We first give a sufficient condition for robust stability when a proper feedback compensator is used. Here it is important to note that no assumption is made about the number of unstable plant poles. Following this we focus attention on how to compute an appropriate robustly stabilizing compensator. This is done in a two step procedure. Initially the solutions of a Diophantine equation generate a class of proper compensators, each of which stabilizes the ‘nominal’ closed loop system. This is followed by a Nyquist Theorem inspired minimization which yields the final choice. An illustrative physical example is given. The compensator thus constructed guarantees stability over . A discussion about how to compute the size of the attainable robust stability region is also included.     

Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.     

A note on fractional-order derivatives of periodic functions

In this paper, it is shown that the fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period. The fractional-order derivative considered here can be obtained based on each of the well-known definitions Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. This concluded point confirms the result of a recently published work proving the non-existence of periodic solutions in a class of fractional-order models. Also, based on this point it can be easily proved the absence of periodic responses in a wider class of fractional-order models. Finally, some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional-order systems.     

A neural network based on the generalized Fischer-Burmeister function for nonlinear complementarity problems

In this paper, we consider a neural network model for solving the nonlinear complementarity problem (NCP). The neural network is derived from an equivalent unconstrained minimization reformulation of the NCP, which is based on the generalized Fischer-Burmeister function ?_p(a,b)=‖(a,b)‖_p-(a+b). We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability as well as exponential stability. It was found that a larger p leads to a better convergence rate of the trajectory. Numerical simulations verify the obtained theoretical results.