Linear State Equations Research Articles

Quintessences compact star with Durgapal potential

A compact star model formed by quintessence and ordinary matter is presented, both sources have anisotropic pressures and are described by linear state equations, also the state equation of the tangential pressure for the ordinary matter incorporates the effect of the quintessence. It is shown that depending on the compactness of the star [Formula: see text] the constant of proportionality [Formula: see text] between the density of the ordinary matter and the radial pressure, [Formula: see text], has an interval of values which is consistent with the possibility that the matter is formed by a mixture of particles like quarks, neutrons and electrons and not only by one type of them. The geometry is described by the Durgapal metric for [Formula: see text] and each one of the pressures and densities is positive, finite and monotonic decreasing, as well as satisfying the condition of causality and of stability [Formula: see text], which makes our model physically acceptable. The maximum compactness that we have is [Formula: see text], so we can apply our solution considering the observational data of mass and radii [Formula: see text], [Formula: see text] km which generate a compactness [Formula: see text] associated to the star PSR J0348[Formula: see text]+[Formula: see text]0432. In this case, the interval of [Formula: see text] and its maximum central density [Formula: see text] and in the surface [Formula: see text] of the star are [Formula: see text] and [Formula: see text], respectively, meanwhile the central density of the quintessence [Formula: see text].

Sensor Fault Reconstruction based on Adaptive Sliding Mode Observer for Forklift Fault-Tolerant Control System

For the problem of multiple sensor fault detection and reconstruction in the forklift fault-tolerant control system, a sliding mode observer (SMO) with adaptive regulation law is proposed. Based on the three-degree-of-freedom (3-DOF) model of forklift, a linear state equation with output disturbance is designed as its equivalent sensor fault model. The sensor fault is converted into an actuator fault by defining an auxiliary state variable as an output signal filter. Then the SMO-based method of sensor fault detection and reconstruction is given. Without knowing the upper bound of an unknown fault, an adaptive sliding mode observer (ASMO) can also be effective through the adaptive algorithm. Finally, experimental results further verify the effectiveness of the method, and provide a foundation for forklift fault-tolerant control.

Asymptotic Behaviours for the Landau-Lifschitz-Bloch Equation

The Landau-Lifshitz-Bloch (LLB) equation is an interpolation between Bloch equation valid for high temperatures and Landau-Lifshitz equation valid for low temperatures. Conversely in this paper, we discuss the behaviours of the solutions of (LLB) equation both as the temperature goes to infinity or 0. Surprisingly in the first case, thebehaviour depends also on the scaling of the damping parameter $\delta$ and the volume exchange parameter $a$. Three cases are considered and accordingly we get either a linear stationary equation, Bloch equation or Stokes equation. As for the small temperature behaviour, $\delta$ and $a$ being independent of the temperature, we show that the limit of (LLB) equation is Landau-Lifshitz-Gilbert equation.

A dual representation result for value functions in stochastic control of infinite dimensional groups

We study here the problem of dual representation of the value functions associated to linear-convex stochastic control problems in infinite dimensional Hilbert spaces. Since the dual state runs backwards in time, it turns out that the dual representation has the meaning of a classical (Markov) control problem only if the primal linear state equation is driven by the generator of a group. In the general case, a dual representation of the value function still holds, but such a representation cannot be reduced to solving a dual Hamilton-Jacobi-Bellman equation.

Topology optimization combined with element‐by‐element solution techniques

AbstractTopology optimization approaches are commonly used for design problems involving physical phenomena related to solid mechanics, acoustics, electromagnetism, fluid mechanics, and combinations thereof. In computational models of these physical phenomena, the field variables are commonly approximated using spatial discretizations within the domain using the Finite Element Method. Even for topology design problems in which the field variables must only satisfy linear state equations, the storage and solution of the resulting global system of equations can become a computational bottleneck. In this contribution, we investigate a topology optimization approach in which the Solid Isotropic Material with Penalization (SIMP) method is combined with element‐wise solution approaches, in order to reduce computational memory requirements.

On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets

We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $\alpha$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $\alpha_-$ and $\alpha_+$, depending on these bounds, and prove that for $\alpha < \alpha_-$, state positivity is guaranteed for all time and all asset-return sequences; i.e., bankruptcy is ruled out and positive solutions of the state equation are continuable indefinitely. On the other hand, for $\alpha > \alpha_+$, we show that there is always a sequence of asset returns for which the state fails to be positive for all time; i.e., along this sequence, bankruptcy is certain and the solution of the state equation ceases to be meaningful after some finite time. Finally, this paper also includes a conjecture which says that for the "gap" interval $\alpha_- \leq \alpha \leq \alpha_+,$ state positivity is also guaranteed for all time. Support for the conjecture, both theoretical and computational, is provided.

Run-Time Efficiency of Bilinear Model Predictive Control Using Variational Methods, With Applications to Hydronic Cooling

Effective real-time execution of nonlinear model predictive control (NMPC) on embedded systems is significantly dependent on the controller formulation. This paper studies the effect of model structure and cost functions on the computation time of scalar bilinear NMPC using variational methods, with hydronic cooling applications. Two algebraically equivalent nonlinear model structures common in literature are primarily considered: a linear state equation with state-dependent control constraints and a bilinear state equation with time-invariant rectangular control constraints. Additionally, the effects of three cost function formulations are also considered: minimum-time, quadratic regulation, and efficient state constraints. High-fidelity computer simulations, hardware-in-the-loop testing, and experiments on a bench-scale hydronic cooling system are used to study sources of computational complexity, rates of convergence, initialization techniques, and overall effectiveness of the different models and costs. These results suggest that NMPC with bilinear state equations, minimizing pump power and a one-sided quadratic state cost, converges sufficiently fast and reliably. This presents an attractive alternative to the traditionally constrained linear quadratic regulator-based NMPC on embedded systems.

A spherically symmetric model of anisotropic fluid for strange quark spheres

In the present work, we try to find a solution without singularity of Einstein’s field equations for the spherically symmetric perfect fluid objects, accurately strange quark spheres, taking into consideration Schwarzschild metric as the outside space-time. An ensemble of inside solutions found on the basis of the simplest linear state equation in the specific form p_r =alpha rho -beta . The energy density rho (r), the radial pressure p_r (r) and the tangential pressure p_t (r) are devoid of any singularity and exhibit a well-behaved nature within the generalized anisotropic solution for compact spherical object. The generalized TOV equation is very much preserved inside the system and all energy conditions are excellent. The stability of the matter distribution of our system is checked by the concept of Herrera’s cracking and the condition of causality is all around fulfilled for our models. The adiabatic index of our specific configuration is greater than 4 / 3 in all interior points of the system and the mass-to-radius ratio in our situation is determined also lies within the Buchdahl limit i.e. hbox {M}/hbox {R}le 4/3,left( {approx 0.444} right) . We explore the physical characteristics based on the analytical model developed for relativistic compact stellar spheres inside the framework of the general theory of relativity. The evaluated mass and radius are in close concurrence with the observational information. We show that various physical characteristics of the known strange spherical object, viz. PSR J1614-2230, Vela X-1, 4U 1608-52, PSR J1903+327, 4U 1820-30, Cen X-3, Her X-1, and SAX J1808.4-3658, can be described by the current model.

Modeling and control of one link flexible arm using mathematical expression extraction from neural network

In recent decades, a number of researches has been focused on development and control of soft robots. There is one approach that uses a neural network to learn a forward model of a robot and control the robot by extracting equivalent equations from the neural network. However, if the training data obtained from the robot only contains partial dynamic characteristics, the linear state equation extracted from the neural network can be barely controlled. In this research, we propose a method to use singular value decomposition in order to reflect deviation of training data. We confirmed that our proposed method can control a flexible arm by simulation.

Dual Faceted Linearization of Nonlinear Dynamical Systems Based on Physical Modeling Theory

A new approach to modeling and linearization of nonlinear lumped-parameter systems based on physical modeling theory and a data-driven statistical method is presented. A nonlinear dynamical system is represented with two sets of differential equations in an augmented space consisting of independent state variables and auxiliary variables that are nonlinearly related to the state variables. It is shown that the state equation of a nonlinear dynamical system having a bond graph model of integral causality is linear, if the space is augmented by using the output variables of all the nonlinear elements as auxiliary variables. The dynamic transition of the auxiliary variables is investigated as the second set of differential equations, which is linearized by using statistical linearization. It is shown that the linear differential equations of the auxiliary variables inform behaviors of the original nonlinear system that the first set of state equations alone cannot represent. The linearization based on the two sets of linear state equations, termed dual faceted linearization (DFL), can capture diverse facets of the nonlinear dynamics and, thereby, provide a richer representation of the nonlinear system. The two state equations are also integrated into a single latent model consisting of all significant modes with no collinearity. Finally, numerical examples verify and demonstrate the effectiveness of the new methodology.

Aggregate modeling and analysis of air conditioning load using coupled\xa0Fokker\u2013Planck equations

Demand response has gained significant attention recently with the increasing penetration of renewable energy sources in power systems. Air conditioning loads are typical thermostatically controlled loads which can play an active role in ancillary services by regulating their aggregated power consumption. The aggregation of air conditioners is essential to the control of air conditioning loads. In this paper, linear state equations are proposed to aggregate air conditioning loads by solving coupled Fokker–Planck equations (CFPEs) using the finite difference method. By analyzing the numerical stability and convergence of the difference scheme, the grid spacings, including temperature step and time step, are properly determined according to the maximal principle. Stationary solutions of the CFPEs are obtained by analytical and numerical methods. Furthermore, a classification method using dimension reduction is proposed to deal with the problem of heterogeneous parameters and interval estimation is applied to describe the stochastic behavior of air conditioning loads. The simulation results verify the effectiveness of the proposed methods.

