In this note, some testable conditions for mean square (i.e., second moment) stability for discrete-time jump linear systems with timehomogenous and time-inhomogenous finite state Markov chain form processes are presented.
This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct observers of a remarkably simple structure: A linear state variable filter followed by a nonlinearity. The observer is established in Sections III-V by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist, and by giving an explicit expression for its nonlinearity. It is demonstrated that the existence conditions are satisfied, and hence our results include a new observer which is not high-gain, for the wide class of smooth systems considered recently in a previous paper by Gauthier, et al. In Section VI, it is shown that the observer can as well be designed to realize an arbitrary, finite accuracy rather than ultimate exactness. On a compact region of the state space, this requires only observability of the system. A corresponding numerical design procedure is described, which is easy to implement and computationally feasible for low order systems.Index Terms-Nonlinear state observer, nonlinear systems, nonsmooth systems.
This paper considers the state observation problem for nonlinear dynamical systems. The proposed framework is a direct generalization of a method introduced in a recent paper for autonomous system. Its characteristic feature is that the dynamic part of the observer is linear and, as a consequence, that convergence takes place globally in the observer coordinates. The observer is completed by a static nonlinearity which maps the observer state in the original state space. An associated observation mapping is introduced and is interpreted in terms of an orthonormal expansion of the input and the output with respect to a certain basis in a suitable Hilbert space. It is shown that, by choosing the observer dimension properly, an observer with arbitrary small asymptotic observation error is obtained, provided that some compactness properties for the subset to be observed and the set of input signals hold. Under a stronger condition, the finite complexity property, an exact observer is achieved. Finally, an integral formula representation for the observer nonlinearity is given.
The linear theory predicts that certain anisotropic velocity distributions will produce unstable extraordinary waves. The development of these unstable waves in the nonlinear regime is investigated and their final amplitudes are estimated. The analysis is restricted to infinite homogeneous plasmas where the background distribution and the wave energy density may be considered as slowly varying functions of time. A set of nonlinear integro-differential equations, which describe the evolution of the system, are derived and discussed. For the physically interesting case where the wave frequency is much less than the cyclotron frequency, the basic behavior of the system and the characteristics of the equilibrium solutions may be estimated without computing the transient behavior of the system. For this case, it is shown that the main diffusion of the background distribution occurs along the v = const lines, in which v is the magnitude of the velocity. Since these lines are also constant energy lines, very little energy is transferred to the waves during the diffusion process. A Maxwellian plasma which has been disturbed by injecting a stream of electrons into it is considered as an example.
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