864 research outputs found
Further Results on Lyapunov-Like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems
We provide several characterizations of convergence to unstable equilibria in
nonlinear systems. Our current contribution is three-fold. First we present
simple algebraic conditions for establishing local convergence of non-trivial
solutions of nonlinear systems to unstable equilibria. The conditions are based
on the earlier work (A.N. Gorban, I.Yu. Tyukin, E. Steur, and H. Nijmeijer,
SIAM Journal on Control and Optimization, Vol. 51, No. 3, 2013) and can be
viewed as an extension of the Lyapunov's first method in that they apply to
systems in which the corresponding Jacobian has one zero eigenvalue. Second, we
show that for a relevant subclass of systems, persistency of excitation of a
function of time in the right-hand side of the equations governing dynamics of
the system ensure existence of an attractor basin such that solutions passing
through this basin in forward time converge to the origin exponentially.
Finally we demonstrate that conditions developed in (A.N. Gorban, I.Yu. Tyukin,
E. Steur, and H. Nijmeijer, SIAM Journal on Control and Optimization, Vol. 51,
No. 3, 2013) may be remarkably tight.Comment: 53d IEEE Conference on Decision and Control, Los-Angeles, USA, 201
Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems
We provide Lyapunov-like characterizations of boundedness and convergence of
non-trivial solutions for a class of systems with unstable invariant sets.
Examples of systems to which the results may apply include interconnections of
stable subsystems with one-dimensional unstable dynamics or critically stable
dynamics. Systems of this type arise in problems of nonlinear output
regulation, parameter estimation and adaptive control.
In addition to providing boundedness and convergence criteria the results
allow to derive domains of initial conditions corresponding to solutions
leaving a given neighborhood of the origin at least once. In contrast to other
works addressing convergence issues in unstable systems, our results require
neither input-output characterizations for the stable part nor estimates of
convergence rates. The results are illustrated with examples, including the
analysis of phase synchronization of neural oscillators with heterogenous
coupling
Invariant template matching in systems with spatiotemporal coding: a vote for instability
We consider the design of a pattern recognition that matches templates to
images, both of which are spatially sampled and encoded as temporal sequences.
The image is subject to a combination of various perturbations. These include
ones that can be modeled as parameterized uncertainties such as image blur,
luminance, translation, and rotation as well as unmodeled ones. Biological and
neural systems require that these perturbations be processed through a minimal
number of channels by simple adaptation mechanisms. We found that the most
suitable mathematical framework to meet this requirement is that of weakly
attracting sets. This framework provides us with a normative and unifying
solution to the pattern recognition problem. We analyze the consequences of its
explicit implementation in neural systems. Several properties inherent to the
systems designed in accordance with our normative mathematical argument
coincide with known empirical facts. This is illustrated in mental rotation,
visual search and blur/intensity adaptation. We demonstrate how our results can
be applied to a range of practical problems in template matching and pattern
recognition.Comment: 52 pages, 12 figure
The Stable Manifold Theorem for Stochastic Differential Equations
We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic
differential equations (sde's) that are driven by spatial Kunita-type
semimartingales with stationary ergodic increments. Both Stratonovich and
It\^o-type equations are treated. Starting with the existence of a stochastic
flow for a sde, we introduce the notion of a hyperbolic stationary trajectory.
We prove the existence of invariant random stable and unstable manifolds in the
neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the
stable and unstable manifolds are dynamically characterized using forward and
backward solutions of the anticipating sde. The proof of the stable manifold
theorem is based on Ruelle-Oseledec multiplicative ergodic theory
Stabilization of cascaded nonlinear systems under sampling and delays
Over the last decades, the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model. Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system
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