Robust control of uncertain systems: H2/H∞ control and computation of invariant sets

This thesis is mainly concerned with robust analysis and control synthesis of linear time-invariant systems with polytopic uncertainties. This topic has received considerable attention during the past decades since it offers the possibility to analyze and design controllers to cope with uncertainties. The most common and simplest approach to establish convex optimization procedures for robust analysis and synthesis problems is based on quadratic stability results, which use a single (parameter-independent) Lyapunov function for the entire uncertainty polytope. In recent years, many researchers have used parameter-dependent Lyapunov functions to provide less conservative results than the quadratic stability condition by working with parameterized Linear Matrix Inequalities (LMIs), where auxiliary scalar parameters are introduced. However, treating the scalar parameters as optimization variables leads to large computational complexity since the scalar parameters belong to an unbounded domain in general. To address this problem, we propose three distinct iterative procedures for H2 and H∞state feedback control, which are all based on true LMIs (without any scalar parameter). The first and second procedures are proposed for continuous-time and discrete-time uncertain systems, respectively. In particular, quadratic stability results can be used as a starting point for these two iterative procedures. This property ensures that the solutions obtained by our iterative procedures with one step update are no more conservative than the quadratic stability results. It is important to emphasize that, to date, for continuous-time systems, all existing methods have to introduce extra scalar parameters into their conditions in order to include the quadratic stability conditions as a special case, while our proposed iterative procedure solves a convex/LMI problem at each update. The third approach deals with the design of robust controllers for both continuous-time and discrete-time cases. It is proved that the proposed conditions contain the many existing conditions as special cases. Therefore, the third iterative procedure can compute a solution, in one step, which is at least as good as the optimal solution obtained using existing methods. All three iterative procedures can compute a sequence of non-increasing upper bounds for H2-norm and H∞-norm. In addition, if no feasible initial solution for the iterative procedures is found for some uncertain systems, we also propose two algorithms based on iterative procedures that offer the possibility of obtaining a feasible initial solution for continuous-time and discrete-time systems, respectively. Furthermore, to address the problem of analysis of H∞-norm guaranteed cost computation, a generalized problem is firstly proposed that includes both the continuous-time and discrete-time problems as special cases. A novel description of polytopic uncertainties is then derived and used to develop a relaxation approach based on the S-procedure to lift the uncertainties, which yields an LMI approach to compute H∞-norm guaranteed cost by incorporating slack variables. In this thesis, one of the main contributions is to develop convex iterative procedures for the original non-convex H2 and H∞ synthesis problems based on the novel separation result. Nonlinear and non-convex problems are general in nature and occur in other control problems; for example, the computation of tightened invariant tubes for output feedback Model Predictive Control (MPC). We consider discrete-time linear time-invariant systems with bounded state and input constraints and subject to bounded disturbances. In contrast to existing approaches which either use pre-defined control and observer gains or optimize the volume of the invariant sets for the estimation and control errors separately, we consider the problem of optimizing the volume of these two sets simultaneously to give a less conservative design.Open Acces

Control of Constrained Dynamical Systems with Performance Guarantees: With Application to Vehicle motion Control

