Computing Reachable Sets as Capture-Viability Kernels in Reverse Time

The set SF(x0;T) of states y reachable from a given state x0 at time T under a set-valued dynamic x’(t)∈F(x (t)) and under constraints x(t)∈K where K is a closed set, is also the capture-viability kernel of x0 at T in reverse time of the target {x0} while remaining in K. In dimension up to three, Saint-Pierre’s viability algorithm is well-adapted;for higher dimensions, Bonneuil’s viability algorithm is better suited. It is used on a large-dimensional example.

Error reachable set based stabilization of switched linear systems with bounded peak disturbances

This paper investigates the error reachable set based stabilization problem for a class of discrete-time switched linear systems with bounded peak disturbances under persistent dwell-time(PDT)constraint.A double-clockdependent control scheme is presented that can split the disturbed switched system into a nominal system and an error system,and assign to each system a controller scheduled by a clock.A necessary and sufficient convex stability criterion is presented for the nominal system,and is further extended to the stabilization controller design with a nominal clock.In the presence of bounded peak disturbances,another stabilization controller with an error clock is developed for the error system,with the purpose of‘‘minimizing’’the reachable set of the error system by the ellipsoidal techniques.It is demonstrated that the disturbed system is also globally exponentially stable in the sense of converging to an over approximation of the reachable set of the error system,i.e.,a union of a family of bounding ellipsoids,that can also be regarded as the cross section of a tube containing the trajectories of the disturbed system.Two numerical examples are provided to verify the effectiveness of the developed results.

Reachable set estimation for discrete-time Markovian jump neural networks with unified uncertain transition probability

This paper focuses on the reachable set estimation for Markovian jump neural networks with time delay.By allowing uncertainty in the transition probabilities,a framework unifies and enhances the generality and realism of these systems.To fully exploit the unified uncertain transition probabilities,an equivalent transformation technique is introduced as an alternative to traditional estimation methods,effectively utilizing the information of transition probabilities.Furthermore,a vector Wirtinger-based summation inequality is proposed,which captures more system information compared to existing ones.Building upon these components,a novel condition that guarantees a reachable set estimation is presented for Markovian jump neural networks with unified uncertain transition probabilities.A numerical example is illustrated to demonstrate the superiority of the approaches.

An Algorithm to Construct Concurrent Reachability Graph of Petri Nets

Reachability graph is a very important tool to analyze the dynamic properties of Petri nets, but the concurrent relation of transitions in Petri nets cannot be represented by reachability graph. Petri net is a concurrent system, while reachability graph is a serial one. However, concurrency is a kind of property which is not only very significant but also difficult to be analyzed and controlled. This paper presents the concepts of concurrent reachable marking and concurrent reachable graph in order to represent and analyze the concurrent system. The algorithm constructing concurrent reachable marking set and concurrent reachability graph is also shown so that we can study the response problems among services in a network computing environment and analyze the throughput of the system. The Dining Philosophers Problem, which is a classic problem of describing the management of concurrent resources, is given as an example to illustrate the significance of concurrent reachability graph.

Reachable set estimation for spacecraft relative motion based on bang-bang principle

For spacecraft formation flight,the information of relative motion reachable set is very important,which can be used to predict the operating boundary of adjacent spacecraft and thus to ensure the safety of spacecraft operation.In this paper,we aim at developing a numerical method to approximate the reachable set for spacecraft relative motion.In particular,we focus on the quality of the approximation and the computational cost.Based on the bang-bang control principle,a polyhedral approximation algorithm is proposed to compute the reachable set of a relative motion spacecraft system.An inner approximation and an outer approximation of the reachable set for the system can be obtained.We prove that the approximation quality measured in Hausdorff distance can be guaranteed.The method is easy to implement and has low computational cost.Finally,the effectiveness of the algorithm is demonstrated by experimental simulation.

Improving flight performance of UAVs by ice shape modulation

Aircraft icing poses a great threat to flight safety.In response to the characteristics of high-power consumption,large volume,and heavy weight of traditional anti-/de-icing technologies,the concept of ice shape modulation is proposed,which is called ice tolerant flight.Firstly,the flight performance of Unmanned Aerial Vehicle(UAV)was compared in three states:no ice,full ice,and modulated ice through flight tests.It was found that ice shape modulation has a significant improvement effect on the aerodynamic performance of aircraft under icing conditions.Under the three modulated ice shape conditions in this experiment,the lift coefficient of the UAV under different ice shape modulation conditions increased by 18%–33%,and the stalling angle was delayed by 3°-5°.Subsequently,the pressure distribution,streamlines in the flow field,and detached vortex distribution of the UAV model in these three states were obtained through numerical simulation,to study the mechanism of ice shape modulation on the aerodynamic performance of aircraft.The simulation found that the reason for the improvement of the wings effect after ice shape modulation is that the modulated area forms a leading-edge protrusion structure similar to a vortex generator.This structure prolongs the mixed flow region on the wings surface and reduces the trend of flow separation,which plays a role in increasing lift and reducing drag for UAVs under icing conditions.Finally,a reverse reachable set that can be used for unexpected state recovery is used as the definition of flight safety boundaries,and an aircraft dynamics model is established to obtain flight safety boundaries for different states.Research has found that the flight safety boundary of the UAV in a no ice state is greater than that in a modulated ice state,and the safety boundary in a modulated ice state is greater than that in a full ice state.Compared with the full ice state,the flight safety boundary after modulation has expanded by 27.0%.The scheme of ice shape modulation can provide a basis for the flight safety of aircraft under icing conditions.

