Model containing coupled subsystems with oscillations of different types

Consideration was given to an autonomous model containing coupled subsystems (MCCS) and, in the absence of couplings between the subsystems, falling down into systems of ordinary differential equations. It is assumed that the subsystems admit different types of nondegenerate single-frequency oscillations. Solved were the problems of oscillations in MCCS, their stability, and stabilization of MCCS oscillation by smooth autonomous coupling controls. It was shown how the developed theory is applied to the coupled Duffing and Van der Pol oscillators.     

Model with coupled subsystems

Consideration was given to the model with coupled subsystems. In the absence of relations between the subsystems, the MIS falls down into independent systems of autonomous ordinary differential equations. In the structure of the entire system, the subsystems make up hierarchical levels. Sun-planets-satellites, interacting moving objects, and so on exemplify the models with coupled subsystems. The problem of studying dynamics of such models was posed. The following natural approach to their analysis was proposed: classification of the subsystems by types (dynamic properties), specification of various bundles of subsystems, and subsequent analysis of these bundles. Realization of the approach to oscillations, stability, stabilization, bifurcation, and resonance was given. These problems were solved for the model with coupled subsystems having two second-kind subsystems in the basic combination of the oscillation modes in the subsystem.     

Feedback control of linear systems with distributed delay

The problem of optimal control of linear systems containing lumped delay, given by differential-difference equations, has been pursued by several authors. However, transportation-lags are better described by distributed delays giving systems that are described by a set of coupled partial and ordinary differential equations. The lumped part of the system is described by ordinary differential equations and the distributed part of the system is described by partial differential equations. The lumped as well as distributed parts are subject to control. The present paper discusses the control of such systems with quadratic performance measures. Riccati-like equations are derived and a technique for their numerical solution is presented.     

Stabilization of oscillations in a coupled periodic system

Consideration was given to a model containing two coupled autonomous planar subsystems of differential equations. It was assumed that the subsystems admitted families of nondegenerate single-frequency oscillations, and coupling was described by time-periodic functions. Solved was the problem of natural stabilization of system oscillations, where by natural stabilization was meant construction of an isolated oscillation and its concurrent stabilization. Constructive conditions for smooth periodic coupling controls providing problem solution were derived. Specific control was proposed for coupled conservative systems with one degree of freedom.     

On the mathematical modelling of physical systems by ordinary differential stochastic equations

This paper deals with the problem of defining and analysing a general mathematical model for the analysis of physical systems described by ordinary stochastic differential equations with random coefficients and initial conditions. The existence, continuity and stability of the evolution process defined by the considered class of evolution equations is here considered. Some actual models of physical systems are also considered and related to the general mathematical model which is proposed and mathematically analyzed in this work.     

Decomposition of Nonlinear Stochastic Differential Systems

Decomposition of a system, i.e., representation of a system as a set of subsystems of lesser dimension, for a multivariate dynamic system described by a nonlinear Ito stochastic differential equation is investigated. The existence and explicit construction of coordinate systems (substitutions of variables) for decomposition are examined. Group-theoretic analysis of ordinary differential equations, which is effective for analyzing deterministic control systems, is also useful in studying the decomposition of stochastic systems. A relationship between the decompositions of the Ito stochastic system and its associated deterministic control system is determined. Examples are given.     

Stabilizing the oscillations of an autonomous system

For a dynamical system that admits a family of oscillations, we propose a small smooth autonomous control that corrects the model itself and at the same time stabilizes the oscillations of the controlled system. We consider a separate system, a set of dynamical systems, and a dynamical model containing weakly coupled subsystem (MCCS). For the MCCS, we give a solution of the stabilization problem for oscillations of the system itself. We use an idea that goes back to Pontryagin’s work on the limit cycle for a system close to Hamiltonian.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

基于策略迭代算法的连续时间线性Markov跳变系统非零和微分反馈Nash控制

针对一类连续时间线性Markov跳变系统,本文提出了一种新的策略迭代算法用于求解系统的非零和微分反馈Nash控制问题.通过求解耦合的数值迭代解,以获得具有线性动力学特性和无限时域二次成本的双层非零和微分策略的Nash均衡解.在每一个策略层,采用策略迭代算法来计算与每一组给定的反馈控制策略相关联的最小无限时域值函数.然后,通过子系统分解将Markov跳变系统分解为N个并行的子系统,并将该算法应用于跳变系统.本文提出的策略迭代算法可以很容易求解非零和微分策略所对应的耦合代数Riccati方程,且对高维系统有效.最后通过仿真示例证明了本文设计方法的有效性和可行性.     

Optimal control of systems with transportation lags

Simultaneous use of first-order partial differential equations and ordinary differential equations, instead of differential-difference equations, is presented to define systems involving transportation lag processes. Variational analysis is carried out for the system to obtain an optimal control law which minimizes a specified objective function, and the strong maximum condition which has to be satisfied by the optimal control is derived and discussed. The theory developed is rather general and may be applied to a class of systems wider than those defined by differential-difference equations.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

Serre’s reduction of linear partial differential systems with holonomic adjoints

Given a linear functional system (e.g., an ordinary/partial differential system, a differential time-delay system, a difference system), Serre’s reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre’s reduction of underdetermined linear systems of partial differential equations with either polynomial, formal power series or locally convergent power series coefficients, and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial differential systems can be defined by means of only one linear partial differential equation. In the case of polynomial coefficients, we give an algorithm to compute the corresponding equation.     

Finite‐Time Stabilization of Coupled Systems on Networks with Time‐Varying Delays via Periodically Intermittent Control

In this paper, finite‐time stabilization of coupled systems on networks with time‐varying delays (CSNTDs) via periodically intermittent control is studied. Both delayed subsystems and delayed couplings are considered; the self‐delays of different subsystems in delayed couplings are not identical. A periodically intermittent controller is designed to stabilize CSNTDs within finite time, and the stabilization duration is closely related to the topological structures of networks. Furthermore, two sufficient criteria are developed to ensure CSNTDs under periodically intermittent control can be stabilized within finite time by using an approach that combines the Lyapunov method with Kirchhoff's Matrix Tree Theorem. Then finite‐time stabilization of coupled oscillators with time‐varying delays is given as a practical application and sufficient criteria is obtained. Finally, a numerical simulation is proposed to support our results and show the effectiveness of the controller.     

Parameter identification and model verification in systems of partial differential equations applied to transdermal drug delivery

The purpose of this paper is to present some numerical tools which facilitate the interpretation of simulation or data fitting results and which allow computation of optimal experimental designs. They help to validate mathematical models describing the dynamical behavior of a biological, chemical, or pharmaceutical system, without requiring a priori knowledge about the physical or chemical background. Although the ideas are quite general, we will concentrate our attention to systems of one-dimensional partial differential equations and coupled ordinary differential equations. A special application model serves as a case study and is outlined in detail. We consider the diffusion of a substrate through cutaneous tissue, where metabolic reactions are included in form of Michaelis–Menten kinetics. The goal is to simulate transdermal drug delivery, where it is supposed that experimental data are available for substrate and metabolic fluxes. Numerical results are included based on laboratory data to show typical steps of a model validation procedure, i.e., the interpretation of confidence intervals, the compliance with physical laws, the identification and elimination of redundant model parameters, the computation of optimum experimental designs and the identifiability of parameters by determining weight distributions.     

Chaos in the brain: Possible roles in biological intelligence

Deterministic dynamic systems are described with sets of coupled ordinary differential equations (ODEs) and differential delay equations (DDEs). By tradition and for convenience these systems are usually operated in the domains of equilibrium and limit cycle attractors (steady states and clocks), which are the conditions that hold for most digital and analog computers, including artificial neural networks (ANN) designed for data processing and logical manipulations. These systems are capable of operations in real and simulated chaotic domains, in which predictability is limited and performance become pseudorandom, but these domains are usually avoided if at all possible. © 1995 John Wiley & Sons, Inc.     

Well-posedness and exponential stability of two-dimensional vibration model of a boundary-controlled curved beam with tip mass

This paper studies the well-posedness and exponential stability of two-dimensional vibration model of a curved beam with tip mass under linear boundary control. The control task is to stabilise the tangential and radial vibrations, which are coupled due to the beam curvature. To reach the main results of the paper, mathematical analyses based on the semigroup theory and Lyapunov approach are conducted, and it is shown that the proposed closed-loop model holds a unique solution that converges to zero exponentially fast. These analyses are based on a hybrid dynamic model that incorporates two coupled partial differential equations and six boundary conditions, including two ordinary differential equations. Simulation results are used to illustrate the efficacy of the suggested method.     

Triangularizing kinematic constraint equations using Gröbner bases for real-time dynamic simulation

Real-time simulation is an essential component of hardware- and operator-in-the-loop applications, such as driving simulators, and can greatly facilitate the design, implementation, and testing of dynamic controllers. Such applications may involve multibody systems containing closed kinematic chains, which are most readily modeled using a set of redundant generalized coordinates. The governing dynamic equations for such systems are differential-algebraic in nature—that is, they consist of a set of ordinary differential equations coupled with a set of nonlinear algebraic constraint equations—and can be difficult to solve in real time. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing a recursively solvable system for calculating the dependent generalized coordinates given values of the independent coordinates. The proposed approach can be used to generate computationally efficient simulation code that avoids the use of iteration, which makes it particularly suitable for real-time applications.     

Optimal control of convection–diffusion process with time-varying spatial domain: Czochralski crystal growth

This paper considers the optimal control of convection–diffusion systems modeled by parabolic partial differential equations (PDEs) with time-dependent spatial domains for application to the crystal temperature regulation problem in the Czochralski (CZ) crystal growth process. The parabolic PDE model describing the temperature dynamics in the crystal region arising from the first principles continuum mechanics is defined on the time-varying spatial domain. The dynamics of the domain boundary evolution, which is determined by the mechanical subsystem pulling the crystal from the melt, are described by an ordinary differential equation for rigid body mechanics and unidirectionally coupled to the convection–diffusion process described by the PDE system. The representation of the PDE as an evolution system on an appropriate infinite-dimensional space is developed and the analytic expression and properties of the associated two-parameter semigroup generated by the nonautonomous operator are provided. The LQR control synthesis in terms of the two-parameter semigroup is considered. The optimal control problem setup for the PDE coupled with the finite-dimensional subsystem is presented and numerical results demonstrate the regulation of the two-dimensional crystal temperature distribution in the time-varying spatial domain.     

Artificial neural networks for solving ordinary and partialdifferential equations

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE), to systems of coupled ODE and also to partial differential equations (PDE). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galerkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.