Implicit Lyapunov-Krasovski Functionals for Time Delay Systems

International audienceThe method of Implicit Lyapunov-Krasovski Functional (ILKF) for stability analysis of time-delay systems is introduced. Theorems on Lyapunov, asymptotic, fiite-time,fixed-time and (hyper-) exponential stability analysis using ILKF are presented. The hyper exponential stabilization algorithm for a time-delay system is developed. The theoretical results are supported by numerical simulations

On Hyper Exponential Stabilization of Linear State-Delay Systems

International audienceA control design algorithm for hyper exponential stabilization of multi-input multi-output linear control system with state-delays is presented based on method of Implicit Lyapunov-Krasovskii Functional (ILKF). The procedure of control parameters tuning is formalized by means of Linear Matrix Inequalities (LMIs). The theoretical results are supported with numerical simulations

On output-based accelerated stabilization of a chain of integrators: Implicit Lyapunov-Krasovskii functional approach

International audienceThe problem of output accelerated stabilization of a chain of integrators is considered. Proposed control law nonlinearly depends on the output and its delayed values, and it does not use an observer to estimate the unmeasured components of the state. It is proven that such a nonlinear delayed control law ensures practical output stabilization with rates of convergence faster than exponential. The effective way of computation of feedback gains is given. It is shown that closed-loop system stability does not depend on the value of artificial delay, but the maximum value of delay determines the width of stability zone. The efficiency of the proposed control is demonstrated in simulations

Output feedback stabilization for uncertain nonlinear time-delay systems subject to input constraints

International audienceRobust stabilization of a class of imperfectly known systems with time-varying time-delays via output feedback is investigated. The systems addressed are composed of a nonlinear nominal sys- tem influenced by nonlinear perturbations which may be time-, state, delayed state, and/or input- dependent. The output of the system is modelled by a nonlinear function, which may depend on the delayed states, and inputs, together with a feed-through term. Using bounding information on the perturbations, in terms of specified growth conditions, classes of unconstrained and constrained output feedback controllers are designed in order to guarantee a prescribed stability property for the closed-loop systems, provided appropriate stability criteria hold. Two stability criterion are given: one in terms of a Linear Matrix Inequality (LMI), the other is algebraic in nature, obtained using a Gersgorin theorem

Practical fixed-time ISS of neutral time-delay systems with application to stabilization by using delays

International audienceThe concept of practical fixed-time input-to-state stability for neutral time-delay systems with exogenous perturbations is introduced. Lyapunov-Krasovskii theorems are formulated in explicit and implicit ways. Further, the problem of static nonlinear output-feedback stabilization of a linear system with parametric uncertainties, external bounded state and output disturbances by using artificial delays is considered. The constructive control design consists in solving linear matrix inequalities with only four tuning parameters to be chosen. It is shown both, theoretically and numerically, that the system governed by the proposed controller converges faster to the given invariant set than in the case of using its linear counterpart

On finite-time stabilization of a class of nonlinear time-delay systems: Implicit Lyapunov-Razumikhin approach

International audienceTheorems on Implicit Lyapunov-Razumikhin functions (ILRF) for asymptotic, exponential, finite-time and nearly fixed-time stability analysis of nonlinear time-delay systems are presented. Based on these results, finite-time stabilization of a special class of such systems is addressed. These systems are represented by a chain of integrators with a time-delay term multiplied by a function of instantaneous state vector. Possible explicit restriction on nonlinear time-delay terms is discussed. Simple procedure of control parameters calculation is given in terms of linear matrix inequalities (LMIs). Some aspects of digital implementations of the presented nonlinear control law are touched upon. Theoretical results are illustrated by numerical simulations

Homogeneity of neutral systems and accelerated stabilization of a double integrator by measurement of its position

