Restrictive Sufficient Conditions Research Articles

Solving Nonlinear Algebraic Loops Arising in Input-Saturated Feedbacks

We propose a dynamic augmentation scheme for the asymptotic solution of the nonlinear algebraic loops arising in well-known input saturated feedbacks typically designed by solving linear matrix inequalities (LMIs). We prove that the existing approach based on dynamic augmentation, which replaces the static loop by a dynamic one through the introduction of a sufficiently small time constant, works under some restrictive sufficient well-posedness conditions, requiring the existence of a diagonal Lyapunov matrix. However it can fail in general, even when the algebraic loop is well-posed. Then, we propose a novel approach whose effectiveness is guaranteed whenever well-posedness holds. We also show how this augmentation allows preserving the guaranteed region of attraction with Lyapunov-based designs, as long as a gain parameter is sufficiently large. We finally propose an adaptive version of the scheme where this parameter is adjusted online. Simulation results show the effectiveness of the proposed solutions.

Improved Sufficient Conditions Based on RIC of Order 2s for IHT and HTP Algorithms

Reconstructing an s-sparse signal x from an un- derdetermined noisy linear model arises in many applications. The iterative hard thresholding (IHT) and hard thresholding pursuit (HTP) algorithms are two popular sparse signal recovery algorithms. Since their recovery performances can be theoreti- cally characterized by their sufficient conditions of stable sparse signal recovery, it is essential to establish less restrictive sufficient conditions. This work develops δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2s</sub> -based sufficient conditions for stable recovery with IHT and HTP. The two new sufficient conditions are significantly less restrictive than the state-of-the- art ones for IHT and HTP.

Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation

AbstractWe introduce a new model ${\mathscr{P}}$-CNAR(s) of sequences of discrete random variables with long memory determined by semibinomial conditionally nonlinear autoregression of order s ∈ ℕ with small number of parameters. Probabilistic properties of this model are studied. For parameters of the model ${\mathscr{P}}$-CNAR a family of consistent asymptotically normal statistical FB-estimates is suggested and the existence of an efficient FB-estimate is proved. Computational advantages of FB-estimate w.r.t. maximum likelihood estimate are shown: less restrictive sufficient conditions for uniqueness, explicit form of FB-estimate, fast recursive computation algorithm under extension of the model ${\mathscr{P}}$-CNAR. Subfamily of “sparse” FB-estimates that use some subset of frequencies of s-tuples is constructed, the asymptotic variance minimization problem within this subfamily is solved.

Complexity and convergence certification of a block principal pivoting method for box-constrained quadratic programs

Active-set (AS) methods for quadratic programming (QP) are particularly suitable for real-time optimization, as they provide a high-quality solution in a finite number of iterations. However, they add or remove one constraint at each iteration, which makes them inefficient when the number of constraints and variables grows. Block principal pivoting (BPP) methods perform instead multiple changes in the working set in a single iteration, resulting in a much faster execution time. The infeasible primal–dual method proposed by Kunisch and Rendl (KR) (Kunisch and Rendl, 2003) is a BPP method for box-constrained QP that is particularly attractive when reducing the time for finding an accurate solution is crucial, such as in linear model predictive control (MPC) applications. However, the method is guaranteed to converge only under very restrictive sufficient conditions, and tight bounds on the worst-case complexity are not available. For a given set of box-constrained QP’s that depend on a vector of parameters, such as those that arise in linear MPC, this paper proposes an algorithm that computes offline the exact number of iterations and flops needed by the KR method in the worst-case, and the region of the parameter space for which the method converges or is proved to cycle.

Statistical Estimation of Parameters for Binary Conditionally Nonlinear Autoregressive Time Series

The problem of statistical parameter estimation is considered for binary GLM-based autoregression with the link function of general form and the base functions (regressors) nonlinear w.r.t. the lagged variables. A new consistent asymptotically normal frequencies-based estimator (FBE) is constructed and compared with the classical MLE. It is shown that the FBE has less restrictive sufficient conditions of uniqueness than the MLE (does not need log-concavity of the inverse link) and can be computed recursively under the model extension. The sparse version of the FBE is proposed and the optimal model-dependent weight matrices (parameterizing the FBE) are found for the FBE and for the sparse FBE. The proposed empirical choice of the subset of s-tuples for the sparse FBE is examined by numerical and analytical examples. Computer experiments for comparison of the FBE versus the MLE are performed on simulated and real (genetic) data.

