Admissible Region Research Articles

Machine Learning Control Based on Approximation of Optimal Trajectories

The paper is devoted to an emerging trend in control—a machine learning control. Despite the popularity of the idea of machine learning, there are various interpretations of this concept, and there is an urgent need for its strict mathematical formalization. An attempt to formalize the concept of machine learning is presented in this paper. The concepts of an unknown function, work area, training set are introduced, and a mathematical formulation of the machine learning problem is presented. Based on the presented formulation, the concept of machine learning control is considered. One of the problems of machine learning control is the general synthesis of control. It implies finding a control function that depends on the state of the object, which ensures the achievement of the control goal with the optimal value of the quality criterion from any initial state of some admissible region. Supervised and unsupervised approaches to solving a problem based on symbolic regression methods are considered. As a computational example, a problem of general synthesis of optimal control for a spacecraft landing on the surface of the Moon is considered as supervised machine learning control with a training set.

Tracklet-to-orbit association for maneuvering space objects using optimal control theory

The maintenance of a catalog of space objects is integral for Space Surveillance and Tracking (SST). In order to enable safe space operations, this catalog shall be complete, accurate and traceable. One of the challenges to keep custody of active satellites comes from their unknown maneuvers plan that adds significantly to the uncertainty on their precise dynamics. This fact can imply the failure of conventional measurement-to-orbit association methods leading at the best to an inconsistent traceability in the catalog (a physical object represented by two or more catalog objects) or, at worst, to the loss of the tracking of an object. This paper presents a method to tackle the measurement-to-orbit association in the presence of unknown maneuvers. It is based on optimal control theory coupled with the admissible region. In particular, the orbit-to-orbit association problem is solved with an energy-optimal indirect approach, which is then generalized to the measurement-to-orbit association problem using the admissible regions concept. Measurements are supposed to be a collection of time-close optical measurements or tracklet that can be represented by an attributable vector. A broad bench of simulated tests is presented in the case of a geostationary (GEO) maneuvering satellite, and the method is finally applied to a real case of a longitude-shift GEO satellite using the data provided by the TAROT telescope network.

New Tools for the Optimized Follow-Up of Imminent Impactors

The solar system is populated with, other than planets, a wide variety of minor bodies, the majority of which are represented by asteroids. Most of their orbits are comprised of those between Mars and Jupiter, thus forming a population named Main Belt. However, some asteroids can run on trajectories that come close to, or even intersect, the orbit of the Earth. These objects are known as Near Earth Asteroids (NEAs) or Near Earth Objects (NEOs) and may entail a risk of collision with our planet. Predicting the occurrence of such collisions as early as possible is the task of Impact Monitoring (IM). Dedicated algorithms are in charge of orbit determination and risk assessment for any detected NEO, but their efficiency is limited in cases in which the object has been observed for a short period of time, as is the case with newly discovered asteroids and, more worryingly, imminent impactors: objects due to hit the Earth, detected only a few days or hours in advance of impacts. This timespan might be too short to take any effective safety countermeasure. For this reason, a necessary improvement of current observation capabilities is underway through the construction of dedicated telescopes, e.g., the NEO Survey Telescope (NEOSTEL), also known as “Fly-Eye”. Thanks to these developments, the number of discovered NEOs and, consequently, imminent impactors detected per year, is expected to increase, thus requiring an improvement of the methods and algorithms used to handle such cases. In this paper we present two new tools, based on the Admissible Region (AR) concept, dedicated to the observers, aiming to facilitate the planning of follow-up observations of NEOs by rapidly assessing the possibility of them being imminent impactors and the remaining visibility time from any given station.

The reachable regions construction for the inelastic anisotropic billiards

Modern methods of the ballistic design of space flights (entry of a spacecraft with non-zero aerodynamic quality into the planet’s atmosphere, carrying out gravity assist maneuvers around planets) are associated with the need to calculate a lot of trajectories (i.e. of the phase beams). For their effective use and determination of dynamically admissible reachability regions, it is necessary to identify and study the structure of accompanying singularities of the phase parameters and to construct the corresponding adequate and adaptive models. In this sense, the analogy of the motion of these systems with the singular motion of billiard systems turns out to be very fruitful. First of all, we are talking about anisotropic billiards with dissipation taken into account. In this work, we set and described the model problems about the flat stones skipping on the surface of the water (the ricochets) and an inelastic ball analogue which is bouncing or rolling on an imperfect surface in the presence of a side deviation effect. The main indicatrices types classification for the singular reflection from the plane surface is carried out. Model problems of the ricocheting pebble on the surface of the water surface and the inelastic ball bouncing or rolling on an imperfect surface in the presence of a side bouncing effect also the attainability region problem of the atmospheric entry are posed and described. The configurations of their maximum reachable regions are described

