Tikhonov Theorem Research Articles

Stability analysis of a linear system coupling wave and heat equations with different time scales

In this paper we consider a system coupling a wave equation with a heat equation through its boundary conditions. The existence of a small parameter in the heat equation, as a factor multiplying the time derivative, implies the existence of different time scales between the constituents of the system. This suggests the idea of applying a singular perturbation method to study stability properties. In fact, we prove that this method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which allows us to approximate the solution of the coupled system using some appropriate uncoupled subsystems.

Linear Active Disturbance Rejection Control for Two-Mass Systems Via Singular Perturbation Approach

This article presents a linear active disturbance rejection control (LADRC) scheme for two-mass systems (TMSs) based on a singular perturbation (SP) approach. In the proposed scheme, the dynamics of TMSs are separated into a quasi-steady-state system at a slow time scale and a boundary-layer system at a fast time scale via the SP method. The quasi-steady-state model is used as the nominal model to design the LADRC, where internal model control (IMC) rules are employed for prescribed tracking performance. This not only reduces the model order but also makes the gain tuning straightforward, which is crucial for practical use. Based on the boundary-layer model, the active damping method is designed for vibration suppression, which aims at achieving a higher stable bandwidth of the IMC to better attenuate the residual disturbance of linear extended state observer such that the tracking accuracy can be improved. Stability of the full system was analyzed via the extended Tikhonov's theorem. Comparative experiments were carried out to verify the effectiveness of the proposed scheme.

Singular Perturbation Analysis of a Coupled System Involving the Wave Equation

This article considers a system coupling an ordinary differential equation with a wave equation through its boundary data. The existence of a small parameter in the wave equation (as a factor multiplying the time derivative) suggests the idea of applying a singular perturbation method to get the stability of the full system by analyzing the stability of some appropriate subsystems given by the method. However, for infinite-dimensional systems, it is known that in some cases, this method does not work. Indeed, one cannot be sure of the stability of the full system even if the given subsystems are stable. In this article, we prove that the singular perturbation method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which is the first of this kind for systems involving the wave equation. Simulations are performed to show the applicability of our results.

A Note on the Tikhonov Theorem on an Infinite Interval

In this note we provide a new proof of the Tikhonov theorem for the infinite time interval and discuss some of its applications.

Justification of quasi-stationary approximation in models of gene expression of a self-regulating protein

We analyse a model of Hes1 gene transcription and protein synthesis with a negative feedback loop. The effect of multiple binding sites in the Hes1 promoter as well as the dimer formation process are taken into account. We consider three, possibly different, time scales connected with: (i) the process of binding to/dissolving from a binding site, (ii) formation and dissociation of dimers, (iii) production and degradation of Hes1 protein and its mRNA. Assuming that the first two processes are much faster than the third one, using the Tikhonov theorem, we reduce in two steps the full model to the classical Hes1 model. In the intermediate step two different models are derived depending on the relation between the time scales of processes (i) and (ii). The asymptotic behaviour of the solutions of systems are studied. We investigate the stability of the positive steady state and perform some numerical experiments showing differences in dynamics of the considered models.

Approximate Dynamic Inversion for Nonaffine Nonlinear Systems With High-Order Mismatched Disturbances and Actuator Saturation

In this paper, a novel disturbance-observer-based approximate dynamic inversion (ADI) approach is developed for pure-feedback nonaffine-in-control nonlinear systems (PFNNSs) in the presence of both high-order mismatched disturbances and actuator saturation. Finite-time disturbance observers (FTDOs) are utilized to estimate the disturbances and their derivatives. Then, we rebuild the system with the outputs of FTDOs. Thereafter, ADI is employed to derive the desired virtual and actual control of the nominal system, where no disturbances and saturation are presented. Furthermore, an augmented intermediate subsystem is constructed for the reduced slow subsystem to compensate for the difference between inputs with and without saturation by approximating its inversion. The stability of the closed-loop system is studied using Tikhonov's theorem. The proposed method is applied to a numerical example and a one-link robotic system with a brush DC motor. The simulation results demonstrate the validity of the presented approach.

Singular perturbation analysis of a two-site model for an epidemic in age-structured population

We formulate a model describing the dynamics for the spatial propagation of an SIS epidemic within a population, with age structure, living in an environment divided into two sites. An analysis of the model is given. We prove the existence of a unique disease free equilibrium (DFE) and its (local and global) stability. Further, we assume that fast infection processes and fast migration processes take place in the above-mentioned model; i.e. such processes last only a few days (less than a week). In opposition to such processes, demographic processes such as birth, death and maturation last quite a lot of years. Such a gap between the time scales gives rise to a multiple time scales model, in particular a singularly perturbed model. Through a singular perturbation analysis, based on Tikhonov theorem, we prove that for certain classes of initial conditions the nonlinear perturbed model can be approximated with very good accuracy by lower-dimensional linear models.

Tikhonov theorem for differential equations with singular impulses

The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. The most general Tikhonov theorem for the impulsive systems s obtained. Illustrative examples with simulations are given to support the theoretical results.

