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O movimento caótico é um tipo de movimento complexo, irregular e imprevisível semelhante ao aleatório que é altamente sensível às condições iniciais e à variação de parâmetros, e que resulta da sobreposição de um número infinito de órbitas periódicas instáveis produzidas por um sistema não-linear determinístico. O caos pode ser desejável ou não dependendo da aplicação. Contudo, um comportamento incerto e desconhecido não é certamente desejado em problemas de engenharia, e nesse sentido, ser capaz de detetar de forma automática transições inesperadas de um movimento regular para caótico torna-se uma tarefa crucial, especialmente em aplicações críticas como em sistemas aeroespaciais e aeronáuticos.
Por outro lado, dado que qualquer sistema não-linear pode vir a exibir um comportamento caótico mesmo que tenha sido projetado para operar num regime bem regular (através de uma ligeira variação de parâmetros ou se o sistema estiver sob o efeito de perturbações externas de características específicas), um sistema de controlo adequado para suprimir eventuais movimentos caóticos tem de garantir robustez contra incertezas paramétricas e perturbações. Além disso, em aplicações altamente exigentes como em sistemas aeroespaciais/aeronáuticos, um sistema de controlo tem de lidar não só com restrições de magnitude nas entradas do sistema mas também com restrições nas taxas devido às limitações físicas dos atuadores, caso contrário o sistema controlado pode-se tornar instável levando possivelmente a um cenário catastrófico.
A presente tese contribui com duas conquistas principais: primeiramente, com um novo algoritmo de deteção de caos em tempo-real e eficaz mesmo na presença de níveis consideráveis de ruído nas medições; e em segundo lugar, com várias extensões de técnicas de controlo linear modernas para o controlo e sincronização de saída robusta de sistemas caóticos contínuos no tempo (sistemas não-lineares por natureza) sujeitos a restrições nas magnitudes e nas taxas dos atuadores.
A ideia-chave por detrás do novo detetor de caos reside no facto de que uma única componente de uma trajetória caótica tende a exibir um número infinito de máximos locais em diferentes intervalos temporais. O facto de que o ruído nas medições não pode ser evitado em sistemas reais torna o problema ainda mais desafiante e este é ultrapassado engenhosamente com recurso a um sistema auxiliar que atua como redutor de ruído. De seguida, recorrendo a um conjunto de operações matemáticas simples é estabelecido um parâmetro que caracteriza o tipo de movimento com base num determinado limiar.
A estratégia de resolução do problema de controlo robusto com restrições no controlo consiste, numa primeira fase, em decompor o sistema não-linear numa parte linear estabilizável mais uma parte não-linear restante. Seguidamente, com a ajuda de um sistema auxiliar, os sinais de referência são gerados em simultâneo com estes termos não-lineares e leis de controlo são projetadas para estabilizar um sistema aumentado resultante. O sistema aumentado é estabilizado através da sua parte linear e os sinais de referência são vistos juntamente com os termos não-lineares como uma única perturbação limitada. Numa primeira abordagem, é considerado restrições simétricas nas magnitudes e nas taxas dos atuadores e estas são impostas por um operador diferencial funcional. Numa segunda abordagem, as restrições são impostas de forma diferente para se conseguir restrições assimétricas nas magnitudes. A terceira abordagem lida com restrições simétricas e a lei de controlo é projetada com base na teoria do controlo 𝐻∞ para garantir, de forma concreta, robustez contra incertezas paramétricas. Uma última abordagem é ainda apresentada, permitindo o controlo/sincronização sem o conhecimento do modelo de referência. Nesta em particular, o sistema aumentado é composto de uma maneira diferente, compreendendo o sistema propriamente dito e o integral do erro da saída, e o controlo robusto é alcançado em duas fases: primeiro, com um controlo baseado numa generalização de uma função de Lyapunov; e de seguida com um Regulador Linear Quadrático (LQR) com um grau de estabilidade especificado.
Simulações numéricas são efetuadas em MATLAB® para validar a eficácia e a robustez das técnicas apresentadas. O detetor de caos e as técnicas de controlo são aplicados a um sistema caótico clássico (sistema de Lorenz) e a sistemas aeroespaciais/aeronáuticos de relevância (movimento de atitude de um veículo espacial; movimento de atitude de um veículo espacial numa órbita elítica; dinâmica de um giróstato eletromecânico; posição de um veículo espacial no problema restrito de três corpos; sistema aeroelástico). Por fim, é apresentado um caso de síntese onde o detetor é aplicado acoplado a um dos controladores para atestar a eficácia de ambos em simultâneo, suprimindo eventuais movimentos de atitude caóticos num veículo espacial induzidos por torques perturbadores existentes no espaço.
No que diz respeito ao detetor de caos, os resultados mostram que a distinção é bastante clara e que a deteção em tempo-real é eficaz mesmo quando os sinais medidos estão corrompidos com Relações de Sinal-Ruído consideravelmente baixas. O detetor proposto é de fácil implementação e bastante eficiente do ponto de vista computacional, contrariamente a outras ferramentas de deteção de caos.
