Ricardo G. Sanfelice

Hybrid dynamical systems

Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.

Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability

This paper shows several versions of the (LaSalles) invariance principle for general hybrid systems. The broad framework allows for nonuniqueness of solutions, Zeno behaviors, and does not insist on continuous dependence of solutions on initial conditions. Instead, only a mild structural property involving graphical convergence of solutions is posed. The general invariance results are then specified to hybrid systems given by set-valued data. Further results involving invariance as well as observability, detectability, and asymptotic stability are given.

Hybrid systems: Generalized solutions and robust stability 1

Abstract Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used. Then a generalized solution definition is given that leads to continuity with respect to initial conditions and perturbations of the system data. This property enables new results on necessary conditions for asymptotic stability in hybrid systems.

Quaternion-Based Hybrid Control for Robust Global Attitude Tracking

It is well known that controlling the attitude of a rigid body is subject to topological constraints. We illustrate, with examples, the problems that arise when using continuous and (memoryless) discontinuous quaternion-based state-feedback control laws for global attitude stabilization. We propose a quaternion-based hybrid feedback scheme that solves the global attitude tracking problem in three scenarios: full state measurements, only measurements of attitude, and measurements of attitude with angular velocity measurements corrupted by a constant bias. In each case, the hybrid feedback is dynamic and incorporates hysteresis-based switching using a single binary logic variable for each quaternion error state. When only attitude measurements are available or the angular rate is corrupted by a constant bias, the proposed controller is observer-based and incorporates an additional quaternion filter and bias observer. The hysteresis mechanism enables the proposed scheme to simultaneously avoid the “unwinding phenomenon” and sensitivity to arbitrarily small measurement noise that is present in discontinuous feedbacks. These properties are shown using a general framework for hybrid systems, and the results are demonstrated by simulation.

Optimal control of Mixed Logical Dynamical systems with Linear Temporal Logic specifications

Recently, linear temporal logic (LTL) has been employed as a tool for formal specification in dynamical control systems. With this formal approach, control systems can be designed to provably accomplish a large class of complex tasks specified via LTL. For this purpose, language generating Buchi automata with finite abstractions of dynamical systems have been used in the literature. In this paper, we take a mathematical programming-based approach to control of a broad class of discrete-time dynamical systems, called mixed logic dynamical (MLD) systems, with LTL specifications. MLDs include discontinuous and hybrid piecewise discrete-time linear systems. We apply these tools for model checking and optimal control of MLD systems with LTL specifications. Our algorithms exploit mixed integer linear programming (MILP) as well as, in the appropriate setting, mixed integer quadratic programming (MIQP) techniques. Our solution approach introduces a general technique useful in representing LTL constraints as mixed-integer linear constraints.

Robust global asymptotic attitude stabilization of a rigid body by quaternion-based hybrid feedback

Global asymptotic stabilization of the attitude of a rigid body is hindered by major topological obstructions. In fact, this task is impossible to accomplish with continuous state feedback. Moreover, when the attitude is parametrized with unit quaternions, it becomes impossible to design a globally stabilizing state feedback (even discontinuous) that is robust to measurement noise. In this paper, we present a quaternion-based hysteretic hybrid feedback that robustly globally asymptotically stabilizes the attitude of a rigid body. The hybrid control laws are derived through Lyapunov analysis in kinematic and dynamic settings. In the dynamic setting, we provide two control laws: one derived from an energybased Lyapunov function and another based on backstepping. Analyzing the change in these Lyapunov functions due to switching of a logic variable yields a straightforward form for state-based hysteresis. A simulation study demonstrates how hysteresis provides robustness to measurement noise and highlights differences between the energy-based and backstepping control laws.

Invariance principles for switching systems via hybrid systems techniques

Invariance principles and sufficient conditions for asymptotic stability for switching systems are given. Multiple Lyapunov-like functions are used, and dwell-time, persistent dwell-time, and weak dwell-time switching signals are considered. The invariance principles are derived from general invariance principles for hybrid systems. Asymptotic stability is concluded under observability assumptions or common bounds on the Lyapunov-like functions.

Robust Source-Seeking Hybrid Controllers for Autonomous Vehicles

We consider the problem of steering an autonomous vehicle to locate a radiation source utilizing measurements of the radiation intensity only. We propose a control algorithm that locates the source through a sequence of line minimizations of the radiation intensity. We implement in a hybrid controller, with sample-and-hold and logic variables, a discretized version of the algorithm suitable for steering a point-mass vehicle. The algorithm confers global convergence and practical stability properties to the closed-loop hybrid system. We discuss these properties and characterize the region of convergence for the vehicle. Convergence and stability results are supplemented with simulations.

A toolbox for simulation of hybrid systems in matlab/simulink: hybrid equations (HyEQ) toolbox

This paper describes the Hybrid Equations (HyEQ) Toolbox implemented in Matlab/Simulink for the simulation of hy- brid dynamical systems. This toolbox is capable of comput- ing approximations of trajectories to hybrid systems given in terms of differential and difference equations with con- straints, called hybrid equations. The toolbox is suitable for the simulation of hybrid systems with different type of trajectories, including those that are Zeno and that have multiple jumps at the same instant. It is also capable of simulating hybrid systems without inputs, with inputs, as well as interconnections of hybrid systems. The structure, components, and usage of the simulation scripts within the toolbox are described. Examples are included to illustrate the main capabilities of the toolbox.

On quaternion-based attitude control and the unwinding phenomenon

The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Often, quaternion-based feedbacks require an additional mechanism that lifts a continuous attitude path to the unit quaternion space. When this mechanism is memoryless, it has a limited domain where it remains injective and leads to discontinuities when used globally. To remedy this limitation, we propose a hybrid-dynamic algorithm for lifting a continuous attitude path to the unit quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets.