LMI techniques for optimization over polynomials in control: A survey

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Plya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers. © 2006 IEEE.published_or_final_versio

Combining Prior Knowledge and Data for Robust Controller Design

We present a framework for systematically combining data of an unknown linear time-invariant system with prior knowledge on the system matrices or on the uncertainty for robust controller design. Our approach leads to linear matrix inequality (LMI) based feasibility criteria which guarantee stability and performance robustly for all closed-loop systems consistent with the prior knowledge and the available data. The design procedures rely on a combination of multipliers inferred via prior knowledge and learnt from measured data, where for the latter a novel and unifying disturbance description is employed. While large parts of the paper focus on linear systems and input-state measurements, we also provide extensions to robust output-feedback design based on noisy input-output data and against nonlinear uncertainties. We illustrate through numerical examples that our approach provides a flexible framework for simultaneously leveraging prior knowledge and data, thereby reducing conservatism and improving performance significantly if compared to black-box approaches to data-driven control

Fixed-structure Control of LTI Systems with Polytopic-type Uncertainty:Application to Inverter-interfaced Microgrids

This thesis focuses on the development of robust control solutions for linear time-invariant interconnected systems affected by polytopic-type uncertainty. The main issues involved in the control of such systems, e.g. sensor and actuator placement, control configuration selection, and robust fixed-structure control design are included. The problem of fixed-structure control is intrinsically nonconvex and hence computationally intractable. Nevertheless, the problem has attracted considerable attention due to the great importance of fixed-structure controllers in practice. In this thesis, necessary and sufficient conditions for fixed-structure H_inf control of polytopic systems with a single uncertain parameter in terms of a finite number of bilinear matrix inequalities (BMIs) are developed. Increasing the number of uncertain parameters leads to sufficient BMI conditions, where the number of decision variables grows polynomially. Convex approximations of robust fixed-order and fixed-structure controller design which rely on the concept of strictly positive realness (SPRness) of transfer functions in state space setting are presented. Such approximations are based on the use of slack matrices whose duty is to decouple the product of unknown matrices. Several algorithms for determination and update of the slack matrices are given. It is shown that the problem of sensor and actuator placement in the polytopic interconnected systems can be formulated as an optimization problem by minimizing cardinality of some pattern matrices, while satisfying a guaranteed level of H_inf performance. The control configuration design is achieved by solving a convex optimization problem whose solution delivers a trade-off curve that starts with a centralized controller and ends with a decentralized or a distributed controller. The proposed approaches are applied to inverter-interfaced microgrids which consist of distributed generation (DG) units. To this end, two important control problems associated with the microgrids are considered: (i) Current control of grid-connected voltage-source converters with L/LCL filters and (ii) Voltage control of islanded microgrids. The proposed control strategies are able to independently regulate the direct and quadrature (dq) components of the converter currents and voltages at the point of common couplings (PCC) in a fully decoupled manner and provide satisfactory dynamic responses. The important problem of plug-and-play (PnP) capability of DGs in the microgrids is also studied. It is shown that an inverter-interfaced microgrid consisting of multi DGs under PnP functionality can be cast as a system with polytopic-type uncertainty. By virtue of this novel description and use of the results from theory of robust control, the stability of the microgrid system under PnP operation of DGs is preserved. Extensive case studies, based on time-domain simulations in MATLAB/SimPowerSystems Toolbox, are carried out to evaluate the performance of the proposed controllers under various test scenarios, e.g., load change, voltage and current tracking. Real-time hardware-in-the-loop case studies, using RT-LAB real-time platform of OPAL-RT Technologies, are also conducted to validate the performance of the designed controllers and demonstrate their insensitivity to hardware implementation issues, e.g., noise and PWM non-idealities. The simulation and experimental results demonstrate satisfactory performance of the designed controllers

