State Parametrization Research Articles

Constraining the equation of state parametrization in Hořava-Lifshitz gravity

In this paper, we investigate the late-time cosmic accelerated expansion using various equations of state parametrizations within the framework of Hořava-Lifshitz gravity. We utilize Markov Chain Monte Carlo (MCMC) analysis to constrain the parameters of each proposed model, employing observational Hubble data and Type Ia supernovae. Additionally, we analyze and plot the deceleration parameters for each model. Our findings suggest that the Universe has recently transitioned from a phase of deceleration to acceleration in all the models considered. We also analyzed the behavior of the energy conditions for each proposed model within the framework of Hořava-Lifshitz gravity, specifically at the present epoch (z=0). To further assess the effectiveness of these models, we apply both the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to compare their performance against the standard ΛCDM model. Our results provide valuable insights into how different models perform relative to ΛCDM, offering a comprehensive evaluation of their viability in describing the Universe's accelerated expansion.

DESI 2024: reconstructing dark energy using crossing statistics with DESI DR1 BAO data

We implement Crossing Statistics to reconstruct in a model-agnostic manner the expansion history of the universe and properties of dark energy, using DESI Data Release 1 (DR1) BAO data in combination with one of three different supernova compilations (PantheonPlus, Union3, and DES-SN5YR) and Planck CMB observations. Our results hint towards an evolving and emergent dark energy behaviour, with negligible presence of dark energy at z ≳ 1, at varying significance depending on data sets combined. In all these reconstructions, the cosmological constant lies outside the 95% confidence intervals for some redshift ranges. This dark energy behaviour, reconstructed using Crossing Statistics, is in agreement with results from the conventional w 0–w a dark energy equation of state parametrization reported in the DESI Key cosmology paper. Our results add an extensive class of model-agnostic reconstructions with acceptable fits to the data, including models where cosmic acceleration slows down at low redshifts. We also report constraints on H 0 r d from our model-agnostic analysis, independent of the pre-recombination physics.

Simulating neutron stars with a flexible enthalpy-based equation of state parametrization in spectre

Numerical simulations of neutron star mergers represent an essential step toward interpreting the full complexity of multimessenger observations and constraining the properties of supranuclear matter. Currently, simulations are limited by an array of factors, including computational performance and input physics uncertainties, such as the neutron star equation of state. In this work, we expand the range of nuclear phenomenology efficiently available to simulations by introducing a new analytic parametrization of cold, beta-equilibrated matter that is based on the relativistic enthalpy. We show that the new enthalpy parametrization can capture a range of nuclear behavior, including strong phase transitions. We implement the enthalpy parametrization in the spectre code, simulate isolated neutron stars, and compare performance to the commonly used spectral and polytropic parametrizations. We find comparable computational performance for nuclear models that are well represented by either parametrization, such as simple hadronic equations of state. We show that the enthalpy parametrization further allows us to simulate more complicated hadronic models or models with phase transitions that are inaccessible to current parametrizations.

Trajectory planning for mechanical systems in the presence of dynamic obstacles

This paper deals with an optimal motion planning method for mechanical systems subjected to velocity and acceleration constraints in the presence of dynamical geometric obstacles. Using a state parametrization method based on time polynomials the trajectory planning problem is reformulated as a constrained nonlinear optimization problem which is solved using a numerical optimization toolbox. Quadrotor motion planning in the horizontal plane is considered as an illustration example.

Improving the convergence order of binary neutron star merger simulations in the Baumgarte- Shapiro-Shibata-Nakamura formulation

High-accuracy numerical relativity simulations of binary neutron star mergers are a necessary ingredient for constructing gravitational waveform templates to analyze and interpret observations of compact object mergers. Numerical convergence in the post-merger phase of such simulations is challenging to achieve with many modern codes. In this paper, we study two ways of improving the convergence properties of binary neutron star merger simulations within the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's equations. We show that discontinuities in a particular constraint damping scheme in this formulation can destroy the post-merger convergence of the simulation. A continuous prescription, in contrast, ensures convergence until late times. We additionally study the impact of the equation of state parametrization on the pre- and post-merger convergence properties of the simulations. In particular, we compare results for a piecewise polytropic parametrization, which is commonly used in merger simulations but suffers unphysical discontinuities in the sound speed, with results using a "generalized" piecewise polytropic parametrization, which was designed to ensure both continuity and differentiability of the equation of state. We report on the differences in the gravitational waves and any spurious pre-merger heating, depending on which equation of state parametrization is used.

Optimal Scheme for Quantum Metrology

AbstractQuantum metrology can achieve far better precision than classical metrology, and is one of the most important applications of quantum technologies in the real world. To attain the highest precision promised by quantum metrology, all steps of the schemes need to be optimized, which include the state preparation, parametrization, and measurement. Here the recent progresses on the optimization of these steps, which are essential for the identification and achievement of the ultimate precision limit in quantum metrology, are reviewed. It is hoped this provides a useful reference for the researchers in quantum metrology and related fields.

Non-linear structure formation for dark energy models with a Steep Equation of State

We study the non-linear regime of large scale structure formation considering a dynamical dark energy (DE) component determined by a Steep Equation of State parametrization (SEoS), w(z)=w0+wi(z/zT)q1+(z/zT)q. In order to perform the model exploration at low computational cost, we modified the publicly available L-PICOLA code. We incorporate the DE model employing the first and second-order matter perturbations in the Lagrangian frame and the expansion parameter. We analyse deviations of dynamical DE models w.r.t. ΛCDM in the non-linear matter power spectrum, Pk, the halo mass function (HMF), and the two-point correlation function (2PCF) . We investigate the effect of a steep (SEoS-I) and smooth transition (CPL-lim) in the DE equation of state (EoS) . In the former case, we found an overall impact in the Pk at the level of ∼ 2–3% while, for the latter, we found ∼ 3–4% differences w.r.t. ΛCDM at 0z=. The HMF shows the possibility to distinguish between the models at the high mass end. For the best-fit model, dubbed SEoS-II, we observe large deviations from ΛCDM . This is explained by the combined effect of the dynamical DE with the different amount of matter, Ωm0, and the higher value for H0. Regarding the 2PCF, our results are relatively robust with deviations w.r.t. ΛCDM or the order ∼ 1–2 % for SEoS-I and CPL-lim, and more significant deviation for SEoS-II throughout the r range, making SEoS-I a competitive model. Finally, we conclude that the search for viable DE models (like the SEoS) must include non-linear growth constraints.

On an Improved State Parametrization for Soft Robots With Piecewise Constant Curvature and Its Use in Model Based Control

Piecewise constant curvature models have proven to be an useful tool for describing kinematics and dynamics of soft robots. However, in their three dimensional formulation they suffer from many issues limiting their range of applicability - as discontinuities and singularities - mainly concerning the straight configuration of the robot. In this work we analyze these flaws, and we show that they are not due to the piecewise constant curvature assumption itself, but that instead they are a byproduct of the commonly employed direction/angle of bending parametrization of the state. We therefore consider an alternative state representation which solves all the discussed issues, and we derive a model based controller based on it. Examples in simulation are provided to support and describe the theoretical results. When using the novel parametrization, the system is able to perform more complex tasks, with a strongly reduced computational burden, and without incurring in spikes and discontinuous behaviors.

Equation of state sensitivities when inferring neutron star and dense matter properties

Understanding the dense matter equation of state at extreme conditions is an important open problem. Astrophysical observations of neutron stars promise to solve this, with NICER poised to make precision measurements of mass and radius for several stars using the waveform modelling technique. What has been less clear, however, is how these mass-radius measurements might translate into equation of state constraints and what are the associated equation of state sensitivities. We use Bayesian inference to explore and contrast the constraints that would result from different choices for the equation of state parametrization; comparing the well-established piecewise polytropic parametrization to one based on physically motivated assumptions for the speed of sound in dense matter. We also compare the constraints resulting from Bayesian inference to those from simple compatibility cuts. We find that the choice of equation of state parametrization and particularly its prior assumptions can have a significant effect on the inferred global mass-radius relation and the equation of state constraints. Our results point to important sensitivities when inferring neutron star and dense matter properties. This applies also to inferences from gravitational wave observations.

Reinforcement Learning-Based Linear Quadratic Regulation of Continuous-Time Systems Using Dynamic Output Feedback.