Relativamente ao controlo, os resultados mostram que as abordagens propostas são realmente eficazes. O controlo/sincronização de saída é alcançado com sucesso, garantindo robustez contra incertezas e sem exceder as restrições de entrada. As leis de controlo são estáticas, facilmente implementáveis e não exigem grandes esforços computacionais dado que os parâmetros dos controladores são todos calculados offline. Além do mais, as soluções apresentadas contribuem efetivamente para as mais avançadas técnicas de controlo abertas à comunidade, na medida que têm em consideração restrições nas magnitudes e nas taxas do controlo, opostamente a outras técnicas de controlo que não consideram qualquer tipo de restrições e que são requeridas particularmente em sistemas aeroespaciais.
The chaotic motion is a complex, irregular and unpredictable random-like type of motion highly sensitive to initial conditions and to parameter changes, which results from a superposition of an infinite number of unstable periodic orbits produced by a deterministic nonlinear system. Chaos may be desirable or not depending on the application. However, an unknown and uncertain behaviour is surely not desired in engineering problems, and in that sense, be able to detect in an automatic way unexpected transitions from a regular- to a chaotic- motion becomes a crucial task, particularly in critical applications such as in aerospace and aeronautical systems. On the other hand, since any nonlinear system may come to exhibit a chaotic behaviour even it has been designed to operate in a well regular regime (through a slight variation of the parameters or if the system is under the effect of external disturbances with specific characteristics), a proper control system to suppress eventual chaotic motions must ensure robustness against parametric uncertainties and disturbances. In addition, in highly demanding applications such as in aerospace/aeronautical systems, a control system must deal not only with magnitude input constraints but also with rate input constraints due to the physical limitations of the actuators, otherwise the controlled system may become unstable leading possibly to a catastrophic scenario. The present thesis contributes with two main achievements: firstly with a new algorithm of chaos detection in real time and effective even in the presence of considerable levels of measurement noise; and secondly with several extensions of modern linear control techniques for robust output control and synchronization of continuous-time chaotic systems (nonlinear systems by nature) subject to magnitude and rate actuator constraints. The key idea behind the new chaos detector lies in the fact that a single component of a chaotic trajectory tends to exhibit an infinite number of local maxima at different timeinstants. The fact that measurement noise cannot be avoided in real-world systems makes the problem even more challenging and is ingeniously overcome by using an auxiliary system acting as a denoiser. Then, resorting to a simple set of mathematical operations it is established a parameter that characterizes the type of motion based on a specified threshold. The strategy to solve the robust and constrained control problem consists, in a first stage, in decomposing the nonlinear system into a stabilizable linear part plus a remaining nonlinear part. Then, with the help of an auxiliary system, the desired reference signals, that is, the signals intended to be tracked, are generated simultaneously with these nonlinear terms and control laws are designed to stabilize a resulting augmented system. The augmented system is stabilized through and only through its linear part and the reference signals are seen together with the nonlinear terms as a unique and bounded disturbance. In a first approach, it is considered symmetric constraints on magnitudes and rates and these are imposed by a functional differential operator. In a second approach, the constraints are enforced in a different way to achieve asymmetric constraints on magnitudes. The third approach deals with symmetric constraints and the control law is designed based on the 𝐻∞ control theory to ensure, in concrete way, robustness against parametric uncertainties. One last approach is yet presented, allowing a control/synchronization without knowledge of the reference model. In this in particular, the augmented system is composed in a different way, comprising the system itself and the integral of the output error, and the robust control is achieved in two steps: first, with a control based on a generalization of the Lyapunov function; and then with a Linear-Quadratic Regulator (LQR) with a prescribed degree of stability. Numerical simulations are performed in MATLAB® to validate the effectiveness and robustness of the proposed techniques. The chaos detector and the control techniques are applied to a classic chaotic system (Lorenz system) and to relevant aerospace/aeronautical systems (attitude motion of a spacecraft; attitude motion of a spacecraft in an elliptical orbit; dynamic of an electromechanical gyrostat; position of a spacecraft in a restricted three-body problem; aeroelastic system). Lastly, it is presented a synthesis case where the detector is applied together with one of the controllers to attest the effectiveness of both, whose purpose is to suppress eventual chaotic attitude motions in a spacecraft induced by perturbing torques existing in space. With respect to the chaos detector, the results show that the distinction is very clear and that the detection, in real time, is effective even when the measured signals are corrupted with considerably low Signal-to-Noise Ratios. The proposed detector is easily-implementable and quite efficient from the computational point of view, as opposed to other chaos detection tools. As regards to the control, the results show that the proposed approaches are truly effective. The output control/synchronization is successfully achieved guaranteeing robustness against uncertainties and without exceeding the input constraints. The control laws are static, easilyimplementable and do not require much computational effort since the controller(s) parameters are all computed offline. Furthermore, the current solutions contribute effectively for the most advanced control techniques open to community, insofar as they take into account both magnitude and rate actuator constraints, as opposed to other control techniques which do not consider any kind of constraints and that are particularly required in aerospace systems.
