Control óptimo inverso para sistemas no lineales en tiempo continuo

Abstract

La optimización aplicada al control automático permite obtener acciones de control que satisfacen no solo el objetivo de control, sino también la minimización de un determinado funcional de costo. Dinámicas complejas dificultan hallar la solución explícita de un problema de control óptimo. El control óptimo inverso evita la solución explícita de la ecuación de Hamilton-Jacobi-Bellman para determinar la ley óptima de control. Ilustrar la potencialidad del control óptimo inverso como alternativa para resolver problemas complejos de optimización en control. Se describe el problema de control óptimo para motivar el control óptimo inverso. Se formulan resultados matemáticos generales y se ilustra su aplicación a través de casos de ejemplo. Las formulaciones matemáticas presentadas son probadas analíticamente en casos del tipo óptimo cuadrático lineal (LQR) y óptimo inverso basado en funciones de control de Lyapuvov (LCF). Es posible formular un problema de control óptimo para sistemas de tipo no lineal, sin abordar la solución explícita del problema de optimización, mediante control óptimo inverso.

Voltage Regulation in Second-Order Dc-Dc Converters Via the Inverse Optimal Control Design with Proportional-Integral Action

“…This control seeks to find all the performance indices for which a given control law is optimal [34]. In addition, this avoids the explicit solution of the Hamilton-Jacobi-Bellman equation (HJB), which establishes the beginning of optimal control, in order to find the optimal control law [35]. Optimization, i.e., either maximizing or minimizing (depending on the system requirements), is carried out through a cost function or performance index associated with system variables such as the error, the control signal, and the state vector.…”

“…If 𝑢(𝑡) is optimal, it is denoted as 𝑢 * . To find the optimal control law, it is necessary to define the associated Hamiltonian of the system, which is given by ( 7) [27], [35]:…”

“…The equation above is the optimal control law and can be written in a formal way using the following theorem [27], [35], [36]:…”

“…Equations above are the simplified equations in terms of the error and can be rewritten in matrix form as is shown in (35):…”

Robust Inverse Optimal Control for a Boost Converter

“…The main problem is that solving the optimal control of nonlinear systems is extremely difficult because it must be solved by Hamilton-Jacobi-Bellman (HJB) equation, and its solution may not exist [29]. The inverse optimal control approach of discrete-time nonlinear systems avoids the need to solve the associate HJB equation through the Control Lyapunov Function (CLF) [25,30,31], and this is because the CLF is a solution to a family of HJB equations [30,32].…”