We propose a methodology for the automatic verification of safety properties of controllers based on dynamical systems, such as those typically used in avionics. In particular, our focus is on proving stability properties of software implementing linear and some non-linear controllers. We develop an abstract interpretation framework that follows closely the Lyapunov methods used in proofs at the model level and describe the corresponding abstract domains, which for linear systems consist of ellipsoidal constraints. These ellipsoidal domains provide abstractions for the values of state variables and must be combined with other domains that model the remaining variables in a program. Thus, the problem of automatically assigning the right type of abstract domain to each variable arises. We provide an algorithm that solves this classification problem in many practical cases and suggest how it could be generalized to more complicated cases. We then find a fixpoint by solving a matrix equation, which in the linear case is just the discrete Lyapunov equation. Contrary to most cases in software analysis, this fixpoint cannot be reached by the usual iterative method of propagating constraints until saturation and so numerical methods become essential. Finally, we illustrate our methodology with several examples.
The study of stability of controllers at the theoretical level is a very mature subject, especially for linear systems. Two techniques are available in that case: eigenvalue analysis and Lyapunov methods. While eigenvalue analysis is more straightforward, Lyapunov methods generalize better to non-linear cases @cite_5 . Thus, most recent work uses the latter. Lyapunov methods give way to the study of propagation of ellipsoids such as that presented in this paper.
@cite_5
1545855501 2104958104 2740824349 1991667713
This paper focuses on a new geometric approach to (fully actuated) control systems on the sphere. Our control laws exploit the basic and intuitive notions of geodesic direction and of distance between points, and generalize the classical proportional plus derivative feedback (PD) without the need of arbitrary local coordinate charts. The stability analysis relies on an appropriate Lyapunov function, where the notion of distance and its properties are exploited. This methodology then applies to spin-axis stabilization of a spacecraft actuated by only two control torques: discarding the rotation about the unactuated axis, a reduced system is considered, whose state is in fact defined on the sphere. For this reduced stabilization problem our approach allows us not only to deal optimally with the inevitable singularity, but also to achieve simplicity, versatility and (coordinate independent) adaptive capabilities. This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper. A data-driven identification of dynamical systems requiring only minimal prior knowledge is promising whenever no analytically derived model structure is available, e.g., from first principles in physics. However, meta-knowledge on the system’s behavior is often given and should be exploited: Stability as fundamental property is essential when the model is used for controller design or movement generation. Therefore, this paper proposes a framework for learning stable stochastic systems from data. We focus on identifying a state-dependent coefficient form of the nonlinear stochastic model which is globally asymptotically stable according to probabilistic Lyapunov methods. We compare our approach to other state of the art methods on real-world datasets in terms of flexibility and stability. We consider an imitation learning approach to model robot point-to-point (also known as discrete or reaching) movements with a set of autonomous Dynamical Systems (DS). Each DS model codes a behavior (such as reaching for a cup and swinging a golf club) at the kinematic level. An estimate of these DS models are usually obtained from a set of demonstrations of the task. When modeling robot discrete motions with DS, ensuring stability of the learned DS is a key requirement to provide a useful policy. In this paper we propose an imitation learning approach that exploits the power of Control Lyapunov Function (CLF) control scheme to ensure global asymptotic stability of nonlinear DS. Given a set of demonstrations of a task, our approach proceeds in three steps: (1) Learning a valid Lyapunov function from the demonstrations by solving a constrained optimization problem, (2) Using one of the-state-of-the-art regression techniques to model an (unstable) estimate of the motion from the demonstrations, and (3) Using (1) to ensure stability of (2) during the task execution via solving a constrained convex optimization problem. The proposed approach allows learning a larger set of robot motions compared to existing methods that are based on quadratic Lyapunov functions. Additionally, by using the CLF formalism, the problem of ensuring stability of DS motions becomes independent from the choice of regression method. Hence it allows the user to adopt the most appropriate technique based on the requirements of the task at hand without compromising stability. We evaluate our approach both in simulation and on the 7 degrees of freedom Barrett WAM arm. Proposing a new parameterization to model complex Lyapunov functions.Estimating task-oriented Lyapunov functions from demonstrations.Ensuring stability of nonlinear autonomous dynamical systems.Applicability to any smooth regression method.