Reachable set modeling and engagement analysis of exoatmospheric interceptor

A novel reachable set（RS） model is developed within a framework of exoatmospheric interceptor engagement analysis. The boost phase steering scheme and trajectory distortion mechanism of the interceptor are firstly explored. A mathematical model of the distorted RS is then formulated through a dimension–reduction analysis. By treating the outer boundary of the RS on sphere surface as a spherical convex hull, two relevant theorems are proposed and the RS envelope is depicted by the computational geometry theory. Based on RS model, the algorithms of intercept window analysis and launch parameters determination are proposed, and numerical simulations are carried out for interceptors with different energy or launch points. Results show that the proposed method can avoid intensive on-line computation and provide an accurate and effective approach for interceptor engagement analysis. The suggested RS model also serves as a ready reference to other related problems such as interceptor effectiveness evaluation and platform disposition.

Reachable Set Analysis for Dynamic Neural Networks with Polytopic Uncertainties

In this paper,the reachable set estimation problem is studied for a class of dynamic neural networks subject to polytopic uncertainties.The problem addressed here is to find a set as small as possible to bound the states starting from the origin by inputs with peak values.The maximal Lyapunov functional is proposed to derive a sufficient condition for the existence of a non-ellipsoidal bound to estimate the states of neural networks.It is theoretically shown that this method is superior to the traditional one based on the common Lyapunov function.Finally,two examples illustrate the advantages of our proposed result.

Reachable Set Estimation and Safety Verification of Nonlinear Systems via Iterative Sums of Squares Programming

In this paper,the problems of forward reachable set estimation and safety verification of uncertain nonlinear systems with polynomial dynamics are addressed.First,an iterative sums of squares(SOS)programming approach is developed for reachable set estimation.It characterizes the over-approximations of the forward reachable sets by sub-level sets of time-varying Lyapunovlike functions that satisfy an invariance condition,and formulates the problem of searching for the Lyapunov-like functions as a bilinear SOS program,which can be solved via an iterative algorithm.To make the over-approximation tight,the proposed approach seeks to minimize the volume of the overapproximation set with a desired shape.Then,the reachable set estimation approach is extended for safety verification,via explicitly encoding the safety constraint such that the Lyapunov-like functions guarantee both reaching and avoidance.The efficiency of the presented method is illustrated by some numerical examples.

Capacity of Sun-driven lunar swingby sequences and their application in asteroid retrieval

For deep-space mission design,the gravity of the Sun and the Moon can be first considered and utilized.Their gravity can provide the energy change for launching spacecraft and retrieving spacecraft as well as asteroids.Regarding an asteroid retrieval mission,it can lead to the mitigation of asteroid hazards and an easy exploration and exploitation of the asteroid.This paper discusses the application of the Sun-driven lunar swingby sequence for asteroid missions.Characterizing the capacity of this technique is not only interesting in terms of the dynamic insights but also non-trivial for trajectory design.The capacity of a Sun-driven lunar swingby sequence is elucidated in this paper with the help of the“Swingby-Jacobi”graph.The capacity can be represented by a range of the Jacobi integral that encloses around 660 asteroids currently cataloged.To facilitate trajectory design,a database of Sun-perturbed Moon-to-Moon transfers,including multi-revolution cases,is generated and employed.Massive trajectory options for spacecraft launch and asteroid capture can then be explored and optimized.Finally,a number of asteroid flyby,rendezvous,sample-return,and retrieval mission options enabled by the proposed technique are obtained.

Finite-Time Observability of Probabilistic Logical Control Systems

This study investigates finite-time observability of probabilistic logical control systems(PLCSs)under three definitions(i.e.,finite-time observability with probability one,finite-time singleinput sequence observability with probability one,and finite-time arbitrary-input observability with probability one).The authors adopt a parallel extension technique to recast the finite-time observability problem of a PLCS as a finite-time set reachability problem.Then,the finite-time set reachability problem can be transferred to stabilization problem of a logic dynamical system by using the state transfer graph reconstruction method.Necessary and sufficient conditions for finite-time observability under the three definitions are derived respectively.Finally,the proposed methods are illustrated by numerical examples.

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class So（Bn） of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f（-, O） of a Loewner chain f（x,t） = etz ＋ ... such that {e-tf（.,t）}t≥o is a normal family on Bn. We show that if f（.,O） is an extreme point （respectively a support point） of So（Bn）, then e-t f（., t） is an extreme point of So（Bn） for t≥0 （respectively a support point of So（Bn） for t C [O, t0] and some to〉 0）. This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to So（Bn） and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.