Modeling, numerical simulation, and nonlinear dynamic behavior analysis of PV microgrid-connected inverter with capacitance catastrophe

The value of the output capacitance (C) should be carefully considered when designing a photovoltaic (PV) inverter since it can cause distortion in the working state of the circuit, and the circuit produces nonlinear dynamic behavior. According to Kirchhoff’s laws and the characteristics of an ideal operational amplifier for a strict piecewise linear state equation, a circuit simulation model is constructed to study the system parameters (time, C) for the current passing through an inductor with an inductance of L and the voltage across the capacitor with a capacitance of C. The developed simulation model uses Runge–Kutta methods to solve the state equations. This study focuses on predicting the fault of the circuit from the two aspects of the harmonic distortion and simulation results. Moreover, the presented model is also used to research the working state of the system in the case of a load capacitance catastrophe. The nonlinear dynamic behaviors in the inverter are simulated and verified.

Differential Stability of a Class of Convex Optimal Control Problems

A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented.

Robust Derivative Unscented Kalman Filter Under Non-Gaussian Noise

A robust derivative unscented Kalman filter is proposed for a nonlinear system with non-Gaussian noise and outliers based on Huber function. In this paper, the time update process can be performed using a Kalman filter (KF), and measurement update process can be carried out utilizing an unscented transform. This novel filtering algorithm differs from the traditional unscented Kalman filter (UKF) which is sensitive to non-Gaussian noise and outliers. Besides, for a nonlinear system with linear state equation, the presented filtering has the merits of both KF and UKF, and it has better performance under non-Gaussian noise and large outliers. Furthermore, the proposed filtering algorithm can not only track the target effectively but also suppress the effects of the non-Gaussian noise and large outliers. Finally, numerical simulations are conducted to verify estimation accuracy and effectiveness of the proposed filtering algorithm.

Reduced Order Modeling for Time-Dependent Optimization Problems with Initial Value Controls

This paper presents a new reduced order model (ROM) Hessian approximation for linear-quadratic optimal control problems where the optimal control is the initial value. Such problems arise in parameter identification and data assimilation, where the parameters to be identified appear in the initial data, and also as subproblems in multiple shooting formulations of more general optimal control problems. The new ROM Hessians can provide a substantially better approximation than the underlying basic ROM approximation, and thus can substantially reduce the computing time needed to solve these optimal control problems. The computation of a Hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the Hessian is applied, followed by the solution of the second order adjoint equation. Projection-based ROMs of these two linear differential equations are used to generate the Hessian approximation while the objective function and gradient are computed exactly using the full model. The challenge is that in general no fixed ROM well-approximates the application of the Hessian to all possible vectors of initial data. The new approach, after having selected a basic ROM, augments this basic ROM by one vector. This vector is either the right-hand side or the vector of initial data to which the Hessian is applied. It is shown that although the size of the ROM increases only by one, this new augmented ROM produces substantially better approximations of the true Hessian vector products as well as of the optimal solution than the basic ROM. The use of these ROM Hessians in a conjugate gradient method is analyzed.

Design of Control System based on Linear State Equations extracted from Neural Networks

Design of Control System based on Linear State Equations extracted from Neural Networks

A Unifying Method to Construct Rational Basis Functions for Linear and Nonlinear Systems

This paper proposes a general and unifying method for constructing rational basis functions (RBFs). The proposal is specifically designed for finding characterizing matrices of linear RBF state equations. Analytic expressions for the realization of such state-space matrices are also developed. As will be shown, the construction method here proposed can be easily applied (but is not limited) to several RBF sets which have been successfully adopted in both linear and nonlinear system identification areas.

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.

A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

AbstractIn the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example confirms the usefulness of the method.

Constrained controllability of fractional linear systems with delays in control

The paper presents finite-dimensional dynamical control systems described by linear fractional-order state equations with multiple delays in control. The constrained controls are considered. The set of admissible control values is assumed to be a compact set containing 0, a convex and compact set containing 0 in its interior, a cone with vertex at zero and a nonempty interior, or a convex and close cone with nonempty interior and vertex at zero. The definitions of various types of constrained controllability of the linear fractional systems with delays in control are formulated. New necessary and sufficient conditions for constrained relative controllability of fractional-order control systems with delayed controls are established and proved. Numerical examples are provided to illustrate the obtained theoretical results.