Controle extremal com equações diferenciais parciais de difusão distribuídas

Abstract

The control of systems described by partial differential equations (PDEs) is a common topic in engineering, especially in industrial processes where the behavior of variables can be described in a distributed manner in time and space. These problems are critical and challenging due to the infinite-dimensional nature and the difficulties in modeling or controlling these systems accurately since it is not always possible to obtain an exact model to represent their behavior due to uncertainties, nonlinearities, and environmental variations. In these scenarios, Extremal Control (ES) becomes a relevant tool, since it allows real-time optimization, seeking the optimum point without requiring a detailed model of the system. However, the application of ES in systems governed by distributed diffusion PDEs presents challenges, especially concerning ensuring system stability, since ES, by itself, does not offer a direct solution for the stability of systems with multiple distributed variables. This work addresses the ES problem for distributed diffusion PDEs, proposing a strategy that combines the ES methodology with the backstepping method, a technique known for its ability to ensure stability in nonlinear and distributed systems. This method allows the design of controllers that ensure the exponential stability of the average system through a Lyapunov functional analysis, while the ES dynamically adjusts the controller parameters, adapting to the system variations and uncertainties. In this way, using the averaging theory for infinite-dimensional systems, it is demonstrated that the trajectories converge to a small neighborhood around the optimum point.