Linear Temporal Logic for Hybrid Dynamical Systems: Characterizations and Sufficient Conditions

This paper introduces operators, semantics, characterizations, and solution-independent conditions to guarantee temporal logic specifications for hybrid dynamical systems. Hybrid dynamical systems are given in terms of differential inclusions -- capturing the continuous dynamics -- and difference inclusions -- capturing the discrete dynamics or events -- with constraints. State trajectories (or solutions) to such systems are parameterized by a hybrid notion of time. For such broad class of solutions, the operators and semantics needed to reason about temporal logic are introduced. Characterizations of temporal logic formulas in terms of dynamical properties of hybrid systems are presented -- in particular, forward invariance and finite time attractivity. These characterizations are exploited to formulate sufficient conditions assuring the satisfaction of temporal logic formulas -- when possible, these conditions do not involve solution information. Combining the results for formulas with a single operator, ways to certify more complex formulas are pointed out, in particular, via a decomposition using a finite state automaton. Academic examples illustrate the results throughout the paper.Comment: 35 pages. The technical report accompanying "Linear Temporal Logic for Hybrid Dynamical Systems: Characterizations and Sufficient Conditions" submitted to Nonlinear Analysis: Hybrid Systems, 201

Sufficient Conditions for Temporal Logic Specifications in Hybrid Dynamical Systems.

In this paper, we introduce operators, semantics, and conditions that, when possible, are solution-independent to guarantee basic temporal logic specifications for hybrid dynamical systems. Employing sufficient conditions for forward invariance and finite time attractivity of sets for such systems, we derive such sufficient conditions for the satisfaction of formulas involving temporal operators and atomic propositions. Furthermore, we present how to certify formulas that have more than one operator. Academic examples illustrate the results throughout the paper

L-2 State Estimation With Guaranteed Convergence Speed in the Presence of Sporadic Measurements

This paper deals with the problem of estimating the state of a nonlinear time-invariant system in the presence of sporadically available measurements and external perturbations. An observer with a continuous intersample injection term is proposed. Such an intersample injection is provided by a linear dynamical system, whose state is reset to the measured output estimation error whenever a new measurement is available. The resulting system is augmented with a timer triggering the arrival of a new measurement and analyzed in a hybrid system framework. The design of the observer is performed to achieve exponential convergence with a given decay rate of the estimation error. Robustness with respect to external perturbations and L2-external stability from plant perturbations to a given performance output are considered. Computationally efficient algorithms based on the solution to linear matrix inequalities are proposed to design the observer. Finally, the effectiveness of the proposed methodology is shown in an example

Forward Invariance of Sets for Hybrid Dynamical Systems (Part I)

In this paper, tools to study forward invariance properties with robustness to dis- turbances, referred to as robust forward invariance, are proposed for hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are given in terms of dif- ferential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required. The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without dis- turbances), including solutions that have persistent flows and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time. Sufficient conditions for sets to enjoy such properties are presented. These conditions are given in terms of the objects defining the hybrid inclusions and the set to be rendered robust forward invariant. In addition, as special cases, these conditions are exploited to state results on nominal forward invariance for hybrid systems without disturbances. Furthermore, results that provide conditions to render the sublevel sets of Lyapunov-like functions forward invariant are established. Analysis of a controlled inverter system is presented as an application of our results. Academic examples are given throughout the paper to illustrate the main ideas.Comment: 39 pages, 7 figures, accepted to TA

Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions

This paper studies set invariance and contractivity in hybrid systems modeled by hybrid inclusions using barrier functions. After introducing the notion of a multiple barrier functions, we investigate the tightest possible sufficient conditions to guarantee different forward invariance and contractivity notions of a closed set for hybrid systems with nonuniqueness of solutions and solutions terminating prematurely. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that reach the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance and contractivity involve infinitesimal conditions in terms of multiple barrier functions. Examples illustrate the results. Keywords: Forward invariance, contractivity, barrier functions, hybrid dynamical systems.Comment: Technical report accompanying the paper entitled: Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions, submitted to Automatica, 201

A symbolic simulator for hybrid equations.

In this paper, the symbolic simulation of hybrid dynamical systems is studied and an algorithm to compute symbolic solutions for such systems is presented. The tasks to perform such simulations are introduced and an algorithm to symbolically calculate a solution to a hybrid system is presented. The symbolic representation allows the proposed simulator to calculate the actual solution to the system. Benefits and drawbacks of symbolic simulation with respect to the numerical approach are presented. These statements are supported and illustrated in several examples throughout the paper

Uniting Nesterov and Heavy Ball Methods for Uniform Global Asymptotic Stability of the Set of Minimizers

We propose a hybrid control algorithm that guarantees fast convergence and uniform global asymptotic stability of the unique minimizer of a smooth, convex objective function. The algorithm, developed using hybrid system tools, employs a uniting control strategy, in which Nesterov's accelerated gradient descent is used "globally" and the heavy ball method is used "locally," relative to the minimizer. Without knowledge of its location, the proposed hybrid control strategy switches between these accelerated methods to ensure convergence to the minimizer without oscillations, with a (hybrid) convergence rate that preserves the convergence rates of the individual optimization algorithms. We analyze key properties of the resulting closed-loop system including existence of solutions, uniform global asymptotic stability, and convergence rate. Additionally, stability properties of Nesterov's method are analyzed, and extensions on convergence rate results in the existing literature are presented. Numerical results validate the findings and demonstrate the robustness of the uniting algorithm.Comment: The technical report accompanying "Uniting Nesterov and Heavy Ball Methods for Uniform Global Asymptotic Stability of the Set of Minimizers", submitted to Automatica, 2022. Revisions made according to first round reviewer feedbac