New Features of P3δ software: Partial Pole Placement via Delay Action

This paper presents the software Partial Pole Placement via Delay Action, or P3$\delta$ for short. P3$\delta$ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays, thanks to two properties of the distribution of quasipolynomials' zeros, called multiplicity-induced-dominancy and coexisting real roots-induced-dominancy. After recalling recent theoretical results on these properties and their use for the feedback stabilization of control systems operating under time delays, the paper presents the main features of the current version of P3$\delta$. We detail, in particular, the assignable admissible region (the set of allowable dominant roots and the corresponding delay), which helps the user in the choice of input information, allowing a reliable stabilizing delayed feedback. We also present the newly set online version of P3$\delta$.

Asynchronous Impulsive Protocols With Asymmetric Feedback Saturation on Leader-Based Formation Control of Multiagent Systems.

Saturation phenomena often exist due to limited system resources, and impulsive protocols can lead to a reduction in communication cost. From these issues, this article investigates a leader-based formation control problem of multiagent systems via asynchronous impulsive protocols with saturated feedback. General linear system models with and without finite time-varying time delays under asymmetric saturated feedback control are concurrently considered. The asynchronous impulsive protocols only permit communication at impulsive instants and each agent has its own communication instants independently. Moreover, to improve system performance, an offset only containing desired formation information is introduced. Finally, because the feedbacks are saturated, admissible regions are proved to exist, which are also estimated by a mean of optimization. Numerical simulations are presented to demonstrate the validity of the proposed schemes.

Weighted Bergman spaces induced by doubling weights in the unit ball of $$\mathbb {C}^n$$

This paper is devoted to the study of the weighted Bergman space $$A_\omega ^p $$ in the unit ball $$\mathbb {B}$$ of $$\mathbb {C}^n$$ with doubling weight $$\omega $$ satisfying $$\begin{aligned} \int _r^1\omega (t)dt<C \int _{\frac{1+r}{2}}^1\omega (t)dt ,\quad 0\le r<1. \end{aligned}$$ The q-Carleson measures for $$A_\omega ^p$$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $$A_\omega ^p$$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $$T_g:A_\omega ^p\rightarrow A_\omega ^q$$ are also investigated in this paper with $$0<p\le q<\infty $$ , where $$\begin{aligned} T_gf(z)=\int _0^1 f(tz)\mathfrak {R}g(tz)\frac{dt}{t}, \quad f\in H(\mathbb {B}), \quad z\in \mathbb {B}. \end{aligned}$$

The Manifold Of Variations: hazard assessment of short-term impactors

When an asteroid has a few observations over a short time span the information contained in the observational arc could be so little that a full orbit determination may be not possible. One of the methods developed in recent years to overcome this problem is based on the systematic ranging and combined with the Admissible Region theory to constrain the poorly-determined topocentric range and range-rate. The result is a set of orbits compatible with the observations, the Manifold Of Variations, a two-dimensional compact manifold parametrised over the Admissible Region. Such a set of orbits represents the asteroid confidence region and is used for short-term hazard predictions. In this paper we present the Manifold Of Variations method and make a detailed analysis of the related probabilistic formalism.

Single-Switching Reachable Operation Points in a DC-DC Buck Converter: An Approximation from Time Optimal Control.

In this paper, we study the time optimal control problem in a DC-DC buck converter in the underdamped oscillatory regime. In particular, we derive analytic expressions for the admissible regions in the state space, satisfying the condition that every point within the region is reachable in optimal time with a single switching action. We then make use of the general result to establish the minimum and maximum variation allowed to the load in two predefined design set-ups that fulfills the time optimal single switching criteria. Finally, we make use of numerical simulations to show the performance of the proposed control under changes in the reference voltage and load resistance.

Data driven robust optimization of grinding process under uncertainty

ABSTRACT Uncertain process parameters present in industrial grinding circuits (IGC) increase the difficulty in modeling IGC. Conventionally, researchers have resorted to box approach for sampling in the uncertain parameter space, mimicking the uncertain parameter realizations, to observe their effects in objective functions and constraints. In case data are scattered in the uncertain parameter space, sampling in the entire range, as done in the box approach, might lead to erroneous results. To mitigate this problem, a sampling technique to generate data points inside the admissible regions is proposed leading to accurate identification of uncertain space. The proposed technique uses neuro-fuzzy c means clustering to create optimal number of clusters in the uncertain parameter space. Data points are generated using SOBOL sampling technique within each cluster boundary obtained by Delaunay triangulations. Using the proposed sampling technique in robust optimization setting and comparing with the box sampling for various sample sizes (500, 1000, 2000, 3000, 4000 and 5000), the efficacy of the proposed method has been established.

