On the Decidability of Reachability in Linear Time-Invariant Systems

We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which control sets are affine subspaces of $\mathbb{R}^n$ is a fundamental result in control theory. Our first result is that reachability is undecidable if the set of controls is a finite union of affine subspaces. We also consider versions of the reachability problem in which (i)~the set of controls consists of a single affine subspace together with the origin and (ii)~the set of controls is a convex polytope. In these two cases we respectively show that the reachability problem is as hard as Skolem's Problem and the Positivity Problem for linear recurrence sequences (whose decidability has been open for several decades). Our main contribution is to show decidability of a version of the reachability problem in which control sets are convex polytopes, under certain spectral assumptions on the transition matrix

Constructive Hybrid Games

Hybrid games are models which combine discrete, continuous, and adversarial dynamics. Game logic enables proving (classical) existence of winning strategies. We introduce constructive differential game logic (CdGL) for hybrid games, where proofs that a player can win the game correspond to computable winning strategies. This is the logical foundation for synthesis of correct control and monitoring code for safety-critical cyber-physical systems. Our contributions include novel static and dynamic semantics as well as soundness and consistency.Comment: 60 pages, preprint, under revie

On the Limits of Collision Detection Performance of a Sense-and-Avoid System for Non-Cooperative Air Traffic

Unmanned Aircraft Systems (UAS) face limitations on their utilization in civil airspace because they do not have the ability to Sense and Avoid (SAA) other air traffic. In recent years, there has been growing interest to provide an effective SAA solution for UAS operations. An effective SAA solution must address both cooperative as well as non-cooperative air traffic. A number of different sensor solutions are being evaluated for SAA pertaining to non-cooperative traffic. Examples of such sensors include electro-optical (EO), on-board radar, passive acoustics, laser radar and ground radar. Using one or more such sensing modalities, it is possible to track a non-cooperative aircraft in the vicinity of the own aircraft otherwise known as an intruder. However, an intruder’s future trajectory is never perfectly known and a SAA system’s performance will always be limited by these uncertainties. One of the components of an SAA system is the logic to decide whether a certain aircraft is on a collision course with the own craft. It, therefore, follows that the performance of this collision detection component will also be limited by the uncertainties in the future trajectory of the intruder. In this paper, we investigate the problem of what is the best that a SAA collision detection system can perform in spite of the future uncertainties in the intruder trajectory

Mode Switching Synthesis for Reachability Specifications

In many control applications, a specific set of output tracking controllers of satisfactory performance have already been designed and must be used. When such a collection of control modes is available, an important problem is to be able to accomplish a variety of high level tasks by appropriately switching between the low-level control modes. In this paper, we define a concept of control modes, and propose a framework for determining the sequence of control modes that will satisfy reachability tasks. Our framework exploits the structure of output tracking controllers in order to extract a finite graph where the mode switching problem can be e#ciently solved, and then implement it using the continuous controllers. Our approach is illustrated on a helicopter example, where we determine the mode switching logic that achieves the high-altitude takeo# task from a hover mode

Low-Cost Visual Tracking of a Landing Place and Hovering Flight Control with a Microcontroller

The growth of civil and military use has recently promoted the development of unmanned miniature aerial vehicles dedicated to surveillance tasks. These flying vehicles are often capable of carrying only a few dozen gramms of payload. To achieve autonomy for this kind of aircraft novel sensors are required, which need to cope with strictly limited onboard processing power. One of the key aspects in autonomous behaviour is target tracking. Our visual tracking approach differs from other methods by not using expensive cameras but a Wii remote camera, i.e. commodity consumer hardware. The system works without stationary sensors and all processing is done with an onboard microcontroller. The only assumptions are a good roll and pitch attitude estimation, provided by an inertial measurement unit and a stationary pattern of four infrared spots on the target or the landing spot. This paper details experiments for hovering above a landing place, but tracking a slowly moving target is also possible