Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

The converging-input converging-state property for Lur'e systems

Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur’e systems, we derive necessary and sufficient conditions for a class of Lur’e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur’e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called “quasi CICS” property for Lur’e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples

Applications of frequency domain stability criteria in the design of nonlinear feedback systems

The Popov criterion for absolute stability of nonlinear feedback systems is applied to several example problems. Model transformations such as pole shifting and zero shifting extend the class of systems to which the criterion applies. Extensions of the criterion having simple graphical interpretations yield stronger results for systems with constant monotonic slope-bounded nonlinearities. Additional extensions lacking simple graphical interpretations in the complex plane are also demonstrated by example. Stability throughout a region in parameter space is discussed, and the Kalman conjecture is verified for a new class of systems. The Popov criterion is also used to prove BIBO stability, process stability, and degree of stability. The conservatism of the criterion, i. e., the margin of actual performance beyond guaranteed performance, is discussed in the light of simulation results. An interactive computer program is developed to make the Popov criterion, along with two of its extensions, a convenient tool for the design of stable systems. The user has the options of completely automatic parameter adjustment or intervention at any stage of the procedure --Abstract, page ii

Aizerman Conjectures for a class of multivariate positive systems

The Aizerman Conjecture predicts stability for a class of nonlinear control systems on the basis of linear system stability analysis. The conjecture is known to be false in general. Here, a number of Aizerman conjectures are shown to be true for a class of internally positive multivariate systems, under a natural generalisation of the classical sector condition and, moreover, guarantee positivity in closed loop. These results are stronger and/or more general than existing results. The paper relates the obtained results to other, diverse, results in the literature

A linear dissipativity approach to incremental input-to-state stability for a class of positive Lur'e systems

Incremental stability properties are considered for certain systems of forced, nonlinear differential equations with a particular positivity structure. An incremental stability estimate is derived for pairs of input/state/output trajectories of the Lur'e systems under consideration, from which a number of consequences are obtained, including the incremental exponential input-to-state stability property and certain input-output stability concepts with linear gain. Incremental stability estimates provide a basis for an investigation into the response to convergent and (almost) periodic forcing terms, and is treated presently. Our results show that an incremental version of the real Aizerman conjecture is true for positive Lur'e systems when an incremental gain condition is imposed on the nonlinear term, as we describe. Our argumentation is underpinned by linear dissipativity theory -- a property of positive linear control systems.Comment: 28 pages, 3 figures (comprising 8 subpanels). Submitting to Arxiv for early dissemination of results. This version was submitted in Dec 2023 to a journal for possible publicatio