Energy- and Entropy-Based Stabilization for Lossless Dynamical Systems via Hybrid Controllers

A novel class of dynamic, energy-based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in lossless dynamical systems. These dynamic controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that the closed-loop system is consistent with basic thermodynamic principles. In particular, the existence of an entropy function for the closed-loop system is established that satisfies a hybrid Clausius-type inequality. Special cases of energy-based and entropy-based hybrid controllers involving state-dependent switching are described.     

von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.     

On reachability and minimum cost optimal control

Questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems, whose solution can be characterized using variants of the Hamilton-Jacobi-Bellman or Isaacs partial differential equations. The formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of viscosity solutions. This paper establishes such a link between reachability, viability and invariance problems and viscosity solutions of a special form of the Hamilton-Jacobi equation. This equation is developed to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon. The main advantage of the proposed approach is that the properties of the value function (uniform continuity) and the form of the partial differential equation (standard Hamilton-Jacobi form, continuity of the Hamiltonian and simple boundary conditions) make the numerical solution of the problem much simpler than other approaches proposed in the literature. This fact is demonstrated by applying our approach to a reachability problem that arises in flight control and using numerical tools to compute the solution.     

高精度三次参数样条曲线的构造

构造参数样条曲线的关键是选取节点，该文讨论了GC^2三次参数样条曲线需满足的连续性方程，提出了构造GC^2三次参数样条曲线的新方法，在讨论了平面有序五点确定一组三次多项式函数曲线，平面有序六点唯一确定一条三次多项式函数曲线的基础上，提出了计算相邻两区间上的节点的算法，构造的插值曲线具有三次多项式函数精,该文还以实例对新方法与其它方法构造的插值曲线的精度进行了比较。     

Limit cycle oscillations in a satellite attitude control system

Limit cycle oscillations in an attitude control system using an integral-pulse-frequency modulator are studied. A previous paper showed, through a study of some geometrical properties of the state transition equations, the necessary and sufficient relationships among the physical parameters for the ultimate behavior to be a two-pulse limit cycle for arbitrary initial states [1]. Here it is shown that two-pulse limit cycles can exist if those relationships are not satisfied provided the initial state lies in a certain region near the origin of the state space. The boundaries of this region are established analytically by geometrical means and are verified by simulator experiments. These experiments also show the existence of four-pulse, six-pulse and higher order limit cycles for certain initial states and combinations of physical parameter values.     

Mixed Methods for a Stream-Function – Vorticity Formulation of the Axisymmetric Brinkman Equations

This paper is devoted to the numerical analysis of a family of finite element approximations for the axisymmetric, meridian Brinkman equations written in terms of the stream-function and vorticity. A mixed formulation is introduced involving appropriate weighted Sobolev spaces, where well-posedness is derived by means of the Babu?ka–Brezzi theory. We introduce a suitable Galerkin discretization based on continuous piecewise polynomials of degree \(k\ge 1\) for all the unknowns, where its solvability is established using the same framework as the continuous problem. Optimal a priori error estimates are derived, which are robust with respect to the fluid viscosity, and valid also in the pure Darcy limit. A few numerical examples are presented to illustrate the convergence and performance of the proposed schemes.     

Finite difference solution of differential equations with stochastic inputs

This paper discusses application of two numerical methods (central difference and predictor corrector) for the solution of differential equations with deterministic as well as stochastic inputs. The methods are applied to a second order linear differential equation representing a series RLC netowrk with step function, sinusoidal and stochastic inputs. It is shown that both methods give correct answers for the step function and sinusoidal inputs. However, the central-difference method of solution is recommended for stochastic inputs. This statement is justified by comparing the auto-correlation and cross-correlation functions of the central-difference solution (with stochastic inputs) with the corresponding theoretical values of a continuous system. It is further shown that the more common predictor-corrector methods, although suitable for solution of differential equations with regular inputs, diverge for stochastic inputs. The reason is that these methods, by the application of several point integral formulas, use a high degree of smoothing on the variable and its derivatives. Inherent in the derivation of these integral formulas is the assumption of the continuity of the variable and its derivatives, a condition which is not satisfied in problems with stochastic inputs.Note that the second order differential equation chosen here for numerical experiments can be solved by classical methods for all of the given inputs, including the probabilistic inputs. The classical methods, however, unlike the numerical solutions, can not be extended to nonlinear differential equations which frequently arise in the digital simulation of engineering problems.     

Categorical and dimensional affect analysis in continuous input: Current trends and future directions

In the context of affective human behavior analysis, we use the term continuous input to refer to naturalistic settings where explicit or implicit input from the subject is continuously available, where in a human–human or human–computer interaction setting, the subject plays the role of a producer of the communicative behavior or the role of a recipient of the communicative behavior. As a result, the analysis and the response provided by the automatic system are also envisioned to be continuous over the course of time, within the boundaries of digital machine output. The term continuous affect analysis is used as analysis that is continuous in time as well as analysis that uses affect phenomenon represented in dimensional space. The former refers to acquiring and processing long unsegmented recordings for detection of an affective state or event (e.g., nod, laughter, pain), and the latter refers to prediction of an affect dimension (e.g., valence, arousal, power). In line with the Special Issue on Affect Analysis in Continuous Input, this survey paper aims to put the continuity aspect of affect under the spotlight by investigating the current trends and provide guidance towards possible future directions.     

