81 research outputs found
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
Dynamical Properties of a Two-gene Network with Hysteresis
A mathematical model for a two-gene regulatory network is derived and several
of their properties analyzed. Due to the presence of mixed continuous/discrete
dynamics and hysteresis, we employ a hybrid systems model to capture the
dynamics of the system. The proposed model incorporates binary hysteresis with
different thresholds capturing the interaction between the genes. We analyze
properties of the solutions and asymptotic stability of equilibria in the
system as a function of its parameters. Our analysis reveals the presence of
limit cycles for a certain range of parameters, behavior that is associated
with hysteresis. The set of points defining the limit cycle is characterized
and its asymptotic stability properties are studied. Furthermore, the stability
property of the limit cycle is robust to small perturbations. Numerical
simulations are presented to illustrate the results.Comment: 55 pages, 31 figures.Expanded version of paper in Special Issue on
Hybrid Systems and Biology, Elsevier Information and Computation, 201
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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
On Minimum-time Paths of Bounded Curvature with Position-dependent Constraints
We consider the problem of a particle traveling from an initial configuration
to a final configuration (given by a point in the plane along with a prescribed
velocity vector) in minimum time with non-homogeneous velocity and with
constraints on the minimum turning radius of the particle over multiple regions
of the state space. Necessary conditions for optimality of these paths are
derived to characterize the nature of optimal paths, both when the particle is
inside a region and when it crosses boundaries between neighboring regions.
These conditions are used to characterize families of optimal and nonoptimal
paths. Among the optimality conditions, we derive a "refraction" law at the
boundary of the regions that generalizes the so-called Snell's law of
refraction in optics to the case of paths with bounded curvature. Tools
employed to deduce our results include recent principles of optimality for
hybrid systems. The results are validated numerically.Comment: Expanded version of paper in Automatic
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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
A Rapidly-Exploring Random Trees Motion Planning Algorithm for Hybrid Dynamical Systems
This paper proposes a rapidly-exploring random trees (RRT) algorithm to solve
the motion planning problem for hybrid systems. At each iteration, the proposed
algorithm, called HyRRT, randomly picks a state sample and extends the search
tree by flow or jump, which is also chosen randomly when both regimes are
possible. Through a definition of concatenation of functions defined on hybrid
time domains, we show that HyRRT is probabilistically complete, namely, the
probability of failing to find a motion plan approaches zero as the number of
iterations of the algorithm increases. This property is guaranteed under mild
conditions on the data defining the motion plan, which include a relaxation of
the usual positive clearance assumption imposed in the literature of classical
systems. The motion plan is computed through the solution of two optimization
problems, one associated with the flow and the other with the jumps of the
system. The proposed algorithm is applied to a walking robot so as to highlight
its generality and computational features.Comment: This paper has been accepted for publication at the 2022 Conference
of Decision and Control (CDC
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
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