Improved L 2–L ∞ filtering for stochastic time-delay systems

The L ₂–L _∞ filtering problem for continuous-time polytopic uncertain stochastic time-delay systems is investigated. The main purpose is to design a full-order filter guaranteeing a prescribed L ₂–L _∞ attenuation level for the filtering error system. A simple alternative proof is given for an enhanced LMI (linear matrix inequality) representation of L ₂–L _∞ performance. Based on the criterion which keeps Lyapunov matrices out of the products of system dynamic matrices, a sufficient condition for the existence of a robust estimator is formulated in terms of LMIs. The corresponding filter design is cast into a convex optimization problem. A numerical example is employed to demonstrate the feasibility and advantage of the proposed design.     

A novel L1/2 regularization shooting method for Cox’s proportional hazards model

Nowadays, a series of methods are based on a L ₁ penalty to solve the variable selection problem for a Cox’s proportional hazards model. In 2010, Xu et al. have proposed a L _1/2 regularization and proved that the L _1/2 penalty is sparser than the L ₁ penalty in linear regression models. In this paper, we propose a novel shooting method for the L _1/2 regularization and apply it on the Cox model for variable selection. The experimental results based on comprehensive simulation studies, real Primary Biliary Cirrhosis and diffuse large B cell lymphoma datasets show that the L _1/2 regularization shooting method performs competitively.     

Exponential L 2-L ∞ filtering for a class of stochastic system with Markovian jump parameters and mixed mode-dependent time-delays

This paper discusses the exponential L ₂-L _∞ filtering problem of a class of nonlinear stochastic systems with Markovian jumping parameters and mixed mode-dependent time-varying delays. By introducing a new multiple mode-dependent Lyapunov-Krasovskii functional, stochastic analysis is conducted. The condition for the existence of mode-dependent L ₂-L _∞ filter, in which the filtering error is guaranteed to be exponentially stable with prescribed L ₂-L _∞ performance, is developed. The developed criterion is delay-range-dependent, mode-dependent and decay-rate-dependent. Based on the derived criterion, the L ₂-L _∞ filtering problems are solved. The mode-dependent filter coefficients can be obtained by solving a set of linear matrix inequalities (LMIs). Numerical simulations are presented to illustrate the effectiveness of the proposed approach.     

LTI approximation of nonlinear systems via signal distribution theory

L₂ and L₁ optimal linear time-invariant (LTI) approximation of discrete-time nonlinear systems, such as nonlinear finite impulse response (NFIR) systems, is studied via a signal distribution theory motivated approach. The use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems, including system identification problems. Specifically, a very explicit solution to the L₂ (least squares) LTI approximation problem for NFIR systems is obtained in this manner. Furthermore, the L₁ (least absolute deviations) LTI approximation problem for NFIR systems is essentially reduced to a linear programming problem. Active LTI modelling emphasizes model quality based on the intended use of the models in linear controller design. Robust stability and LTI approximation concepts are studied here in a nonlinear systems context. Numerical examples are given illustrating the performance of the least squares (LS) method and the least absolute deviations (LAD) method with LTI models against nonlinear unmodelled dynamics.     

L 2-gain analysis and anti-windup design of discrete-time switched systems with actuator saturation

This paper investigates L2-gain analysis and anti-windup compensation gains design for a class of discrete-time switched systems with saturating actuators and L2 bounded disturbances by using the switched Lyapunov function approach.For a given set of anti-windup compensation gains,we firstly give a sufficient condition on tolerable disturbances under which the state trajectory starting from the origin will remain inside a bounded set for the corresponding closed-loop switched system subject to L2 bounded disturbances.Then,the upper bound on the restricted L2-gain is obtained over the set of tolerable disturbances.Furthermore,the antiwindup compensation gains aiming to determine the largest disturbance tolerance level and the smallest upper bound of the restricted L2-gain are presented by solving a convex optimization problem with linear matrix inequality(LMI) constraints.A numerical example is given to illustrate the effectiveness of the proposed design method.     

L2–L∞ fuzzy control for Markov jump systems with neutral time-delays

The L₂–L_∞ fuzzy control problem is considered for nonlinear stochastic Markov jump systems with neutral time-delays. By means of Takagi–Sugeno fuzzy models, the fuzzy controller systems and the overall closed-loop fuzzy dynamics are constructed. A sufficient condition is firstly established on the stochastic stability using stochastic Lyapunov–Krasovskii functional. Then in terms of linear matrix inequalities techniques, the sufficient conditions on the existence of mode-dependent state feedback L₂–L_∞ fuzzy controller are presented and proved respectively for constant and time varying case. Finally, the design problems are formulated as optimization algorithms. Simulation results are exploited to illustrate the effectiveness of the developed techniques.     