Testing for discontinuity or type of distribution

In this paper, we propose a simple test for the continuity of a distribution function or of the type of distribution. The main advantage of our test in comparison to others, as used in earnings-management studies, for example, is that no assumptions regarding the underlying distribution function are necessary. Nonetheless, by use of the Chebyshev inequality we are able to define the upper limit of probabilities of test values. Results of Monte Carlo simulations indicate the robustness of the test in that the hypothesis of continuity for distribution functions with jumps is rejected whilst for continuous distributions it is not rejected. We also show that the test appropriately rejects/does not reject hypotheses regarding the type of distribution that a set of data follows. The test is particularly reliable for samples of more than 5000 observations. Applications employing such tests, for example in the earnings-management literature, typically exceed this threshold.     

Predictability and unpredictability in Kalman filtering

The authors study the dynamical behavior of the Kalman filter when the given parameters are allowed to vary in a way which does not necessarily correspond to an underlying stochastic system. This may correspond to situations in which the basic parameters are chosen incorrectly through estimates. The authors show that, as has been suggested by Kalman, the filter equations converge to a limit (corresponding to a steady-state filter) for a subset of the parameter space which is much larger than that corresponding to bona fide stochastic systems. More surprisingly, in the complement of this subset, the filtering equations behave in both a regular and an unpredictable manner, representative of some of the basic aspects of chaotic dynamics. This interesting dynamical behavior occurs already for one-dimensional filters, and a complete phase portrait in this case is given     

Solving systems of linear equations via gradient systems with discontinuous righthand sides: application to LS-SVM

A gradient system with discontinuous righthand side that solves an underdetermined system of linear equations in the L/sub 1/ norm is presented. An upper bound estimate for finite time convergence to a solution set of the system of linear equations is shown by means of the Persidskii form of the gradient system and the corresponding nonsmooth diagonal type Lyapunov function. This class of systems can be interpreted as a recurrent neural network and an application devoted to solving least squares support vector machines (LS-SVM) is used as an example.     

Adaptive control design for a boost inverter

In this paper, a novel control strategy for a nonlinear boost inverter is proposed. The idea is based on generating an autonomous oscillator that does not need an external reference signal. This aim is achieved by using energy-shaping methodology with a suitable Hamiltonian function which defines the desired system behavior. A phase controller is added to the control law in order to achieve 180°-synchronization between both parts of the circuit as well as synchronize the voltage output with a pre-specified signal, e.g. synchronization with the electrical grid. An adaptive control is designed for dealing with the common problem of unknown load. In order to analyze the stability of the full system, singular perturbation approach is used. The resulting control is tested by means of simulations.     

Existence of limit cycle and stabilization of induction motor via new nonlinear state observer

In this paper the boundedness of the solutious of the bilinear and nonlinear differential equations, which describe the dynamic behavior of an ideal three-phase squirrel cage induction motor, is shown using a Lyapunov function. It is then proved by sampling combined with a digital simulation that an unstable machine has limit cycle. Utilizing these results a new bilinear and nonlinear reduced-order state observer, which is globally asymptotically stable, is constructed to estimate the unmeasurable state variables. By using this observer a new two-step procedure for stabilizing an unstable machine, which has a limit cycle, is proposed. This scheme can be easily implemented resulting in an asymptotically stable overall system. These results are numerically verified by simulation.     

On the Uniqueness of the Solutions of some Equations of Optimal Control Theory†

The uniqueness of the solution of the differential equations derived from Pontryagin's Maximum Principle is considered for a general control problem with a freo right end of the trajectory and fixed terminal time. Under several restrictive conditions on the controlled system and performance criterion, it is proved that those equations have a unique solution. This is done by embedding the problem into a class of similar problems by means of the introduction of a positive parameter ? multiplying the performance criterion, and then making use of the uniqueness properties of an associated autonomous system together with the continuity of the solutions of the differential equations with respect to the parameter ? 

An example that illustrates the range of applicability of the theorems proved is also presented.     

Modelling thrombopoiesis regulation—II: Mathematical investigation of the model

The authors investigate the mathematical properties of the thrombopoiesis model presented in Part I, which is described by a differential system involving Von Foerster type partial differential equations. Apart from the existence, uniqueness and continuity properties of nonnegative solutions they establish conditions for the existence of a unique equilibrium and for its asymptotic stability, too.     

Sensitivity and robust stability of general input-output systems

The author considers a general model of an input-output system that is governed by nonlinear operator equations which relate the input, the state, and the output of the system. This model encompasses feedback systems as a special case. Assuming that the governing equations depend on a parameter A which is allowed to vary in a neighborhood of a nominal value A₀ in a linear space, the author studies the dependence of the system behavior on A. A system is considered insensitive if, for any fixed input, the output depends continuously on A. Similarly, the system is robust if it is stable for each A in a neighborhood of A₀. Stability is defined as an appropriate continuity of the input-output operator. The results give various sufficient conditions for insensitivity and robustness. Applications of the theory are discussed, including the estimation of the difference of operator inverses, and the insensitivity and robust stability of a Hilbert network, a feedback-feedforward system, a traditional feedback system, and a time-varying dynamical system described by a linear vector differential equation on (0, ∞)     

Bézier representation of geometrically continuous splines

As an intrinsic measure of smoothness,geometric continuity is an important problem in the fields of computer aided geometric design.It can afford more degrees of freedom for manipulating the shape of curve.However,piecewise polynomial functions of geometrically continuous splines are difficult to be constructed.In this paper,the conversion matrix between geometrically continuous spline basis functions and Bézier representation is analyzed.Based on this,construction of arbitrary degree geometrically continuous spline basis functions can be translated into a solution of linear system of equations.The original construction of geometrically continuous spline is simplified.