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The article (Bazanella et al., 2019) presented the first results on generic identifiability of dynamic networks with partial excitation and partial measurements. All previous papers assumed that either all nodes are excited or all nodes are measured. One key contribution of that paper was to establish a set of necessary conditions on the excitation and measurement pattern (EMP) that guarantee generic identifiability: all sources must be excited and all sinks measured, and all other nodes must be either excited or measured. In this article, we show that two other types of nodes, which are defined by the local topology of the network, play an essential rôle in the search for a valid EMP, i.e., one that guarantees generic identifiability. We have called these nodes dources and dinks. We show that a network is generically identifiable only if, in addition to the abovementioned conditions, all dources are excited and all dinks are measured. We also show that sources and dources are the only nodes in a network that always need to be excited, and that sinks and dinks are the only nodes that need to be measured for an EMP to be valid.

This article is concerned with the development and use of duality theory for a nonlinear filtering model with white noise observations. The main contribution of this article is to introduce a stochastic optimal control problem as a dual to the nonlinear filtering problem. The mathematical statement of the dual relationship between the two problems is given in the form of a duality principle. The constraint for the optimal control problem is the backward stochastic differential equation introduced in the companion paper. The optimal control solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The proposed duality is shown to be an exact extension of the classical Kalman–Bucy duality, and different from other types of optimal control and variational formulations given in literature.

This technical note addresses the problem of transforming a single-input–single-output discrete-time system into the extended observer form, which comprise a linear time-invariant observable component, and a nonlinear injection term, which depends on the input, output, and their forward shifts up to a finite order. Intrinsic necessary and sufficient conditions are provided for obtaining the extended observer form via a parametrized state transformation. The conditions are formulated directly in terms of the state equations and do not rely on input–output equations as in the earlier papers. Further, an algorithm for obtaining the required transformation is presented. Unlike the existing results on observer forms, the results are not restricted to reversible systems but to more general submersive systems, i.e., to systems, which are reversible via static state feedback.

Kernel-based methods have been successfully introduced in system identification to estimate the impulse response of a linear system. Adopting the Bayesian viewpoint, the impulse response is modeled as a zero mean Gaussian process whose covariance function (kernel) is estimated from the data. The most popular kernels used in system identification are the tuned-correlated (TC), the diagonal-correlated (DC) and the stable spline (SS) kernel. TC and DC kernels admit a closed-form factorization of the inverse. The SS kernel induces more smoothness than TC and DC on the estimated impulse response, however, the aforementioned property does not hold in this case. In this paper we propose a second-order extension of the TC and DC kernel, which induces more smoothness than TC and DC, respectively, on the impulse response and a generalized-correlated kernel, which incorporates the TC and DC kernels and their second order extensions. Moreover, these generalizations admit a closed-form factorization of the inverse and thus they allow to design efficient algorithms for the search of the optimal kernel hyperparameters. We also show how to use this idea to develop higher oder extensions. Interestingly, these new kernels belong to the family of the so called exponentially convex local stationary kernels: such a property allows to immediately analyze the frequency properties induced on the estimated impulse response by these kernels.

Reliable quasi-static object manipulation and robotic locomotion require the verification of the stability of equilibria under unilateral contacts and friction. In a recent paper, Posa et al. (2016) demonstrated that sums-of-squares (SOS) programming can be used to verify the Lyapunov stability of planar systems via Lyapunov’s direct method if impacts are inelastic. However, this method appears to be too conservative to verify the stability of some remarkably simple examples. In this article, an extension of Lyapunov’s direct method is proposed, which makes use of a piecewise smooth Lyapunov function and allows a temporary increase of the Lyapunov function along a motion trajectory. In addition, a modified SOS formulation enables the investigation of planar systems with partially elastic contacts. The proposed method remains compatible with SOS programming techniques. The improved stability test is successfully applied to a point mass on a slope and to a rigid body with two contact points. For the latter, several mechanisms of instability have been demonstrated experimentally, but the exact conditions of Lyapunov stability are unknown.

In this article, stabilization of systems by delayed noisy states is investigated. The time delays in the stabilizing noisy states are extended into the general form. To support this novelty, the familiar Doob martingale inequality in the continuous version is improved; the equivalence principle, which says that the exponential stability in moment of an anhysteretic stochastic system infers the same property of the corresponding hysteretic system, is extended to the cases with the newly proposed semi-Lipschitz condition; the implication theorem, which indicates that the exponential stability in moment implies almost sure exponential stability, is generalized to the stochastic systems with time delays of general forms only under the linear growth condition. Technically, variable substitution is applied to time to map the function spaces for the functional differential equation models with unbounded time delays into those with bounded ones. Unified framework for the stability analysis for both bounded and unbounded time delays is built with the mapping, the decay rates, including those for polynomial stability, for estimates of the solutions with time delays of general form are formulated. At the end of the article, typical strategies, including the divided state feedback with unbounded time delays, for the stabilizing noise are presented, a numerical example is proposed to illustrate the method and to show the efficiency of the results of the paper.