Alexander Yu. Pogromsky

Convergent dynamics, a tribute to Boris Pavlovich Demidovich

We review and pay tribute to a result on convergent systems by the Russian mathematician Boris Pavlovich Demidovich. In a sense, Demidovichs approach forms a prelude to a field which is now called incremental stability of dynamical systems. Developments on incremental stability are reviewed from a historical perspective.

Partial synchronization: from symmetry towards stability

Abstract In this paper we study the existence and stability of linear invariant manifolds in a network of coupled identical dynamical systems. Symmetry under permutation of different units of the network is helpful to construct explicit formulae for linear invariant manifolds of the network, in order to classify them, and to examine their stability through Lyapunov’s direct method.

Time scaling for observer design with linearizable error dynamics

In this paper, we consider the problem of observer design for dynamical systems with scalar output by linearization of the error dynamics via coordinate change, output injection, and time scaling. We present necessary and sufficient conditions which guarantee the existence of a coordinate change and output-dependent time scaling, such that in the new coordinates and with respect to the new time the system has linear error dynamics.

On global properties of passivity-based control of an inverted pendulum

The paper adresses the problem of stabilization of a specific target position of underactuated Lagrangian or Hamiltonian systems. We propose to solve the problem in two steps: first to stabilize a set with the target position being a limit point for all trajectories originating in this set and then to switch to a locally stabilizing controller. We illustrate this approach by the well-known example of inverted pendulum on a cart. Particularly, we design a controller which makes the upright position of the pendulum and zero displacement of the cart a limit point for almost all trajectories. We derive a family of static feedbacks such that any solution of the closed loop system except for those originating on some two-dimensional manifold approaches an arbitrarily small neighbourhood of the target position. The proposed technique is based on the passivity properties of the inverted pendulum. A possible extension to a more general class of underactuated mechanical systems is discussed.

On Diffusion Driven Oscillations in Coupled Dynamical Systems

The paper deals with the problem of destabilization of diffusively coupled identical systems. It is shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillat ...

On convergence properties of piecewise affine systems

In this paper convergence properties of piecewise affine (PWA) systems are studied. In general, a system is called convergent if all its solutions converge to some bounded globally asymptotically stable steady-state solution. The notions of exponential, uniform and quadratic convergence are introduced and studied. It is shown that for non-linear systems with discontinuous right-hand sides, quadratic convergence, i.e., convergence with a quadratic Lyapunov function, implies exponential convergence. For PWA systems with continuous right-hand sides it is shown that quadratic convergence is equivalent to the existence of a common quadratic Lyapunov function for the linear parts of the system dynamics in every mode. For discontinuous bimodal PWA systems it is proved that quadratic convergence is equivalent to the requirements that the system has some special structure and that certain passivity-like condition is satisfied. For a general multimodal PWA system these conditions become sufficient for quadratic convergence. An example illustrating the application of the obtained results to a mechanical system with a one-sided restoring characteristic, which is equivalent to an electric circuit with a switching capacitor, is provided. The obtained results facilitate bifurcation analysis of PWA systems excited by periodic inputs, substantiate numerical methods for computing the corresponding periodic responses and help in controller design for PWA systems.

Observer-Based Robust Synchronization of Dynamical Systems

The paper deals with the problem of robust synchronization of dynamical systems. The design procedure is based on the concept of observers with absolutely stable error dynamics. In the general case of nonlinear time-varying error dynamics the procedure requires exact knowledge of a Lyapunov function while in case of the linearizable error dynamics frequency domain conditions which ensure existence of such a function can be employed. Two examples are considered: synchronization of two Lorenz systems and Rossler systems.

Convergent piecewise affine systems: analysis and design Part II: discontinuous case

In this paper convergence properties of piecewise affine (PWA) systems with discontinuous right-hand sides are studied. It is shown that for discontinuous PWA systems existence of a common quadratic Lyapunov function is not sufficient for convergence. For discontinuous bimodal PWA systems necessary and sufficient conditions for quadratic convergence, i.e. convergence with a quadratic Lyapunov function, are derived.

A partial synchronization theorem.

When synchronization sets in, coupled systems oscillate in a coherent way. It is possible to observe also some intermediate regimes characterized by incomplete synchrony which are referred to as partial synchronization. The paper focuses on analysis of partial synchronization in networks of linearly coupled oscillators.

An observer for phase synchronization of chaos

In this Letter we design, using only a single scalar output, a possible class of observers to detect whether a dynamical system perturbed by an unknown harmonic disturbance exhibits phase synchronization with the disturbance.