Reliable quasi-static object manipulation and robotic locomotion require the verification of the stability of equilibria under unilateral contacts and friction. In a recent paper, Posa et al. (2016) ...demonstrated that sums-of-squares (SOS) programming can be used to verify the Lyapunov stability of planar systems via Lyapunov's direct method if impacts are inelastic. However, this method appears to be too conservative to verify the stability of some remarkably simple examples. In this article, an extension of Lyapunov's direct method is proposed, which makes use of a piecewise smooth Lyapunov function and allows a temporary increase of the Lyapunov function along a motion trajectory. In addition, a modified SOS formulation enables the investigation of planar systems with partially elastic contacts. The proposed method remains compatible with SOS programming techniques. The improved stability test is successfully applied to a point mass on a slope and to a rigid body with two contact points. For the latter, several mechanisms of instability have been demonstrated experimentally, but the exact conditions of Lyapunov stability are unknown.
•Simulated Type I errors rates of ANOVA based on GLM and LM models for CATA data.•Simulated testing powers of ANOVA based on the GLM and LM models for CATA data.•Predicted precisions using GLM and LM ...models for CATA data.•The claim that logistic regressions violate Type I error rates is not generally true.•Liberty or conservativeness is not a criterion of validity or invalidity of a test.
Some discussions about statistical models used for analysis of CATA data appear in recent issues of this journal. This paper is a further discussion on the topic, following Bi and Kuesten (2022) (Food Quality and Preference, 95), related to Meyners and Hasted (2021, 2022) (Food Quality and Preference, 92, 95). This paper presents some statistical analyses for a real consumer CATA dataset using a generalized linear model (GLM) and a linear model (LM), respectively. The main objectives are to simulate Type I error levels of ANOVA; to simulate the empirical testing powers of ANOVA; and to compare testing results and predicted precisions based on different models. Meyners and Hasted (2022) claim that logistic regressions violate Type I error rates. The simulation results show that the claim is not generally true and suggest that violating Type I error rates is due to two-way ANOVA with a special type of Sums of Squares (Type2SS), not due to the GLM. Meyners and Hasted (2022) conclude that the GLM or logistic regression is invalid and flawed for analysis of CATA data. We disagree with the conclusion because we cannot find any convincing reasoning supporting the conclusion. It is true that the test of two-way ANOVA with Type2SS based on the GLM is more liberal than that with Type1SS and that based on the LM. Liberty and conservativeness are a characteristic of a test, not a criterion to judge validity or invalidity of a test. Meyners and Hasted (2022) advocate that GLM should be precluded for the use of CATA data. Our position is that the GLM deserves to be a standard and first selected practice, at least one of the useful and valid methods for analysis of CATA data. The advocacy of precluding the GLM for analysis of CATA data is unacceptable.
We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type S×R with S={x¯∈Rn|g1(x¯)≥0,…,gs(x¯)≥0} such that the quadratic module generated by g1,…,gs in ...RX1,…,Xn is archimedean, and we provide a degree bound for the representation of a polynomial f∈RX1,…,Xn,Y which is positive on S×R as an explicit element of the quadratic module generated by g1,…,gs in RX1,…,Xn,Y. We also include an example to show that an additional assumption is necessary for Putinar's Positivstellensatz to hold on cylinders of this type.
Sums of squares in Macaulay2 Cifuentes, Diego; Kahle, Thomas; Parrilo, Pablo
The journal of software for algebra and geometry,
3/2020, Letnik:
10, Številka:
1
Journal Article
We consider the problem of generating motion plans for a robot that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, and disturbances. Furthermore, we ...consider scenarios where these plans must be generated in real time, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Our approach is to pre-compute a library of “funnels” along different maneuvers of the system that the state is guaranteed to remain within (despite bounded disturbances) when the feedback controller corresponding to the maneuver is executed. We leverage powerful computational machinery from convex optimization (sums-of-squares programming in particular) to compute these funnels. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances.
