Input-to-state stability of infinite-dimensional control systems

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.Comment: 33 page

Stability of interconnected impulsive systems with and without time-delays using Lyapunov methods

In this paper we consider input-to-state stability (ISS) of impulsive control systems with and without time-delays. We prove that if the time-delay system possesses an exponential Lyapunov-Razumikhin function or an exponential Lyapunov-Krasovskii functional, then the system is uniformly ISS provided that the average dwell-time condition is satisfied. Then, we consider large-scale networks of impulsive systems with and without time-delays and we prove that the whole network is uniformly ISS under a small-gain and a dwell-time condition. Moreover, these theorems provide us with tools to construct a Lyapunov function (for time-delay systems - a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function) and the corresponding gains of the whole system, using the Lyapunov functions of the subsystems and the internal gains, which are linear and satisfy the small-gain condition. We illustrate the application of the main results on examples

Numerical Construction of LISS Lyapunov Functions under a Small Gain Condition

In the stability analysis of large-scale interconnected systems it is frequently desirable to be able to determine a decay point of the gain operator, i.e., a point whose image under the monotone operator is strictly smaller than the point itself. The set of such decay points plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system and in the numerical construction of a LISS Lyapunov function. We provide a homotopy algorithm that computes a decay point of a monotone op- erator. For this purpose we use a fixed point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. The advantage to an earlier algorithm is demonstrated. Furthermore an example is given which shows how to analyze a given perturbed interconnected system.Comment: 30 pages, 7 figures, 4 table

Experimental evaluation of two complementary decentralized event-based control methods

To appear in Control Engineering PracticeInternational audienceEvent-based control aims at the reduction of the feedback communication effort among the sensors, controllers and actuators in control loops. The feedback communication is invoked by some well-defined triggering condition. This paper presents a new method for the decentralized event-based control of physically interconnected systems and shows its experimental evaluation. The novel method is based on two complementary approaches, called the global and the local approach, which jointly ensure the ultimate boundedness of the closed-loop system. The global approach steers the state of each subsystem into a target region, whereas the local approach makes the state remain in this set in spite of exogenous disturbances and the effect of the interconnections to other subsystems. This event-based control method is applied to a continuous flow process to show its practical implementation and to evaluate the analytical results on the basis of experiments

Input-to-state stability of infinite-dimensional control systems

We define the notion of local ISS-Lyapunov function and prove, that existence of a local ISS-Lyapunov function implies local ISS (LISS) of the system. Then we consider infinite-dimensional systems generated by differential equations in Banach spaces. We prove, that an interconnection of such systems is ISS if all the subsystems are ISS and the small-gain condition holds. Next we show that a system is LISS provided its linearization is ISS. In the second part of the thesis we deal with infinite-dimensional impulsive systems. We prove, that existence of an ISS Lyapunov function (not necessarily exponential) for an impulsive system implies ISS of the system over impulsive sequences satisfying nonlinear fixed dwell-time condition. Also we prove, that an impulsive system, which possesses an exponential ISS Lyapunov function is uniform ISS over impulse time sequences, satisfying the generalized average dwell-time condition. Then we generalize small-gain theorems to the case of impulsive systems

The Global Risks Report 2016, 11th Edition

Now in its 11th edition, The Global Risks Report 2016 draws attention to ways that global risks could evolve and interact in the next decade. The year 2016 marks a forceful departure from past findings, as the risks about which the Report has been warning over the past decade are starting to manifest themselves in new, sometimes unexpected ways and harm people, institutions and economies. Warming climate is likely to raise this year's temperature to 1° Celsius above the pre-industrial era, 60 million people, equivalent to the world's 24th largest country and largest number in recent history, are forcibly displaced, and crimes in cyberspace cost the global economy an estimated US$445 billion, higher than many economies' national incomes. In this context, the Reportcalls for action to build resilience – the "resilience imperative" – and identifies practical examples of how it could be done.The Report also steps back and explores how emerging global risks and major trends, such as climate change, the rise of cyber dependence and income and wealth disparity are impacting already-strained societies by highlighting three clusters of risks as Risks in Focus. As resilience building is helped by the ability to analyse global risks from the perspective of specific stakeholders, the Report also analyses the significance of global risks to the business community at a regional and country-level

Stability, observer design and control of networks using Lyapunov methods

We investigate different aspects of the analysis and control of interconnected systems. Different tools, based on Lyapunov methods, are provided to analyze such systems in view of stability, to design observers and to control systems subject to stabilization. All the different tools presented in this work can be used for many applications and extend the analysis toolbox of networks. Considering systems with inputs, the stability property input-to-state dynamical stability (ISDS) has some advantages over input-to-state stability (ISS). We introduce the ISDS property for interconnected systems and provide an ISDS small-gain theorem with a construction of an ISDS-Lyapunov function and the rate and the gains of the ISDS estimation for the whole system. This result is applied to observer design for single and interconnected systems. Observers are used in many applications where the measurement of the state is not possible or disturbed due to physical reasons or the measurement is uneconomical. By the help of error Lyapunov functions we design observers, which have a so-called quasi ISS or quasi-ISDS property to guarantee that the dynamics of the estimation error of the systems state has the ISS or ISDS property, respectively. This is applied to quantized feedback stabilization. In many applications, there occur time-delays and/or instantaneous jumps of the systems state. At first, we provide tools to check whether a network of time-delay systems has the ISS property using ISS-Lyapunov-Razumikhin functions and ISS-Lyapunov-Krasovskii functionals. Then, these approaches are also used for interconnected impulsive systems with time-delays using exponential Lyapunov-Razumikhin functions and exponential Lyapunov-Krasovskii functionals. We derive conditions to assure ISS of an impulsive network with time-delays. Controlling a system in a desired and optimal way under given constraints is a challenging task. One approach to handle such problems is model predictive control (MPC). In this thesis, we introduce the ISDS property for MPC of single and interconnected systems. We provide conditions to assure the ISDS property of systems using MPC, where the previous result of this thesis, the ISDS small-gain theorem, is applied. Furthermore, we investigate the ISS property for MPC of time-delay systems using the Lyapunov-Krasovskii approach. We prove theorems, which guarantee ISS for single and interconnected systems using MPC