Global stabilization of nonlinear cascade systems

Cartesian Impedance Control of Flexible Joint Robots: A Decoupling Approach

“…Then the question arises, under which conditions also the coupled system (1)-(2) will be asymptotically stable. Locally this is always true, see the references in (Seibert & Suarez, 1990). In order to ensure that this holds also globally it was proven in (Seibert & Suarez, 1990) that it is sufficient to show that all solutions of the coupled system remain bounded.…”

“…Locally this is always true, see the references in (Seibert & Suarez, 1990). In order to ensure that this holds also globally it was proven in (Seibert & Suarez, 1990) that it is sufficient to show that all solutions of the coupled system remain bounded. This is formulated in a more general form in the following theorem, taken from (Seibert & Suarez, 1990 .…”

“…In order to ensure that this holds also globally it was proven in (Seibert & Suarez, 1990) that it is sufficient to show that all solutions of the coupled system remain bounded. This is formulated in a more general form in the following theorem, taken from (Seibert & Suarez, 1990 . Notice that Theorem 1 also handles the case of global asymptotic stability, if the region of attraction of (2) is the whole state space 2 n ℜ .…”

“…Then, an additional outer control loop implements the desired compliant behaviour (Ott et al, 2003). The stability theory for cascaded control systems (Seibert & Suarez, 1990;Loria, 2001) can be applied to analyze the closed-loop system. When dealing with a robot model with flexible joints, the maybe most obvious approach for the design of an impedance controller is the singular perturbation approach (Spong, 1987;De Luca, 1996;Ott et al, 2002).…”

Generic saddle-node bifurcation for cascade second order ODEs on manifolds

“…Systems of the form (1.3), where the second equation is independent of the first one, are called cascade systems. In [2] we have studied nonautonomous cascade systems of second order differential equations of the form (1.4)ẍ + R(y, t)x = f (x, y, t),ÿ + S(y)y = 0, are studied from the point of view of their feedback stabilization (u is a control parameter) in [8].…”