International audienceA new theory of homogeneity for neutral type systems with application to fast stabilization of the 2nd-order integrator is proposed. It is assumed that only the position is available for measurements, and the designed feedback uses the output and its delayed values without an estimation of velocity. It is shown that by selecting the closed-loop system to be homogeneous with negative or positive degree, it is possible to accelerate the rate of convergence in the system at the price of a small steady-state error. Robustness of the developed stabilization strategy with respect to exogenous perturbations is investigated. The efficiency of the proposed control is demonstrated in simulations

Systems Structure and Control

The title of the book System, Structure and Control encompasses broad field of theory and applications of many different control approaches applied on different classes of dynamic systems. Output and state feedback control include among others robust control, optimal control or intelligent control methods such as fuzzy or neural network approach, dynamic systems are e.g. linear or nonlinear with or without time delay, fixed or uncertain, onedimensional or multidimensional. The applications cover all branches of human activities including any kind of industry, economics, biology, social sciences etc

Dynamic of particular classes of time-delay systems defined over finite time interval

U ovoj doktorskoj disertaciji su razmatrani problemi dinamičke analize posebnih klasa sistema sa čistim vremenskim kašnjenjem, kao i njihovo ponašanje na konačnom. U uvodnom delu akcenat izlaganja stavljen je na proučavanje suštinskih osobina singularnih sistema, sistema sa kašnjenjem i singularnih sistema sa kašnjenje, kao i na njihove diskretene analogane. U tom smislu razmatrana su pitanja koja se tiču egzistencije i jedinstvenosti rešenja, problema impulsnih ponašanja i konzistentnih početnih uslova, kauzalnosti i funkcija početnih uslova razmatranog sistema. Detaljan pregled do sada postignutih rezultata na polju izučavanja neljapunovske stabilnosti oličene u konceptu stabilnosti na konačnom vremenskom intervalu i praktične stabilnosti na ove klase sisteme, iscrpno je dat u odgovarajućem poglavlju. Disertacija je prvenstveno posvećena osnovnom pianju koje je vezano za teoriju a posebno za primenu automatskog upravljanja u praksi, tj. za pitanje stabilnosti u tzv. neljapunovskom smislu, a to pitanje je bilo rešavano sa dva stanovišta: metode koji koristi deskriptivni prilaz i postupke koji se zasnivaju na primeni klasičnih algebarskih matričnih nejednakosti (Jensenova i Kopelova nejednačina), imajući u vidu da poslednje pomenuti prilaz redukuje probleme upravljanja na rešavanje jednostavnih algebarskih nejednačina, koje se lako sprovode standardnim numeričkim procedurama, a oba prilaza vode ka samo dovoljnim uslovima stabilnosti primenjenog koncepta, što je sasvim prihvatljivo sa inženjerske tačke gledišta. U prvom slučaju, za izvoĎenje dovoljnih uslova stabilnosti na konačnom vremenskom intervalu, korišćene su funkcionali tipa Ljapunov-Krsovski. Za razliku od nekih ranijih rezultata, ovi funkcionali ne moraju da zadovoljavaju neke stroge matematičke uslove, kao što je pozitivna odreĎeneost u celom prostoru stanja, kao i da ne poseduju negativnu odreĎenost njihovih izvoda duž kretanja sistema. U svim slučajevima od interesa, numeričkim primerima datim u ovoj disertaciji, dodatno je potvrĎena primena predloženih novih metodologija, kao i analitičko sračunavanje i iznalaženje uslova stabilnosti. Konačno, utvrĎeno je da su izvedeni dovoljni uslovi manje restriktivni u poreĎenju sa ranije izvedenim rezultatima. Analogni zaključci mogu se izvesti i za rezultate dobijene na polju izučavanja praktične stabilnosti. Još neki manje značajni doprinosi pruženi su u sferi proučavanja osobina robusnosti sistema i njihove stabilizacije...In this doctoral thesis the problems of dynamical analysis of particular class of control time delay systems were considered, as well as their behavior on finite and time intervals. Following the introduction disscusion emphasis has been put on the peculiar properties of singular, time delay and singular time delay systems, as well as on theirs discrete counterparts. In that sense the questions, concerning the existence and uniqueness of the solutions, the problems of impulsive behavior, consistent initial conditions, causality and functions of initial conditions of the system itself. On overview of the modern stability frameworks has been presented, starting from the so called non-Lyapunov concepts: finite time stability and practical stability in particular. A historical overview of ideas, concepts and results has been presented and the key contributions have been highlighted through key papers from the modern literature. This dissertation is mostly dedicated to the main question of control engineering, eg., to the question of stabilitly in Non-Lyapunov sense, two main lines of research were established: the qvasy descriptive methodology and the approach based on classical matrix algebraic inequalites (Jensen's and Coppel inequality), the latter being known to reduce control tasks to simple algebraic conditions easily solvable by numerical computation, in both casses leading to the sufficient stability conditions, only, what is more than acceptable for the engineering point ov view. In the first case for the derivation of the finite time stability sufficient conditions, the Lyapunov-Krassovski functionals were used. Unlike in the previously reported results, the functionals did not have to satisfy some strict mathematical conditions, such as positivity in the whole state space and possession of the negative derivatives along the system state trajectories. In all cases, of interest, the numerical examples presented in this study additionally clarified the implementation of the new methodologies, and the calculations and analitical determination of the stability conditions. Finally, it was found that the proposed sufficient conditions were less restrictive compared to the ones previously reported. The analogous results have been derived for practical stability. Some others contributions has been given through some disscussion of concept of stability robustness and stabilization procedure..

PROBLEMS IN DISTRIBUTED CONTROL SYSTEMS, CONSENSUS AND FLOCKING NETWORKS

An important variant of the linear model is the delayed one where it is discussed in great detail under two theoretical frameworks: a variational stability analysis based on fixed point theory arguments and a standard Lyapunov-based analysis. The investigation revisits scalar variation unifying the behavior of old biologically inspired model and extends to the multi-dimensional (consensus) alternatives. We compare the two methods and assess their applicability and the strength of the results they provide whenever this is possible. The obtained results are applied to a number of nonlinear consensus networks. The first class of networks regards couplings of passive nature. The model is considered on its delayed form and the linear theory is directly applied to provide strong convergence results. The second class of networks is a generally nonlinear one and the study is carried through under a number of different conditions. In additions the non-linearity of the models in conjunction with delays, allows for new type of synchronized solutions. We prove the existence and uniqueness of non-trivial periodic solutions and state sufficient conditions for its local stability. The chapter concludes with a third class of nonlinear models. We introduce and study consensus networks of neutral type. We prove the existence and uniqueness of a consensus point and state sufficient conditions for exponential convergence to it. The discussion continues with the study of a second order flocking network of Cucker-Smale or Motsch-Tadmor type. Based on the derived contraction rates in the linear framework, sufficient conditions are established for these systems' solutions to exhibit exponentially fast asymptotic velocity. The network couplings are essentially state-dependent and non-uniform and the model is studied in both the ordinary and the delayed version. The discussion in flocking models concludes with two noisy networks where convergence with probability one and in the r-th square mean is proved under certain smallness conditions. The linear theory is, finally, applied on a classical problem in electrical power networks. This is the economic dispatch problem (EDP) and the tools of the linear theory are used to solve the problem in a distributed manner. Motivated by the emerging field of Smart Grid systems and the distributed control methods that are needed to be developed in order to t their architecture we introduce a distributed optimization algorithm that calculates the optimal point for a network of power generators that are needed to operate at, in order to serve a given load. In particular, the power grid of interconnected generators and loads is to be served at an optimal point based on the cost of power production for every single power machine. The power grid is supervised by a set of controllers that exchange information on a different communication network that suffers from delays. We define a consensus based dynamic algorithm under which the controllers dynamically learn the overall load of the network and adjust the power generator with respect to the optimal operational point