Robust Constrained Predictive Fault-Tolerant Control With Generalized Input Parameterization and Event-Triggered Regulation: Design and Experimental Results

This article proposes some new improvements to the traditional constrained predictive fault-tolerant control (FTC) method by enlarging the maximal fault-tolerant admissible region and reducing the online computational complexity. In general, the generalized parameterization of predictive input and event-triggered predictive regulation are integrated into a framework of fault compensation-based dual-mode tube predictive FTC. This predictive FTC scheme has three notable characteristics: 1) The dual-mode FTC consisting of fixed-mode fault compensation and freely predictive adjustments allows most of the calculations to be predetermined offline, while only a simple optimization problem requires to be solved online; 2) the fault-tolerant ability is enhanced by utilizing generalized parameterization of predictive inputs to enlarge the region of attraction of a faulty system; 3) the real-time computational effort of predictive FTC is further reduced by constructing an event-triggered condition to avoid unnecessary online predictive optimization. A case study of a practical synchronous dc-dc Buck converter is given to validate the effectiveness of the proposed predictive FTC method.

Possibilistic admissible region using outer probability measure theory

The admissible region method provides a reliable orbit initialization framework for space objects with extremely limited observations. In general, the uncertainty of the admissible region is either ignored or treated as a probability distribution. However, the probabilistic characterization requires that one possess exact knowledge of the distribution of all random errors, e.g., the observation process, the constraint parameters, and the observer related information, which may not be available in realistic scenarios. In this paper, we explore the use of Outer Probability Measure (OPM) theory to achieve a more truthful characterization of the uncertainty of the admissible region, dubbed the Possibilistic Admissible Region (PAR+). Considering that the uncertainty of the constraints is epistemic and modeled by possibility functions, a rigorous possibilistic uncertainty representation is achieved without the need of known random statistics of the system. To wit, the method presented herein can be appropriately interpreted as initializing our inference of a dynamical system with a so-called uninformative prior. The developed PAR+ models the uncertainty of the admissible region as an OPM represented by a collection of particles. It provides a method for space object tracking in the presence of assumed ignorance and unknown randomness (i.e. uninformative prior). The PAR+ for the commonly used angles-only and radar measurements are developed and illustrated using simulated scenarios.

A complementary quantitative feedback theory solution to the 2 × 2 tracking error problem

SummaryThis article presents a solution to the 2 × 2 multivariable tracking error problem. Current quantitative feedback theory methods commonly employ plant‐inverting splittings in order to arrive at an approximately decoupled design on the feedback control elements. This results in suitable design regions in the high‐gain, low‐frequency range, but can result in conservative design at and above the gain‐crossover frequency, resulting in overdesigned feedback controllers. Our method aims to reduce this conservatism by supplementing a plant‐inverting design with a non‐plant‐inverting design, which tends to perform well at higher frequencies. Decoupling the design of the controller elements is facilitated by making use of appropriate existence conditions. The union of the two resulting design regions then describes a larger solution space, and by enlarging the admissible design regions for the feedback controller, lower gain feedback controller design at all design frequencies of interest may be possible. A benchmarking example is presented, showing the viability of the proposed method.

MPC strategies based on the equivalent hydraulic model for the fast charge of commercial Li-ion batteries

In this work, we propose and demonstrate two low-computational MPC-based strategies to charge low-capacity commercial Li-ion batteries in the shortest time possible, while limiting the occurrence of degradation mechanisms. To do so, we use a low-complexity electrochemical model (the Equivalent Hydraulic Model, EHM). To reduce the computational burden of the resulting non-convex optimization problem, two approximations of the admissible region are considered: a time-invariant convexification, and a time-varying convex embedding. These approximations are experimentally tested on commercial LCO batteries (Turnigy 160mAh), under controlled ambient temperatures (10, 20, and 30 ∘C) and a non-controlled scenario. They are compared with the Constant Current-Constant Voltage (CCCV) protocol. Experimental results show that the proposed schemes achieve relevant reductions of the charging time with respect to the CCCV in thermal non-controlled and controlled environments. Results also highlight that the time-varying convex embedding does not significantly impact the computational time while further reducing the charging time.

Maximum Principle of Discrete Stochastic Control System Driven by Both Fractional Noise and White Noise

In this paper, we investigate the necessary optimality conditions of the discrete stochastic optimal control problems driven by both fractional noise and white noise. Here, the admissible control region is not necessarily convex. The corresponding variational inequalities are obtained by applying the classical variation method and Malliavin calculus. We also apply the stochastic maximum principle to a linear-quadratic optimal control problem to illustrate the main result.

A Stochastic Smoothing Method for Nonsmooth Global Optimization

The paper presents the results of testing the stochastic smoothing method for global optimization of a multiextremal function in a convex feasible subset of Euclidean space. Preliminarily, the objective function is extended outside the admissible region so that its global minimum does not change, and it becomes coercive. The smoothing of a function at any point is carried out by averaging the values of the function over some neighborhood of this point. The size of the neighborhood is a smoothing parameter. Smoothing eliminates small local extrema of the original function. With a sufficiently large value of the smoothing parameter, the averaged function can have only one minimum. The smoothing method consists in replacing the original function with a sequence of smoothed approximations with vanishing to zero smoothing parameter and optimization of the latter functions by contemporary stochastic optimization methods. Passing from the minimum of one smoothed function to a close minimum of the next smoothed function, we can gradually come to the region of the global minimum of the original function. The smoothing method is also applicable for the optimization of nonsmooth nonconvex functions. It is shown that the smoothing method steadily solves test global optimization problems of small dimensions from the literature.

Probabilistic data association: the orbit set

This paper presents a novel method to obtain the solution to the initial orbit determination problem for optical observations as a continuum of orbits—namely the orbit set—that fits the set of acquired observations within a prescribed accuracy. Differential algebra is exploited to analytically link the uncertainty in the observations to the state of the orbiting body with truncated power series, thus allowing for a compact analytical description of the orbit set. The automatic domain splitting tool controls the truncation error of the polynomial approximation by patching the uncertainty domain with different polynomial expansions, effectively creating a mesh. The algorithm is tested for different observing strategies to understand the working boundaries, thus defining the region for which the admissible region is necessary to extract meaningful information from observations and highlight where the new method can achieve a smaller uncertainty region, effectively showing that for some observing strategies it is possible to extract more information from a tracklet than the attributable. Consequently, the method enables comparison of orbit sets avoiding sampling when looking for correlation of different observations. Linear regression is also implemented to improve the uncertainty estimation and study the influence of the confidence level on the orbit set size. This is shown both for simulated and real observations obtained from the TFRM observatory.

Short-Arc Orbital Uncertainty Propagation with Arbitrary Polynomial Chaos and Admissible Region

This paper addresses the short-arc orbit uncertainty propagation by combining the admissible region with a new arbitrary polynomial chaos (APC) method. With short-arc optical observations, the admissible region in the range and range rate plane can be identified from various physical constraints. The initial uncertainty of the space object can be depicted by samples of the admissible region. The challenge is that such an initial uncertainty distribution is hard to characterize analytically. Hence, the parametric representation, such as the conventional polynomial chaos based on a priori assumption of the distribution, cannot be directly used. The proposed APC does not need complete knowledge or even existence of the probability density function of the uncertainty. It requires only a finite number of moments, which are used to construct the orthogonal polynomial basis functions of APC and generate the stochastic collocation points/weights to determine the APC coefficients. The moments can be easily calculated using sampling data in the admissible region. Furthermore, the multielement APC is used to improve the accuracy and computation efficiency of APC for the long-term propagation. The APC is especially efficient when it is applied in the two-dimensional admissible region because it does not need to propagate a large number of collocation points.

Tail Risk Monotonicity Under Temporal Aggregation in GARCH(1,1) Models

The stationary distribution of a GARCH(1,1) process has a power law decay, under broadly applicable conditions. We study the change in the exponent of the tail decay under temporal aggregation of parameters, with the distribution of innovations held fixed. This comparison is motivated by the fact that GARCH models are often fit to the same time series at different frequencies. The resulting models are not strictly compatible so we seek more limited properties we call forecast consistency and tail consistency. Forecast consistency is satisfied through a parameter transformation. Tail consistency leads us to derive conditions under which the tail exponent increases under temporal aggregation, and these conditions cover most relevant combinations of parameters and innovation distributions. But we also prove the existence of counterexamples near the boundary of the admissible parameter region where monotonicity fails. These counterexamples include several standard choices for innovation distributions, including the normal case.

Performance analysis for time-delay systems: application to the control of an active mass damper

This paper proposes a method to analyze, beyond stability, the performances of linear time-delay systems. Using robust analysis techniques, a sufficient condition that analyzes the location of eigenvalues in the complex plane is presented. More precisely, a set of quadratic inequality constraints are designed to define an admissible region for the infinitely many eigenvalues of a time-delay system and the quadratic separation theorem is applied to assess that the eigenvalues are effectively belonging to that stability region. This method is then used for the control of an active mass damper. A standard state feedback control is replaced with a static output feedback plus a static delayed output feedback. This strategy avoids the full measurement of the state and shows that delays in the dynamic may be helpful for stabilization. The closed-loop system is then expressed as a time-delay system and the performance criterion is exploited to analyze the stability and the damping properties. Simulations and experimental tests support the approach.