Multi‐input control design for a constrained bilinear biquadratic regulator with external excitation

Abstract

Summary This study addresses an optimal control problem that is defined for a type of bilinear excited plant, multiple control inputs, a biquadratic finite horizon performance index, and control trajectory constraints. A novel sequence of improving functions, which suits this problem, is derived, and the corresponding Krotov's algorithm is formulated. The significance of the results is illustrated by a structural control design for a structure that is subjected to a harmonic ground acceleration input and semi‐active control constraints.

The Meaning and Accuracy of the Improving Functions in the Solution of the CBQR by Krotov’s Method

“…It is a well-known instrument for constructing optimal control for quantum systems [12,13]. Additionally, its efficiency was demonstrated for a class of structural control problems [14][15][16], iron and steel manufacturing processes [17], and biological systems [18].…”

Solution of the Continuous Time Bilinear Quadratic Regulator Problem by Krotov's Method

“…Bilinear state-space models are simple nonlinear models, useful for capturing dynamic attributes of systems in various fields, such as quantum mechanics [1], chaotic dynamics [2], biology [3]- [5], mechanical damping [6]- [9] and structural control [10]- [14]. Even when more complex nonlinear plants are addressed, they can sometime be well approximated by bilinear models [1], [15], [16].…”

“…The ongoing research, conducted on such systems, has been yielding diverse results, including optimal control design tools. Among the published works, discussing finite dimensional bilinear systems, one can find results concerning homogeneous [10]- [12], [17]- [21] or inhomogeneous [13], [22], [23] plants, continuous [10]- [12], [17], [18], [20], [22], [23] or discrete [16], [19], [21] time, problems with control constraints [10]- [14], [16], [21], [23] and quadratic [10], [11], [17], [19]- [21], [23] or biquadratic [12], [14] performance index. These solutions are provided in a form of algorithm.…”

“…These solutions are provided in a form of algorithm. Some are founded on necessary conditions such as Pontryagin's minimum principle [10]- [14], [22], while others rely on sufficient conditions such as Hamilton-Jacobi-Bellman equation [24], [25] and two-sided algorithms [21].…”