Pierdomenico Pepe

A Lyapunov–Krasovskii methodology for ISS and iISS of time-delay systems

Abstract This paper presents a Lyapunov–Krasovskii methodology for studying the input-to-state stability and the integral input-to-state stability of nonlinear time-delay systems. An integral input-state estimate which takes into account non-zero initial conditions is also proposed.

A new approach to state observation of nonlinear systems with delayed output

The article presents a new approach for the construction of a state observer for nonlinear systems when the output measurements are available for computations after a nonnegligible time delay. The proposed observer consists of a chain of observation algorithms reconstructing the system state at different delayed time instants (chain observer). Conditions are given for ensuring global exponential convergence to zero of the observation error for any given delay in the measurements. The implementation of the observer is simple and computer simulations demonstrate its effectiveness.

Brief paper: On Liapunov-Krasovskii functionals under Carathéodory conditions

In [Driver, R. D. (1962). Existence and stability of solutions of a delay-differential system. Archive for Rational Mechanics and Analysis 10, 401-426] a proper definition, not involving the solution, of the derivative of the Liapunov-Krasovskii functional for retarded functional differential equations with continuous right side is given and it is showed that this definition coincides with the non-constructive one given in Krasovskii [1956. On the application of the second method of A. M. Lyapunov to equations with time delays (in Russian). Prikladnaya Matematika i Mekhanika 20, 315-327] involving the solution, for functionals V which are locally Lipschitz (and not only continuous, as it is considered in most literature). In this paper, the result by Driver is extended to a general class of retarded functional differential equations coupled with continuous time difference equations with more general right sides, verifying the Caratheodory conditions. Such result is applied to build a new Liapunov-Krasovskii theorem for studying the input-to-state stability of time-invariant neutral functional differential equations with linear difference operator. An example taken from the literature, concerning transmission lines, is reported, showing the effectiveness of the methodology.

A New Lyapunov-Krasovskii Methodology for Coupled Delay Differential Difference Equations

In this paper a new Lyapunov-Krasovskii methodology for nonlinear coupled delay differential difference equations is proposed. This methodology is based on the concept of input-to-state stability applied to the difference equation, for which a sufficient Lyapunov criterion is given, and on previous methodologies developed in the literature for linear delay descriptor systems

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hales form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hales form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.

The Problem of the Absolute Continuity for Lyapunov–Krasovskii Functionals

The condition of non-positivity, almost everywhere, of the upper right-hand Dini derivative of a (simply) continuous function is not a sufficient condition for such function to be non-increasing. That condition is sufficient for the non-increasing property if the function is locally absolutely continuous. Therefore, if the time function obtained by the evaluation of a Liapunov-Krasovskii functional on the solution of a time-delay system is not locally absolutely continuous, but simply continuous, and its upper right-hand Dini derivative is almost everywhere non-positive, then the conclusion that such function is non-increasing cannot be drawn. And, as a consequence, related stability conclusions cannot be drawn. In this paper such problem is investigated for input-to-state stability concerns of time invariant time-delay systems forced by measurable locally essentially bounded inputs. It is shown that, if the Liapunov-Krasovskii functional is locally Lipschitz with respect to the norm of the uniform topology, then the problem of the absolute continuity is overcome

Input-to-State Stability of Time-Delay Systems: A Link With Exponential Stability

The main contribution of this technical note is to establish a link between the exponential stability of an unforced system and the input-to-state stability (ISS) via the Liapunov-Krasovskii methodology. It is proved that a system which is (globally, locally) exponentially stable in the unforced case is (globally, locally) input-to-state stable when it is forced by a measurable and locally essentially bounded input, provided that the functional describing the dynamics in the unforced case is (globally, on bounded sets) Lipschitz and the functional describing the dynamics in the forced case satisfies a Lipschitz-like hypothesis with respect to the input. Moreover, a new feedback control law is provided for delay-free linearizable and stabilizable time-delay systems, whose dynamics is described by locally Lipschitz functionals, by which the closed-loop system is ISS with respect to disturbances adding to the control law, a typical problem due to actuator errors.

Symbolic models for nonlinear time-delay systems using approximate bisimulations

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.

Local asymptotic stability for nonlinear state feedback delay systems

In previous papers the authors presented an elementary theory for feedback control of nonlinear delay systems, in which methods of standard nonlinear analysis were used to solve control problems such as output regulation and tracking, disturbance decoupling and model matching for a class of nonlinear delay systems. Output control was obtained by means of state feedback control laws, but nothing was said about the behavior of the system state. In this paper some results have been obtained about this problem. It is proved that if the output and its derivatives up to a given order are driven to zero, and if the system owns a certain Lipschitz property in a suitable neighborhood of the origin, and the initial state is inside such neighborhood, then the system state asymptotically goes to zero. Simulations on nonlinear delay systems unstable in open loop match the theoretical results.

A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals

This paper considers interconnected retarded nonlinear systems. Integral input-to-state stable subsystems and the construction of Lyapunov-Krasovskii functionals for their interconnections are focused on. Both discrete and distributed time-delays in the subsystems and the communication channels are covered. This paper provides a sufficient small-gain type condition for the stability of the interconnected systems with respect to external inputs in the framework of Lyapunov-Krasovskii functionals. Global asymptotic stability is addressed as a special case which deals with time-varying delays in communication channels effectively.