Global asymptotic stability of nonlinear cascade systems

AbstractIn this paper, we give a new, quick proof for a known result on the global asymptotic stability of continuous-time nonlinear cascade systems. Next, we state and prove a similar result for the global asymptotic stability of discrete-time nonlinear cascade systems

Analysis of the Qi three dimensional chaotic system

Abstract: This paper applies the center manifold theorem to reduce the dimensions of the Qi three-dimensional system. Local bifurcation phenomena are analyzed, including the pitchfork and Hopf bifurcations of the chaotic system. The Poincaré map is also investigated. The analyses demonstrate the rich dynamics of the Qi chaotic system. Finally, the frequency spectral analysis shows that the system has a broad frequency bandwidth, which is desirable for engineering applications such as secure communications

Surge-varying LOS based path following of under actuated surface vehicles

1048-1055Subject to harsh ocean environment, a novel path following control scheme with accurate guidance and high anti-disturbance ability for under actuated surface vehicles is proposed. The innovative work is as follow: (1) Based on the traditional line-of-sight (LOS), a surge-varying LOS (SVLOS) guidance law is designed to achieve double guidance of speed and heading, which enhances the flexibility and precision of the previous LOS; (2) Unknown disturbances are exactly estimated by an exact disturbance observer (EDO), wherein the limitations of bounded and asymptotic observations can be avoided; (3) The EDO-based robust tracking controllers enable accurate disturbance compensation and guided signal tracking in harsh ocean environment. Rigorous theoretical analysis and significant simulation comparison have been done to demonstrate superiority of the EDO-SVLOS scheme

A Consensus Approach to Distributed Convex Optimization in Multi-Agent Systems

In this thesis we address the problem of distributed unconstrained convex optimization under separability assumptions, i.e., the framework where a network of agents, each endowed with local private convex cost and subject to communication constraints, wants to collaborate to compute the minimizer of the sum of the local costs. We propose a design methodology that combines average consensus algorithms and separation of time-scales ideas. This strategy is proven, under suitable hypotheses, to be globally convergent to the true minimizer. Intuitively, the procedure lets the agents distributedly compute and sequentially update an approximated Newton-Raphson direction by means of suitable average consensus ratios. We consider both a scalar and a multidimensional scenario of the Synchronous Newton-Raphson Consensus, proposing some alternative strategies which trade-off communication and computational requirements with convergence speed. We provide analytical proofs of convergence and we show with numerical simulations that the speed of convergence of this strategy is comparable with alternative optimization strategies such as the Alternating Direction Method of Multipliers, the Distributed Subgradient Method and Distributed Control Method. Moreover, we consider the convergence rates of the Synchronous Newton-Raphson Consensus and the Gradient Descent Consensus under the simplificative assumption of quadratic local cost functions. We derive sufficient conditions which guarantee the convergence of the algorithms. From these conditions we then obtain closed form expressions that can be used to tune the parameters for maximizing the rate of convergence. Despite these formulas have been derived under quadratic local cost functions assumptions, they can be used as rules-of-thumb for tuning the parameters of the algorithms. Finally, we propose an asynchronous version of the Newton-Raphson Consensus. Beside having low computational complexity, low communication requirements and being interpretable as a distributed Newton-Raphson algorithm, the technique has also the beneficial properties of requiring very little coordination and naturally supporting time-varying topologies. Again, we analytically prove that under some assumptions it shows either local or global convergence properties. Through numerical simulations we corroborate these results and we compare the performance of the Asynchronous Newton-Raphson Consensus with other distributed optimization methods

Formalismo para la síntesis de sensores virtuales basados en un modelo maestro de base fenomenológica

