Output-input stability and minimum-phase nonlinear systems

This paper introduces and studies the notion of output-input stability, which represents a variant of the minimum-phase property for general smooth nonlinear control systems. The definition of output-input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the ``input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of output-input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.Comment: Revised version, to appear in IEEE Transactions on Automatic Control. See related work in http://www.math.rutgers.edu/~sontag and http://black.csl.uiuc.edu/~liberzo

Modal Test of the NASA Mobile Launcher at Kennedy Space Center

The NASA Mobile Launcher (ML), located at Kennedy Space Center (KSC), has recently been modified to support the launch of the new NASA Space Launch System (SLS). The ML is a massive structureconsisting of a 345-foot tall tower attached to a two-story base, weighing approximately 10.5 million poundsthat will secure the SLS vehicle as it rolls to the launch pad on a Crawler Transporter, as well as provide a launch platform at the pad. The ML will also provide the boundary condition for an upcoming SLS Integrated Modal Test (IMT). To help correlate the ML math models prior to this modal test, and allow focus to remain on updating SLS vehicle models during the IMT, a ML-only experimental modal test was performed in June 2019. Excitation of the tower and platform was provided by five uniquely-designed test fixtures, each enclosing a hydraulic shaker, capable of exerting thousands of pounds of force into the structure. For modes not that were not sufficiently excited by the test fixture shakers, a specially-designed mobile drop tower provided impact excitation at additional locations of interest. The response of the ML was measured with a total of 361 accelerometers. Following the random vibration, sine sweep vibration, and modal impact testing, frequency response functions were calculated and modes were extracted for three different configurations of the ML in 0 Hz to 12 Hz frequency range. This paper will provide a case study in performing modal tests on large structures by discussing the Mobile Launcher, the test strategy, an overview of the test results, and recommendations for meeting a tight test schedule for a large-scale modal test

Stochastic drift in discrete waves of nonlocally interacting particles

In this paper, we investigate a generalized model of N particles undergoing second-order nonlocal interactions on a lattice. Our results have applications across many research areas, including the modeling of migration, information dynamics, and Muller's ratchet—the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as (ln N)-2 and (ln N)-3, respectively—much slower than the usual N-1/2 factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications

Dynamics of Information Networks

We explore a simple model of network dynamics which has previously been applied to the study of information flow in the context of epidemic spreading. A random rooted network is constructed that evolves according to the following rule: at a constant rate pairs of nodes (i,j) are randomly chosen to interact, with an edge drawn from i to j (and any other out-edge from i deleted) if j is strictly closer to the root with respect to graph distance. We characterise the dynamics of this random network in the limit of large size, showing that it instantaneously forms a tree with long branches that immediately collapse to depth two, then it slowly rearranges itself to a star-like configuration. This curious behaviour has consequences for the study of the epidemic models in which this information network was first proposed

Stochastic drift in discrete waves of non-locally interacting-particles

In this paper, we investigate a generalised model of $N$ particles undergoing second-order non-local interactions on a lattice. Our results have applications across many research areas, including the modelling of migration, information dynamics and Muller's ratchet -- the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as $(\log N)^2$ and $(\log N)^{3}$, respectively -- much slower than the usual $\sqrt{N}$ factor. As a result, the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population are processes that occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.Comment: 13 pages, 9 figure

An ISS Small-Gain Theorem for General Networks

We provide a generalized version of the nonlinear small-gain theorem for the case of more than two coupled input-to-state stable (ISS) systems. For this result the interconnection gains are described in a nonlinear gain matrix and the small-gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than one. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated artificial dynamical system.Comment: 26 pages, 3 figures, submitted to Mathematics of Control, Signals, and Systems (MCSS