Singularly Perturbed Systems Research Articles

Ultimate Boundedness Control for Networked Singularly Perturbed Systems With Deception Attacks: A Markovian Communication Protocol Approach

In this study, the ultimate boundedness controlfor a type of networked singularly perturbed systems (SPSs) with communication constraints and deception attacks is explored. To improve the observer performance, the measurement outputs are quantized with the aid of a logarithmic quantizer. Meanwhile, the Markovian communication protocol (MCP) is forwarded to schedule the transmission sequence of the quantized signals. Unlike the conventional MCP, a novel nonhomogeneous MCP is proposed to further alleviate the communication bandwidth usage, whose transition probabilities are time-varying. By virtue of the hidden Markov model, the mismatches between the transmission mode over nonhomogeneous MCP and its detected ones are revealed, and an asynchronous observer-based controller is formed. Then, by establishing a novel singular-perturbation-parameter-based polytope Lyapunov-Krasovskii functional, the ultimate boundedness of the augmented networked SPS with randomly occurring deception attacks is elicited, and the local minimization of the controlled output is guaranteed. Finally, the effectiveness and applicability of the propounded control strategy are validated by two simulation examples.

Renormalization Group Method for Singular Perturbation System with Fractional Gaussian Noise

Renormalization Group Method for Singular Perturbation System with Fractional Gaussian Noise

Distributed generalized Nash equilibrium seeking: A singular perturbation-based approach

In this paper, a distributed optimization algorithm is proposed for aggregative game with coupled constraints. Based on the singular perturbation system, the generalized Nash equilibrium is sought by a group of agents. By employing the average consensus method in the fast manifold, the aggregates in the object function can be estimated via simple information exchanges, as well as the aggregate of dual variables, which provides necessary information for the fully distributed algorithm design. Moreover, the exponential convergence of the proposed algorithm is explored based on the Lyapunov method, the properties of the variational inequality and the characteristic of the singular perturbation system. Application to the resource competition problem in smart grid verifies the effectiveness of the proposed algorithm.

Network-based quantized [formula omitted] control for T–S fuzzy singularly perturbed systems with persistent dwell-time switching mechanism and packet dropouts

This paper focuses on the H∞ control problem for a class of discrete-time persistent dwell-time (PDT) switched nonlinear singularly perturbed systems (SPSs) under T–S fuzzy model. In order to make the system more consistent with the actual situation, the signal quantization associated with a logarithmic quantizer and the packet dropouts governed by a Bernoulli variable are considered simultaneously. And, the persistent dwell-time switching mechanism is employed to describe the fast and slow switching among several subsystems in the investigated systems alternatively. Based on the Lyapunov function approach and Takagi–Sugeno (T–S) fuzzy technique, some sufficient conditions are obtained, which can ensure the exponentially mean-square stable (EMSS) and H∞ performance of the closed-loop system. In addition, a unified fuzzy controller design methodology is established for the nonlinear SPSs, so that the fuzzy controller can be designed depending on whether the fast states of the systems are available or not. Finally, a numerical example and a series DC motor model are provided to demonstrate the superiority and practicability of the obtained theoretical results.

Suboptimal reduced control of unknown nonlinear singularly perturbed systems via reinforcement learning

AbstractIn this paper, a suboptimal reduced control method is proposed for a class of nonlinear singularly perturbed systems (SPSs) with unknown dynamics. By using singular perturbation theory, the original system is reduced to a reduced system, by which a policy iterative method is proposed to solve the corresponding reduced Hamilton–Jacobi–Bellman (HJB) equation with convergence guaranteed. A reinforcement learning (RL) algorithm is proposed to implement the policy iterative method without using any knowledge of the system dynamics. In the RL algorithm, the unmeasurable state of the virtual reduced system is reconstructed by the slow state measurements of the original system, the controller and cost function are approximated by actor‐critic neural networks (NNs) and the method of weighted residuals is utilized to update the NN weights. The influence introduced by state reconstruction error and NN function approximation on the convergence, suboptimality of the reduced controller and stability of the closed‐loop SPSs are rigorously analyzed. Finally, the effectiveness of our proposed method is illustrated by examples.

