Positive Semi-definite Function Research Articles

Adaptive Superpixel Generation for SAR Images With Linear Feature Clustering and Edge Constraint

Due to the speckle noise and complex geometric distortions within SAR images, it is still a challenge to develop a stable method that can produce superpixels with both high boundary adherence and visual compactness with low computational costs at the same time. In this paper, we propose an adaptive superpixel generation approach with linear feature clustering and edge constraint for synthetic aperture radar (SAR) images, which consists of three stages. First, the local gradient ratio pattern of each pixel in SAR imagery is extracted as features, which was previously proposed by us for SAR target recognition and has been proven to be insensitive to speckle noise. Second, we propose to use the feature-ratio-based edge detector with Gauss-shaped window instead of the traditional rectangle-shaped window to obtain the edge strength map and final edges for SAR images. Finally, a modified normalized cut (Ncut)-based superpixel generation strategy is adopted using a distance metric that simultaneously measures both the feature similarity and space proximity. In this strategy, we approximate the similarity measure through a positive semidefinite kernel function rather than directly using the traditional eigen-based algorithm. Therefore, the objective functions of weighted local K- means and Ncuts can achieve the same optimum point by appropriately weighting each point in this feature space, which greatly reduces the computation cost. During the linear feature clustering, the coefficient of variation is used to automatically determine the tradeoff factor between the feature similarity and space proximity, which helps change the superpixel shape and size adaptively according to the image homogeneity. Furthermore, the edge information is also introduced to constrain the clustering for the sake of high boundary adherence. By bridging the local K-means clustering and Ncuts, as well as the benefits of edge constraint, our method not only produces superpixels with good boundary adherence but also captures the global image structure information. Experimental results with simulated and real SAR images demonstrate the effectiveness of our proposed method, which performs better than other state-of-the-art algorithms.

Trajectory Convergence From Coordinate-Wise Decrease of Quadratic Energy Functions, and Applications to Platoons

We consider trajectories where the sign of the derivative of each entry is opposite to that of the corresponding entry in the gradient of an energy function. We show that this condition guarantees convergence when the energy function is quadratic and positive definite and partly extend that result to some classes of positive semi-definite quadratic functions including those defined using a graph Laplacian. We show how this condition allows establishing the convergence of a platoon application in which it naturally appears, due to deadzones in the control laws designed to avoid instabilities caused by inconsistent measurements of the same distance by different agents.

Modified vector controlled DFIG wind energy system based on barrier function adaptive sliding mode control

Increased penetration of wind energy systems has serious concerns on power system stability. In spite of several advantages, doubly fed induction generator (DFIG) based wind energy systems are very sensitive to grid disturbances. DFIG system with conventional vector control is not robust to disturbances as it is based on PI controllers. The objective of this paper is to design a new vector control that is robust to external disturbances. To achieve this, inner current loop of the conventional vector control is replaced with sliding mode control. In order to avoid chattering effect and achieve finite time convergence, the control gains are selected based on positive semi-definite barrier function. The proposed barrier function adaptive sliding mode (BFASMC) is evaluated by testing it on a benchmark multi-machine power system model under various operating conditions. The simulated results show that the proposed method is robust to various disturbances.

On the Aizerman Problem for Systems of Two Differential Equations

The stability of equilibria of systems of nonlinear ordinary differential equations is studied. Acriterion for the reducibility of a second-order linear systemto a scalar differential equation is given. Both positive definite and semidefinite Lyapunov functions are used to obtain sufficient conditions for the asymptotic stability (global stability) of second-order nonlinear differential equations. It is proved that the Aizerman problem has a positive solution with respect to the roots of the characteristic equation of two-dimensional systems of differential equations.

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

Abstract In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2- DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.

Discrete-Time Local Value Iteration Adaptive Dynamic Programming: Convergence Analysis

In this paper, convergence properties are established for the newly developed discrete-time local value iteration adaptive dynamic programming (ADP) algorithm. The present local iterative ADP algorithm permits an arbitrary positive semidefinite function to initialize the algorithm. Employing a state-dependent learning rate function, for the first time, the iterative value function and iterative control law can be updated in a subset of the state space instead of the whole state space, which effectively relaxes the computational burden. A new analysis method for the convergence property is developed to prove that the iterative value functions will converge to the optimum under some mild constraints. Monotonicity of the local value iteration ADP algorithm is presented, which shows that under some special conditions of the initial value function and the learning rate function, the iterative value function can monotonically converge to the optimum. Finally, three simulation examples and comparisons are given to illustrate the performance of the developed algorithm.