In control engineering, models of the system are commonly used for controller design. A standard control design problem consists of steering the given system output (or states) towards a predefined reference. Such a problem can be solved by employing feedback control strategies. By utilizing the knowledge of the model, these strategies compute the control inputs that shrink the error between the system outputs and their desired references over time. Usually, the control inputs must be computed such that the system output signals are kept in a desired region, possibly due to design or safety requirements. Also, the input signals should be within the physical limits of the actuators. Depending on the constraints, their violation might result in unacceptable system failures (e.g. deadly injury in the worst case). Thus, in safety-critical applications, a controller must be robust towards the modelling uncertainties and provide a priori guarantees for constraint satisfaction. A fundamental tool in constrained control application is the robust control invariant sets (RCI). For a controlled dynamical system, if initial states belong to RCI set, control inputs always exist that keep the future state trajectories restricted within the set. Hence, RCI sets can characterize a system that never violates constraints. These sets are the primary ingredient in the synthesis of the well-known constraint control strategies like model predictive control (MPC) and interpolation-based controller (IBC). Consequently, a large body of research has been devoted to the computation of these sets. In the thesis, we will focus on the computation of RCI sets and the method to generate control inputs that keep the system trajectories within RCI set. We specifically focus on the systems which have time-varying dynamics and polytopic constraints. Depending upon the nature of the time-varying element in the system description (i.e., if they are observable or not), we propose different sets of algorithms.The first group of algorithms apply to the system with time-varying, bounded uncertainties. To systematically handle the uncertainties and reduce conservatism, we exploit various tools from the robust control literature to derive novel conditions for invariance. The obtained conditions are then combined with a newly developed method for volume maximization and minimization in a convex optimization problem to compute desirably large and small RCI sets. In addition to ensuring invariance, it is also possible to guarantee desired closed-loop performance within the RCI set. Furthermore, developed algorithms can generate RCI sets with a predefined number of hyper-planes. This feature allows us to adjust the computational complexity of MPC and IBC controller when the sets are utilized in controller synthesis. Using numerical examples, we show that the proposed algorithms can outperform (volume-wise) many state-of-the-art methods when computing RCI sets.In the other case, we assume the time-varying parameters in system description to be observable. The developed algorithm has many similar characteristics as the earlier case, but now to utilize the parameter information, the control law and the RCI set are allowed to be parameter-dependent. We have numerically shown that the presented algorithm can generate invariant sets which are larger than the maximal RCI sets computed without exploiting parameter information.Lastly, we demonstrate how we can utilize some of these algorithms to construct a computationally efficient IBC controller for the vehicle motion control. The devised IBC controller guarantees to meet safety requirements mentioned in ISO 26262 and the ride comfort requirement by design

Learning Tube-Certified Control Using Robust Contraction Metrics

Control design for general nonlinear robotic systems with guaranteed stability and/or safety in the presence of model uncertainties is a challenging problem. Recent efforts attempt to learn a controller and a certificate (e.g., a Lyapunov function or a contraction metric) jointly using neural networks (NNs), in which model uncertainties are generally ignored during the learning process. In this paper, for nonlinear systems subject to bounded disturbances, we present a framework for jointly learning a robust nonlinear controller and a contraction metric using a novel disturbance rejection objective that certifies a universal $\mathcal L_\infty$ gain bound using NNs for user-specified variables. The learned controller aims to minimize the effect of disturbances on the actual trajectories of state and/or input variables from their nominal counterparts while providing certificate tubes around nominal trajectories that are guaranteed to contain actual trajectories in the presence of disturbances. Experimental results demonstrate that our framework can generate tighter tubes and a controller that is computationally efficient to implement.Comment: 8 pages, 4 figure

Intelligent control of a class of nonlinear systems

The objective of this study is to improve and propose new fuzzy control algorithms for a class of nonlinear systems. In order to achieve the objectives, novel stability theorems as well as modeling techniques are also investigated. Fuzzy controllers in this work are designed based on the fuzzy basis function neural networks and the type-2 Takagi-Sugeno fuzzy models. For a class of single-input single-output nonlinear systems, a new stability condition is derived to facilitate the design process of proportional-integral Mamdani fuzzy controllers. The stability conditions require a new technique to calculate the dynamic gains of nonlinear systems represented by fuzzy basis function network models. The dynamic gain of a fuzzy basis function network can be approximated by finding the maximum of norm values of the locally linearized systems or by solving a non-smooth optimal control problem. Based on the new stability theorem, a multilevel fuzzy controller with self-tuning algorithm is proposed and simulated in a tower crane control system. For a class of multi-input multi-output nonlinear systems with measurable state variables, a new method for modeling unstructured uncertainties and robust control of unknown nonlinear dynamic systems is proposed by using a novel robust Takagi-Sugeno fuzzy controller. First, a new training algorithm for an interval type-2 fuzzy basis function network is presented. Next, a novel technique is derived to convert the interval type-2 fuzzy basis function network to an interval type-2 Takagi-Sugeno fuzzy model. Based on the interval type-2 Takagi-Sugeno and type-2 fuzzy basis function network models, a robust controller is presented with an adjustable convergence rate. Simulation results on an electrohydraulic actuator show that the robust Takagi-Sugeno fuzzy controller can reduce steady-state error under different conditions while maintaining better responses than the other robust sliding mode controllers can. Next, the study presents an implementation of type-2 fuzzy basis function networks and robust Takagi-Sugeno fuzzy controllers to data-driven modeling and robust control of a laser keyhole welding process. In this work, the variation of the keyhole diameter during the welding process is approximated by a type-2 fuzzy-basis-function network, while the keyhole penetration depth is modelled by a type-1 fuzzy basis function network. During the laser welding process, a CMOS camera integrated with the welding system was used to provide a feedback signal of the keyhole diameter. An observer was implemented to estimate the penetration depth in real time based on the adaptive divided difference filter and the feedback signal from the camera. A robust Takagi-Sugeno fuzzy controller was designed based on the fuzzy basis function networks representing the welding process with uncertainties to adjust the laser power to ensure that the penetration depth of the keyhole is maintained at a desired value. Experimental results demonstrated that the fuzzy models provided an accurate estimation of both the welding geometry and its variations due to uncertainties, and the robust Takagi-Sugeno fuzzy controller successfully reduced the penetration depth variation and improved the quality of the welding process

