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A thorough review of the literature on the model reduction of linear, time-invariant, dynamical systems in both the frequency and time domains is presented. Particular attention is paid to the least-squares extension of the classical method of Padé approximation. An account is given of the development of apparently different approaches of least-squares parameter- matching Padé model reduction applied to continuous-time and discrete-time systems. These approaches are shown to be related via a unifying theory. From the formulation it is possible to show several interesting features of the least-squares approach which lead to a fuller understanding of exactly how the reduced model approximates the full system. An error index is derived in the general continuous-time case and it is shown that a range of system parameter preservation options are available. Using the theory developed in the continuous-time case a non-uniqueness property of the method is proven. An ‘optimal’ least-squares method based on the approach and the introduction of weighting for the system parameters are both investigated. The unifying theory is extended to the discrete-time case where an important new stability preservation property is proved and is shown to provide the basis for a new least- squares Padé method. This method uses transformations between the z- and 5-planes to guarantee stable reduced order models approximating stable high order continuous-time systems. The application of least-squares Padé approximation is further extended to the multivariable case with particular attention given to the factors affecting the levels of order reduction achieved. Appropriate numerical examples are used to illustrate the main points of the thesis and graphs of the impulse and step responses are used throughout to visually highlight the accuracy of approximation.

Orientador: Edvaldo Assunção / Banca: Marcelo Carvalho Minhoto Teixeira / Banca: Alfredo Del Sole Lordelo / Resumo: Uma técnica de projeto de controladores robustos para sistemas não-lineares contínuos no tempo é proposta neste trabalho. É suposto que a planta não-linear está sujeita a falhas estruturais, que podem ser consideradas como incertezas politópicas. Os sistemas não-lineares são representados por modelos fuzzy propostos por Takagi-Sugeno e uma formulação para o tratamento das incertezas politópicas é apresentado para o projeto dos controladores. Este trabalho aborda projetos de controle usando a realimentação dos estados e a realimentação da derivada dos estados. O projeto do controlador é realizado através de condições baseadas em Desigualdades Matriciais Lineares (em inglês, Linear Matrix Inequalities (LMI)), que podem ser resolvidas facilmente utilizando técnicas de programação convexa. Essa metodologia permite a inclusão de restrições de desempenho no projeto, tais como: taxa de decaimento e restrição na entrada. Ao final, exemplos numéricos e suas simulações ilustram a eficiência da técnica proposta. / Abstract: A technique of robust controllers design for nonlinear continuous-time systems is proposed in this work. It is supposed that the nonlinear plant is subject to structural failures, which can be considered as polytope uncertainties. The nonlinear systems are represented through fuzzy models proposed by Takagi-Sugeno and a formulation for the treatment of polytope uncertain is presented for the controllers design. This work focuses control designs using state feedback and state-derivative feedback. The controllers design is made through conditions based in Linear Matrix Inequalities (LMIs), which can be easily solved using convex programming techniques. This methodology allows the inclusion of performance restrictions on design, such as: decay rate and input constraint. Finally, numeric examples and their simulations show the efficiency of the proposed method. / Mestre

Continuous-time system identification deals with the problem of building continuous-time models of dynamical systems from sampled input and output data. In this field, there are two main approaches: indirect and direct. In the indirect approach, a suitable discrete-time model is first determined, and then it is transformed into continuous-time. On the other hand, the direct approach obtains a continuous-time model directly from the sampled data. In both approaches there exists a dichotomy between discrete-time data and continuous-time models, which can induce robustness issues and complications in the theoretical analysis of identification algorithms. These difficulties are addressed in this thesis. First, we consider the indirect approach to continuous-time system identification. For a zero-order hold sampling mechanism, this approach usually leads to a transfer function estimate with relative degree one, independent of the relative degree of the strictly proper true system. Inspired by the indirect prediction error method, we propose an indirect-approach estimator that enforces the desired number of poles and zeros in the continuous-time transfer function estimate, and show that the estimator is consistent and asymptotically efficient. A robustification of this method is also developed, by which the estimates are also guaranteed to deliver stable models. In the second part of the thesis, we analyze asymptotic properties of the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC), which is one of the most popular direct identification methods. This algorithm applies an adaptive prefiltering to the sampled input and output that requires assumptions on the intersample behavior of the signals. We present a comprehensive analysis on the consistency and asymptotic efficiency of the SRIVC estimator while taking into account the intersample behavior of the input signal. Our results show that the SRIVC estimator is generically consistent when the intersample behavior of the input is known exactly and subsequently used in the implementation of the algorithm, and we give conditions under which consistency is not achieved. In terms of statistical efficiency, we compute the asymptotic Cramér-Rao lower bound for an output error model structure with Gaussian noise, and derive the asymptotic covariance of the SRIVC estimates. We conclude that the SRIVC estimator is asymptotically efficient under mild conditions, and that this property can be lost if the intersample behavior of the input is not carefully accounted for in the SRIVC procedure. Moreover, we propose and analyze the statistical properties of an extension of SRIVC that is able to deal with input signals that cannot be interpolated exactly via hold reconstructions. The proposed estimator is generically consistent for any input reconstructed using zero or first-order-hold devices, and we show that it is generically consistent for continuous-time multisine inputs as well. Comparisons with the Maximum Likelihood technique and an analysis of the iterations of the method are provided, in order to reveal the influence of the intersample behavior of the output and to propose new robustifications to the SRIVC algorithm. / <p>QC 20200511</p>

Made available in DSpace on 2014-06-11T19:22:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-02-20Bitstream added on 2014-06-13T18:49:28Z : No. of bitstreams: 1 silva_erp_me_ilha.pdf: 509609 bytes, checksum: 37c083ec485dbc3db2cf95730b404b40 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Uma técnica de projeto de controladores robustos para sistemas não-lineares contínuos no tempo é proposta neste trabalho. É suposto que a planta não-linear está sujeita a falhas estruturais, que podem ser consideradas como incertezas politópicas. Os sistemas não-lineares são representados por modelos fuzzy propostos por Takagi-Sugeno e uma formulação para o tratamento das incertezas politópicas é apresentado para o projeto dos controladores. Este trabalho aborda projetos de controle usando a realimentação dos estados e a realimentação da derivada dos estados. O projeto do controlador é realizado através de condições baseadas em Desigualdades Matriciais Lineares (em inglês, Linear Matrix Inequalities (LMI)), que podem ser resolvidas facilmente utilizando técnicas de programação convexa. Essa metodologia permite a inclusão de restrições de desempenho no projeto, tais como: taxa de decaimento e restrição na entrada. Ao final, exemplos numéricos e suas simulações ilustram a eficiência da técnica proposta. / A technique of robust controllers design for nonlinear continuous-time systems is proposed in this work. It is supposed that the nonlinear plant is subject to structural failures, which can be considered as polytope uncertainties. The nonlinear systems are represented through fuzzy models proposed by Takagi-Sugeno and a formulation for the treatment of polytope uncertain is presented for the controllers design. This work focuses control designs using state feedback and state-derivative feedback. The controllers design is made through conditions based in Linear Matrix Inequalities (LMIs), which can be easily solved using convex programming techniques. This methodology allows the inclusion of performance restrictions on design, such as: decay rate and input constraint. Finally, numeric examples and their simulations show the efficiency of the proposed method.

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第18993号 / 工博第4035号 / 新制||工||1621(附属図書館) / 31944 / 京都大学大学院工学研究科電気工学専攻 / (主査)教授 萩原 朋道, 教授 松尾 哲司, 准教授 古谷 栄光 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

Neste trabalho estudamos as noções de alcançabilidade e controlabilidade para sistemas lineares a tempo contínuo com perturbações aditivas e saltos nos parâmetros sujeitos a uma cadeia de Markov geral. Definimos conceitos de alcançabilidade e controlabilidade médios de maneira natural exigindo que os valores esperados dos gramianos correspondentes sejam definidos positivos. Visando obter uma condição testável para ambos os conceitos, introduzimos conjuntos de matrizes de alcançabilidade e de controlabilidade para esta classe de sistemas e usamos certas propriedades de invariância para mostrar que: o sistema é alcançável em média, e, analogamente, controlável em média, se e somente se as matrizes respectivas, de alcançabilidade e de controlabilidade, têm posto completo. Usamos alcançabilidade média de sistemas para mostrar que a matriz de segundo momento do estado é definida positiva com uma margem uniforme. Uma consequência deste resultado no problema de estimação linear do estado é que a matriz de covariância do erro de estimação é positiva definida em média, no sentido que existe um nível mínimo de ruído nas estimativas. Na sequência, para estimadores lineares markovianos, estudamos a limitação do valor esperado da matriz de covariância do erro para mostrar que o filtro é estável num certo sentido, sendo esta uma propriedade desejável em aplicações reais. Quanto às aplicações da controlabilidade média, usamos este conceito para estabelecer condições necessárias e suficientes que garantem a existência de um processo de controle que leva a componente contínua do estado do sistema para a origem em tempo finito e com probabilidade positiva. / In this work we study the reachability and controllability notions for continuous-time linear systems with exogenous inputs and jump parameters driven by a quite general Markov chain. We define a rather natural average reachability and controllability concepts by requiring that the associated gramians are average positive definite, respectively. Aiming at testable conditions for each concept, we introduce certain sets of matrices linked with the gramians, and employ some invariance properties to find rank-based conditions. We show for average reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, for linear markovian filters we study the average boundedness of the error covariance matrix to show that the filter is stable in an appropriate sense, which consists in a property that is desirable in real applications. Regarding the average controllability concept, we show that it is a necessary and sufficient condition for the feasibility of the following control problem: find a control process that drives the continuous component of the state to zero in finite time with positive probability.

Cette thèse traite des problèmes d’estimation et de contrôle de cohérence par l’utilisation des techniques ensemblistes. L’objectif est la mise en place d’une démarche méthodologique pour la surveillance et la détection d’anomalies au sein des systèmes où le déterminisme des indicateurs relatifs à l’état de santé du système est une condition sine qua non. Une fois placé dans un contexte ensembliste, l’évolution de chaque variable du système est représentée par une enveloppe traduisant les incertitudes internes et externes ; cette enveloppe représente le seuil au delà duquel le comportement observé représente un écart anormal par rapport à son comportement nominal, et pouvant conduire à une incapacité pour accomplir les objectifs de sa mission. Les techniques développées sont appliquées à un procédé hydraulique de laboratoire . / This work deals with the development of set-membership methods for set esti- mation and consistency checks for nonlinear continuous-time systems. The main objective is to setup a methodology for fault detection and isolation for the systems where the deter- minism of faults indicators on the health system is a necessary condition. Once placed in a set-membership framework, the evolution of each variable is represented by an envelope re?ecting the internal and external uncertainties. This envelope corresponds to the threshold beyond which the observed behavior is an abnormal discrepancy over its nominal behav- ior, thus preventing the accomplishment the mission objectives. The proposed methods are applied on a hydraulic laboratory process.

Neste trabalho estudamos as noções de alcançabilidade e controlabilidade para sistemas lineares a tempo contínuo com perturbações aditivas e saltos nos parâmetros sujeitos a uma cadeia de Markov geral. Definimos conceitos de alcançabilidade e controlabilidade médios de maneira natural exigindo que os valores esperados dos gramianos correspondentes sejam definidos positivos. Visando obter uma condição testável para ambos os conceitos, introduzimos conjuntos de matrizes de alcançabilidade e de controlabilidade para esta classe de sistemas e usamos certas propriedades de invariância para mostrar que: o sistema é alcançável em média, e, analogamente, controlável em média, se e somente se as matrizes respectivas, de alcançabilidade e de controlabilidade, têm posto completo. Usamos alcançabilidade média de sistemas para mostrar que a matriz de segundo momento do estado é definida positiva com uma margem uniforme. Uma consequência deste resultado no problema de estimação linear do estado é que a matriz de covariância do erro de estimação é positiva definida em média, no sentido que existe um nível mínimo de ruído nas estimativas. Na sequência, para estimadores lineares markovianos, estudamos a limitação do valor esperado da matriz de covariância do erro para mostrar que o filtro é estável num certo sentido, sendo esta uma propriedade desejável em aplicações reais. Quanto às aplicações da controlabilidade média, usamos este conceito para estabelecer condições necessárias e suficientes que garantem a existência de um processo de controle que leva a componente contínua do estado do sistema para a origem em tempo finito e com probabilidade positiva. / In this work we study the reachability and controllability notions for continuous-time linear systems with exogenous inputs and jump parameters driven by a quite general Markov chain. We define a rather natural average reachability and controllability concepts by requiring that the associated gramians are average positive definite, respectively. Aiming at testable conditions for each concept, we introduce certain sets of matrices linked with the gramians, and employ some invariance properties to find rank-based conditions. We show for average reachable systems that the state second moment is positive definite. One consequence of this result in the context of linear estimation for reachable systems is that the expectation of the error covariance matrix is positive definite. Moreover, for linear markovian filters we study the average boundedness of the error covariance matrix to show that the filter is stable in an appropriate sense, which consists in a property that is desirable in real applications. Regarding the average controllability concept, we show that it is a necessary and sufficient condition for the feasibility of the following control problem: find a control process that drives the continuous component of the state to zero in finite time with positive probability.

