Case studies on passivity-based stabilisation of closed sets

Abstract

We present a novel control design procedure for the passivity-based stabilisation of closed sets which leverages recent theoretical advances. The procedure involves using part of the control freedom in order to enforce a detectability property, while the remaining part is used for passivity-based stabilisation. The procedure is illustrated in four case studies of path following coordination for one or two kinematic unicycles, and variations of these problems. Among other things, we present a smooth global path following controller making the unicycle converge to an arbitrary closed and strictly convex curve, and a coordinated path following controller for two unicycles.

Quasi‐passivity‐based adaptive stabilization for switched nonlinearly parameterized systems

“…There exist a positive definite smooth function W i (z) for the i-th subsystem of system (21), a smooth function η 0 (z) with η 0 (0) = 0, constants α i ≥ 0, μ ≥ 1, class K ∞ functions β 1 , β 2 and constants λ 1 > 0 and λ 2 > 0 such that (24) hold for i, j ∈ I. Assumption 1.…”

“…Switched systems have attracted a great amount of attention because of their importance from both theoretical and practical points of view. [24][25][26][27] Many systems, especially physical or biological systems, can be expected to be passive with respect to a set or semipassive, such as the Lorenz system 28 and many neuronal oscillators. 2,4-6 Stability analysis and stabilization problems for switched systems have been extensively studied.…”

“…First, when all the subsystems have any same relative degree, the global practical stability is achieved by combining the recursive feedback quasipassification design technique with a switched adaptive control technique. [24][25][26][27] Many systems, especially physical or biological systems, can be expected to be passive with respect to a set or semipassive, such as the Lorenz system 28 and many neuronal oscillators. Second, when the system states are unavailable for measurements, adaptive output feedback controllers are designed to stabilize the system using quasi-passivity.…”

“…Similar to the concept of strict quasi-passivity, the concepts of passivity with respect to a set and semipassivity were proposed to derive a stability property of the set. [24][25][26][27] Many systems, especially physical or biological systems, can be expected to be passive with respect to a set or semipassive, such as the Lorenz system 28 and many neuronal oscillators. 29 When a system is not passive, a feedback controller can be designed to render it passive, which is called feedback…”

Geometric path following control of a rigid body based on the stabilization of sets

“…An alternative approach is based on stabilization of invariant manifolds in state space based on feedback linearization [Nielsen et al (2009), Hladio et al (2013)] or passivebased control Maggiore (2011),El-Hawwary andMaggiore (2013)]. Simply speaking, a transformation generating an attractor in state space is selected for the initial system.…”