In this paper, we propose a model-free solution to the linear quadratic regulation (LQR) problem of continuous-time systems based on reinforcement learning using dynamic output feedback. The design objective is to learn the optimal control parameters by using only the measurable input-output data, without requiring model information. A state parametrization scheme is presented which reconstructs the system state based on the filtered input and output signals. Based on this parametrization, two new output feedback adaptive dynamic programming Bellman equations are derived for the LQR problem based on policy iteration and value iteration (VI). Unlike the existing output feedback methods for continuous-time systems, the need to apply discrete approximation is obviated. In contrast with the static output feedback controllers, the proposed method can also handle systems that are state feedback stabilizable but not static output feedback stabilizable. An advantage of this scheme is that it stands immune to the exploration bias issue. Moreover, it does not require a discounted cost function and, thus, ensures the closed-loop stability and the optimality of the solution. Compared with earlier output feedback results, the proposed VI method does not require an initially stabilizing policy. We show that the estimates of the control parameters converge to those obtained by solving the LQR algebraic Riccati equation. A comprehensive simulation study is carried out to verify the proposed algorithms.

A numerical method based on Cardan polynomials for solving optimal control problems

This paper presents, an iterative numerical scheme for solving a class of optimal control problems. The solution is based on the state parametrization such that, the state variable can be considered as a linear combination of Cardan polynomials with the unknown coefficients. This converts the optimal control problem to a non-linear polynomial optimization problem. Illustrative examples are included to demonstrate the efficiency of the proposed method.

State Parametrization Method Based on Shifted Legendre Polynomials for Solving Fractional Optimal Control Problems

In this paper, the parametric optimization method is used to solve a class of fractional optimal control problems. The solution is based on state parametrization, as a linear combination of shifted Legendre polynomials with unknown coefficients. An iterative algorithm is proposed for computing optimal coefficients. In this method the state and control variables can be approximated as a function of time. Convergence of the algorithm is investigated and to show the efficiency of the presented method three examples are solved and the results are compared with exact solution.

Finite Element Modeling and Motion Planning of an Adaptive Elastic Wingsail

A systematic motion planning methodology for elastomechanic structures is introduced using a combination of flatness based feedforward control and state feedback control. The concept is applied to a so-called wingsail for yachts, this structure can be described as a complex flexible structure based on two curved carbon sail areas with embedded actuators allowing to generate deflections to accomplish an adaptive sail structure. Due to the complexity of the structure the finite element method (FEM) is applied to determine the equations of motion (EOMs) in terms of a high dimensional system of ordinary differential equations (ODEs). The usage of model order reduction by means of modal truncation leads to a design system of feasible dimension and also directly represents a suitable form for the determination of the flatness-based state and input parametrization. This yields an efficient approach for motion planning and feedforward control, which is amended by a state feedback controller to address model uncertainties or disturbances. Simulation results illustrate the performance of the presented two-degrees-of-freedom control approach.

Black-oil minimal fluid state parametrization for constrained reservoir control optimization

Abstract We propose to solve a black-oil reservoir optimal control problem with the Direct Multiple Shooting Method (MS). MS allows for parallelization of the simulation time and the handling of output constraints. However, it requires continuity constraints on state variables to couple simulation intervals. The black-oil fluid model, considering volatile oil or wet gas, requires a change of primary variables for simulation. This is a consequence of the absence of a fluid phase due to dissolution or vaporization. Therefore, reservoir simulators parametrize the states with an augmented vector and select primary variables accordingly. However, the augmented state vector and the corresponding change of primary variables are not suitable for the application of MS because the optimization problem formulation must change according to the change of variables. Thus, we propose a minimal state-space variable representation that prevents this shortcoming. We show that there is a bijective mapping between the proposed state-space representation and the augmented state-space. The minimal representation is used for optimization and the augmented representation for simulation, thereby keeping the simulator implementation unchanged. Therefore, the proposed solution is not invasive. Finally, the application of the method is exemplified with benchmark cases involving live oil or wet gas. Both examples emphasize the requirement of output constraints which are efficiently dealt with the MS method.

Flatness of a class of linear Volterra partial integro–differential equations

Flatness–based trajectory planning is addressed for a parabolic–like linear Volterra partial integro–differential equation (PIDE) with boundary control. For this purpose the PIDE is formally integrated which results in an implicit state and input parametrization. Successive approximation is used to explicitly resolve the implicit result in terms of a recursion scheme. Absolute and uniform convergence is verified by restricting the flat output trajectory to Gevrey class functions of a particular order. Simulation results confirm the applicability and illustrate the tracking behavior for finite time transitions between operating points along predefined transition paths.

