Nonlinear Analysis-Hybrid Systems最新文献

In this paper, we provide necessary and sufficient Lyapunov conditions for discrete-time switching systems to be globally exponentially stable, when the switching signal obeys to a switches digraph and is subject to dwell-time constraints. In order to best exploit the information on switching-dwelling constraints, conditions are given by means of multiple Lyapunov functions. The number of involved Lyapunov functions is equal to the number of switching modes. To avoid a pileup of Lyapunov functions, we do not introduce dummy vertices that account for dwell-time ranges. For example, in the linear case, such a pileup corresponds to a pileup of decision matrices related to some linear matrix inequalities. A link between global exponential stability and exponential input-to-state stability is provided. The following result is proved: if, in the case of zero input, the discrete-time switching system is globally exponentially stable, and the functions describing the dynamics of the subsystems, with input, are suitably globally Lipschitz, then the switching system is exponentially input-to-state stable. Finally, exploiting the well known relationship between discrete-time systems with delays and discrete-time switching systems, the provided results are shown for the former systems, in the linear case. In particular, linear matrix inequalities, by which the global exponential stability of linear discrete-time systems with constrained time delays can be possibly established, are provided. The utility of these linear matrix inequalities is shown with a numerical example taken from the literature.

This paper investigates the $p$th moment exponential stability of impulsive stochastic delayed systems (ISDS). New stability criteria for ISDS are developed by applying the Razumikhin approach and a novel concept of average random impulsive estimation (ARIE). Stochastic impulsive strength, the time-varying delay in continuous dynamics, and the impulsive delay are considered simultaneously, and there are no imposed restrictions on their sizes. Utilizing ARIE, the proposed control scheme with stabilizing impulse can still ensure the stability of the whole system under the presence of impulsive perturbations at some impulsive instants, that is, it allows some impulsive intensities at impulsive instants are greater than 1 or a certain threshold which is more than 1. In addition, the requirement for the sign of time-derivative of Razumikhin function is also relaxed. Several illustrated examples are given to verify the effectiveness of our proposed results.

By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The obtained results are comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Dumortier and Roussarie in Mem. Am. Math. Soc. 1996, and Krupa and Szmolyan in J. Differ. Equ. 2001. In addition, some novel slow–fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle–node bifurcation of canard limit cycles, one involving headless canard cycles, and the other involving canard cycles with head. This configuration also occurs in smooth systems with N-shaped fast nullcline. However, it has not been previously reported in the Van der Pol system. Our results provide justification for this observation.

In 1983, Andersen proposed the RATTLE algorithm as an extension of the SHAKE algorithm. The RATTLE algorithm is a well-established method for simulating mechanical systems with perfect bilateral constraints. This paper further extends RATTLE for simulating nonsmooth mechanical systems with frictional unilateral constraints (i.e. frictional contact). With that, it satisfies the need for higher-order integration methods within the framework of nonsmooth contact dynamics in phases where the contact status does not change (i.e. no collisions/constant sliding states). In particular, the proposed method can simulate impact-free motions, such as persistent frictional contact, with second-order accurate positions and velocities and prohibits penetration by unilateral constraints on position level.

Stochastic and robust optimization of uncertain contact-rich systems is relatively unexplored. This paper presents a chance-constrained formulation for robust trajectory optimization during manipulation. In particular, we present chance-constrained optimization of Stochastic Discrete-time Linear Complementarity Systems (SDLCS). The optimization problem is formulated as a Mixed-Integer Quadratic Program with Chance Constraints (MIQPCC). In our formulation, we explicitly consider joint chance constraints for complementarity variables and states to capture the stochastic evolution of dynamics. Additionally, we demonstrate the use of our proposed approach for designing a Stochastic Model Predictive Controller (SMPC) with complementarity constraints for a planar pushing system. We evaluate the robustness of our optimized trajectories in simulation on several systems. The proposed approach outperforms some recent approaches for robust trajectory optimization for SDLCS.