An alternative approach for stability analysis of discrete time nonlinear dynamical systems

The paper endeavours to solve the problem of the necessary and sufficient conditions for testing asymptotic stability of the equilibrium state without using a positive definite or semi-definite Lyapunov function for time-invariant nonlinear discrete-time dynamical systems. The solution is based on the concept of the -functions introduced in this paper. As a result, new necessary and sufficient conditions for asymptotic stability of such systems and an estimation or the exact determination of the asymptotic stability domain of the state are established. Examples are worked out to illustrate the results.

Adaptive Dynamic Programming for Discrete-Time Zero-Sum Games.

In this paper, a novel adaptive dynamic programming (ADP) algorithm, called "iterative zero-sum ADP algorithm," is developed to solve infinite-horizon discrete-time two-player zero-sum games of nonlinear systems. The present iterative zero-sum ADP algorithm permits arbitrary positive semidefinite functions to initialize the upper and lower iterations. A novel convergence analysis is developed to guarantee the upper and lower iterative value functions to converge to the upper and lower optimums, respectively. When the saddle-point equilibrium exists, it is emphasized that both the upper and lower iterative value functions are proved to converge to the optimal solution of the zero-sum game, where the existence criteria of the saddle-point equilibrium are not required. If the saddle-point equilibrium does not exist, the upper and lower optimal performance index functions are obtained, respectively, where the upper and lower performance index functions are proved to be not equivalent. Finally, simulation results and comparisons are shown to illustrate the performance of the present method.

On (conditional) positive semidefiniteness in a matrix-valued context

In a nutshell, we intend to extend Schoenberg's classical theorem connecting conditionally positive semidefinite functions $F\colon \mathbb{R}^n \to \mathbb{C}$, $n \in \mathbb{N}$, and their positive semidefinite exponentials $\exp(tF)$, $t > 0$, to the case of matrix-valued functions $F \colon \mathbb{R}^n \to \mathbb{C}^{m \times m}$, $m \in \mathbb{N}$. Moreover, we study the closely associated property that $\exp(t F(- i \nabla))$, $t>0$, is positivity preserving and its failure to extend directly in the matrix-valued context.

Value Iteration Adaptive Dynamic Programming for Optimal Control of Discrete-Time Nonlinear Systems

In this paper, a value iteration adaptive dynamic programming (ADP) algorithm is developed to solve infinite horizon undiscounted optimal control problems for discrete-time nonlinear systems. The present value iteration ADP algorithm permits an arbitrary positive semi-definite function to initialize the algorithm. A novel convergence analysis is developed to guarantee that the iterative value function converges to the optimal performance index function. Initialized by different initial functions, it is proven that the iterative value function will be monotonically nonincreasing, monotonically nondecreasing, or nonmonotonic and will converge to the optimum. In this paper, for the first time, the admissibility properties of the iterative control laws are developed for value iteration algorithms. It is emphasized that new termination criteria are established to guarantee the effectiveness of the iterative control laws. Neural networks are used to approximate the iterative value function and compute the iterative control law, respectively, for facilitating the implementation of the iterative ADP algorithm. Finally, two simulation examples are given to illustrate the performance of the present method.

Consistency of Probability Measure Quantization by Means of Power Repulsion–Attraction Potentials

This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure $$\omega $$ to be quantized. Then we show that the discrete functionals, defining the discrete quantizers as their minimizers, actually $$\Gamma $$ -converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-definite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which already found applications in the asymptotic analysis of corresponding gradient flows. To model situations where the given probability is affected by noise, we further consider a modified energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of $$\Gamma $$ -convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses.

Infinite horizon self-learning optimal control of nonaffine discrete-time nonlinear systems.

