Control of systems modeled by hyperbolic partial diferential equations

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2017.Sistemas com parâmetros distribuídos representam uma vasta gama de processos da engenharia. Neste caso, as variáveis do sistema irão conter termos dependentes do tempo assim como gradientes espaciais e, portanto, é natural representa-los por equações diferenciais parciais. Exemplos podem ser encontrados em diversas áreas: desde processos químicos e térmicos, sistemas de produção e distribuição de energia, e problemas relacionados ao transporte de fluidos e ciência médica. Esta tese trata dois tipos de problemas: estabilização de equações diferenciais parciais lineares hiperbólicas com variável de controle na condição de contorno e controle regulatório de sistemas descritos por equações diferenciais parciais quasi-lineares hiperbólicas com variável de controle no domínio. Com relação ao primeiro, estudaram-se duas metodologias de controle: (i) uma lei de controle estática que garante convergência do sistema para o ponto de equilíbrio desejado. A metodologia de controle utiliza uma função de Lyapunov para encontrar os valores dos parâmetros do controlador que garantem estabilidade exponencial em malha fechada. Resultados de simulação para o problema de supressão de golfadas em sistemas de produção de petróleo são apresentados para ilustrar a eficiência do método; (ii) uma lei de controle baseada nas ferramentas clássicas do domínio da frequência. Neste caso, aplicamos a transformada de Laplace na equação diferencial parcial para obter uma função de transferência irracional e então, ferramentas clássicas do domínio da frequência são usadas para projetar o controlador, de maneira similar aos sistemas de dimensão finita com função de transferência racional. Estes resultados foram aplicados experimentalmente no problema de controle de oscilações termoacústicas do tubo de Rijke, mostrando a efetividade do método. Para o segundo problema, utiliza-se o método das características combinado com a técnica de controle por modos deslizantes. O método das características é usado para transformar o sistema de equações diferenciais parciais em um conjunto de equações diferenciais ordinárias que descrevem o sistema original. O projeto de controle é então realizado a partir deste conjunto de equações diferenciais ordinárias através de resultados bem conhecidos da teoria de equações diferenciais ordinárias. Os resultados obtidos foram testados experimentalmente em dois sistemas de escala industrial: uma planta solar e um fotobiorreator tubular.Abstract : Distributed parameter systems represent a wide range of engineeringprocesses. In this case, the system variables will contain temporally dependentterms as well spatial gradients and, therefore, it is natural to representthem by partial dierential equations. Examples can be found in manyelds: chemical and thermal processes, production and distribution energysystems, and problems related to uid transport and medical science.This thesis deals with two dierent problems: stabilization of linear hyperbolicpartial dierential equations with boundary control and regulatorycontrol of systems described by quasilinear hyperbolic partial dierentialequations with in domain control. Concerning the boundary control problem,we studied two control methodologies: (i) a static control law thatguarantees convergence of the system to the desired equilibrium point. Thiscontrol methodology uses a Lyapunov function to nd the values of thecontrol parameters that guarantee closed-loop exponential stability. Simulationresults for the slugging control problem in oil production facilities arepresented to illustrate the eciency of the methodology; (ii) a control lawbased on the frequency domain tools. In this case, we applied the Laplacetransform on the partial dierential equation to obtain an irrational transferfunction and then classical frequency domain tools are used to designthe control law. These results were applied experimentally to the controlproblem of thermoacoustic oscillations in the Rijke tube, showing the effectivenessof the method. Regarding the regulatory control problem, weuse the method of characteristics together with the sliding mode controlmethodology. The method of characteristics is used to transform the partialdierential equations into a system of ordinary dierential equations thatdescribes the original system without any kind of approximation. Then,the control design is performed on the ordinary dierential equations withwell-known results of the theory of lumped parameter systems. The resultswere validated experimentally in two industrial scale systems: a solar powerplant and a tubular photobioreactor

Active Control of Thermoacoustical Instabilities.

This dissertation presents some advances in active control of thermoacoustic instabilities in combustion chambers. Large-size gaseous and liquid fueled swirl stabilized combustors were used during the studies. Active control was implemented using different types of actuators. Proportional (loudspeakers and fuel valves) and discrete actuators (open-close automotive fuel injectors) were investigated. Acoustic and fuel modulation control were successfully applied. In large-scale combustors, flame stabilization techniques such as swirl add three dimensional characteristics to the flow. Moreover, the induced turbulence creates highly nonlinear interactions in the system. Thus, in order to capture these characteristics nonlinear partial differential equations have to be used. Alternatively, the main dynamics of the combustion process can be modeled experimentally. This approach was chosen. Time and frequency domain linear identification techniques were used for this purpose. Several model-based control strategies such as LQG, Hinfinity Disturbance Rejection and Hinfinity Loop-Shaping techniques were tested experimentally with success. A simple controller whose parameters were optimized on-line is also introduced. An evolution algorithm was developed to perform its parameter optimization achieving good convergence to optimal values. The improvements with these proposed control techniques over classical phase-delay control are demonstrated experimentally. A new control configuration was suggested from heat-release visualizations of the flame. In this new configuration, control actuation is directly focused onto the main area of heat-release in the flame front. Consequently, a more efficient actuation is achieved. It is shown that with just a small amount of modulated fuel, phase-delay control can substantially attenuate the pressure oscillations. Finally, during the development of Hinfinity controllers, there were cases where the stability of the resulting controllers restricted the closed-loop performance. A control design strategy to solve the Hinfinity Strong Stabilization problem is then presented. The proposed design strategy pursues to overcome the conservativeness of existing formulations. Examples show its potential for future applications

Self-Tuning Control of a Nonlinear Model of Combustion Instabilities

We present a self-tuning scheme for adapting the parameters of a PI controller proposed by Fung and Yang for stabilization of a Culick-type model of nonlinear acoustic oscillations in combustion chambers. Our adaptation criterion is Lyapunov-based and its objective is the regulation of nonlinear pressure oscillations to zero. We focus on a two-mode model and first develop a design based on an assumption that the amplitudes of the two modes are available for measurement. The adaptation mechanism is designed to stabilize both modes and prevent the phenomenon observed by Candel and coworkers whose adaptive controller stabilizes the first but (under some conditions) apparently destabilizes the second mode. We also prove that the adaptation mechanism is robust to a time delay inherent to the actuation approach via heat release. In order to avoid requirements for sophisticated sensing of the mode amplitudes needed for feedback, we also develop an adaptation scheme which employs only one pres..