Abstract
In this work, a new method for solving an optimal semi-active control problem is formulated. It extends previously published results by addressing a continuous-time bilinear quadratic regulator control problem, defined for a structure that is controlled by multiple controlled hydraulic dampers and subjected to external, a priori known deterministic excitation input. The dynamic system is modeled by a linear state-space model, accompanied by constraints on the control forces that reflect the semi-active nature of the prescribed dampers. Next, the control force is written in an equivalent bilinear form, thereby leading to a bilinear state-space model. The optimal control problem is defined by this dynamic bilinear representation, control force constraints, and a performance index that is quadratic in the states and the equivalent dam** gains. Namely, as each device has only two allowable operation modes, its control signal’s range is a two-object set. Krotov’s method is used for solving this problem. This method requires to formulate a function’s sequence with special properties and a suitable minimizing feedback. By doing so, a process improvement becomes possible and an improving sequence can be formulated. This sequence of improving functions and the suitable minimizing feedback, which enable the use of Krotov’s method in this case, are formulated in the context of the addressed problem. The results are encapsulated in a convergent algorithm whose outcome is an approximation of a candidate optimal feedback. Numerical example is given to demonstrate the use of the suggested results for a CBQR control design of a structure, subjected to seismic input.
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Notes
- 1.
Recall that M > 0, K > 0 iff z T Mz > 0, z T Kz > 0 for all \({\mathbf {z}}\in \mathbb {R}^{n_z}\), z ≠ 0 and C d ≥ 0 iff z T C d z ≥ 0 for all \({\mathbf {z}}\in \mathbb {R}^{n_z}\).
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Halperin, I., Agranovich, G., Ribakov, Y. (2021). Optimal Feedback for Structures Controlled by Hydraulic Semi-active Dampers. In: Mariano, P.M. (eds) Variational Views in Mechanics. Advances in Mechanics and Mathematics(), vol 46. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-90051-9_8
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