Direct Data-Driven Discrete-time Bilinear Biquadratic Regulator

TL;DR

This paper introduces a novel data-driven method for designing optimal control policies for unknown bilinear systems with complex performance indices, using transformations and the Hamiltonian approach to derive solvable linear matrix equalities.

Contribution

It develops a direct data-driven algorithm that transforms the control problem into linear matrix equalities, enabling optimal control without system identification.

Findings

Successfully applied to numerical examples demonstrating effectiveness.

Provides a new approach to handle nonlinear performance indices.

Enables control design directly from data without explicit system models.

Abstract

We present a novel direct data-driven algorithm that learns an optimal control policy for the Bilinear Biquadratic Regulator (BBR) for an unknown bilinear system. The BBR is difficult to solve owing to the presence of the nonlinear biquadratic performance index and the bilinear cross-term in the dynamics. To address these difficulties, we apply several transformations on the state decision variables to obtain a nonlinear optimization problem with a linear performance index and affine (in the parameterized control) state-dependent equality. The adroit use of the Hamiltonian and Pontryagin's Minimum Principle allows us to derive a pair of first-order necessary conditions that, at each point in time, are easily solvable linear matrix equalities (LMEs) which give the optimal state-dependent control law. We then use the marginal sample autocorrelation of the collected data to obtain a direct data-driven equivalent of these LMEs. We demonstrate the performance of the proposed algorithm via illustrative numerical examples.