Relaxed LMI conditions for control of nonlinear Takagi-Sugeno models

Los problemas de optimización de desigualdades matriciales lineales en control borroso se han convertido en la herramienta más utilizada en dicha área desde los años 90. Muchos sistemas no lineales pueden ser modelados como sistemas borrosos de modo que el control borroso puede considerarse como una técnica de control no lineal. Aunque se han obtenido muchos y buenos resultados, quedan algunas fuentes de conservadurismo cuando se comparan con otros enfoques de control no lineal. Esta tesis discute dichas cuestiones de conservadurismo y plantea nuevos enfoques para resolverlas. La principal ventaja de la formulación mediante desigualdades matriciales lineales es la posibilidad de asegurar estabilidad y prestaciones de un sistema no lineal modelado como un sistema borroso Takagi-Sugeno. Estos modelos están formados por un conjunto de modelos lineales eligiendo el sistema a aplicar mediante el uso de unas reglas borrosas. Estas reglas se traducen en funciones de interpolación o de pertenecía que nos indican el grado de validez de un modelo lineal respecto del resto. El mayor problema que presentan estas técnicas basadas en desigualdades matriciales lineales es que las funciones de pertenencia no están incluidas en las condiciones de estabilidad del sistema, lo que significa que se prueba la estabilidad y prestaciones para cualquier forma de interpolación entre los diferentes modelos lineales. Esto genera una fuente de conservadurismo que sería conveniente limitar. En la tesis doctoral se presentan varias metodologías capaces de trasladar la información de las funciones de pertenencia del sistema al problema basado en desigualdades matriciales lineales de estabilidad y prestaciones. Las dos principales aportaciones propuestas se basan, respectivamente, en introducir una serie de matrices de relajación que permitan incorporar esta información y en aprovechar la descripción de una amplia clase de sistemas borrosos en productos tensoriales de...Ariño Latorre, CV. (2008). Relaxed LMI conditions for control of nonlinear Takagi-Sugeno models [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8301Palanci

Linear Parameter Varying Approaches as Advanced Control Techniques: Application to Vehicle Dynamics

TCC(graduação) - Universidade Federal de Santa Catarina. Centro Tecnológico. Engenharia de Controle e Automação.Ce travail de Fin-d’études présente plusieurs techniques de modélisation, identification et de la commande avancée appliqués a l’étude des systèmes de suspensions semi-actifs. Ce travail est divisé en trois domaines principaux: développement et l’application des techniques LPV pour l’identification des défauts sur les actionneurs dans les systèmes de suspension; développement et mise-en-œuvre d’un système de contrôle prédictif basé sur modèle appliqué en temps réel sur des suspensions semi-actifs; développement des techniques LPV de reconfiguration pour la commande tolerant aux défauts des systèmes de suspension. Les stratégies de commande développées sont analysées par simulation et validation et se montrent satisfaisantes.This End-of-Studies Work presents a range of techniques of Modeling, Identification and Advanced Control applied to the study of Semi-Active Suspensions in Vehicular Systems. This work is divided into three main branches: i) development and application of LPV fault identification techniques on actuators of suspension systems; ii) development and implementation of a real-time model predictive control scheme applied the control of semi- active suspensions; iii) development and application of LPV reconfiguration techniques for fault tolerant control of suspension system. The developed control strategies are analysed through simulation and validation on a mechatronic test-bench and prove themselves satisfactory.Este Trabalho de Conclusão de Curso apresenta diversas técnicas de Modelagem, Identifi- cação e Controle Avançado aplicadas ao estudo de Suspensões Semi-Ativas em Sistemas Veiculares. Este trabalho é divido em três eixos principais: i) Desenvolvimento e aplicação de técnicas LPV de Identificação de Falhas em amortecedores de sistemas de suspensão; ii) Desenvolvimento e implementação de um sistema de Controle Preditivo baseado em modelo aplicado em tempo-real para o controle de suspensões semi-ativas; iii) Desenvolvimento e aplicação de técnicas de reconfiguração LPV para o Controle Tolerante a Falhas de sistemas de suspensão. As técnicas e o desenvolvimento feito são analisados através de simulação e validação em uma plataforma mecatrônica experimental e demonstram-se satisfatórios