En esta tesis se propone una estructura de estimación de estado y parámetros para procesos químicos, la cual se basa en la información a priori que se tiene del proceso y en la combinación de estimadores clásicos cuyo diseño es de fácil comprensión. La estructura propuesta permite realizar los diseños del observador de estado y los estimadores de parámetros de forma independiente, lo cual simplifica el cálculo de las ganancias que ellos requieren. El método de diseño se basa principalmente en el análisis del modelo del sistema, más que en la solución de una gran estructura matemática de estimación de estado y parámetros, lo cual permite diseñar con un mejor criterio el tipo de estimador de estado requerido y definir los parámetros críticos que se deben estimar. El análisis de la estructura propuesta se realiza con un modelo no lineal de un reactor de tanque agitado en continuo (CSTR) que tiene tres variables de estado; sin embargo, los resultados obtenidos pueden ser fácilmente extendidos a otro tipo de procesos con estructuras de modelos similares. De hecho, en este trabajo también se aplica la metodología en un modelo de gasificador de carbón en lecho fluidizado presurizado con 10 variables de estado. Los resultados obtenidos con la estructura propuesta son satisfactorios y mejores que los arrojados por estimadores de estado clásicos. / Abstract: In this thesis is proposed a structure of state and parameters estimation for chemical processes, which is based on a priori information that is had of the process, and in the combination of classic estimators which design is of easy understanding. The proposed structure allows realizing the designs of the state observer and the estimators of parameters of independent form, which simplifies the calculation of the gains that they require. The design method is based mainly on the model system analysis, more than in the solution of a large mathematical structure parameters state estimation, which allows to design with, a better criterion, the type of required state estimator and to define the critical parameters considered for estimation. The analysis of the proposed structure is realized with a nonlinear model of a Continuous Stirred Tank Reactor (CSTR) that has three state variables; nevertheless, the obtained results can easily be extended to other type of processes with similar model structures. In fact, in this work, the methodology is also applied to a model of coal gasifier with 10 state variables. The results obtained with the proposed structure are satisfactory and better than the obtained with classic state estimators.Doctorad

A modular approach to analyzing biological networks

This monograph addresses the decomposition of biochemical networks into functional modules that preserve their dynamic properties upon interconnection with other modules, which permits the inference of network behavior from the properties of its constituent modules. The modular decomposition method developed here also has the property that any changes in the parameters of a chemical reaction only affect the dynamics of a single module. To illustrate our results, we define and analyze a few key biological modules that arise in gene regulation, enzymatic networks, and signaling pathways. We also provide a collection of examples that demonstrate how the behavior of a biological network can be deduced from the properties of its constituent modules, based on results from control systems theory. We then use this modular decomposition method to analyze the p53 core regulation network, which plays a key role in tumor suppression in many organisms. By decomposing the network into modules, we study the evolution of the p53 core regulation network and conduct a formal analysis of the different network configurations that emerge in the evolutionary path to complexity from putative primordial organisms to more evolved vertebrates. We develop an algorithm to solve the system of equations that describe the network behavior by interconnecting the network modules systematically, as these equations are typically difficult to solve using standard numerical solvers. In the process, we qualitatively compare the distinct types of switching behaviors that each network can exhibit. We demonstrate how our novel model for the core regulation network matches experimentally observed phenomena, and contrast this with the plausible behaviors that primordial network configurations can admit. Specifically, we show that the complexity of the p53 network in humans and evolved vertebrates permits a wide range of behaviors that can bring about distinct cell fate decisions, but that this is not the case for primordial organisms