Articulated pose estimation using tangent space approximations

We describe an algorithm to estimate the pose of a generic articulated object. Our algorithm takes as input a description of the object and a potentially incomplete series of observations; it outputs an on-line estimate of the object’s configuration. This task is challenging because: (1) the distribution of object states is often multi-modal; (2) the object is not assumed to be under our control, limiting our ability to predict its motion; and (3) rotational joints make the state space highly non-linear. The proposed method represents three principal contributions to address these challenges. First, we use a particle filter implementation which is unique in that it does not require a reliable state transition model. Instead, the method relies primarily on observations during particle proposal, using the state transition model only at singularities. Second, our particle filter formulation explicitly handles missing observations via a novel proposal mechanism. Although existing particle filters can handle missing observations, they do so only by relying on good state transition models. Finally, our method evaluates noise in the observation space, rather than state space. This reduces the variability in performance due to choice of parametrization and effectively handles non-linearities caused by rotational joints. We compare our method to a baseline implementation without these techniques and demonstrate, for a fixed error, more than an order-of-magnitude reduction in the number of required particles, an increase in the number of effective particles, and an increase in frame rate. We examine the effects of errors in the kinematic model and demonstrate a reduced dependence on state parametrization. The novel use of a precision matrix allows observations which do not provide complete 6-DOF pose information to be processed. Source code for the method is available at http://rvsn.csail.mit.edu/articulated.

A Numerical Approach for Solving Optimal Control Problems Using the Boubaker Polynomials Expansion Scheme

In this paper, we present a computational method for solving optimal control problems and the controlled Duffing oscillator. This method is based on state parametrization. In fact, the state variable is approximated by Boubaker polynomials with unknown coefficients. The equation of motion, performance index and boundary conditions are converted into some algebraic equations. Thus, an optimal control problem converts to a optimization problem, which can then be solved easily. By this method, the numerical value of the performance index is obtained. Also, the control and state variables can be approximated as functions of time. Convergence of the algorithms is proved. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.

Application of a parametrization method to problem of optimal control

A new approach to the problem of optimal control of linear dynamic systems is proposed that makes use of a method of input and state parametrization to transform the original problem into a problem of the Calculus of Variations. In contrast to the standard approaches for this class of problems, two salient features of the new approach are that no Lagrange multiplier functions need to be invoked and that the class of inputs can be restricted to the – relatively small – class of continuous functions, even for problems with fixed end-states. The resulting necessary conditions of optimality, i.e., the Euler–Lagrange equation and the boundary conditions for the transformed problem, are proved to be equivalent to those resulting from the standard method of First Variations. In case of quadratic cost functionals, the new approach provides a simpler alternative to the well known, but equally difficult, Riccati differential equation approach and results in a simple dynamic state-feedback implementation of the optimal control.

Trajectory planning for boundary controlled parabolic PDEs with varying parameters defined on a parallelepiped

AbstractThe trajectory planning and feedforward tracking control problem is considered for a boundary controlled diffusion‐reaction system with a spatially and time varying reaction parameter defined on a 3‐dimensional parallelepiped. For this, an implicit state and input parametrization in terms of a basic output via a Volterra‐type integral equation with operator kernel is determined, which is solved recursively by means of a series ansatz. The absolute and uniform convergence of the resulting series is verified by restricting the reaction parameter and the basic output to a certain but broad Gevrey class. Hence, assigning an admissible desired trajectory for the basic output directly yields the respective feedforward control, which is required to realize a desired spatio‐temporal transition path. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Trajectory Planning for Boundary Controlled Parabolic PDEs With Varying Parameters on Higher-Dimensional Spatial Domains

The flatness-based design of a feedforward tracking control is considered for the solution of the trajectory planning problem for a boundary controlled diffusion-convection-reaction system with spatially and temporally varying parameters defined on a 1 les m-dimensional parallelepipedon with the nonlinear input being restricted to a (m-1) -dimensional hyperplane. For this, an implicit state and input parametrization in terms of a basic output is determined via a Volterra-type integral equation with operator kernel. By recursively computing successive series coefficients, a series solution of the integral equation is obtained, whose absolute and uniform convergence is verified by restricting the system parameters and the basic output to a certain but broad Gevrey class. Hence, prescribing an admissible desired trajectory for the basic output directly yields the feedforward control by evaluating the input parametrization. This results in a systematic procedure for trajectory planning and feedforward control design for boundary controlled parabolic distributed-parameter systems defined on higher-dimensional domains.