Dual-Disturbance Observers-Based Control for a Class of Singularly Perturbed Systems

Singularly perturbed systems (SPSs) wildly exist in practical engineering practice; however, the disturbance rejection in SPSs is still a problem. As well-known disturbance rejection methods, composite hierarchical anti-disturbance control (CHADC) can effectively compensate and suppress disturbance, which improves the control performances of complex systems. To enhance the capability of anti-disturbance and control precision, this article proposes a novel composite hierarchical anti-disturbance (CHAD) dynamic surface control (DSC) for the fast subsystem considering highly dynamic disturbance, and CHAD proportion-integration (PI) control for the slow subsystem with slow-varying harmonic disturbance. The disturbances are supposed to be generated by an exogenous system based on partially known information with model uncertainty, and the stabilities of both fast and slow closed-loop subsystems are analyzed. Meanwhile, the compensability of disturbances is discussed. Finally, simulations for an F-8 aircraft are given to demonstrate the effectiveness of the proposed methods.

A Novel Equivalent Input Disturbance-Based Adaptive Sliding Mode Control for Singularly Perturbed Systems

This paper develops a novel equivalent input disturbance (EID)-based adaptive sliding mode control (SMC) method for singularly perturbed systems (SPSs). Firstly, the block diagonalization approach is introduced to decompose the full-order SPSs exactly, and slow and fast subsystems are obtained by solving the upper and lower triangular matrices individually. Secondly, an EID is constructed to estimate the unknown disturbances with the observer gain and error system convergence analyzed. Then, depending on the decoupled reduced-order system models, a Lyapunov equation-based solution is adopted to construct a composite sliding surface. Finally, combined with the EID estimation, an adaptive SMC law is proposed to compensate the adverse effect of disturbances and the reachability condition is proven. The presented control strategy is free of any priori disturbances information while the satisfactory system performance can be guaranteed. Simulation results on two examples illustrate its superiority over the existing methods.

Стійкість двоконтурних систем керування напругою DC-DC перетворювача

The paper is devoted to research and development of the cascaded DC-link voltage control systems for DC-DC boost converters whose mathematical model is highly nonlinear and non-minimum phase. The relevance of the analysis method for classic control systems for DC-DC boost converters is substantiated, which are similar to vector-controlled electric drives systems if the speed controller is replaced by DC-link voltage controller. It is shown that the reduced-order solution of initial nonlinear system dynamics can be obtained giving the time-scale separation between the input current and the DC-link voltage control processes based on the singular perturbation systems theory. In its turn, it is concluded that the reduced-order system dynamics is locally (asymptotically) stable if cascaded control algorithm is applied. The similarity of the time-scale separation conditions for the cascaded control systems with proportional and proportional-integral DC-link voltage controller is shown. The time-scale separation of the control processes is achieved if inner current regulation loop is much faster than the outer voltage regulation loop. The simulation study demonstrates that if the time-scale separation conditions between the input current and the DC-link voltage regulation are met then the full-order system dynamics of the DC-link voltage regulation of DC-DC boost converters is close to the reduced-order system dynamics. The control algorithm has a typical structure of the modern controlled converters applied in hybrid energy storage systems for electric vehicles. From the results of the simulation study, it follows that the influence of the inner resistance of the input inductance on the system stability and quality indicators of the DC-link voltage control process is insignificant for the investigated DC-DC boost converter.

Event-triggered H ∞ filtering for singularly perturbed systems with external disturbance

The problem of event-triggered filter design for singularly perturbed systems (SPSs) with external disturbance is investigated in this paper. A scheme consisting of an event-triggered mechanism (ETM) and an filter is considered. Firstly, -dependent sufficient conditions are derived via constructing an ϵ-dependent Lyapunov–Krasovskii functional, where the conditions ensure that the SPSs are asymptotically stable avoiding Zeno phenomenon. Then, the ETM and the filter are designed simultaneously on the basis of the aforementioned criteria. Finally, an RC circuit system is presented to demonstrate the effectiveness and superiority of the obtained method.

A distributed normalized Nash equilibrium seeking algorithm for power allocation among micro-grids

In this paper, a power allocation problem based on the Cournot game and generalized Nash game is proposed. After integrating dynamic average consensus algorithm and distributed projection neural network through singular perturbation systems, a normalized Nash equilibrium seeking algorithm is presented to solve the proposed power allocation problem in a distributed way. Combine Lyapunov stability with the singular perturbation analysis, the convergence of the proposed algorithm is analyzed. A simulation on IEEE 118-bus confirms that the proposed distributed algorithm can adjust the power allocation according to different situations, while keeping the optimal solution within the feasible set.

Canard-induced mixed-mode oscillations and bifurcation analysis in a reduced 3D pyramidal cell model

This study entails investigation of mixed-mode oscillations (MMOs) with a conductance-based pyramidal cell (PC) model located in the entorhinal cortex layer V. This six-dimensional neuron model was reduced to three dimensions by analysis of the voltage-dependent timescales to illustrate a regime in which the MMOs are generated. Additionally, the mechanism of generation of the MMOs under antiepileptic drug conditions was illustrated in the 3D PC model. Combined with geometric singular perturbation theory (GSPT), this work shows that there is a range of parameters under which the reduced model explains the emergence of MMOs caused by an underlying canard phenomenon. In particular, we theoretically calculate the number of subthreshold oscillations using the relationship with the eigenvalue ratio of the singular perturbation system at the folded singular node, which is consistent with numerical simulations. Furthermore, a slow–fast dynamics analysis of the 3D PC model is performed, where two slow/one fast and one slow/two fast systems with the layer problem and the reduced problem are considered to explain the trajectory on the critical manifold. General one- and two-parameter bifurcation types are also discussed in this work. The first Lyapunov coefficient of the Hopf bifurcation can decide whether the bifurcation is supercritical or subcritical. Bogdanov–Takens (BT) bifurcation was also analyzed in this study and associated with three bifurcation curves near the BT point. Finally, studies on the GSPT and bifurcation analysis are of great importance for further understanding the complex dynamic behaviors and crucial roles of the signal transmission and information processing pathways of the biological nervous system.

Fault‐tolerant control of singularly perturbed systems with actuator faults and disturbances

SummaryThe article proposes several fault‐tolerant control (FTC) laws for singularly perturbed systems (SPS) with actuator faults and disturbances. One of the main challenges in this context is that the fast‐slow decomposition is not available for actuator faults and disturbances. In view of this, some conditions for the asymptotic stability of the closed‐loop dynamics are investigated by amending the composite Lyapunov approach. On top of this, a closed‐form expression of the upper bound of singular perturbation parameter (SPP) is provided. Moreover, we design several SPP‐independent composite FTC laws, which can be applied when this parameter is unknown. Finally, the chattering phenomenon is eliminated by using the continuous approximation technique. We also emphasize that, for linear SPSs, the FTC design can be formulated as a set of linear matrix inequalities, while the SPP upper bound can be obtained by solving a convex optimization problem. Two numerical examples are given to illustrate the effectiveness of the proposed methodology.

Feedback Linearization of Nonlinear Singularly Perturbed Systems with State-dependent Coefficients

The article examines a feedback linearization (FL) problem for a nonlinear singular perturbed (SP) system in a state-dependent coefficients form (SDC-form). The combination of the composite control technique of singular perturbation theory and a canonical similarity transformation approach for systems in a SDC-form are explored in this research. As a result, two FL problems for a fast state variables subsystem and a slow state variables subsystem are solved separately. The transformation matrix and feedback linearizing control for the entire nonlinear SP system are designed as a composition of solutions of these two FL problems. The composite stabilizing controller, based on feedback linearization and a pole placement method, is proposed for a nonlinear SP system in a SDC-form. Practical implementation of the proposed feedback linearizing composite stabilizing control is shown through an example (an inverted pendulum, controlled by a direct current motor).

Adaptive composite suboptimal control for linear singularly perturbed systems with unknown slow dynamics

SummaryIn this article, using singular perturbation theory and adaptive dynamic programming (ADP) approach, an adaptive composite suboptimal control method is proposed for linear singularly perturbed systems (SPSs) with unknown slow dynamics. First, the system is decomposed into fast‐ and slow‐subsystems and the original optimal control problem is reduced to two subproblems in different time‐scales. Afterward, the fast subproblem is solved based on the known model of the fast‐subsystem and a fast optimal control law is designed by solving the algebraic Riccati equation corresponding to the fast‐subsystem. Then, the slow subproblem is reformulated by introducing a system transformation for the slow‐subsystem. An online learning algorithm is proposed to design a slow optimal control law by using the information of the original system state in the framework of ADP. As a result, the obtained fast and slow optimal control laws constitute the adaptive composite suboptimal control law for the original SPSs. Furthermore, convergence of the learning algorithm, suboptimality of the adaptive composite suboptimal control law and stability of the whole closed‐loop system are analyzed by singular perturbation theory. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed methods.