We demonstrate and validate our method using extensive hardware experiments on a small fixed-wing airplane avoiding obstacles at high speed (~12 mph), along with thorough simulation experiments of ground vehicle and quadrotor models navigating through cluttered environments. To our knowledge, these demonstrations constitute one of the first examples of provably safe and robust control for robotic systems with complex nonlinear dynamics that need to plan in real time in environments with complex geometric constraints.
The global and semi-global analytic hypoellipticity on the torus is proved for two classes of sums of squares operators, introduced by P. Albano, A. Bove, and M. Mughetti, satisfying the Hörmander ...condition and which fail to be either locally or microlocally analytic hypoelliptic.
A sign pattern matrix is a matrix whose entries are from the set {+,−,0}. For a sign pattern matrix A, the qualitative class of A, denoted Q(A), is the set of all real matrices whose entries have ...signs given by the corresponding entries of A. An n×n sign pattern matrix A requires all distinct eigenvalues if every real matrix in Q(A) has n distinct eigenvalues. Li and Harris (2002) 13 characterized the 2×2 and 3×3 irreducible sign pattern matrices that require all distinct eigenvalues, and established some useful general results on n×n sign patterns that require all distinct eigenvalues. In this paper, we characterize 4×4 irreducible sign patterns that require four distinct eigenvalues. This is done by characterizing 4×4 irreducible sign patterns that require four distinct real eigenvalues, that require four distinct nonreal real eigenvalues, or that require two distinct real eigenvalues and a pair of conjugate nonreal eigenvalues. The last case turns out to be much more involved. Some interesting open problems are presented. Three important tools that are used in the paper are the following: the discriminant of a polynomial; the fact that if a square sign pattern matrix A requires all distinct eigenvalues then A requires a fixed number of real eigenvalues; and the known result that if A is a “k-cycle” sign pattern then for each B∈Q(A), the k nonzero eigenvalues of B are evenly distributed on a circle in the complex plane centered at the origin.
Sums of squares I: Scalar functions Korobenko, Lyudmila; Sawyer, Eric
Journal of functional analysis,
03/2023, Letnik:
284, Številka:
6
Journal Article
Recenzirano
This is the first in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. A result of C. Fefferman and D. H. Phong shows that every C3,1 ...nonnegative function on Rn can be written as a finite sum of squares of C1,1 functions, and was used by them to improve Gårding's inequality, and subsequently by P. Guan to prove regularity for certain degenerate operators.
In this paper we investigate sharp criteria sufficient for writing a smooth nonnegative function f on Rn as a finite sum of squares of C2,δ functions for some δ>0, and we denote this property by saying f is SOSregular. The emphasis on C2,δ, as opposed to C1,1, arises because of applications to hypoellipticity for smooth infinitely degenerate operators in the spirit of M. Christ, which are pursued in the third paper of this series.
Thus we consider the case where f is smooth and flat at the origin, and positive away from the origin. Our sufficient condition for such an f to be SOSregular is that f is ω-monotone for some modulus of continuity ωs(t)=ts, 0<s≤1, where ω-monotone meansf(y)≤Cω(f(x)),y∈Bx,and where Bx=B(x2,|x|2) is the ball having a diameter with endpoints 0 and x (this is the interval (0,x) in dimension n=1). On the other hand, we show that if ω is any modulus of continuity with limt→0ω(t)ωs(t)=∞ for all s>0, then there exists a smooth nonnegative function f that is flat at the origin, and positive away from the origin, that is notSOSregular, answering in particular a question left open by Bony.
Refinements of these results are given for f∈C4,2δ, and the related problem of extracting smooth positive roots from such smooth functions is also considered.
We present an example of a strictly positive polynomial with rational coefficients that can be decomposed as a sum of squares of polynomials over R but not over Q. This answers an open question by C. ...Scheiderer posed as the second question in 14, Section 5.1. We verify that the example we construct defines a nonsingular projective hypersurface, giving also a positive answer to the third question posed in that section.
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