A singularly perturbed differential equation with piecewise constant argument of generalized type

The paper considers the extension of Tikhonov Theorem for singularly perturbed differential equation with piecewise constant argument of generalized type. An approximate solution of the problem has been obtained. A new phenomenon of humping has been observed in the boundary layer area. An illustrative example with simulations is provided.

Singular-Perturbation-Based Nonlinear Hybrid Control of Redundant Parallel Robot

High-speed robots are being applied more and more widely in the scenarios of aviation manufacturing, electronic packaging, and three-dimensional printing, etc. To achieve high speed, the robots are typically designed to be lightweight, hence suffer from undesired vibration, so that the dynamic performance would deteriorate. In light of this, this paper considers the dynamic modeling and active control of an emerging redundant parallel robot. First, the recursively dynamic model is modularly established by means of Lagrangian multipliers method. For completeness, the model of permanent magnet synchronous motor is further integrated with the model of mechanical system to formulate the electromechanically coupled dynamic model (ECDM). To facilitate dynamic control, the ECDM is cleverly projected into the null space of constraint Jacobian matrix eliminating the need to compute Lagrangian multipliers explicitly. Based on singular perturbation approach and Tikhonov theorem, the ECDM is decomposed into two subsystems in different timescales. On this basis, a nonlinear hybrid control scheme is proposed for both trajectory tracking and vibration suppression. Numerical simulations suggest that the nonlinear hybrid control exhibits overwhelming performance than the conventional joint-based proportional and differential feedback control.

Stable biased sampling

This paper analyzes an indirect evolutionary model of sampling biases in probability estimates, which combines the sampling best response dynamics with the replicator dynamics. The arrival rate of revision opportunities in the best response dynamics is high, so that the resulting joint dynamical system is a slow-fast system and we can use Tikhonov's theorem to study its solutions, employing practical asymptotic stability as a stability criterion. For two-strategy population games with a unique Nash equilibrium that is in mixed strategies, we find that the stable sampling bias is generically non-zero and that it is highest when the equilibrium is most asymmetric, yet that the stable sampling bias vanishes in the sample size.

A Simple Trapping and Manipulation Method of Biological Cell Using Robot-Assisted Optical Tweezers: Singular Perturbation Approach

Optical tweezers have been widely utilized in biological sciences and biomedical engineering because of the attractive feature of manipulating microobjects without physical contacts. The integration of optical tweezers and robotic technologies has also led to the emergence of many robot-assisted optical manipulation systems, and a variety of motion control schemes have been reported to automated the process and improve the efficiency. However, the performance of existing control methods for optical tweezers is commonly limited by several issues: 1) presence of spatially varying trapping stiffness, which is difficult to model and identify; 2) requirements of high-order state variables, which are not measurable; and 3) effectiveness of trapping is only valid locally around the center of laser beam. These issues lead to the constructions of controllers, which are computationally involved. This paper presents a simple tracking control method for optical manipulation, which enables the laser beam to automatically trap and then manipulate the cell to track a time-varying trajectory, without high-order derivatives or construction of observers. The development of the proposed controller is based on the singular perturbation approach, by treating the fast manipulator or stage dynamics as a perturbation of the slow cell dynamics, such that the lowest control complexity is achieved. The exponential stability of the overall system that consists of the fast and slow subsystems is proved by using Tikhonov's theorem. Experimental results are presented to illustrate the performance of the proposed controller.

Quasi-steady-state model of a class of nonlinear singularly perturbed system in a bond graph approach

A particular class of nonlinear singularly perturbed systems by using a bond graph approach is proposed. The class of nonlinear systems is defined by terms of product of state variables whose bond graph representation is described by MTF or MGY elements modulated by state variables. The singular pertubations applied to the system permit the separation of time scales. When the assumptions of Tikhonov Theorem are satisfied, then a new bond graph called singularly perturbed nonlinear bond graph (SPNBG) to determine the quasi-steady-state model of the system is presented. The SPNBG is characterized by having an integral causality assignment for the storage elements that represent the slow dynamics, and the storage elements of the fast dynamics have a derivative causality assignment. Finally, the proposed methodology is applied to an illustrative example.

Singular perturbation for an abstract non-densely defined Cauchy problem

This work is devoted to the study of a class of singularly perturbed non-densely defined abstract Cauchy problems. We extend the Tikhonov theorem for ordinary differential equations to the case of abstract Cauchy problems. Roughly speaking we prove that the solutions rapidly evolve and stay in some neighbourhood of the slow manifold. As a consequence we conclude that the solutions of the problem converge on each compact time interval, as the singular parameter goes to zero, towards the solutions of the so-called reduced problem. These results are applied to an example of age-structured model as well as to a class of functional differential equations.

