Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Topics in Automotive Rollover Prevention: Robust and Adaptive Switching Strategies for Estimation and Control

The main focus in this thesis is the analysis of alternative approaches for estimation and control of automotive vehicles based on sound theoretical principles. Of particular importance is the problem rollover prevention, which is an important problem plaguing vehicles with a high center of gravity (CG). Vehicle rollover is, statistically, the most dangerous accident type, and it is difficult to prevent it due to the time varying nature of the problem. Therefore, a major objective of the thesis is to develop the necessary theoretical and practical tools for the estimation and control of rollover based on robust and adaptive techniques that are stable with respect to parameter variations. Given this background, we first consider an implementation of the multiple model switching and tuning (MMST) algorithm for estimating the unknown parameters of automotive vehicles relevant to the roll and the lateral dynamics including the position of CG. This results in high performance estimation of the CG as well as other time varying parameters, which can be used in tuning of the active safety controllers in real time. We then look into automotive rollover prevention control based on a robust stable control design methodology. As part of this we introduce a dynamic version of the load transfer ratio (LTR) as a rollover detection criterion and then design robust controllers that take into account uncertainty in the CG position. As the next step we refine the controllers by integrating them with the multiple model switched CG position estimation algorithm. This results in adaptive controllers with higher performance than the robust counterparts. In the second half of the thesis we analyze extensions of certain theoretical results with important implications for switched systems. First we obtain a non-Lyapunov stability result for a certain class of linear discrete time switched systems. Based on this result, we suggest switched controller synthesis procedures for two roll dynamics enhancement control applications. One control design approach is related to modifying the dynamical response characteristics of the automotive vehicle while guaranteeing the switching stability under parametric variations. The other control synthesis method aims to obtain transient free reference tracking of vehicle roll dynamics subject to parametric switching. In a later discussion, we consider a particular decentralized control design procedure based on vector Lyapunov functions for simultaneous, and structurally robust model reference tracking of both the lateral and the roll dynamics of automotive vehicles. We show that this controller design approach guarantees the closed loop stability subject to certain types of structural uncertainty. Finally, assuming a purely theoretical pitch, and motivated by the problems considered during the course of the thesis, we give new stability results on common Lyapunov solution (CLS) existence for two classes of switching linear systems; one is concerned with switching pair of systems in companion form and with interval uncertainty, and the other is concerned with switching pair of companion matrices with general inertia. For both problems we give easily verifiable spectral conditions that are sufficient for the CLS existence. For proving the second result we also obtain a certain generalization of the classical Kalman-Yacubovic-Popov lemma for matrices with general inertia

Trajectory tracking in switched systems: an internal model principle approach: the elliptical billiard system as a benchmark for theory

Sistemi dinamici caratterizzati dall'interazione tra dinamiche continue e discrete sono detti sistemi ibridi. Un sistema switched è un particolare sistema ibrido costituito da una famiglia di sottosistemi a tempo continuo e da una legge che ne regola le transizioni. Questi sistemi hanno numerose applicazioni nel controllo di sistemi meccanici, nell'industria automobilistica e aeronautica, nel controllo del traffico aereo, nell'elettronica di potenza, etc. Questa tesi sarà incentrata sul problema dell'inseguimento asintotico di traiettoria per sistemi switched. Nella prima parte, il problema di inseguimento è stato propriamente definito e risolto prendendo in esame il sistema biliardo ellittico. Al fine di definire una classe di traiettorie di riferimento ammissibili per il sistema biliardo un problema di pianificazione di traiettoria è stato approntato e risolto attraverso l'utilizzo di risultati della teoria dei polinomi non negativi e tecniche LMI. Il problema di inseguimento in presenza di incertezze nei parametri del sistema è stato considerato e risolto sia nel caso di feedback dallo stato che dalla sola posizione. Nella seconda parte della tesi i risultati ottenuti per il sistema biliardo sono stati generalizzati per una classe di sistemi switched con dinamica lineare in ogni modo operazionale, mappe di reset lineari e dimensione dello spazio di stato possibilmente variabile tra i vari modi. In tutti i casi la strategia di controllo proposta è basata su una versione discontinua del principio del modello interno.Dynamical systems that are described by an interaction between continuous and discrete dynamics are called hybrid systems. Their evolution is generally given by equations of motion containing mixtures of logic, discrete-valued or digital dynamics, and continuous-variable or analog dynamics. A switched system is a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. These systems have numerous applications in control of mechanical systems, automotive industry, aircraft and air traffic control, switching power converters, and many others. This thesis will focus on the problem of asymptotic trajectory tracking for switched systems. First, the tracking control problem is properly stated and solved for a controlled elliptical billiard system. In order to find an admissible class of reference trajectories inside the billiards a motion planning problem has been solved by using results from the theory of non-negative polynomials and LMIs techniques. The trajectory tracking problem in presence of uncertainties on the plant parameters has been also considered and solved in both cases of state-feedback and output-feedback. In the second part, the results obtained for the billiard system are generalized for a class of switched systems having linear dynamics in each operating mode, linear reset maps and possible nonuniform state space among the different modes. In all cases the proposed control strategy is based on a dynamic compensator, whose state is subject to discontinuities and whose structure is based on a nonsmooth version of the internal model principle

Mobile Network Data Analytics for Intelligent Transportation Systems

In this dissertation, we explore how the interplay between transportation and mobile networks manifests itself in mobile network billing and signaling data, and we show how to use this data to estimate different transportation supply and demand models. To perform the necessary simulation studies for this dissertation, we present a simula- tion scenario of Luxembourg, which allows the simulation of vehicular Long-Term Evolu- tion (LTE) connectivity with realistic mobility. We first focus on modeling travel time from Cell Dwell Time (CDT), and show – on a synthetic data set– that we can achieve a prediction Mean Absolute Percentage Error (MAPE) below 12%. We also encounter proportionality between the square of the mean CDT and the number of handovers in the system, which we confirmed in the aforementioned simulation scenario. This motivated our later studies of traffic state models generated from mobile network data. We also consider mobile network data for supporting synthetic population generation and demand estimation. In a study on Call Detail Records (CDR) data from Senegal, we estimate CDT distributions to allow generating the duration of user activities, and validate them at a large scale against a data set from China. In a different study, we show how mobile network signaling data can be used for initializing the seed Origin- Destination (O-D) matrix in demand estimation schemes, and show that it increases the rate of convergence. Finally, we address the traffic state estimation problem, by showing how handovers can be used as a proxy metric for flows in the underlying urban road network. Using a traffic flow theory model, we show that clusters of mobile network cells behave characteristically, and with this model we reach a MAPE of 11.1% with respect to floating-car data as ground truth. The presented model can be used in regions without traffic counting infrastructure, or complement existing traffic state estimation systems

Stabilization of cascaded nonlinear systems under sampling and delays

Over the last decades, the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model. Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system