Sampled-Data Control of Invariant Systems on Exponential Lie Groups

This thesis examines the dynamics and control of a class of systems furnished by kinematic systems on exponential matrix Lie groups, when the plant evolves in continuous-time, but whose controller is implemented in discrete-time. This setup is called sampled-data and is ubiquitous in applied control. The class of Lie groups under consideration is motivated by our previous work concerning a similar class of kinematic systems on commutative Lie groups, whose local dynamics were found to be linear, which greatly facilitated control design. This raised the natural question of what class of systems on Lie groups, or class of Lie groups, would admit global characterizations of stability based on the linear part of their local dynamics. As we show in this thesis, the answer is---or at least includes---left- or right-invariant systems on exponential Lie groups, which are necessarily solvable, nilpotent, or commutative. We examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: 1) its Lie series expansion enjoys a type of strong convergence; 2) the origin is an equilibrium; 3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of dynamics at the origin, in particular, global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability. We then study the synchronization of networks of identical continuous-time kinematic agents on a matrix Lie group, controlled by discrete-time controllers with constant sampling periods and directed, weighted communication graphs with a globally reachable node. We present a smooth, distributed, nonlinear discrete-time control law that achieves global synchronization on exponential matrix Lie groups, which include simply connected nilpotent Lie groups as a special case. Synchronization is generally asymptotic, but if the Lie group is nilpotent, then synchronization is achieved at an exponential rate. We first linearize the synchronization error dynamics at the identity, and show that the proposed controller achieves local exponential synchronization on any Lie group. Building on the local analysis, we show that, if the Lie group is exponential, then synchronization is global. We provide conditions for deadbeat convergence when the communication graph is unweighted and complete. Lastly, we examine a regulator problem for a class of fully actuated continuous-time kinematic systems on Lie groups, using a discrete-time controller with constant sampling period. We present a smooth discrete-time control law that achieves global regulation on simply connected nilpotent Lie groups. We first solve the problem when both the plant state and exosystem state are available for feedback. We then present a control law for the case where the plant state and a so-called plant output are available for feedback. The class of plant outputs considered includes, for example, the quantity to be regulated. This class of output allows us to use the classical Luenberger observer to estimate the exosystem states. In the full-information case, the regulation quantity on the Lie algebra is shown to decay exponentially to zero, which implies that it tends asymptotically to the identity on the Lie group

Nonlinear and Geometric Controllers for Rigid Body Vehicles

In this thesis we investigate the motion control problem for a class of vehicles C V , which includes satellites, quadrotors, underwater vehicles, and tailsitters. Given a globally represented model of C V , and a curve, the motion control problem entails following the curve using control inputs. In this thesis the motion control problem is viewed under two settings, 1) as a local path following problem, 2) as a geometric trajectory tracking problem. We provide solutions to both problems by designing controllers based on the concept of feedback linearization. In the local path following problem, the C V class of vehicles is represented by a local chart. The problem is solved in a monolithic control setting, and the path that needs to be followed is treated as a set to be stabilized. The nonlinear model under study is first dynamically extended and then converted into a fully linear form through a coordinate transformation and smooth feed- back. This approach achieves path invariance. We also design a fault tolerant local controller that ensure path following and path invariance in the presence of a one rotor failure for a quadrotor. The second major problem addressed is the geometric trajectory tracking problem, which is treated in an inner-outer loop setting. Specifically, we design a controller class for the attitude dy- namics of the C V class of vehicles. The novel notion of Lie algebra valued functions are defined on the Special Orthogonal group SO(3), which constitutes a family of functions. This family of functions induces a novel geometric controller class, which consists of almost globally stable and locally stable controllers. This class is designed using the idea of feedback linearization, and is proven to be asymptotically stable through a Lyapunov-like argument. This allows the system to perform multiple flips. We also design geometric controllers for the position loop, which are demonstrated to work with the attitude controller class through simulations with noisy sensor data