Machine Learning Aided Stochastic Elastoplastic and Damage Analysis of Functionally Graded Structures

The elastoplastic and damage analyses, which serve as key indicators for the nonlinear performances of engineering structures, have been extensively investigated during the past decades. However, with the development of advanced composite material, such as the functionally graded material (FGM), the nonlinear behaviour evaluations of such advantageous materials still remain tough challenges. Moreover, despite of the assumption that structural system parameters are widely adopted as deterministic, it is already illustrated that the inevitable and mercurial uncertainties of these system properties inherently associate with the concerned structural models and nonlinear analysis process. The existence of such fluctuations potentially affects the actual elastoplastic and damage behaviours of the FGM structures, which leads to the inadequacy between the approximation results with the actual structural safety conditions. Consequently, it is requisite to establish a robust stochastic nonlinear analysis framework complied with the requirements of modern composite engineering practices. In this dissertation, a novel uncertain nonlinear analysis framework, namely the machine leaning aided stochastic elastoplastic and damage analysis framework, is presented herein for FGM structures. The proposed approach is a favorable alternative to determine structural reliability when full-scale testing is not achievable, thus leading to significant eliminations of manpower and computational efforts spent in practical engineering applications. Within the developed framework, a novel extended support vector regression (X-SVR) with Dirichlet feature mapping approach is introduced and then incorporated for the subsequent uncertainty quantification. By successfully establishing the governing relationship between the uncertain system parameters and any concerned structural output, a comprehensive probabilistic profile including means, standard deviations, probability density functions (PDFs), and cumulative distribution functions (CDFs) of the structural output can be effectively established through a sampling scheme. Consequently, by adopting the machine learning aided stochastic elastoplastic and damage analysis framework into real-life engineering application, the advantages of the next generation uncertainty quantification analysis can be highlighted, and appreciable contributions can be delivered to both structural safety evaluation and structural design fields

