Stability analysis of interconnected systems using computer generated Lyapunov functions

This paper suggests frequency-domain stability criteria for homogeneous multiconnected automatic control systems that allow an easy stability analysis of high-order systems of a large dimension. The results are confirmed by mathematical modeling.

This paper is concerned with the stability analysis of interconnected systems composed of locally and globally exponentially stable subsystems. Sufficient conditions for not necessarily asymptotic and exponentially asymptotic stability are derived. The problem of estimating the stability region of the composite system is also considered.

Uncertainty of parameters in control systems was described by interval variables, and some idea of the non-probabilistic reliability method that we proposed in previous papers was used for stability analysis and synthesis of uncertain systems. A new method, called robust reliability method, was presented for quadratic stability analysis and robust stabilization of dynamic systems with interval parameters. In the method, the performance function for robust reliability calculation was established by criterion of quadratic stability, and a solving method in the form of optimization for robust reliability was presented. The proposed stabilization method can take both the required energy supply for control and the control robustness into account. This makes the reasonable strategy of compromising between the control cost and system reliability become possible. By the method, robust reliability measure of the stability degree of uncertain control systems can be provided, and the maximum allowable ranges of the basic parameters can be obtained. The presented method is based on the linear matrix inequality (LMI) approach, and is convenient to implement Three numerical examples were provided to illustrate the validity and efficiency of the presented method.

Algorithms that extract the principal or minor components of a signal are widely used in signal processing and control applications. This paper explores new frameworks for generating learning rules for iteratively computing the principal and minor components (or subspaces) of a given matrix. Stability analysis using Lyapunov theory and La Salle invariance principle is provided to determine regions of attraction of these learning rules. Among many derivations, it is specifically shown that Oja's rule and many variations of it are asymptotically globally stable. Lyapunov stability theory is also applied to weighted learning rules. Some of the essential features for the proposed MCA/PCA learning rules are that they are self normalized and can be applied to non-symmetric matrices. Exact solutions for some nonlinear dynamical systems are also provided

The stability analysis of dynamical systems is a critical aspect of understanding their behavior and ensuring reliable performance in various applications. Lyapunov functions play a pivotal role in this analysis, offering a powerful mathematical tool to assess the stability of equilibrium points. Despite the fact that at the moment there is no general method for finding the Lyapunov function, this review explores the diverse landscape of Lyapunov functions and their contributions to achieving global stability in dynamical systems. Examples of epidemic models and their Lyapunov functions with the graphical implementations in Python in different reproduction numbers are included for deeper understanding. In addition, the history of the Lyapunov global asymptotic stability theorem was considered shortly. Possible further research ideas in this area are included at the end.

Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback–Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.

SummaryThe H∞ filtering problem for interconnected systems is addressed in this paper. In order to facilitate the stability analysis of interconnected systems under the conditions of failed interconnections, a strategy related to switched systems is introduced. Assume that the interconnections among these subsystems disconnect based on the mode‐dependent average dwell time (MDADT), a switching signal is implemented to denote the failed interconnections cases. In the aid of the MDADT method, a filter is designed such that the fault detection (FD) system is asymptotically stable and satisfies the weighted H∞ performance. Finally, a numerical simulation is provided to show the effectiveness of the proposed filter.

In this letter, an analytical framework is proposed to examine stability of two stable, linear time invariant systems interconnected in positive feedback where the systems have "mixed" properties of negative-imaginary and small-gain. Using the notion of dissipativity, the interconnection of systems is guaranteed to be finite-gain stable under the condition that the dc loop gain is contractive. This work builds on Griggs, and exploits a new set of frequency dependent triplets that was introduced in above reference to "mix" two unconditional stability statements, i.e., small-gain and passivity. Unlike the above reference the present work explores the important question of how a conditional stability statement as needed when two negative-imaginary systems are connected in a feedback loop can be "mixed" with an unconditional stability statement as needed when two contractive systems are connected in a feedback loop. The usefulness of the proposed analytical framework is demonstrated via a numerical example.

Stability analysis of interconnected systems by means of aggregatre subsystem models

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We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1, consisting of a block (usually the plant) which is described by an operator L and of a finite-dimensional block described by a system of ordinary differential equations (usually the controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of these results are phrased in terms of the I/O properties of L and in terms of the Lyapunov stability properties of the subsystem described by the indicated ordinary differential equations. The applicability of our results is demonstrated by means of general specific examples (involving C 0 -semigroups, partial differential equations, or integral equations which determine L ).

In this paper, we study the output consensus problems for heterogeneous multi‐agent systems (MASs) with a high‐dimensional leader and directed topology, where the followers are subject to unknown uncertainties. First, an observer is designed to estimate leader's unmeasurable states. Then, by using neural network (NN) approximation theory, a neuro‐adaptive controller consists of a compensation part and a cooperative term is proposed. Unlike most existing neuro‐adaptive controllers that utilize some discontinuous functions to eliminate the effects of NN approximation errors, a new class of continuous function is used and thereby the chattering effect is avoided. By using tools from graph theory and stability analysis of dynamical systems, it is proven that asymptotical output consensus can be achieved if the underlying network topology contains a directed spanning tree rooted at the leader and the control parameters are suitably chosen. Finally, a numerical simulation is presented to verify the theoretical result.

On the convergence of two-step methods generated by point-to-point operators

We analyze the stability of a class of thin-shell wormholes with spherical symmetry evolving in flat FRW spacetimes. The wormholes considered here are supported at the throat by a perfect fluid with equation of state [Formula: see text] and have a physical radius equal to aR, where a is a time-dependent function describing the dynamics of the throat and R is the background scale factor. The study of wormhole stability is done by means of the stability analysis of dynamic systems.

Introduction. Minimization of ravine convex functions, both smooth and non-smooth, arises in many problems of planning, control, stability analysis of dynamic systems, artificial intelligence, and machine learning. Therefore, the development of new and improvement of existing methods is an important task, taking into account the fact that more and more frequently the functions to be minimized depend on a large number of variables. Special attention should be paid to the methods using the operation of linear transformation of space, which allow to improve the properties of the objective function and significantly accelerate its minimization. Known methods of this type, in particular, ellipsoid methods and r-algorithms, require recalculation of the transformation matrix at each iteration. Therefore, the development of methods with a one-time transformation of space, which have a lower cost of iteration, is definitely an urgent task. The purpose of the article is to review modifications of subgradient method with Polyak’s step that use a scalar parameter m(1 and one-time space transformation operation. Supplement the justification of the convergence and convergence rate of the modifications described. Give recommendations regarding determination of parameter m and space transformation matrix B. Results. Subgradient method with Polyak’s step in transformed space and parameter m>1 is an effective method for minimizing smooth and non-smooth convex functions that have a ravine structure. Determination of an appropriate value of the parameter m(1 and space transformation matrix B allows to significantly accelerate this method and use it for minimization functions that depend on a large number of variables. Conclusions. The development of fast methods for minimization of non-smooth convex functions of many variables with a ravine structure makes it possible to effectively solve modern problems of artificial intelligence, in particular, the problems of machine learning, image recognition, big data analysis, etc. Keywords: subgradient method, space transformation, ravine functions.