Design and experimentation of an observer‐based linear adaptive control applied to an electropneumatic actuator

This study proposes an observer-based linear adaptive control scheme for the position control of a pneumatic actuator system. The proposed controller does not rely on a physical model of the pneumatic actuator system to be controlled; it only needs the position measurement for its implementation, which is really a novelty in the field of such applications. First, based on the implicit function theory, the existence of an ideal position controller is demonstrated. Then, the objective is to construct this unknown ideal position controller using a linear control with a stable adaptation law and an extended observer. This latter is used to estimate both the tracking error vector and an unknown function of the error between the implicit ideal control and the actual control which is used in the adaptation mechanism. The stability of the overall closed-loop system is studied by using singular perturbation theory and Tikhonov's theorem. Finally, real-time experimental results are provided to show the performances of the proposed control scheme.

Приближенная линеаризация обратной связью на основе сингулярно возмущенного подхода

One of the most common methods of synthesis of the nonlinear control systems is the method of a feedback linearization (FL). The idea of this method consists in conversion of the original nonlinear system into a linear one by means of a state feedback and coordinate transformation. Then, the methods of control theory for the linear systems are used for the system design. If the original nonlinear system cannot be linearized exactly by the state feedback, the method of the approximate feedback linearization (AFL) is used. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system (not of the entire system). In this paper, the author proposes a method of an approximate feedback linearization control of the nonlinear singularly perturbed (SP) systems. The proposed method is based on a decomposition of the original SP system and construction of AFL control in the form of composite FL controls for the slow and fast subsystems. In general, a nonlinear SP system cannot be easily separated into slow and fast subsystems, because the conditions of Tikhonov theorem are not complied. In order to overcome this, the author proposes to perform the feedback linearization method at first for the system's part, which describes the fast state variables. Thus, a fast control is chosen, so that the conditions of Tikhonov theorem would be met. Then, using a standard singular perturbation technique, we obtain a slow subsystem. Further the problem of FL control for a slow subsystem is solved. The resulting AFL control is obtained in the form of a composite control. Application of the proposed approach is illustrated with two examples.

Tikhonov theorem for linear hyperbolic systems

A class of linear hyperbolic systems of conservation laws with multiple time scales which are modeled by a perturbation parameter is considered in this paper. By setting the perturbation parameter to zero, two subsystems, the reduced subsystem standing for the slow dynamics and the boundary-layer subsystem representing the fast dynamics, are computed. It is first proved that the exponential stability of the full system implies the stability of both subsystems. Secondly, a counterexample is given to indicate that the stability of the two subsystems does not ensure the full system’s stability. Moreover a new Tikhonov theorem for this class of infinite dimensional systems is stated. The solution of the full system can be approximated by that of the reduced subsystem, and this is proved by Lyapunov techniques. An application to boundary feedback stabilization of gas transport model is used to illustrate the results.

A singularly perturbed HIV model with treatment and antigenic variation.

We study the long term dynamics and the multiscale aspects of a within-host HIV model that takes into account both mutation and treatment with enzyme inhibitors. This model generalizes a number of other models that have been extensively used to describe the HIV dynamics. Since the free virus dynamics occur on a much faster time-scale than cell dynamics, the model has two intrinsic time scales and should be viewed as a singularly perturbed system. Using Tikhonov's theorem we prove that the model can be approximated by a lower dimensional nonlinear model. Furthermore, we show that this reduced system is globally asymptotically stable by using Lyapunov's stability theory.

Dynamic Modeling and Control of Parallel Robots With Elastic Cables: Singular Perturbation Approach

In this paper, control of fully–constrained parallel cable robots with elastic cables is studied in detail. In the modeling process, longitudinal vibration of cables is considered as their dominant dynamics, and the governing equations of motion are rewritten to the standard form of singular perturbation. The proposed composite controller consists of two main components. A rigid controller is designed based on the slow or rigid model of the system and a corrective term is added to guarantee asymptotic stability of the fast dynamics. Then, by using Tikhonov theorem, slow and fast variables are separated and incorporated into the stability analysis of the overall closed–loop system, and a set of sufficient conditions for the stability of the total system is derived. Finally, the effectiveness of the proposed control law is verified through simulations.

A singularly perturbed age structured SIRS model with fast recovery

Age structure of a population often plays a significant role in the spreading of a disease among its members. For instance, childhood diseases mostly affect the juvenile part of the population, while sexually transmitted diseases spread mostly among the adults. Thus, it is important to build epidemiological models which incorporate the demography of the affected populations. Doing this we must be careful as many diseases act on a time scale different from that of the vital processes. For many diseases, e.g. measles, influenza, the typical time unit is one day or one week, whereas the proper time unit for the vital processes is the average lifespan in the population; that is, 10-100 years. In such a case, the epidemiological model with vital dynamics becomes a multiple time scale model and thus it often can be significantly simplified by various asymptotic methods. The presented paper is concerned with an SIRS type disease spreading in a population with a continuous age structure modelled by the McKendrick-von Foerster equation. We consider a disease with a quick recovery rate in a large population. Though it is not too surprising that in such a model the introduced disease quickly vanishes, the result is mathematically interesting as the error estimates are uniform on the whole infinite time interval, in contrast to the typical results based on the Tikhonov theorem and classical asymptotic expansions.