912 research outputs found

    Stability of Nonlinear Systems with Parameter Uncertainty

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    A novel approach is introduced to assess stability of nonlinear systems in the presence of parameter uncertainty. The idea is to consider the deterministic dynamics of the system as a function of parameter values, where the parameter-dependent initial condition may for example be the output of a particular finite-time disturbance. The goal is to numerically determine the boundary in parameter space, referred to as the recovery boundary, between parameter values which lead to recovery and those which lead to a failure to recover to an initial stable equilibrium point. Critical parameter values, which are defined to be those parameter values whose corresponding initial conditions lie on the boundary of the region of attraction of their corresponding stable equilibrium points, have the potential to provide an explicit connection between the recovery boundary in parameter space and the region of attraction boundary in state space that can be exploited for algorithm design. However, examples are provided to illustrate that the recovery boundary may not contain critical parameter values when the boundary of the region of attraction of the stable equilibrium point varies discontinuously with parameter. Fortunately, it is shown that, for a large class of vector fields possessing stable equilibrium points, the boundaries of the regions of attraction of these equilibrium points vary continuously with respect to small variations in parameter values. This region of attraction boundary continuity ensures that the recovery boundary consists entirely of critical parameter values and that the nearest critical parameter value to any non-critical parameter value lies on the recovery boundary. Two classes of theoretically motivated algorithms are developed to compute critical parameter values by exploiting the structure and behavior of the region of attraction boundary under parameter variation. The system trajectory corresponding to a critical parameter value converges to an invariant set and, therefore, spends an infinite amount of time in any neighborhood of that invariant set. A first class of algorithms proceed by varying parameter values so as to maximize the time in a neighborhood of the invariant set. Under reasonable assumptions, the system trajectory corresponding to a critical parameter value becomes infinitely sensitive to small changes in parameter value. A second class of algorithms proceed by varying parameter values so as to maximize the trajectory sensitivities to parameters. Theoretical motivation is provided for both classes of algorithms, and shows that under reasonable assumptions they will drive parameter values to their critical values. Both of these approaches transform the abstract problem of finding critical parameter values into concrete numerical optimization problems. Based on these approaches, algorithms are developed to find the closest parameter value on the recovery boundary in the case of one-dimensional parameter space, trace the recovery boundary in two dimensional parameter space, and find the nearest point on the recovery boundary in parameter space of arbitrary dimension. The algorithms are applied to assess fault vulnerability in power systems. Results from the test cases of a modified IEEE 37-bus feeder and a modified IEEE 39-bus system illustrate the algorithm performance. The emphases in these test cases are, respectively, to explore the onset of induction motor stalling during Fault Induced Delayed Voltage Recovery, and to analyse stability of a system of synchronous machines under high levels of load uncertainty.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155156/1/fishermw_1.pd

    Lie Algebras and the Stability of Nonlinear Systems

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    Lie algenras and the Cartan decomposition are used to study the stability of "pseudo-linear" systems of differential equations

    Model-based networked control for finite-time stability of nonlinear systems: the stochastic case

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    In this paper we analyze model-based networked control systems for a discrete-time nonlinear plant model, operating in the presence of stochastic dropout of state observations. The dropout is modeled as a Markov chain, and sufficient conditions for finite-time stochastic stability are provided using the stochastic version of Lyapunov second method. In a companion paper we model the dropout as a deterministic sequence

    On absolute stability of nonlinear systems with small delays

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    Nonlinear nonautonomous retarded systems with separated autonomous linear parts and continuous nonlinear ones are considered. It is assumed that deviations of the argument are sufficiently small. Absolute stability conditions are derived. They are formulated in terms of eigenvalues of auxiliary matrices

    Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

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    This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions

    Neural Contraction Metrics for Robust Estimation and Control: A Convex Optimization Approach

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    This letter presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between perturbed and unperturbed system trajectories. We demonstrate how to exploit NCMs to design an online optimal estimator and controller for nonlinear systems with bounded disturbances utilizing their duality. The performance of our framework is illustrated through Lorenz oscillator state estimation and spacecraft optimal motion planning problems
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