In this paper, a novel iterative adaptive dynamic programming (ADP)-based infinite horizon self-learning optimal control algorithm, called generalized policy iteration algorithm, is developed for nonaffine discrete-time (DT) nonlinear systems. Generalized policy iteration algorithm is a general idea of interacting policy and value iteration algorithms of ADP. The developed generalized policy iteration algorithm permits an arbitrary positive semidefinite function to initialize the algorithm, where two iteration indices are used for policy improvement and policy evaluation, respectively. It is the first time that the convergence, admissibility, and optimality properties of the generalized policy iteration algorithm for DT nonlinear systems are analyzed. Neural networks are used to implement the developed algorithm. Finally, numerical examples are presented to illustrate the performance of the developed algorithm.

Neuro-optimal tracking control for a class of discrete-time nonlinear systems via generalized value iteration adaptive dynamic programming approach

In this paper, a novel value iteration adaptive dynamic programming (ADP) algorithm, called "generalized value iteration ADP" algorithm, is developed to solve infinite horizon optimal tracking control problems for a class of discrete-time nonlinear systems. The developed generalized value iteration ADP algorithm permits an arbitrary positive semi-definite function to initialize it, which overcomes the disadvantage of traditional value iteration algorithms. Convergence property is developed to guarantee that the iterative performance index function will converge to the optimum. Neural networks are used to approximate the iterative performance index function and compute the iterative control policy, respectively, to implement the iterative ADP algorithm. Finally, a simulation example is given to illustrate the performance of the developed algorithm.

Finite-approximation-error-based discrete-time iterative adaptive dynamic programming.

In this paper, a new iterative adaptive dynamic programming (ADP) algorithm is developed to solve optimal control problems for infinite horizon discrete-time nonlinear systems with finite approximation errors. First, a new generalized value iteration algorithm of ADP is developed to make the iterative performance index function converge to the solution of the Hamilton-Jacobi-Bellman equation. The generalized value iteration algorithm permits an arbitrary positive semi-definite function to initialize it, which overcomes the disadvantage of traditional value iteration algorithms. When the iterative control law and iterative performance index function in each iteration cannot accurately be obtained, for the first time a new "design method of the convergence criteria" for the finite-approximation-error-based generalized value iteration algorithm is established. A suitable approximation error can be designed adaptively to make the iterative performance index function converge to a finite neighborhood of the optimal performance index function. Neural networks are used to implement the iterative ADP algorithm. Finally, two simulation examples are given to illustrate the performance of the developed method.

On Hilbert's 17th problem and Pfister's multiplicative formulae for the ring of real analytic functions

In this work, we present infinite multiplicative formulae for countable collections of sums of squares (of meromorphic functions on R-n). Our formulae generalize the classical Pfister's ones concerning the representation as a sum of 2(r) squares of the product of two elements of a field K which are sums of 2(r) squares. As a main application, we reduce the representation of a positive semidefinite analytic function on R-n as a sum of squares to the representation as sums of squares of its special factors. Recall that roughly speaking a special factor is an analytic function on R-n which has just one complex irreducible factor and whose zeroset has dimension between 1 and n - 2.

Nearly Optimal Control Scheme for Discrete-Time Nonlinear Systems With Finite Approximation Errors Using Generalized Value Iteration Algorithm

In this paper, a new generalized value iteration algorithm is developed to solve infinite horizon optimal control problems for discrete-time nonlinear systems. The idea is to use iterative adaptive dynamic programming (ADP) to obtain the iterative control law which makes the iterative performance index function reach the optimum. The generalized value iteration algorithm permits an arbitrary positive semi-definite function to initialize it, which overcomes the disadvantage of traditional value iteration algorithms. When the iterative control law and iterative performance index function in each iteration cannot be accurately obtained, a new design method of the convergence criterion for the generalized value iteration algorithm with finite approximation errors is established to make the iterative performance index functions converge to a finite neighborhood of the lowest bound of all performance index functions. Simulation results are given to illustrate the performance of the developed algorithm.

Multiplication operators on the energy space

This paper studies the $\mathcal{H}_{\mathcal{E}}$ (the Hilbert space of functions of finite energy, aka the Dirichlet-finite functions) on an infinite network (weighted connected graph), from the point of view of the multiplication operators $M_f$ associated to functions $f$ on the network. We show that the multiplication operators $M_f$ are not Hermitian unless $f$ is constant, and compute the adjoint $M_f^\star$ in terms of a reproducing kernel for $\mathcal{H}_{\mathcal{E}}$. A characterization of the bounded multiplication operators is given in terms of positive semidefinite functions, and we give some conditions on $f$ which ensure $M_f$ is bounded. Examples show that it is not sufficient that $f$ be bounded or have finite energy. Conditions for the boundedness of $M_f$ are also expressed in terms of the behavior of the simple random walk on the network. We also consider the bounded elements of $\mathcal{H}_{\mathcal{E}}$ and the (possibly unbounded) multiplication operators corresponding to them. In a previous paper, the authors used functional integration to construct a type of boundary for infinite networks. The boundary is described here in terms of a certain subalgebra of these multiplication operators, and is shown to embed into the Gel'fand space of that subalgebra. In the case when the only harmonic functions of finite energy are constant, we show that the Gel'fand space is the 1-point compactification of the underlying network.

Aircraft Multi-Constrained Generalized Dynamic Inversion Control

The new and evolving paradigm of generalized dynamic inversion (GDI) is applied to aircraft maneuvering control design, through the development of a new multiple constraint formulation. The flexibility of GDI in utilizing nullspace of the controls coefficient matrix is exploited to eliminate the problems of excessive control effort and limited performance that are associated with classical dynamic inversion. Scalar deviation functions of the aircraft state error variables are differentiated along the aircraft mathematical model solution trajectories, and linear time varying stable dynamics in these deviation functions are manipulated to obtain linear algebraic constraint equations in the control vector. The constraint equations are inverted using the Greville formula, resulting in the control law. The particular part of the control law drives the states to their desired values, and the auxiliary part acts to stabilize the vehicles inner dynamics by means of the null-control vector. The null-control vector is constructed via a novel class of positive semidefinite Lyapunov functions and nullprojected Lyapunov equations. The closed loop control system is assessed by performing a fully coordinated simultaneous heading-velocity change maneuver. The new type of GDI control laws shows to require low control effort, in addition to providing excellent transient response and reasonable steady-state tracking accuracy.

A family of robust adaptive excitation controllers for synchronous generators with steam valve via Hamiltonian function method

Using the Hamiltonian function method, this paper proposes a family of robust adaptive excitation controllers for synchronous generators with steam valve control. First, a parameterization method of robust adaptive controllers is investigated for dissipative Hamiltonian systems. This method avoids solving Hamilton-Jacobi-Issacs inequalities. The parameters in the family of controllers thus obtained are determined by a locally positive semidefinite function, which has only 2n independent variables, twice as many as the one used to characterize the state feedback. Then, with the parameterization method and the structural properties of dissipative Hamiltonian systems, a family of robust adaptive excitation controllers is designed for synchronous generators with steam valve control, disturbances and unknown parameters. Simulations illustrate the effectiveness and feasibility of the excitation control strategy proposed in the paper.

Graph kernels for chemoinformatics – a critical discussion

We analyze the use, advantages, and drawbacks of graph kernels in chemoin-formatics, including a comparison of kernel-based approaches with other methodology, as well as examples of applications. Kernel-based machine learning [1], now widely applied in chemoinformatics, delivers state-of-the-art performance [2] in tasks like classification and regression. Molecular graph kernels [3] are a recent development where kernels are defined directly on the molecular structure graph. This allows the adaptation of methods from graph theory to structure graphs and their direct use with learning algorithms. The main advantage of learning, the so-called kernel trick, allows for a systematic, computationally feasible, and often globally optimal search for non-linear patterns, as well as the direct use of non-numerical inputs such as strings and graphs. A drawback is that solutions are expressed indirectly in terms of similarity to training samples, and runtimes that are typically quadratic or cubic in the number of training samples. Graph kernels [3] are positive semidefinite functions defined directly on graphs. The most important types are based on random walks, subgraph patterns, optimal assignments, and graphlets. Molecular structure graphs have strong properties that can be exploited [4], e.g., they are undirected, have no self-loops and no multiple edges, are connected (except for salts), annotated, often planar in the graph-theoretic sense, and their vertex degree is bounded by a small constant. In many applications, they are small. Many graph kernels are general-purpose, some are suitable for structure graphs, and a few have been explicitly designed for them. We present three exemplary applications of the iterative similarity optimal assignment [5], which was designed for the comparison of small structure graphs: The discovery of novel agonists of the peroxisome proliferator-activated receptor γ [6] (ligand-based virtual screening), the estimation of acid dissociation constants [7] (quantitative structure-property relationships), and molecular de novo design [8].