This paper studies the finite-time stability via $\mathrm{GKL}$-functions ($\mathrm{GKL}$-FTS) for impulsive dynamical systems (IDS). The notions of $\mathrm{GKL}$-functions, $\mathrm{GKL}$-FTS, and event-$\mathrm{GKL}$-FTS are proposed for IDS. The $\mathrm{GKL}$-FTS is a type of well-defined finite-time stability which is expressed via $\mathrm{GKL}$-functions. The $\mathrm{GKL}$-FTS is decomposed into specific types through the decomposition of $\mathrm{GKL}$-functions. By establishing the comparison principles of FTS including $\mathrm{GKL}$-FTS and event-$\mathrm{GKL}$-FTS, and by using the decompositions of $\mathrm{GKL}$-functions, the criteria on $\mathrm{GKL}$-FTS and event-$\mathrm{GKL}$-FTS are derived for IDS. And with the help of the decompositions of $\mathrm{GKL}$-FTS, the settling time of the $\mathrm{GKL}$-FTS is effectively calculated. Moreover, two types of specific $\mathrm{GKL}$-FTS with fixed settling time, i.e., $\mathrm{GKL}$-FTS via resetting state to zero, and event-$\mathrm{GKL}$-FTS via Zeno behaviour, are provided. And four examples with numerical simulations are presented to demonstrate the effectiveness of the results. It is shown that the $\mathrm{GKL}$-FTS criteria are less conservative in relaxing the FTS conditions of IDS in the literature. And specific effects of impulses on FTS are given in these $\mathrm{GKL}$-FTS criteria, including that an unstable continuous system may obtain FTS under a finite number of impulses.

Optimal control (OC) algorithms such as differential dynamic programming (DDP) take advantage of the derivatives of the dynamics to control physical systems efficiently. Yet, these algorithms are prone to failure when dealing with non-smooth dynamical systems. This can be attributed to factors such as the existence of discontinuities in the dynamics derivatives or the presence of non-informative gradients. On the contrary, reinforcement learning (RL) algorithms have shown better empirical results in scenarios exhibiting non-smooth effects (contacts, frictions, etc.). Our approach leverages recent works on randomized smoothing (RS) to tackle non-smoothness issues commonly encountered in optimal control and provides key insights on the interplay between RL and OC through the prism of RS methods. This naturally leads us to introduce the randomized Differential Dynamic Programming (RDDP) algorithm accounting for deterministic but non-smooth dynamics in a very sample-efficient way. The experiments demonstrate that our method can solve classic robotic problems with dry friction and frictional contacts, where classical OC algorithms are likely to fail, and RL algorithms require, in practice, a prohibitive number of samples to find an optimal solution.

This paper studies formal synthesis of controllers for continuous-space systems with unknown dynamics to satisfy requirements expressed as linear temporal logic formulas. Formal abstraction-based synthesis schemes rely on a precise mathematical model of the system to build a finite abstract model, which is then used to design a controller. The abstraction-based schemes are not applicable when the dynamics of the system are unknown. We propose a data-driven approach that computes a growth bound of the system using a finite number of trajectories. The computed growth bound together with the sampled trajectories are then used to construct the abstraction and synthesise a controller.

Our approach casts the computation of a growth bound as a robust convex optimisation program (RCP). Since the unknown dynamics appear in the optimisation, we formulate a scenario convex program (SCP) corresponding to the RCP using a finite number of sampled trajectories. We establish a sample complexity result that gives a lower bound for the number of sampled trajectories to guarantee the correctness of the growth bound computed from the SCP with a given confidence. Our sample complexity result requires knowing a possibly conservative bound on the Lipschitz constant of the system. We also provide a sample complexity result for the satisfaction of the specification on the system in closed loop with the designed controller for a given confidence. Our data-driven synthesised controller can provide guarantees on satisfaction of both finite and infinite-horizon specifications. We show that our data-driven approach can be readily used as a model-free abstraction refinement scheme by modifying the formulation of the system’s growth bounds and providing similar sample complexity results. The performance of our approach is shown on three case studies.

This work deals with the problem of designing a feedback compensator that forces the output of a linear system with abrupt discontinuities in the state evolution and polytopic uncertainties to match that of a given model with the same features. First, the case in which the system and the model are initialized at zero and output matching is required to be exact is considered. Then, the case in which, for arbitrary initialization, output matching is required to be asymptotic for sufficiently slow sequences of the time instants wherein the state exhibits abrupt discontinuities is studied. In addition, on the assumption that the model is stable for sufficiently slow jump time sequences, also the further requirement that asymptotic output matching be achieved with stability of the compensated system is investigated. Constructive, directly checkable, solvability conditions for the problems addressed are derived by leveraging on appropriate structural notions and geometric tools. Algorithmic procedures for the synthesis of the compensators, when the solvability conditions are met, are devised. Some illustrative examples conclude the work.

With the development of information technology, the amount of data transmitted in the network is getting larger and larger, and how to effectively utilize the communication resources has become the focus of a lot of research. In the framework of Finite Impulse Response (FIR) systems, this paper proposes a congruential summation-triggered communication scheme based on the cumulative effect of binary-valued observations to address the parameter identification problem, and its communication rate is derived. Then, the parameter estimation algorithm is designed under periodic input, and its strong convergence is proved, while an index is established to characterize the convergence performance of the algorithm. Furthermore, the trade-off problem between algorithm performance and communication resource is modeled as an optimization problem with constraints, and its explicit solution is given. Finally, numerical simulations are performed to verify the correctness and reasonableness of the algorithm.