Advances in gain-scheduling and fault tolerant control techniques

This thesis presents some contributions to the state-of-the-art of the fields of gain-scheduling and fault tolerant control (FTC). In the area of gain-scheduling, the connections between the linear parameter varying (LPV) and Takagi-Sugeno (TS) paradigms are analyzed, showing that the methods for the automated generation of models by nonlinear embedding and by sector nonlinearity, developed for one class of systems, can be easily extended to deal with the other class. Then, two measures, based on the notions of overboundedness and region of attraction estimates, are proposed in order to compare different models and choose which one can be considered the best one. Later, the problem of designing state-feedback controllers for LPV systems has been considered, providing two main contributions. First, robust LPV controllers that can guarantee some desired performances when applied to uncertain LPV systems are designed, by using a double-layer polytopic description that takes into account both the variability due to the varying parameter vector and the uncertainty. Then, the idea of designing the controller in such a way that the required performances are scheduled by the varying parameters is explored, which provides an elegant way to vary online the behavior of the closed-loop system. In both cases, the problem reduces to finding a solution to a finite number of linear matrix inequalities (LMIs), which can be done efficiently using the available solvers. In the area of fault tolerant control, the thesis first shows that the aforementioned double-layer polytopic framework can be used for FTC, in such a way that different strategies (passive, active and hybrid) are obtained depending on the amount of available information. Later, an FTC strategy for LPV systems that involves a reconfigured reference model and virtual actuators is developed. It is shown that by including the saturations in the reference model equations, it is possible to design a model reference FTC system that automatically retunes the reference states whenever the system is affected by saturation nonlinearities. In this way, a graceful performance degradation in presence of actuator saturations is incorporated in an elegant way. Finally, the problem of FTC of unstable LPV systems subject to actuator saturations is considered. In this case, the design of the virtual actuator is performed in such a way that the convergence of the state trajectory to zero is assured despite the saturations and the appearance of faults. Also, it is shown that it is possible to obtain some guarantees about the tolerated delay between the fault occurrence and its isolation, and that the nominal controller can be designed so as to maximize the tolerated delay.Aquesta tesi presenta diverses contribucions a l'estat de l'art del control per planificació del guany i del control tolerant a fallades (FTC). Pel que fa al control per planificació del guany, s'analitzen les connexions entre els paradigmes dels sistemes lineals a paràmetres variants en el temps (LPV) i de Takagi-Sugeno (TS). Es demostra que els mètodes per a la generació automàtica de models mitjançant encastament no lineal i mitjançant no linealitat sectorial, desenvolupats per una classe de sistemes, es poden estendre fàcilment per fer-los servir amb l'altra classe. Es proposen dues mesures basades en les nocions de sobrefitació i d'estimació de la regió d'atracció, per tal de comparar diferents models i triar quin d'ells pot ser considerat el millor. Després, es considera el problema de dissenyar controladors per realimentació d'estat per a sistemes LPV, proporcionant dues contribucions principals. En primer lloc, fent servir una descripció amb doble capa politòpica que té en compte tant la variabilitat deguda al vector de paràmetres variants i la deguda a la incertesa, es dissenyen controladors LPV robustos que puguin garantir unes especificacions desitjades quan s'apliquen a sistemes LPV incerts. En segon lloc, s'explora la idea de dissenyar el controlador de tal manera que les especificacions requerides siguin programades pels paràmetres variants. Això proporciona una manera elegant de variar en línia el comportament del sistema en llaç tancat. En tots dos casos, el problema es redueix a trobar una solució d'un nombre finit de desigualtats matricials lineals (LMIs), que es poden resoldre fent servir algorismes numèrics disponibles i molt eficients. En l'àrea del control tolerant a fallades, primerament la tesi mostra que la descripció amb doble capa politòpica abans esmentada es pot utilitzar per fer FTC, de tal manera que, en funció de la quantitat d'informació disponible, s'obtenen diferents estratègies (passiva, activa i híbrida). Després, es desenvolupa una estratègia de FTC per a sistemes LPV que fa servir un model de referència reconfigurat combinat amb la tècnica d'actuadors virtuals. Es mostra que mitjançant la inclusió de les saturacions en les equacions del model de referència, és possible dissenyar un sistema de control tolerant a fallades que resintonitza automàticament els estats de referència cada vegada que el sistema es veu afectat per les no linealitats de la saturació en els actuadors. D'aquesta manera s'incorpora una degradació elegant de les especificacions en presència de saturacions d'actuadors. Finalment, es considera el problema de FTC per sistemes LPV inestables afectats per saturacions d'actuadors. En aquest cas, es porta a terme el disseny de l'actuador virtual de tal manera que la convergència a zero de la trajectòria d'estat està assegurada tot i les saturacions i l'aparició de fallades. A més, es mostra que és possible obtenir garanties sobre el retard tolerat entre l'aparició d'una fallada i el seu aïllament, i que el controlador nominal es pot dissenyar maximitzant el retard tolerat