Petri nets and regular languages

It is shown that the regularity problem for firing sequence sets of Petri nets is decidable. For the proof, new techniques to characterize unbounded places are introduced. In the class $\text{L}$₀ of terminal languages of labelled Petri nets the regularity problem in undecidable. In addition some lower bounds for the undecidability of the equality problems in $\text{L}$₀ and $\text{L}$ are given. L₀^λ is shown to be not closed under complementation without reference to the reachability problem.     

Noise-robust semi-supervised learning via fast sparse coding

This paper presents a novel noise-robust graph-based semi-supervised learning algorithm to deal with the challenging problem of semi-supervised learning with noisy initial labels. Inspired by the successful use of sparse coding for noise reduction, we choose to give new L₁-norm formulation of Laplacian regularization for graph-based semi-supervised learning. Since our L₁-norm Laplacian regularization is explicitly defined over the eigenvectors of the normalized Laplacian matrix, we formulate graph-based semi-supervised learning as an L₁-norm linear reconstruction problem which can be efficiently solved by sparse coding. Furthermore, by working with only a small subset of eigenvectors, we develop a fast sparse coding algorithm for our L₁-norm semi-supervised learning. Finally, we evaluate the proposed algorithm in noise-robust image classification. The experimental results on several benchmark datasets demonstrate the promising performance of the proposed algorithm.     

A theoretical limit for safety verification techniques with regular fix-point computations

In computer aided verification, the reachability problem is particularly relevant for safety analyses. Given a regular tree language L, a term t and a relation R, the reachability problem consists in deciding whether there exist a positive integer n and terms t₀,t₁,…,tn such that t₀∈L, tn=t and for every 0?i<n, (ti,ti+1)∈R. In this case, the term t is said to be reachable, otherwise it is said unreachable. This problem is decidable for particular kinds of relations, but it is known to be undecidable in general, even if L is finite. Several approaches to tackle the unreachability problem are based on the computation of an R-closed regular language containing L. In this paper we show a theoretical limit to this kind of approaches for this problem.     

A geometric programming approach for bivariate cubic L1 splines

Bivariate cubic L₁ splines provide C¹-smooth, shape-preserving interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude. The minimization principle for bivariate cubic L₁ splines results in a nondifferentiable convex optimization problem. This problem is reformulated as a generalized geometric programming problem. A geometric dual with a linear objective function and convex cubic constraints is derived. A linear system for dual-to-primal conversion is established. The results of computational experiments are presented.     

Evolution strategies based adaptive Lp LS-SVM

Not only different databases but two classes of data within a database can also have different data structures. SVM and LS-SVM typically minimize the empirical ?-risk; regularized versions subject to fixed penalty (L₂ or L₁ penalty) are non-adaptive since their penalty forms are pre-determined. They often perform well only for certain types of situations. For example, LS-SVM with L₂ penalty is not preferred if the underlying model is sparse. This paper proposes an adaptive penalty learning procedure called evolution strategies (ES) based adaptive L_p least squares support vector machine (ES-based L_p LS-SVM) to address the above issue. By introducing multiple kernels, a L_p penalty based nonlinear objective function is derived. The iterative re-weighted minimal solver (IRMS) algorithm is used to solve the nonlinear function. Then evolution strategies (ES) is used to solve the multi-parameters optimization problem. Penalty parameterp, kernel and regularized parameters are adaptively selected by the proposed ES-based algorithm in the process of training the data, which makes it easier to achieve the optimal solution. Numerical experiments are conducted on two artificial data sets and six real world data sets. The experiment results show that the proposed procedure offer better generalization performance than the standard SVM, the LS-SVM and other improved algorithms.     

Author index : Volume 10 (1980)

An explicit bound is given for a solution of the DOL sequence equivalence problem; that is, given two DOL systems G₁ and G₂ we compute explicitly positive integer constant G(G₁, G₂) such that the sequences generated by G₁ and G₂ are equal if and only if the corresponding first C(G₁, G₂) elements of the sequences are equal.     

On Approximating Polygonal Curves in Two and Three Dimensions

Given a polygonal curve P =[p₁, p₂, . . . , p_n], the polygonal approximation problem considered calls for determining a new curve P′ = [p′₁, p′₂, . . . , p′_m] such that (i) m is significantly smaller than n, (ii) the vertices of P′ are an ordered subset of the vertices of P, and (iii) any line segment [p′_A, p′_{A + 1} of P′ that substitutes a chain [p_B, . . . , p_C] in P is such that for all i where B ≤ i ≤ C, the approximation error of p_i with respect to [p′_A, p′_{A + 1}], according to some specified criterion and metric, is less than a predetermined error tolerance. Using the parallel-strip error criterion, we study the following problems for a curve P in R^d, where d = 2, 3: (i) minimize m for a given error tolerance and (ii) given m, find the curve P′ that has the minimum approximation error over all curves that have at most m vertices. These problems are called the min-# and min-ϵ problems, respectively. For R² and with any one of the L₁, L₂, or L_∞ distance metrics, we give algorithms to solve the min-# problem in O(n²) time and the min-ϵ problem in O(n² log n) time, improving the best known algorithms to date by a factor of log n. When P is a polygonal curve in R³ that is strictly monotone with respect to one of the three axes, we show that if the L₁ and L_∞ metrics are used then the min-# problem can be solved in O(n²) time and the min-ϵ problem can be solved in O(n³) time. If distances are computed using the L₂ metric then the min-# and min-ϵ problems can be solved in O(n³) and O(n³ log n) time, respectively. All of our algorithms exhibit O(n²) space complexity. Finally, we show that if it is not essential to minimize m, simple modifications of our algorithms afford a reduction by a factor of n for both time and space.     

Decentralized adaptive stabilization for interconnected systems with dynamic input-output and nonlinear interactions

This paper considers the problem of robust decentralized adaptive output feedback stabilization for a class of interconnected systems with dynamic input and output interactions and nonlinear interactions by using MT-filters and the backstepping design method. It is shown that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for the parameter estimates can be regulated to zero asymptotically, and the L₂ and L_∞ norms of the system outputs are also be bounded by functions of design parameters. The scheme is demonstrated by a simulation example.     

A linear-time algorithm for linearL 1 approximation of points

In this paper we present a linear-time algorithm for approximating a set ofn points by a linear function, or a line, that minimizes theL ₁ norm. The algorithmic complexity of this problem appears not to have been investigated, although anO(n ³) naive algorithm can be easily obtained based on some simple characteristics of an optimumL ₁ solution. Our linear-time algorithm is optimal within a constant factor and enables us to use linearL ₁ approximation of many points in practice. The complexity ofL ₁ linear approximation of a piecewise linear function is also touched upon.     

Stochastic H 2 /H ∞-control for a dynamical system with internal noises multiplicative with respect to state, control, and external disturbance

We consider an optimal control problem for a dynamical system under the influence of disturbances of both deterministic and stochastic nature. The system is defined on a finite time interval, and its diffusion coefficient depends on the control signal. The controller in the feedback circuit is assumed to be static, nonstationary, linear in the state vector, and satisfying the condition ‖L‖_∞ < γ that bounds the norm of operator L: v ?z for the transition of external disturbance to the controllable output signal. Solving the optimization H ₂/H _∞-control problem, we get three matrix functions satisfying a system of two differential equations of Riccati type and one matrix algebraic equation. In the special case of a stochastic system whose diffusion coefficient does not depend on the control signal, the system is reduced to two related Riccati equations.     

Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems

For elliptic boundary value problems of the form ?ΔU+F(x, U, U _x )=0 on Ω,B[U]=0 on ?Ω, with a nonlinearityF growing at most quadratically with respect to the gradientU _x and with a mixed-type linear boundary opeatorB, a numerical method is presented which can be used to prove the existence of a solution within a “close”H _1,4(Ω)-neighborhood of some approximate solution ω∈H ₂(Ω) satisfying the boundary condition, provided that the defect-norm ∥?Δω +F(·, ω, ω _x )∥₂ is sufficiently small and, moreover, the linearization of the given problem at ω leads to an invertible operatorL. The main tools are explicit Sobolev imbeddings and eigenvalue bounds forL or forL*L. All kinds of monotonicity or inverse-positivity assumptions are avoided.     

Algorithms for unconstrained L1 simple linear regression

Many algorithms have been proposed for L₁ simple linear regression computations, and several codes implementing these algorithms are now available. Some of these algorithms are based on a linear programming formulation while others propose a descent approach.We report on a computational study comparing L₁ computer programs for solving the simple linear regression problem.     

Optimization of the boundary control by shift or elastic force at one end of string in a sufficiently long arbitrary time

The problem of the boundary control optimization by shift or elastic force at one end of the string over a sufficiently long arbitrary time interval was solved by minimizing the L _p -norm of the derivative of control by shift or the L _p -norm of control by elastic force. Solution was established in an explicit analytical form which is valid for all p ? 1.     

A propos du lemme de substitution

We establish a new version of the Greibach syntactic lemma regarding substitutions between full trios, namely:Let ?₁ be a bifaithful trio, ?₂ a full trio, L₁, L₂ languages over disjoint alphabets. Then L₁↑?₂∈?₁□?₂ implies L₁∈?₁ or (L₂c)^*∈?₂.