Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay

The observer design problem for nonlinear time-delay systems becomes more and more a subject of research in constant evolution Germani et al. (2002), Germani & Pepe (2004), Aggoune et al. (1999), Raff & Allgöwer (2006), Trinh et al. (2004), Xu et al. (2004), Zemouche et al. (2006), Zemouche et al. (2007). Indeed, time-delay is frequently encountered in various practical systems, such as chemical engineering systems, neural networks and population dynamic model. One of the recent application of time-delay is the synchronization and information recovery in chaotic communication systems Cherrier et al. (2005). In fact, the time-delay is added in a suitable way to the chaotic system in the goal to increase the complexity of the chaotic behavior and then to enhance the security of communication systems. On the other hand, contrary to nonlinear continuous-time systems, little attention has been paid toward discrete-time nonlinear systems with time-delay. We refer the readers to the few existing references Lu & Ho (2004a) and Lu & Ho (2004b), where the authors investigated the problem of robust H∞ observer design for a class of Lipschitz time-delay systems with uncertain parameters in the discrete-time case. Their method show the stability of the state of the system and the estimation error simultaneously. This chapter deals with observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main result lies in the use of a new structure of the proposed observer inspired from Fan & Arcak (2003). Using a Lyapunov-Krasovskii functional, a new nonrestrictive synthesis condition is obtained. This condition, expressed in term of LMI, contains more degree of freedom than those proposed by the approaches available in literature. Indeed, these last use a simple Luenberger observer which can be derived from the general form of the observer proposed in this paper by neglecting some observer gains. An extension of the presented result to H∞ performance analysis is given in the goal to take into account the noise which affects the considered system. A more general LMI is established. The last section is devoted to systems with differentiable nonlinearities. In this case, based on the use of the Differential Mean Value Theorem (DMVT), less restrictive synthesis conditions are proposed

Qualitative Validation and Generalization in Nonlinear System Identification

Cell to cell mapping for global analysis of nonlinear systems is adopted to enable the qualitative validation of identified nonlinear systems. The method provides a framework of the global analysis of a diverse range of nonlinear systems, including continuous and discrete time systems and nonlinear identified models. In the present study, the method is used to reveal the dynamic properties of a nonlinear system, including the fixed points, periodic, aperiodic solutions or chaotic behaviour and the corresponding stability properties. The orthogonal least squares algorithm (OLS) is then used to identify a parametric NARMAX model of the system. The resulting model is analysed using the same framework and the dynamic properties of the model are qualitatively compared with thos of the original system. Based on the results of the validation, a modified selection criterion for the OLS algorithm is proposed, which incorporates the nonlinear degree of the terms in the model complexity. The effectiveness of the new algorithm is demonstrated using examples

Robust economic model predictive control: recursive feasibility, stability and average performance

This thesis is mainly concerned with designing algorithms for Economic Model Predictive Control (EMPC), and analysis of its resulting recursive feasibility, stability and asymptotic average performance. In particular, firstly, in order to extend and unify the formulation and analysis of economic model predictive control for general optimal operation regimes, including steady-state or periodic operation, we propose the novel concept of a “control storage function” and introduce upper and lower bounds to the best asymptotic average performance for nonlinear control systems based on suitable notions of dissipativity and controlled dissipativity. As a special case, when the optimal operation is periodic, we present a new approach to formulate terminal cost functions. Secondly, aiming at designing a robust EMPC controller for nonlinear systems with general optimal regimes of operation, we present a tube-based robust EMPC algorithm for discrete-time nonlinear systems that are perturbed by disturbance inputs. The proposed algorithm minimizes a modified economic objective function, which considers the worst cost within a tube around the solution of the associated nominal system. Recursive feasibility and an a-priori upper bound to the closed-loop asymptotic average performance are ensured. Thanks to the use of dissipativity of the nominal system with a suitable supply rate, the closed-loop system under the proposed controller is shown to be asymptotically stable, in the sense that it is driven to an optimal robust invariant set. Thirdly, for the purpose of combining robust EMPC design with a state observer in a single pure economic optimization problem, we consider homothetic tube-based EMPC synthesis for constrained linear discrete time systems. Since, in practical systems, full state measurement is seldom available, the proposed method integrates a moving horizon estimator to achieve closed-loop stability and constraint satisfaction despite system disturbances and output measurement noise. In contrast to existing approaches, the worst cost within a single homothetic tube around the solution of the associated nominal system is minimized, which at the same time tightens the bound on the set of potential states compatible with past output and input data. We show that the designed optimization problem is recursively feasible and adoption of homothetic tubes leads to less conservative economic performance bounds. In addition, the use of strict dissipativity of the nominal system guarantees asymptotic stability of the resulting closed-loop system. Finally, to deal with the unknown nonzero mean disturbance and the presence of plant-model error, we propose a novel economic MPC algorithm aiming at achieving optimal steady-state performance despite the presence of plant-model mismatch or unmeasured nonzero mean disturbances. According to the offset-free formulation, the system's state is augmented with disturbances and transformed into a new coordinate framework. Based on the new variables, the proposed controller integrates a moving horizon estimator to determine a solution of the nominal system surrounded by a set of potential states compatible with past input and output measurements. The worst cost within a single homothetic tube around the nominal solution is chosen as the economic objective function which is minimized to provide a tightened upper bound for the accumulated real cost within the prediction horizon window. Thanks to the combined use of the nominal system and homothetic tube, the designed optimization problem is recursively feasible and less conservative economic performance bounds are achieved.Open Acces

Variance-constrained multiobjective control and filtering for nonlinear stochastic systems: A survey

The multiobjective control and filtering problems for nonlinear stochastic systems with variance constraints are surveyed. First, the concepts of nonlinear stochastic systems are recalled along with the introduction of some recent advances. Then, the covariance control theory, which serves as a practical method for multi-objective control design as well as a foundation for linear system theory, is reviewed comprehensively. The multiple design requirements frequently applied in engineering practice for the use of evaluating system performances are introduced, including robustness, reliability, and dissipativity. Several design techniques suitable for the multi-objective variance-constrained control and filtering problems for nonlinear stochastic systems are discussed. In particular, as a special case for the multi-objective design problems, the mixed H 2 / H ∞ control and filtering problems are reviewed in great detail. Subsequently, some latest results on the variance-constrained multi-objective control and filtering problems for the nonlinear stochastic systems are summarized. Finally, conclusions are drawn, and several possible future research directions are pointed out

A survey on gain-scheduled control and filtering for parameter-varying systems

Copyright © 2014 Guoliang Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper presents an overview of the recent developments in the gain-scheduled control and filtering problems for the parameter-varying systems. First of all, we recall several important algorithms suitable for gain-scheduling method including gain-scheduled proportional-integral derivative (PID) control, H 2, H ∞ and mixed H 2 / H ∞ gain-scheduling methods as well as fuzzy gain-scheduling techniques. Secondly, various important parameter-varying system models are reviewed, for which gain-scheduled control and filtering issues are usually dealt with. In particular, in view of the randomly occurring phenomena with time-varying probability distributions, some results of our recent work based on the probability-dependent gain-scheduling methods are reviewed. Furthermore, some latest progress in this area is discussed. Finally, conclusions are drawn and several potential future research directions are outlined.The National Natural Science Foundation of China under Grants 61074016, 61374039, 61304010, and 61329301; the Natural Science Foundation of Jiangsu Province of China under Grant BK20130766; the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning; the Program for New Century Excellent Talents in University under Grant NCET-11-1051, the Leverhulme Trust of the U.K., the Alexander von Humboldt Foundation of Germany

Discrete Breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm

3 sampled-data control of nonlinear systems

This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research

Mathematical control of complex systems

Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited