Singularly Perturbed System Research Articles

Robust Fault Estimation and Fault‐Tolerant Control for a Class of Fuzzy Singularly Perturbed Systems With State Time Delay Based on Learning Observer

ABSTRACTThis article studies the problem of fault estimation (FE) and dynamic output fault‐tolerant control (DOFTC) for a class of Takagi–Sugeno (T–S) fuzzy singularly perturbed systems (SPSs) which subject to time delay, external disturbance and actuator fault. A new fault estimation and fault‐tolerant control scheme was proposed and designed for the influence of the change of perturbation parameter on the singularly perturbed system. This scheme adopts the Takagi–Sugeno (T–S) fuzzy model, learning observer and dynamic output feedback fault‐tolerant mechanism, so that the system has multi‐time‐scale dynamic stability. The results show that this method has more accurate estimation effect, faster convergence speed, and very fast steady‐state response of fault‐tolerant control when faults occur. Furthermore, when constructing the Lyapunov function, the improved matrix was selected to ensure that the closed‐loop system is stable for all , and at the same time, the conservativeness of the results was reduced. Finally, the feasibility and correctness of the proposed design method were illustrated through two numerical examples.

Robust PI multiobserver for discrete nonlinear singularly perturbed system with unmeasurable premise variables

This article deals with the observability bound problem of a discrete-time nonlinear singularly perturbed systems (SPSs) subject to disturbances and noises. This nonlinear SPS is represented by a coupled state multimodel (MM) with unmeasurable premise variables. A [Formula: see text] proportional-integral multiobserver ([Formula: see text] PI multiobserver), known by its robustness, is designed to accomplish this task. The proposed method is based on the [Formula: see text] technique to minimize the effect of the disturbances, the noises, and all the unmeasurable premise variables. To design this observer, sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are developed as a first step to ensure the robust stability bound of the considered system represented by a coupled MM for a bound of the singular perturbation parameter based on a quadratic Lyapunov function and the [Formula: see text] gain. Then, sufficient conditions are developed to ensure the robust stability bound of the state estimation error between the MM and the PI multiobserver for a bound of the singular perturbation parameter that is less than or equal to the considered system’s robust stability bound. Hence, the observability bound of the considered system represented by a coupled state MM is determined. Two simulation examples are then given to validate the proposed strategy.

Model‐based event‐triggered control of singularly perturbed system with dual event‐triggering mechanism

AbstractIn this article, a new dual event‐triggering mechanism is proposed to implement a distributed model‐based event‐triggered control for a singularly perturbed system (SPS) having structured state space uncertainty. The proposed approach significantly reduces the amount of necessary information exchanged to control the system. Dual event‐triggering mechanism allows the measurements of slow and fast states to be transmitted asynchronously at different rates which is more effective due to the two time‐scale nature of SPSs. Approximate slow and fast systems of the nominal plant are used as the model to reduce the computational demand. The proposed control strategy guarantees that the system states converge to an adjustable region around the equilibrium point and the inter‐event intervals are lower‐bounded by a positive constant. A numerical simulation example is used to illustrate the theoretical results of the article.

Reduced-dimensional reinforcement learning control using singular perturbation approximations

We present a set of model-free, reduced-dimensional reinforcement learning (RL) based optimal control designs for linear time-invariant singularly perturbed (SP) systems. We first present a state feedback and an output feedback based RL control design for a generic SP system with unknown state and input matrices. We take advantage of the underlying time-scale separation property of the plant to learn a linear quadratic regulator (LQR) for only its slow dynamics, thereby saving significant amount of learning time compared to the conventional full-dimensional RL controller. We analyze the sub-optimality of the designs using SP approximation theorems, and provide sufficient conditions for closed-loop stability. Thereafter, we extend both designs to clustered multi-agent consensus networks, where the SP property reflects through clustering. We develop both centralized and cluster-wise block-decentralized RL controllers for such networks, in reduced dimensions. We demonstrate the details of the implementation of these controllers using simulations of relevant numerical examples, and compare them with conventional RL designs to show the computational benefits of our approach.

Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

Composite control of nonlinear singularly perturbed systems via approximate feedback linearization

This article is devoted to the problem of composite control design for continuous nonlinear singularly perturbed (SP) system using approximate feedback linearization (AFL) method. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system. According to AFL approach, we suggest to solve feedback linearization problems for continuous nonlinear SP system by reducing it to two feedback linearization problems for slow and fast subsystems separately. The resulting AFL control is constructed in the form of asymptotic composition (composite control). Standard procedure for the composite control design consists of the following steps: 1) system decomposition, 2) solution of control problem for fast subsystem, 3) solution of control problem for slow subsystem, 4) construction of the resulting control in the form of the composition of slow and fast controls. The main difficulty during system decomposition is associated with dynamics separation condition for nonlinear SP system. To overcome this, we propose to change the sequence of the design procedure: 1) solving the control problem for fast state variables part, 2) system decomposition, 3) solving the control problem for slow state variables part, 4) construction of the resulting composite control. By this way, fast feedback linearizing control is chosen so that the dynamics separation condition would be met and the fast subsystem would be stabilizable. The application of the proposed approach is illustrated through several examples.

