Finite element-based observer design for nonlinear systems with delayed measurements

summary:This paper presents a computational procedure for the design of an observer of a nonlinear system. Outputs can be delayed, however, this delay must be known and constant. The characteristic feature of the design procedure is computation of a solution of a partial differential equation. This equation is solved using the finite element method. Conditions under which existence of a solution is guaranteed are derived. These are formulated by means of theory of partial differential equations in $L^2$-space. Three examples demonstrate viability of this approach and provide a comparison with the solution method based on expansions into Taylor polynomials

Consensus of multi-agent systems and stabilization of large-scale systems with time delays and nonlinearities - a comparison of both problems

summary:The problem of stabilization of large-scale systems and the consensus problem of multi-agent systems are related, similar tools for their solution are used. Therefore, they are occasionally confused. Although both problems show similar features, one can also observe important differences. A comparison of both problems is presented in this paper. In both cases, attention is paid to the explanation of the effects of the time delays. The most important fact is that, if the time delays are heterogeneous, full synchronization of the multi-agent systems cannot be achieved; however, stabilization of the large-scale network is reachable. In the case of nonlinear systems, we show that the stabilization of a large-scale nonlinear system is possible under more restrictive assumptions compared to the synchronization of a nonlinear multi-agent system

Numerical method for the solution of the regulator equation with application to nonlinear tracking

Abstract A numerical method to solve the so-called regulator equation is presented here. This equation consists of partial differential equations combined with algebraic ones and arises when solving the output-regulation problem. Solving the regulator equation is becoming difficult especially for the nonminimum phase systems where reducing variables against algebraic part leads to a potentially unsolvable differential part. The proposed numerical method is based on the successive approximation of the differential part of the regulator equation by the finite-element method while trying to minimize a functional expressing the error of its algebraical part. The method is analyzed to obtain theoretical estimates of its convergence and it is tested on an example of the "two-carts with an inverted pendulum" system. Simulations are included to illustrate the suggested approach

Regulatory network of drug-induced enzyme production: parameter estimation based on the periodic dosing response measurement

summary:The common goal of systems pharmacology, i.e. systems biology applied to the field of pharmacology, is to rely less on trial and error in designing an input-output systems, e.g. therapeutic schedules. In this paper we present, on the paradigmatic example of a regulatory network of drug-induced enzyme production, the further development of the study published by Duintjer Tebbens \textit{et al}. (2019) in the Applications of Mathematics. Here, the key feature is that the nonlinear model in form of an ODE system is controlled (or periodically forced) by an input signal being a drug intake. Our aim is to test the model features under both periodic and nonrecurring dosing, and eventually to provide an innovative method for a parameter estimation based on the periodic dosing response measurement

Biochemical network of drug-induced enzyme production: Parameter estimation based on the periodic dosing response measurement

summary:The well-known bottleneck of systems pharmacology, i. e., systems biology applied to pharmacology, refers to the model parameters determination from experimentally measured datasets. This paper represents the development of our earlier studies devoted to inverse (ill-posed) problems of model parameters identification. The key feature of this research is the introduction of control (or periodic forcing by an input signal being a drug intake) of the nonlinear model of drug-induced enzyme production in the form of a system of ordinary differential equations. First, we tested the model features under periodic dosing, and subsequently, we provided an innovative method for a parameter estimation based on the periodic dosing response measurement. A numerical example approved the satisfactory behavior of the proposed algorithm

Generalized synchronization in the networks with directed acyclic structure

summary:Generalized synchronization in the direct acyclic networks, i.e. the networks represented by the directed tree, is presented here. Network nodes consist of copies of the so-called generalized Lorenz system with possibly different parameters yet mutually structurally equivalent. The difference in parameters actually requires the generalized synchronization rather than the identical one. As the class of generalized Lorenz systems includes the well-known particular classes such as (classical) Lorenz system, Chen system, or Lü system, all these classes can be synchronized using the presented approach as well. The main theorem is rigorously mathematically formulated and proved in detail. Extensive numerical simulations are included to illustrate and further substantiate these theoretical results. Moreover, during these numerical experiments, the so-called duplicated system approach is used to double-check the generalized synchronization

Sum-of-squares based observer design for polynomial systems with a known fixed time delay

summary:An observer for a system with polynomial nonlinearities is designed. The system is assumed to exhibit a time delay whose value is supposed to be constant and known. The design is carried out using the sum-of-squares method. The key point is defining a suitable Lyapunov-Krasovskii functional. The resulting observer is in form of a polynomial in the observable variables. The results are illustrated by two examples

Consensus of a multi-agent systems with heterogeneous delays

summary:The paper presents an algorithm for the solution of the consensus problem of a linear multi-agent system composed of identical agents. The control of the agents is delayed, however, these delays are, in general, not equal in all agents. The control algorithm design is based on the $H_\infty$-control, the results are formulated by means of linear matrix inequalities. The dimension of the resulting convex optimization problem is proportional to the dimension of one agent only but does not depend on the number of agents, hence this problem is computationally tractable. It is shown that heterogeneity of the delays in the control loop can cause a steady error in the synchronization. Magnitude of this error is estimated. The results are illustrated by two examples

Outputs and Results

This manuscript aims to deliver a survey of results obtained during the solution of the project No. GA19-07635S of the Czech Science Foundation. The timespan dedicated to the work on this project was 1.3.2019 - 30.6.2022. The main area dealt with were\nnonlinear multi-agent systems and their synchronization, further, attention was paid to some auxiliary results in the area of nonlinear observers. This Report briefly introduces the Project, provides a summary of the results obtained and also sketches an outline how these results will be applied and extended in future

A NUMERICAL METHOD FOR THE SOLUTION OF THE NONLINEAR OBSERVER PROBLEM

The central part in the process of solving the observer problem for nonlinear systems is to nd a solution of a partial differential equation of first order. The original method proposed to solve this equation used expansions into Taylor polynomials, however, it suffers from rather restrictive assumptions while the approach proposed here allows to generalize these requirements. Its characteristic feature is that it is based on the application of the Finite Element\nMethod. An illustrating example is provided