Continuous-time system identification is an important subject with applications within many fields. Many physical processes are continuous in time. Therefore, when identifying a continuous-time model, we can use our insight into the system to decide the system structure and have a direct interpretation of the parameters. Furthermore, in systems such as network control systems and sensor networks, there is a common feature that the output data is quantized meaning we can only represent our data with a limited amount of distinct values. When performing continuous-time system identification of a system with quantized output data, we have errors from process and measurement noise and also a quantization error. This will make it more difficult to estimate the system parameters. This thesis aims to evaluate if it is possible to obtain accurate estimates of continuous-time systems with quantized output data using the indirect inference method. Indirect Inference is a simulation-based method that first estimates a misspecified auxiliary model to the observed data and in the second step, the parameters of the true system are estimated by simulations. Experiments are done both on one linear and two non-linear Hammerstein systems with quantized output data. The indirect inference estimator is shown to have the means to yield accurate estimates on both linear systems as well as non-linear Hammerstein systems with quantized output. The method performs better than the simplified refined instrumental variable method for continuous-time systems (SRIVC), which is commonly used for system identification of continuous-time systems, on a linear system. Furthermore, it performed significantly better compared to the Hammerstein Simplified Refined Instrumental Variable method for continuous-time systems (HSRIVC) for one of the non-linear systems and slightly better for the second one. The downside is that indirect inference is computationally expensive and time-consuming, hence not a good choice when computation time is a critical factor / Identifiering av Tidskontinuerlig system är ett viktigt ämne med användningsområde inom många områden. De flesta fysiska processer är tidskontinuerliga och när vi identifierar tidskontinuerliga modeller av dessa system kan vi använda vår insikt av systemet för att bestämma systemstrukturen och även direkt tolka dessa parametrar. I nätverkssystem och sensor-nätverk är det vanligt att vår utdata är kvantiserad, därav kan vi endast representera vår data med ett begränsat antal distinka värden. När vi identifierar tidskontinuerliga system med kvantiserad utdata, har vi därför både fel som ett resultat av process och mätbrus ovh ett kvantiseringsfel. Detta gör det svårare att identifiera parametrarna av systemet. I detta projekt var målet att utvärdera om det är möjligt att erhålla bra estimat för ett tidskontinuerligt system med kvantiserad utdata genom att använda metoden indrect inference. Indirect inference är en simuleringsbaserad metod som först estimerar en misspecificerad model från det observerade datat och i nästa steg, estimerar paramtrarna av det sanna systemet via simulering. Experiment utfördes både på ett linjärt och två olinära Hammerstein system med kvantiserad utdata. Indirect inference metoden visas ha potential att genere bra estimat på både linjära och icke-linära Hammerstein system med kvantiserad utdata. Metoden presterar bättre än SimplifiedRefined Instrumental Variable Method for continuous-time systems (SRIVC) på det linjära systemet och även mycket bättre än Hammerstein Simplified Refined InstrumentalVariable method for continuous-time systems (HSRIVC) för ett av det olinjära systemen och lite bättre för det andra. En nackdel med indirect inference är att det är beräkningstungt och att det tar lång tid att generera estimaten. Därav är denna metod inte att rekomendera när tid är en kritisk faktor.