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.

On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

The paper aims at the development of tools for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems with long run average optimality criteria. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional (ID) linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with a numerical example.

Averaging and Linear Programming in Some Singularly Perturbed Problems of Optimal Control

The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite time horizon. We mostly focus on problems with time discounting criteria but a possibility of the extension of results to periodic optimization problems is discussed as well. Our consideration is based on earlier results on averaging of SP control systems and on linear programming formulations of optimal control problems. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with two numerical examples.

Physical Continuous Model of Heating, Evaporation and Ignition of Polydisperse Fuel Droplets

A new modeling of heating and evaporation of fuel droplets and ignition of a fuel vapour/ air mixture in continuousform is suggested. The size distribution of fuel droplets is assumed to be continuous and found from thesolution of the kinetic equation for the probability density function (PDF). The semi-transparency of droplets, thedifference between the gas temperature and the external temperature are take into account. The model represent indimensionless from, and the dynamics of the system is present in term of the dynamics of a multi-scale, singularlyperturbed system (SPS)

Uniform Difference Scheme on the Singularly Perturbed System

This paper is concerned with the numerical solution for singular perturbation system of two coupled second ordinary differential equations with initial and boundary conditions, respectively. Fitted finite difference scheme on a uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed.

Additive Fault Tolerant Control Applied to Delayed Singularly Perturbed System

The additive fault tolerant control (FTC) for delayed system is studied in this work. To design the additive control, two steps are necessary; the first one is the estimation of the sensor fault amplitude using a Luenberger observer with delay, and the second one consists to generate the additive fault tolerant control law and to add it to the nominal control of delayed system. The additive control law must be in function of fault term, then, in the absence of fault the expression of additive control equal to zero. The generation of nominal control law consist to determinate the state feedback gain by using the Lambert W method. Around all these control tools, we propose an extension of the additive FTC to delayed singularly perturbed systems (SPS). So, this extension consists to decompose the delayed SPS in two parts: delayed slow subsystem (delayed SS) and fast subsystem (FS) without time delay. Next, we consider that the delayed SPS is affected at its steady-state, and we apply the principal of FTC to the delayed SS and finally we combine them with the feedback gain control of FS by using the principal of composite control.

On transformation to the singularly perturbed system

A rapid progress in hard- and software development of computational facilities as well as in numerical methods has increased the role of numerical simulations in the quantitative system analysis of many engineering problems. At the same time, the system complexity (in terms of dimensionality and non-linearity) has grown considerably increasing demand for automatic methods of analysis of qualitative system behavior. For instance, nowadays, definition of key system parameters controlling the system dynamics and finding critical regimes automatically have become crucial issue of numerical system analysis. In the present paper a transformation to the Singularly Perturbed System (SPS) as a main theoretical framework to cope with the complexity and high dimensionality will be discussed in detail. Both simple but famous and meaningful model example of Van der Pol oscillator and an example of application to numerical analysis of chemical kinetics mechanisms will be used to show the potential of the suggested framework.

Effective solution of a singularly perturbed system of nonlinear differential equations

The authors consider a boundary-value problem for a singularly-perturbed system of three nonlinear ordinary differential equations arising in modeling the interaction of physical fields in a semiconducting diode. They suggest a higher-accuracy method for solution of this problem based on the combination of the Newton operator method and the continuation method with respect to a parameter. In this case, the Frechet derivative, the Green function of the corresponding differential equation, and the initial approximation are found in explicit analytical form. The numerical implementation confirms the higher effectivity and the superexponential rate of convergence of the proposed method.

The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system

This paper is concerned with the numerical solution for singular perturbation system of two coupled ordinary differential equations with first and second orders and with initial and boundary conditions, respectively. Finite difference scheme on a special non-uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed. Numerical results supporting the theory are presented.

Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H∞ state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H∞ problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H∞ problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H∞ problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H∞ problem is decomposed into two lower-dimensional parameter-independent H∞ subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H∞ problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.

Qualitative Analysis of a Singularly-Perturbed System of Differential Equations Related to the Van der Pol Equations

Qualitative Analysis of a Singularly-Perturbed System of Differential Equations Related to the Van der Pol Equations

Stability margins of singular perturbation systems via state-feedback control

The gain and phase margins of singular perturbation systems are analysed under unmodelled high-frequency dynamics control, composite control, and the original full-order linear quadratic (LQ) control. The analysis is on the basis that there is a good relation between the minimum singular value of return difference transfer matrix and the stability margins. We begin with the examination of stability margins of subsystems and then show that state-feedback control design of subsystems could preserve gain and phase margins for the original full-order singularly perturbed system if the singular perturbation parameter epsiv; is sufficiently small. The effectiveness of e on stability margins is formulated and determined. It is found that the effectiveness can be evaluated by a simple method. Two examples are exploited to illustrate the analytic results.

Existence, uniqueness and asymptotic expansion of the solution of a multipoint boundary-value problem with parameters for a singularly-perturbed system of integro-differential equations

SynopsisAn asymptotic expansion in powers of the small parameter of the solution of a singularly-perturbed system of integro-differential equations with a non-linear boundary condition set at several points of the interval considered is constructed and verified.