arXiv subjects

Ricardo G. Sanfelice

Publications and source records attributed to Ricardo G. Sanfelice.

49 records · Page 3Linked to original sources

arXiv2020

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.

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arXiv2019

A Hybrid Control Algorithm for Gradient-Free Optimization using Conjugate Directions

The problem of steering a particular class of $n$-dimensional continuous-time dynamical systems towards the minima of a function without gradient information is considered. We propose an hybrid controller, implementing a discrete-time Direct Search algorithm based on conjugate directions, able to solve the optimization problem for the resulting closed loop system in an almost global sense. Furthermore, we show that Direct Search algorithms based on asymptotic step size reduction are not robust to measurement noise, and, to achieve robustness, we propose a modified version by imposing a lower bound on the step size and able to achieve robust practical convergence to the optimum. In this context we show a bound relating the supremum norm of the noise signal to the step size by highlighting a trade-off between asymptotic convergence and robustness.

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arXiv2019

Robust Global Exponential Stabilization on the $n$-Dimensional Sphere with Applications to Trajectory Tracking for Quadrotors

In this paper, we design a hybrid controller that globally exponentially stabilizes a system evolving on the n-dimensional sphere, denoted by Sn. This hybrid controller is induced by a synergistic collection of potential functions on Sn. We propose a particular construction of this class of functions that generates flows along geodesics of the sphere, providing convergence to the desired reference with minimal path length. We show that the proposed strategy is suitable to the exponential stabilization of a quadrotor vehicle.

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arXiv2018

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.

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arXiv2016

Convergence of Nonlinear Observers on R^n with a Riemannian Metric (Part II)

In [1], it is established that a convergent observer with an infinite gain margin can be designed for a given nonlinear system when a Riemannian metric showing that the system is differentially detectable (i.e., the Lie derivative of the Riemannian metric along the system vector field is negative in the space tangent to the output function level sets) and the level sets of the output function are geodesically convex is available. In this paper, we propose techniques for designing a Riemannian metric satisfying the first property in the case where the system is strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property) or where it is strongly differentially observable (i.e. the mapping state to output derivatives is an injective immersion) or where it is Lagrangian. Also, we give results that are complementary to those in [1]. In particular, we provide a locally convergent observer and make a link to the existence of a reduced order observer. Examples illustrating the results are presented.

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arXiv2015

Convergence of Nonlinear Observers on R^n with a Riemannian Metric (Part I)

We study how convergence of an observer whose state lives in a copy of the given system's space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally negative. Moreover, we establish that the existence of this metric is related to the observability of the system's linearization along its solutions. Moreover, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. Conversely, we establish that, if a complete Riemannian metric has a Lie derivative along the system vector field that is conditionally negative and is such that the output function has a monotonicity property, then there exists an observer with an infinite gain margin.

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arXiv2015

Robust Asymptotic Stability of Desynchronization in Impulse-Coupled Oscillators

The property of desynchronization in an all-to-all network of homogeneous impulse-coupled oscillators is studied. Each impulse-coupled oscillator is modeled as a hybrid system with a single timer state that self-resets to zero when it reaches a threshold, at which event all other impulse-coupled oscillators adjust their timers following a common reset law. In this setting, desynchronization is considered as each impulse-coupled oscillator's timer having equal separation between successive resets. We show that, for the considered model, desynchronization is an asymptotically stable property. For this purpose, we recast desynchronization as a set stabilization problem and employ Lyapunov stability tools for hybrid systems. Furthermore, several perturbations are considered showing that desynchronization is a robust property. Perturbations on both the continuous and discrete dynamics are considered. Numerical results are presented to illustrate the main contributions.

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arXiv2015

Interconnected Observers for Robust Decentralized Estimation with Performance Guarantees and Optimized Connectivity Graph

Motivated by the need of observers that are both robust to disturbances and guarantee fast convergence to zero of the estimation error, we propose an observer for linear time-invariant systems with noisy output that consists of the combination of N coupled observers over a connectivity graph. At each node of the graph, the output of these interconnected observers is defined as the average of the estimates obtained using local information. The convergence rate and the robustness to measurement noise of the proposed observer's output are characterized in terms of $\mathcal{KL}$ bounds. Several optimization problems are formulated to design the proposed observer so as to satisfy a given rate of convergence specification while minimizing the $H_\infty$ gain from noise to estimates or the size of the connectivity graph. It is shown that that the interconnected observers relax the well-known tradeoff between rate of convergence and noise amplification, which is a property attributed to the proposed innovation term that, over the graph, couples the estimates between the individual observers. Sufficient conditions involving information of the plant only, assuring that the estimate obtained at each node of the graph outperforms the one obtained with a single, standard Luenberger observer are given. The results are illustrated in several examples throughout the paper.

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arXiv2014

Distance function design and Lyapunov techniques for the stability of hybrid trajectories

The comparison between time-varying hybrid trajectories is crucial for tracking, observer design and synchronisation problems for hybrid systems with state-triggered jumps. In this paper, a systematic way of designing an appropriate distance function is proposed that can be used for this purpose. The so-called "peaking phenomenon", which occurs when using the Euclidean distance to compare two hybrid trajectories, is circumvented by taking the hybrid nature of the system explicitly into account in the design of the distance function. Based on the proposed distance function, we define the stability of a trajectory of a hybrid system with state-triggered jumps and present sufficient Lyapunov-type conditions for stability of a hybrid trajectory. A constructive design method for the distance function is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane. For this case, the mentioned Lyapunov-type stability conditions can be verified using linear matrix conditions. Finally, for this class of systems, we present a tracking controller that asymptotically stabilises a given hybrid reference trajectory, and we illustrate our results with examples.

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arXiv2014

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.

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arXiv2014

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.

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arXiv2013

A Hybrid Model of a Genetic Regulatory Network in Mammalian Sclera

Myopia in human and animals is caused by the axial elongation of the eye and is closely linked to the thinning of the sclera which supports the eye tissue. This thinning has been correlated with the overproduction of matrix metalloproteinase (MMP-2), an enzyme that degrades the collagen structure of the sclera. In this short paper, we propose a descriptive model of a regulatory network with hysteresis, which seems necessary for creating oscillatory behavior in the hybrid model between MMP-2, MT1-MMP and TIMP-2. Numerical results provide insight on the type of equilibria present in the system.

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arXiv2013

Robust Supervisory Control for Uniting Two Output-Feedback Hybrid Controllers with Different Objectives

The problem of robustly, asymptotically stabilizing a point (or a set) with two output-feedback hybrid controllers is considered. These control laws may have different objectives, e.g., the closed-loop systems resulting with each controller may have different attractors. We provide a control algorithm that combines the two hybrid controllers to accomplish the stabilization task. The algorithm consists of a hybrid supervisor that, based on the values of plant's outputs and (norm) state estimates, selects the hybrid controller that should be applied to the plant. The accomplishment of the stabilization task relies on an output-to-state stability property induced by the controllers, which enables the construction of an estimator for the norm of the plant's state. The algorithm is motivated by and applied to robust, semi-global stabilization problems uniting two controllers.

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