Stabilisation of state-and-input constrained nonlinear systems via diffeomorphisms: A Sontag's formula approach with an actual application
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In this work, we provide a new and constructive outlook for the control of state-and-input constrained nonlinear systems. Previously, explicit solutions have been mainly focused on the finding of a barrier-like Lyapunov function, whereas we propose the construction of a diffeomorphism to map all the trajectories of the constrained dynamics into an unconstrained one. Careful analysis has revealed that only some foundations of differential geometry and a technical assumption are necessary to construct the proposed methodology based on the well-established theories of control Lyapunov functions and Sontag's universal formulae. Altogether, it allows us to obtain an explicit solution that even includes bounded constraints in the control action, giving the designer a way to decide (to some extent) the trade-off between control saturations and robustness. Moreover, this approach does not rely on the own structure of the system dynamics, therefore covering a broad class of nonlinear systems. The main advantage of this approach is that the use of a diffeomorphism allows the splitting of the mathematical treatment of the constraint and the Lyapunov controller design. The result has been successfully applied to solve the dynamic positioning of an actual ship, where the nonlinear state constraints describe a strait. This approach enabled us to design a control Lyapunov function and thereby use Sontag's formula to solve the stabilisation problem. Realistic simulations have been executed in a real scenario on the simulator owned by an international shipbuilding company.
CitationAcosta, J. Á., Doria-Cerezo, A., Fossas, E., E. Fossas, E. Fossas-Colet, Enric Fossas, E. F. Stabilisation of state-and-input constrained nonlinear systems via diffeomorphisms: A Sontag's formula approach with an actual application. "International journal of robust and nonlinear control", 10 Setembre 2018, vol. 28, núm. 13, p. 4032-4044.