arXiv subjects
Publications and source records attributed to Pierdomenico Pepe.
We present a candidate counterexample to a conjecture stating that the existence of a Lyapunov-Krasovskii functional with a pointwise dissipation rate is sufficient for the input-to-state stability of time-delay systems. The counterexample has been derived through interactions with large language models.
This paper gives further insights about the Lyapunov-Krasovskii characterization of input-tostate stability (ISS) for switching retarded systems on the basis of the results in [I. Haidar and P. Pepe. Lyapunov-krasovskii characterization of the input-to-state stability for switching retarded systems. SIAM Journal on Control and Optimization, 59(4):2997-3016, 2021]. We give new characterizations of the ISS property through the existence of a relaxed common Lyapunov-Krasovskii functional. More precisely, we show that the existence of a continuous Lyapunov-Krasovskii functional whose upper right-hand Dini derivative satisfies a dissipation inequality almost everywhere is necessary and sufficient for the ISS of switching retarded systems with measurable inputs and measurable switching signals. Different characterization results, using different derivative notions, are also given.
At the occasion of Eduardo D. Sontag's 70 th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov-Krasosvkii, Lyapunov-Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.
For time-delay systems, it is known that global asymptotic stability is guaranteed by the existence of a Lyapunov-Krasovskii functional that dissipates in a point-wise manner along solutions, namely whose dissipation rate involves only the current value of the solution's norm. So far, the extension of this result to global exponential stability (GES) holds only for systems ruled by a globally Lipschitz vector field and remains largely open for the input-to-state stability (ISS) property. In this paper, we rely on the notion of exponential ISS to extend the class of systems for which GES or ISS can be concluded from a point-wise dissipation. Our results in turn show that these properties still hold in the presence of a sufficiently small additional term involving the whole state history norm. We provide explicit estimates of the tolerable magnitude of this extra term and show through an example how it can be used to assess robustness with respect to modeling uncertainties.
For time-invariant finite-dimensional systems, it is known that global asymptotic stability (GAS) is equivalent to uniform global asymptotic stability (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e. under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or Holder spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equivalences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces.
This paper introduces sufficient Lyapunov conditions guaranteeing exponential mean square stability of discrete-time systems with markovian delays. We provide a transformation of the discrete-time system with markovian delays into a discrete-time Markov jump system. Then, we extend sufficient Lyapunov conditions existing for the global asymptotic stability of discrete-time systems with delays digraphs to the mean square stability of discrete-time systems with markovian delays. Finally, an example is provided to illustrate the efficiency and advantage of the proposed method.
The communication channels used to convey information between the components of wireless networked control systems (WNCSs) are subject to packet losses due to time-varying fading and interference. We consider a wireless networked control scenario, where the packet loss occurs in both the sensor-controller link (sensing link) and the controller-actuator link (actuation link). Moreover, we consider one time-step delay mode observations of the actuation link. While the problems of state feedback optimal control and stabilizability conditions for systems with one time-step delay mode observations of the actuation link have been already solved, we study the optimal output feedback control problem, and we derive a separation principle for the aforementioned wireless networked control scenario. Particularly, we show that the optimal control problem (with one time-step delay in the mode observation of actuation link state) and the optimal filtering problem can be solved independently under a TCP-like communication scheme.
In this paper, we deal with the problem of the stabilization in the sample-and-hold sense, by emulation of continuous-time, observer-based, global stabilizers. Fully nonlinear time-delay systems are studied. Sufficient conditions are provided such that the Euler approximation of continuous-time, observer-based, global stabilizers, for nonlinear time-delay systems, yields stabilization in the sample-and-hold sense. Submitted (in an extended version) to Automatica.
The paradigm of Cyber-Physical Systems of Systems (CPSoS) is becoming rather popular in the control systems research community because of its expressive power able to properly handle many engineered complex systems of interest. Decentralized control techniques offer a promising approach in taming the inherent complexity of CPSoS, also connected with the design of needed communication infrastructures and computing units. In this paper, we propose decentralized control of networks of discrete-time nonlinear control systems, enforcing complex specifications expressed in terms of regular languages, within any desired accuracy. As discussed in the paper, regular languages, while being traditionally studied in the research community of discrete-event systems, also provide a useful mean to model a rather wide variety of complex specifications for control systems. The design of decentralized controllers is based on formal methods and in particular, on the use of discrete abstractions. Efficient synthesis of such controllers is derived by resorting to on-the-fly algorithmic techniques that also allow the use of parallel computing architectures. Advantages and disadvantages of the decentralized approach over a centralized one, also in terms of computational complexity, are discussed. An illustrative example is presented, which shows the applicability and effectiveness of the results proposed.
Symbolic models have recently spurred the interest of the research community because they offer a correct-by-design approach to the control of embedded and cyber-physical systems. In this paper we address construction of symbolic models for networks of discrete-time nonlinear control systems. The main result of the paper shows that under some small gain theorem-type conditions, a network of symbolic models can be constructed which approximates a network of incrementally stable control systems in the sense of approximate bisimulation with any desired accuracy. Compositional design of quantization parameters of the symbolic models is also derived and based on the topological properties of the network.
In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on the Arstein's theory of control Liapunov functions and related Sontag's formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distributed time-delays, of any size, as long as they are constant and, in general, known.
In this paper we show that the existence of a Lyapunov-Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map in the right-hand side of the equation is Lipschitz on bounded sets. A link between global exponential stability and input-to-state stability is also provided. @ The extended version of this paper has been submitted to the International Journal of Control, Taylor & Francis.
Time-delay systems are an important class of dynamical systems that provide a solid mathematical framework to deal with many application domains of interest. In this paper we focus on nonlinear control systems with unknown and time-varying delay signals and we propose one approach to the control design of such systems, which is based on the construction of symbolic models. Symbolic models are abstract descriptions of dynamical systems where one symbolic state and one symbolic input correspond to an aggregate of states and an aggregate of inputs. We first introduce the notion of incremental input-delay-to-state stability and characterize it by means of Lyapunov-Krasovskii functionals. We then derive sufficient conditions for the existence of symbolic models that are shown to be alternating approximately bisimilar to the original system. Further results are also derived which prove the computability of the proposed symbolic models in a finite number of steps.
Time-delay systems are an important class of dynamical systems which provide a solid mathematical framework to deal with many application domains of interest ranging from biology, chemical, electrical, and mechanical engineering, to economics. However, the inherent complexity of such systems poses serious difficulties to control design, when control objectives depart from the standard ones investigated in the current literature, e.g. stabilization, regulation, and etc. In this paper we propose one approach to control design, which is based on the construction of symbolic models, where each symbolic state and each symbolic label correspond to an aggregate of continuous states and to an aggregate of input signals in the original system. The use of symbolic models offers a systematic methodology for control design in which constraints coming from software and hardware, interacting with the physical world, can be integrated. The main contribution of this paper is in showing that incrementally input-to-state stable time-delay systems do admit symbolic models that are approximately bisimilar to the original system, with a precision that can be rendered as small as desired. An algorithm is also presented which computes the proposed symbolic models. When the state and input spaces of time-delay systems are bounded, which is the case in many realistic situations, the proposed algorithm is shown to terminate in a finite number of steps.
In this work stability results for systems described by coupled Retarded Functional Differential Equations (RFDEs) and Functional Difference Equations (FDEs) are presented. The results are based on the observation that the composite system can be regarded as the feedback interconnection of a subsystem described by RFDEs and a subsystem described by FDEs. Recent Small-Gain results and Lyapunov-like characterizations of the Weighted Input-to-Output Stability property for systems described by RFDEs and FDEs are employed. It is shown that the stability results provided in this work can be used to study stability for systems described by neutral functional differential equations and systems described by hyperbolic partial differential equations.
In this work characterizations of notions of output stability for uncertain time-varying systems described by retarded functional differential equations are provided. Particularly, characterizations by means of Lyapunov and Razumikhin functions of uniform and non-uniform in time Robust Global Asymptotic Output Stability and Input-to-Output Stability are given. The results of this work have been developed for systems with outputs in abstract normed linear spaces in order to allow outputs with no delay, with discrete or distributed delay or functional outputs with memory.