Maximal monotonicity is explored as a generalization of the linear theory of passivity, aiming at an algorithmic input/output analysis of physical models. The theory is developed for maximal monotone one-port circuits, formed by the series and parallel interconnection of basic elements. An algorithmic method is presented for solving the periodic output of a periodically driven circuit using a maximal monotone splitting algorithm, which allows computation to be separated for each circuit component. A new splitting algorithm is presented, which applies to any monotone circuit defined as a port interconnection of monotone elements.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.
Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on L 2 is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function and quadratic constraints, allowing system properties to be verified or disproved. Interconnections of systems correspond to graphical manipulations of their SRGs. This is used to provide a simple, graphical proof of the classical incremental passivity theorem.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.
Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of non-Riemannian metrics. We provide open loop and sampled data controllers. The sampled data control construction presented here does not require real time computation of globally shortest paths, simplifying computation. A Lyapunov function characterizes the stability of a system and is related to the intuitive idea of energy decaying in stable systems. A control Lyapunov function (CLF) is necessary and sufficient for controllability of a system [5], [6], and for large classes of systems (those affine in controls), the construction of a controller given a CLF is simple [7]. However, control Lyapunov functions are in general difficult to find [8].The Control Contraction Metric (CCM) method of control synthesis, introduced by Manchester and Slotine [9], simplifies the search for a Lyapunov function. Rather than explicitly search for a Lyapunov function, a convex search is performed for a CCM which measures distance between trajectories. The CCM may be thought of as inducing an infinite family of local Lyapunov functions. Online computation involves a search for a minimal path and integration of a local, differential control law along this minimal path. CCM controllers have more efficient online computation than nonlinear Model Predictive Control [10], and have been applied in several application areas, including mechanical systems [11], decentralized and distributed control [12], [13], [14] and collision-free motion planning [15].CCMs are based closely on contraction analysis, introduced by Lohmiller and Slotine [16]. The central idea is that if all nearby trajectories converge to each other, then all trajectories converge to one nominal trajectory and the system is stable. The idea that global properties can be inferred from the local behaviour of trajectories does away with the need to construct global functions.
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