Continuity Properties of the Inf-Sup Constant for the Divergence

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary.

Representations of knot groups into SLn(ℂ) and twisted Alexander polynomials

Let $\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $\Gamma$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for the existence of irreducible deformations. We also give a more restrictive sufficient condition for the existence of irreducible deformations. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.

Razumikhin-type stability criteria for differential equations with delayed impulses

This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.

Sufficient conditions for expected utility to imply drawdown-based performance rankings

The least restrictive sufficient condition for expected utility to imply Sharpe ratio rankings is the location and scale (LS) property (see Sinn, 1983; Meyer, 1987). The normal, the extreme value, and many other distributions commonly used in finance satisfy this property. We argue that the LS property is also sufficient for expected utility to imply drawdown-based performance measure rankings, because for investment funds satisfying the LS condition, the Sharpe ratio and drawdown-based performance measures result in identical rankings. Hence, the same conditions that provide an expected utility foundation for the Sharpe ratio also provide a foundation for drawdown-based performance measures. We conclude that from a decision-theoretic perspective, drawdown-based performance measures are as good as the Sharpe ratio.

Sufficient Conditions for Expected Utility to Imply Drawdown-Based Performance Rankings

The least restrictive sufficient condition for expected utility to imply Sharpe ratio rankings is the location and scale (LS) condition (see Meyer, 1987). The LS condition includes the normal and many other (asymmetric and leptokurtic) distributions commonly used in finance. In this paper we argue that this condition is also sufficient for expected utility to imply drawdown-based performance measure rankings because for investment funds satisfying the LS condition, it does not matter whether funds are ranked with the Sharpe ratio or with a drawdown-based performance measure as the rankings are identical. Hence, the same conditions that provide an expected utility foundation for the Sharpe ratio also provide an expected utility foundation for drawdown-based performance measures. Theoretically drawdown-based performance measures are as good as the Sharpe ratio.

Generalized quasilinearization method for nonlinear boundary value problems with integral boundary conditions

The quasilinearization method coupled with the method of upper and lower solutions is used for a class of nonlinear boundary value problems with integral boundary conditions. We obtain some less restrictive sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result.

Multi-path nonlinear dynamic compensation for rudder roll stabilization

The design of rudder roll stabilizers (RRS) for ships is complicated by yaw-axis coupling, non-minimum phase shift, and the angle and rate limits of the rudder. These saturations tend to limit the available feedback and restrict RRS implementation to vessels with very fast rudders. This paper describes the application of a Nyquist-stable RRS with a multiple feedback path nonlinear dynamic compensator to provide high performance with slower steering mechanisms used on larger vessels. The restrictive sufficient condition of absolute stability is satisfied for angle or rate saturation. Computer simulation results are included to show the efficacy of the approach.

Delay-dependent robust stability of linear systems with non-commensurate time varying delays

In this article we investigate the robust stability of uncertain systems with non-commensurate time varying delays. Two types of perturbations are considered. First, we consider time varying and unknown but norm bounded uncertainties. Next systems with polytopic-type uncertainties are studied. Based on Finsler's lemma we propose a new method for achieving less restrictive sufficient conditions for delay-dependent robust stability. The results are presented in terms of linear matrix inequalities. Three examples are given for illustration and comparison with previous results.

Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time‐delay systems

AbstractIn this paper, an improved linear matrix inequality (LMI)‐based robust delay‐dependent stability test is introduced to ensure a larger upper bound for time‐varying delays affecting the state vector of an uncertain continuous‐time system with norm‐bounded‐type uncertainties. A quasi‐full‐size Lyapunov–Krasovskii functional is chosen and free‐weighting matrix approach is employed. Less restrictive sufficient conditions are derived for robust stability of time‐varying delay systems with norm‐bounded‐type uncertainties. Moreover, the investigation of the stabilization problem with memoryless state‐feedback control is presented such that the stabilizability criteria are obtained in terms of matrix inequalities, which can be solved via utilizing a cone complementarity minimization algorithm. Finally, the problem of output feedback stabilization for square systems is also taken into consideration. The output feedback stabilizability criteria are derived in the form of linear matrix inequalities, which are convex and can be easily solved using interior point algorithms. A plenty of numerical examples are presented indicating that the proposed stability and stabilization methods effectively improve the existing results. Copyright © 2006 John Wiley &amp; Sons, Ltd.

A sufficient condition for asymptotic normality of the normalized quadratic form [formula omitted

Many sufficient conditions of asymptotic normality of the normalized quadratic form Ψ n ( f , g ) have been proposed since 1958. The less restrictive was given in the paper of L. Giraitis and D. Surgailis (1990). Using a linear operator approach, it is possible to produce an even less restrictive sufficient condition on the couple of functions ( f , g ) . To cite this article: V. Solev, L. Gerville-Reache, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

On nonlinear stability of baroclinic fronts

We study the nonlinear stability of baroclinic fronts in a channel with variable bottom topography in the framework of the model proposed by Swaters and Flierl (1991). Sufficient conditions for nonlinear stability of steady flows in a channel with arbitrary bottom topography are obtained. We consider quite general perturbations which include non-zero perturbations of the circulations of the velocity along the rigid boundaries. The special case of zonal (x-independent) topography is treated in more detail. Here we make use of an additional invariant of the governing equations (the momentum). This allows us to obtain a less restrictive (than in the general situation) sufficient condition for nonlinear stability.

Some extensions of the Hewitt-Savage zero-one law

Let ℳ denote the class of infinite product probability measures μ=μ1×μ2×⋯ defined on an infinite product of replications of a given measurable space (X, A), and let ℋ denote the subset of ℳ for which μ(A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member μ, of ℳ belongs to ℋ. In the present paper, the class ℋ is shown to possess several closure properties. E.g., if μ∈ℋ and μ0≪μn for some n ≧1, then μ0×μ1×μ2×...∈ℋ. While the current results do not permit a complete characterization of ℋ they demonstrate conclusively that ℋ is a much larger subset of ℳ than previous results indicated. The interesting special case X={0,1} is discussed in detail.

Separability, Welfare Economics and the Theory of Second Best

This paper reconsiders the Lipsey-Lancaster [4] theory of second best and formulates less restrictive sufficient conditions for the preservation of the usual first-order conditions for the non-deviant commodities. In particular, we derive a more general class of separable objective and constraint functions which yield the result that, even in the presence of a Lipsey-Lancaster constraint for deviant commodities, the usual Paretian optimum conditions for the non-deviants continue to be optimal. Consider the maximization problem: max F(x1, ..., xn)

Stability of orbiting gyrostats in the elementary cases

The stability of the elementary cases of equilibrium of an orbiting gyrostat have been examined previously by two other authors using two different methods. In the first instance only two of the three equilibrium cases has been analysed, while in the second the author arrives at unnecessarily restrictive sufficient conditions for stability. The present paper systematically reexamines the stability question using the two previous methods and a third which makes a direct appeal to the Thomson-Tait-Chetaev theorem (not a logically independent method, but one having a considerably different guise). It extends the first technique to the missing case and significantly sharpens all previous results. In comparing the three approaches it establishes that all three, when properly used, lead to precisely the same set of stability inequalities, but that the third method involves substantially less labor. The stability results are interpreted in a certain inertia phase plane.

Solutions of Partial-Wave Dispersion Relations. II

The study of the partial-wave dispersion relation (PWDR) is continued, being extended to the relativistic problem in the presence of inelasticity. Less restrictive sufficient conditions on the left-hand-cut discontinuity $\ensuremath{\Delta}T$ are obtained to guarantee a solution of the PWDR problem when $\ensuremath{\Delta}T$ has both positive and negative parts. The effect of Castillejo-Dalitz-Dyson poles on solutions is also discussed.