The chaotic motion is a complex, irregular and unpredictable random-like type of motion highly sensitive to initial conditions and to parameter changes, which results from a superposition of an infinite number of unstable periodic orbits produced by a deterministic nonlinear system. Chaos may be desirable or not depending on the application. However, an unknown and uncertain behaviour is surely not desired in engineering problems, and in that sense, be able to detect in an automatic way unexpected transitions from a regular- to a chaotic- motion becomes a crucial task, particularly in critical applications such as in aerospace and aeronautical systems. On the other hand, since any nonlinear system may come to exhibit a chaotic behaviour even it has been designed to operate in a well regular regime (through a slight variation of the parameters or if the system is under the effect of external disturbances with specific characteristics), a proper control system to suppress eventual chaotic motions must ensure robustness against parametric uncertainties and disturbances. In addition, in highly demanding applications such as in aerospace/aeronautical systems, a control system must deal not only with magnitude input constraints but also with rate input constraints due to the physical limitations of the actuators, otherwise the controlled system may become unstable leading possibly to a catastrophic scenario. The present thesis contributes with two main achievements: firstly with a new algorithm of chaos detection in real time and effective even in the presence of considerable levels of measurement noise; and secondly with several extensions of modern linear control techniques for robust output control and synchronization of continuous-time chaotic systems (nonlinear systems by nature) subject to magnitude and rate actuator constraints. The key idea behind the new chaos detector lies in the fact that a single component of a chaotic trajectory tends to exhibit an infinite number of local maxima at different timeinstants. The fact that measurement noise cannot be avoided in real-world systems makes the problem even more challenging and is ingeniously overcome by using an auxiliary system acting as a denoiser. Then, resorting to a simple set of mathematical operations it is established a parameter that characterizes the type of motion based on a specified threshold. The strategy to solve the robust and constrained control problem consists, in a first stage, in decomposing the nonlinear system into a stabilizable linear part plus a remaining nonlinear part. Then, with the help of an auxiliary system, the desired reference signals, that is, the signals intended to be tracked, are generated simultaneously with these nonlinear terms and control laws are designed to stabilize a resulting augmented system. The augmented system is stabilized through and only through its linear part and the reference signals are seen together with the nonlinear terms as a unique and bounded disturbance. In a first approach, it is considered symmetric constraints on magnitudes and rates and these are imposed by a functional differential operator. In a second approach, the constraints are enforced in a different way to achieve asymmetric constraints on magnitudes. The third approach deals with symmetric constraints and the control law is designed based on the 𝐻∞ control theory to ensure, in concrete way, robustness against parametric uncertainties. One last approach is yet presented, allowing a control/synchronization without knowledge of the reference model. In this in particular, the augmented system is composed in a different way, comprising the system itself and the integral of the output error, and the robust control is achieved in two steps: first, with a control based on a generalization of the Lyapunov function; and then with a Linear-Quadratic Regulator (LQR) with a prescribed degree of stability. Numerical simulations are performed in MATLAB® to validate the effectiveness and robustness of the proposed techniques. The chaos detector and the control techniques are applied to a classic chaotic system (Lorenz system) and to relevant aerospace/aeronautical systems (attitude motion of a spacecraft; attitude motion of a spacecraft in an elliptical orbit; dynamic of an electromechanical gyrostat; position of a spacecraft in a restricted three-body problem; aeroelastic system). Lastly, it is presented a synthesis case where the detector is applied together with one of the controllers to attest the effectiveness of both, whose purpose is to suppress eventual chaotic attitude motions in a spacecraft induced by perturbing torques existing in space. With respect to the chaos detector, the results show that the distinction is very clear and that the detection, in real time, is effective even when the measured signals are corrupted with considerably low Signal-to-Noise Ratios. The proposed detector is easily-implementable and quite efficient from the computational point of view, as opposed to other chaos detection tools. As regards to the control, the results show that the proposed approaches are truly effective. The output control/synchronization is successfully achieved guaranteeing robustness against uncertainties and without exceeding the input constraints. The control laws are static, easilyimplementable and do not require much computational effort since the controller(s) parameters are all computed offline. Furthermore, the current solutions contribute effectively for the most advanced control techniques open to community, insofar as they take into account both magnitude and rate actuator constraints, as opposed to other control techniques which do not consider any kind of constraints and that are particularly required in aerospace systems.
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Palavras-chave
Sistemas caóticos - Controlo robusto Sistemas caóticos - Detecção do caos Veículo espacial - Sistemas de controlo Veículo espacial - Técnicas de controlo