Contributions to Control of Electronic Power Converters

This thesis deals with the control of electronic power converters. In its development two main parts have been differentiated. On the one hand, the problem of the voltage balance in the capacitors of the dc-link in a three-level NPC converter is addressed. On the other hand, given that the techniques used in the first part to model the converters need to make certain assumptions and, with the intention of avoiding averaged models, in the second part, switched affine models have been developed to design the control of the output voltage in DC-DC boost type converters. In this way, in the first part several control laws have been developed using an averaged model formulated by duty cycles for each level in each phase. This formulation allows to consider, in the controllers design stage, the degree of freedom associated with the homopolar voltage injection. Therefore, the controllers are designed as well as a part of the modulation, so that control and modulation are integrated in the same stage. In this way, three controllers have been designed where, apart from the objective of the voltage balance of the capacitors, other objectives such as the number of commutations or the quality of the output signal have also been improved. In the second part of the thesis, four methods have been developed for the design of control laws taking advantage of the modeling of converters as switched affine systems given their hybrid behaviour. Thus, the first two laws take advantage of this modeling using the delta operator to avoid numerical problems when using systems where the sampling time is very low. The first of these controllers is based on Lyapunov’s function while the second is independent of this function, thus obtaining less conservative results. The other two laws developed for switched affine systems use an alternative model to that performed in the first two controllers, so certain existing disadvantages are avoided using again a design not based on Lyapunov’s function. Thus, the first law presents a basic control but, even so, improves the results of other existing laws in the literature. Finally, a design method to deal with systems with variations in their parameters has been presented.La presente tesis trata sobre el control de convertidores electrónicos de potencia. En su desarrollo se han diferenciado dos partes principales. Por un lado, se trata el problema del balance de tensiones en los condensadores que forman el dc-link en un convertidor NPC de tres niveles. Por otro lado, dado que las técnicas utilizadas en la primera parte para modelar los convertidores necesitan realizar determinadas suposiciones y, con la intención de evitar modelos promediados, en la segunda parte se han desarrollado modelos afines conmutados para diseñar el control de la tensión de salida en convertidores DC-DC tipo boost. De esta forma, en la primera parte se han desarrollado varias leyes de control utilizando un modelo promediado formulado mediante ciclos de trabajo para cada nivel en cada fase. Esta formulación permite considerar en la fase de diseño de los controladores, un grado de libertad asociado a la inyección de tensión homopolar. Por lo tanto, se diseñan los controladores a la vez que una parte de la modulación, de forma que se integra control y modulación en una misma fase. De esta forma, se han diseñado tres controladores donde, a parte del objetivo de balancear la tensión de los condensadores, se ha ido buscando mejorar también otros objetivos como el número de conmutaciones o la calidad de la señal de salida. En la segunda parte de la tesis, se han desarrollado cuatro leyes de control aprovechando el modelado de convertidores como sistemas afines conmutados dada su naturaleza híbrida. De esta forma, las dos primeras leyes, aprovechan dicho modelado usando el operador delta para evitar problemas numéricos al utilizar sistemas donde el tiempo de muestreo es muy bajo. El primero de dichos controladores está basado en la función de Lyapunov mientras que el segundo es independiente de dicha función obteniendo así resultados menos conservadores. Las otras dos leyes desarrolladas para sistemas afines conmutados utilizan un modelado alternativo al realizado en las dos primeras, de forma que se evitan ciertas desventajas existentes y mantienen un diseño no basado en la función de Lyapunov. Así, la primera ley presenta un control más básico pero que, aun así, mejora los resultados de otras leyes existentes en la literatura. Por último, se ha presentado un procedimiento de diseño que hace frente a sistemas con variaciones en sus parámetros

Adaptive Control

Adaptive control has been a remarkable field for industrial and academic research since 1950s. Since more and more adaptive algorithms are applied in various control applications, it is becoming very important for practical implementation. As it can be confirmed from the increasing number of conferences and journals on adaptive control topics, it is certain that the adaptive control is a significant guidance for technology development.The authors the chapters in this book are professionals in their areas and their recent research results are presented in this book which will also provide new ideas for improved performance of various control application problems

Control and filtering of time-varying linear systems via parameter dependent Lyapunov functions

The main contribution of this dissertation is to propose conditions for linear filter and controller design, considering both robust and parameter dependent structures, for discrete time-varying systems. The controllers, or filters, are obtained through the solution of optimization problems, formulated in terms of bilinear matrix inequalities, using a method that alternates convex optimization problems described in terms of linear matrix inequalities. Both affine and multi-affine in different instants of time (path dependent) Lyapunov functions were used to obtain the design conditions, as well as extra variables introduced by the Finsler\u27s lemma. Design problems that take into account an H-infinity guaranteed cost were investigated, providing robustness with respect to unstructured uncertainties. Numerical simulations show the efficiency of the proposed methods in terms of H-infinity performance when compared with other strategies from the literature

Energy efficient wireless sensor network protocols for monitoring and prognostics of large scale systems

In this work, energy-efficient protocols for wireless sensor networks (WSN) with applications to prognostics are investigated. Both analytical methods and verification are shown for the proposed methods via either hardware experiments or simulation. This work is presented in five papers. Energy-efficiency methods for WSN include distributed algorithms for i) optimal routing, ii) adaptive scheduling, iii) adaptive transmission power and data-rate control --Abstract, page iv