We consider the problem of asymptotic convergence to invariant sets in interconnected nonlinear dynamic systems. Standard approaches often require that the invariant sets be uniformly attracting. e.g. stable in the Lyapunov sense. This, however, is neither a necessary requirement, nor is it always useful. Systems may, for instance, be inherently unstable (e.g. intermittent, itinerant, meta-stable) or the problem statement may include requirements that cannot be satisfied with stable solutions. This is often the case in general optimization problems and in nonlinear parameter identification or adaptation. Conventional techniques for these cases rely either on detailed knowledge of the system’s vector-fields or require boundeness of its states. The presently proposed method relies only on estimates of the input-output maps and steady-state characteristics. The method requires the possibility of representing the system as an interconnection of a stable, contracting, and an unstable, exploratory part. We illustrate with examples how the method can be applied to problems of analyzing the asymptotic behavior of locally unstable systems as well as to problems of parameter identification and adaptation in the presence of nonlinear parametrizations. The relation of our results to conventional small-gain theorems is discussed.

We consider a class of systems influenced by perturbations that are nonlinearly parameterized by unknown constant parameters, and develop a method for estimating the unknown parameters within an arbitrarily large parameter space. The method applies to systems where the states are available for measurement, and perturbations with the property that an exponentially stable estimate of the unknown parameters can be obtained if the whole perturbation is known. The main contribution is to introduce a conceptually simple, modular design that gives freedom to the designer in accomplishing the main task, which is to construct an update law to asymptotically invert a nonlinear equation. Compensation for the perturbations in the system equations is considered for a class of systems with uniformly globally bounded solutions and for which the origin is uniformly globally asymptotically stable when no perturbations are present. We also consider the case when the parameters can only be estimated when the controlled state is bounded away from the origin, and show that we may still be able to achieve convergence of the controlled state. We illustrate the method through examples, and apply it to the problem of downhole pressure estimation during oil well drilling.

estimation and control for a class of

We consider the problem of asymptotic reconstruction of the state and parameter values for dynamical systems that cannot be transformed into the canonical adaptive observer form. A solution to this problem is proposed for a class of systems for which the unknowns are allowed to be nonlinearly parameterized functions of state and time. Going beyond asymptotic Lyapunov stability, we provide for this class of systems a reconstruction technique, based on the concepts of weakly attracting sets, non-uniform convergence, and Poisson stability. Key words: Adaptive observers, nonlinear parametrization, weakly attracting sets, unstable attractors, nonlinear systems, Poisson stability 1

We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of canonic models governing the dynamics of evoked potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo, and Morris-Lecar models. We consider the problem of state and parameter reconstruction for these models within the classical framework of observer design. This framework offers computationallyefficient solutions to the problem of state and parameter reconstruction of a system of nonlinear differential equations, provided that these equations are in the so-called adaptive observer canonic form. We show that despite typical neural oscillators being locally observable they are not in the adaptive canonic observer form. Furthermore, we show that no parameter-independent diffeomorphism exists such that the original equations of these models can be transformed into the adaptive canonic observer form. We demonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models, parameter-dependent coordinate transformations can be used to render these systems into the adaptive observer canonical form. This allows reconstruction, at least partially and up to a (bi)linear transformation, of unknown state and parameter values with exponential rate of convergence. In order to avoid the problem of only partial reconstruction