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We consider the Linear-Quadratic-Regulator (LQR) problem in terms of optimizing a real-valued matrix function over the set of feedback gains. Such a setup facilitates examining the implications of a natural initial-state independent formulation of LQR in designing first order algorithms. We characterize several analytical properties (smoothness, coerciveness, quadratic growth) that are crucial in the analysis of gradient-based algorithms. We then examine three types of well-posed flows for LQR: gradient flow, natural gradient flow and the quasi-Newton flow. The coercive property suggests that these flows admit unique solutions while gradient dominated property indicates that the corresponding Lyapunov functionals decay at an exponential rate; quadratic growth on the other hand guarantees that the trajectories of these flows are exponentially stable in the sense of Lyapunov.

In this paper, we present the first averaging-based extremum seeking controller able to achieve semi-global practical fixed-time asymptotic stability in static maps, where by fixed-time asymptotic stability we mean convergence via a KL bound that has a finite-time convergence property with a uniformly bounded settling time. In general, this property cannot be achieved by standard smooth extremum seeking algorithms having a Lipschitz continuous average system. The extremum seeking dynamics are based on an underlying average system that is a perturbed version of a continuous gradient flow with prescribed finite-time convergence properties, recently studied in the literature. In order to study the stability properties of the ES dynamics, we make use of averaging tools for nonsmooth dynamical systems, which allow us to link the KL bound of the average system with the KL bound that characterizes the convergence properties of the ES dynamics. Numerical simulations illustrate our results.

The design of apodization and windowing functions forms the predominant part of the beamforming process in ultrasound imaging. However, the detailed analysis of apodization in the case of beamforming for flow imaging is very limited in the literature. This paper introduces the concept of dual apodization technique in vector triangulation for ultrasound-based vector flow imaging. The approach utilizes the idea of multiple apodization to induce a steering effect at receive along with sidelobe suppression for the delay compensated radio-frequency (RF) signals. The method is investigated using extensive simulations for transverse flows with different flow profiles at different velocities, which shows an improvement in resolution and better clutter suppression. The proposed approach is further analyzed using various conventional and data adaptive apodization techniques for a gradient flow profile. The error variance in velocity magnitude estimate is as low as $5.0251\times 10^{-4}\ [m/s]^{2}$ and the mean angle error is +0.7358° using Hanning-Gaussian apodization for a transverse gradient flow with a peak velocity of 0.25 m/s.

A technique for real-time dynamic grasping force optimization is presented in this article. The force optimization is divided into online and offline two parts. The improved linear constrained gradient flow algorithm reduces the dimension of the description matrix based on the traditional gradient flow, and the non-negative linear combination method is introduced to optimize the initial force of the algorithm. Relatively optimal initial grasping force and dimensionality reduction gradient flow algorithm improve the calculating efficiency online. The corresponding illustrative examples are made to verify that the proposed method to obtain both optimal grasping force and faster convergence rate in the dynamic grasp process is feasible.

In this paper, we review the concept of the infinitesimal narrow weak rigidity (INWR). Then, we apply the concept to a formation control problem with the gradient flow law to explore almost global convergence on (d + 1)-agent formations in ℝd. We also analyze the stability of incorrect equilibrium points of the control problem. In particular, we show that all of incorrect equilibrium points are unstable. Consequently, we show that a formation with (d + 1) agents in ℝd converges to a desired equilibrium point (desired formation) exponentially fast. Finally, a simulation is presented to validate the result.

Model-based cost functions are developed to build optimal stabilizing controllers for linear and nonlinear systems. The prescribed model-based cost functions can be selected to impose a specific desired performance of the closed-loop system. The proposed approach differs from the classical optimal control formulation in the sense that the objective function and the system dynamics are fused in one criterion. As a result, the optimal control law is finally built as the solution to an unconstrained optimization problem. For systems that are not affine in the control input, dynamic feedbacks are proposed as the gradient flow of model-based performance indices.

In this paper, we examine the global stability of gradient flow dynamics associated with the problem of finding the best rank-one approximation of a given matrix. We partition the state-space into an infinite family of invariant manifolds over which the dynamics reduce to the special case of approximating a symmetric matrix. This allows us to employ a Lyapunov-based argument to explicitly characterize the region of attraction for the stable equilibrium points. This characterization proves an almost everywhere convergence for the gradient flow dynamics to the minimizers of the corresponding rank-one approximation problem.

In this paper we present a new balanced flowthrough viscosity sensor working with a torsionally resonating pipe. The whole system is machined in aluminum and is designed to be placed in the pipes of a chemical production plant. The quality factor Q and the resonance frequency fr are used as measures for the viscosity and density. We describe the sensor setup, its working principle for different resonant modes, the actuation and the readout of the sensor and present first measurement results for liquids featuring different viscosities.

This article proposes “Hexpo”, an activation function that has ability to scale the gradient and hence overcome the vanishing gradient problem. Unlike rectified linear units family which produces identity mapping on positive inputs, Hexpo has scalable limits on both positive and negative values. With parametrization, the active domain of Hexpo and the output it maps to is flexible. Thus it can alleviate the vanishing gradient problem from both gradient flow and local gradient aspect, while preserving the upper and lower bound of output. Parametrization also offers Hexpo the ability to produce the output that is close to zero. On experiments involving MNIST hand digit recognition dataset, Hexpo outperforms rectified linear units family(rectified linear unit and exponential linear unit) by the accuracy and speed of learning. On the experiments involving CIFAR-10 tiny image recognition dataset and convolutional layers, Hexpo outperforms rectified linear unit and performs similar in comparison with exponential linear unit.

The propose of this study is the calculation of optimal approximants for complete order digital filters, by the use of three different schemes based on special projections of frequency Weighted model order reduction (MOR) algorithms, based on namely, the Gradient Flow without Frequency Weighted (GF), the Frequency Weighted Balanced Truncation (FWBT), and the Gradient Flow with Frequency Weighted (FWGF). The obtained low order approximants are expected to approximate the original, complete order digital filter. To see how well each technique performs, an illustrative example is presented and by means of two robust performances criteria, the L 2 norm, and the integral-square-error (ISE) index error, the distance between original complete order filter and its approximants is measured. Moreover, some important results and conclusion are highlighted when comparing with a robust MOR technique.

We consider recovering a signal x ∈ ℝn from the magnitudes of Gaussian measurements by minimizing a second order yet non-smooth loss function. By exploiting existing concentration results of the loss function, we show that the non-convex loss function satisfies several quadratic geometrical properties. Based on these geometrical properties, we characterize the linear convergence of the sequence of function graph generated by the gradient flow on minimizing the loss function. Furthermore, we propose an accelerated version of the gradient flow, and establish an in-exact linear convergence of the generated sequence of function graph by exploiting the quadratic geometries of the loss function. Then, we verify the numerical advantages of the proposed algorithms over other state-of-art algorithms.

Robust pole assignment problem for linear systems with uncertainty is studied in this paper. The proposed gradient flow optimization algorithm is used to solve the Sylvester equations, in order that the close-loop control systems have the desired robust poles, namely, the uniformly asymptotically stable performance. The feedback gain matrix of the synthesized system can be derived from the gradient flow optimization algorithm. Robust pole assignment for linear system via the gradient flow optimization algorithm is examined in illustrative example that shows that the proposed approach has an important effect on continuous linear system with uncertainty.