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On the relationship between the energy shaping and the Lyapunov constraint based methods. (English) Zbl 1373.93248

Summary: In this paper, we make a review of the Controlled Hamiltonians (CH) method and its related matching conditions, focusing on an improved version recently developed by D. E. Chang. Also, we review the general ideas around the Lyapunov Constraint Based (LCB) method, whose related Partial Differential Equations (PDEs) were originally studied for underactuated systems with only one actuator, and then we study its PDEs for an arbitrary number of actuators. We analyze and compare these methods within the framework of Differential Geometry, and from a purely theoretical point of view. We show, in the context of control systems defined by simple Hamiltonian functions, that the LCB method and the Chang’s version of the CH method are equivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we show that Chang’s improvement of the energy shaping method is precisely the LCB method. As a by-product, coordinate-free and connection-free expressions of Chang’s matching conditions are obtained.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D20 Asymptotic stability in control theory
93C10 Nonlinear systems in control theory
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