This paper presents a design framework to obtain a robust multivariable Proportional-Integral-Derivative (PID) controller for second-order linear vibrating systems. A Proportional-Integral-Derivative plus acceleration (PIDA) controller is also proposed to deal with the regularization problem. Relevant control challenges, such as modeling error, regulatory performance optimization, regional pole placement, saturation avoidance, and constant reference tracking are handled within the proposed Linear Matrix Inequality (LMI) design approach. The design strategy is obtained from a linear transformation that can be applied to achieve constant reference tracking for an actuated subspace of underactuated systems. Moreover, the integral action has two additional objectives: (1) to improve regulatory performance in the presence of constant disturbance and (2) to increase the design degree of freedom in order to robustly achieve closed-loop specifications. Three simulation case studies are used to highlight the benefits of the PID and PIDA controllers.
This paper deals with the PI observer synthesis applied to state-feedback control. The objectives are to propose a synthesis formulation and to compare the PI observer with the classical Luenberger observer. We propose a H ∞ /H 2 multiobjective optimization problem formulation to compute an observer that achieves a trade-off between minimization of the error due to disturbances, that requires higher observer gains, and minimization of the variance of the estimation error due to measurement noises, that requires lower observer gains. Minimizing the variance of the estimation error results in the corresponding minimization of the variance of the control action that is based on the estimated state variables. We consider a case study to analyze if the PI observer outperforms the classical Luenberger observer for state estimation applied to state-feedback control. The results showed that it is possible to design a PI observer with the same performance as the Luenberger observer but with smaller gains, which results in better attenuation of measurement noise. Resumo: Neste trabalhoé proposto um método de síntese de observador Proporcional Integral (PI) aplicado ao controle por realimentação de estados. O objetivoé propor uma formulação de síntese e comparar o observador PI com o observador clássico de Luenberger. Um problema de otimização multiobjetivo H ∞ /H 2é formulado para projetar um observador que alcance um compromisso entre minimização do erro na presença de perturbações de entrada, o que exige maiores ganhos do observador e a minimização da variância do erro de estimação devido a ruídos de medição, que requer ganhos mais baixos do observador. A minimização da variação do erro de estimação resulta na correspondente minimização da variação da ação de controle com base nas variáveis de estado estimadas. Consideramos um estudo de caso para analisar se o observador PI supera o observador clássico de Luenberger quando a estimação de estadosé aplicada ao controle por realimentação de estados. Os resultados mostraram queé possível projetar um observador PI com mesmo desempenho do observador de Luenberger porém com ganhos menores, o que resulta em melhor atenuação dos ruídos de medição.
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