1. What are the contributions in "Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one" ?
The authors prove in this paper that for a class of mechanical systems with underactuation degree one the partial differential equations can be explicitly solved.. Furthermore, the authors introduce a suitable parametrization of assignable energy functions that provides the designer with a handle to address transient performance and robustness issues.
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2. What are the future works mentioned in the paper "Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one" ?
In particular, the authors present an almost globally stabilizing controller for the VTOL aircraft that ensures asymptotic regulation from any initial condition to an arbitrary position with zero roll angle and zero speed ; and a controller for the pendulum on the cart that can swing-up the pendulum from any position in the ( open ) upper half plane and stop the cart at any desired location.. Current research is under way to extend the present work in the following directions.. More precisely, the total energy function can be effectively shaped via the selection of the scaling matrix, the constant matrix in the inertia matrix ( 20 ) and the choice of the function in the potential energy ( 25 ).. An additional tuning parameter is the damping injection gain that may be any positive definite ( possibly state-dependent ) matrix.
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3. What is the proof of the equilibrium set of the closed-loop system?
Some simple calculations establish that the equilibrium set of the closed-loop system isTo prove stability of the desired equilibrium the authors note that ensures is (locally) positive definite in , therefore to qualify as a Lyapunov function candidate the authors only need to prove that satisfies the minimum condition (8).
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4. What is the simplest way to determine the asymptotic stability claim?
Remark 6: To quantify the domain of attraction, e.g., to obtain an (almost) global version of the asymptotic stability claim, the authors need to rule out the existence of limit cycles in the whole space as well as stable equilibria, different from the desired one.
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