Extended Dissipativity-Based Control for Hidden Markov Jump Singularly Perturbed Systems Subject to General Probabilities

This article deals with the extended dissipativity-based control issue for singularly perturbed systems (SPSs) with Markov jump parameters, in which the partial information issues of the Markov chain are fully considered. A comprehensive hidden Markov model (HMM) is established for the partial information issues on Markov chain, in which the transition probabilities of the hidden Markov state and the observation probabilities of the observed state are general, that is, the uncertainty and the unknown peculiarity of them may be encountered simultaneously. By using the HMM with general probabilities, a comprehensive criterion is derived to analyze the extended stochastic dissipativity of the hidden Markov jump SPSs with the different partial information issues on the Markov chain. Based on the derived criterion, an explicit expression to acquire the desired HMM-based controller is presented. An illustrative example and a vehicle active suspension system are, finally, show the validity of the established theoretical results.

Reliable Sliding Mode Control of Fast Sampling Singularly Perturbed Systems: A Redundant Channel Transmission Protocol Approach

This paper endeavors to investigate the output-feedback sliding mode control (SMC) issue of the networked singularly perturbed systems (SPSs) under fast sampling. In order to improve the reliability of network communication, a redundant channel transmission protocol is introduced in the SMC design. Based on the measurement outputs, a sliding function is constructed with the consideration of the transmission protocol. With the aid of some appropriate Lyapunov functions, the sufficient conditions are derived to ensure the mean-square exponentially ultimately boundedness of the sliding mode dynamics and the reachability of the specified sliding surface. Moreover, a convex optimization algorithm is formulated to solve the output-feedback SMC law by searching the available upper bound of the singularly perturbed parameter. Finally, an operational amplifier circuit is exploited to explore the influences from the redundant channel transmission protocol to the output-feedback SMC performance and the estimated $\varepsilon$ -bound.

Passivity analysis and disturbance observer‐based adaptive integral sliding mode control for uncertain singularly perturbed systems with input non‐linearity

This study addresses the problems of passivity analysis and disturbance observer-based adaptive integral sliding mode control (ISMC) for uncertain singularly perturbed systems (SPSs) with input non-linearity. Firstly, the passivity condition enabling system exponentially stable is presented in terms of linear matrix inequalities (LMIs). Secondly, a singular perturbation parameter e-dependent disturbance observer is constructed, and the estimate is incorporated in the design of the adaptive ISMC with the reachability condition guaranteed. Then, a set of LMIs are obtained to determine the controller gain of the ISMC, and an e-bound estimation approach is derived, such that the closed-loop SPSs are passive and exponentially stable. Compared with the existing results, the proposed approaches have less conservatism because of adopting a more general storage function, which leads to a superior transient performance and a bigger e-bound. Finally, the advantages and effectiveness of the proposed methods are demonstrated by three examples.

Active suspension control by output feedback through extended high-gain observers

The paper addresses extended high-gain observer (EHGO) design for a vehicle suspension system. The different way from previous reports is that a regional input-to-state stabilization (including exponential stabilization) controller is presented. To pursue this aim, first the original suspension dynamical equation is transformed into a normal form. And then, a feedback linearization controller is designed as the state feedback control, which makes the closed-loop system regionally input-to-state stable. Based on this, the output feedback control by using EHGO is proposed. The stability of the singular perturbation system composed by the state closed-loop system and the scaled error equation is demonstrated. Simulation examples are illustrated, which validate that provided small high-gain parameter the output feedback control will recover the performance of RISS under state feedback control and the responses of suspension performance behave better than those under the later.

Dynamic Event-Triggered Sliding Mode Control: Dealing With Slow Sampling Singularly Perturbed Systems

This brief endeavors to investigate the sliding mode control (SMC) issue of the networked singularly perturbed systems (SPSs) under slow sampling. For the energy saving purpose in network communication, a dynamic event-triggered mechanism is introduced to SMC design. By considering the structure characteristics of the controlled system, a novel sliding function is constructed with taking the singular perturbed matrix $E_\varepsilon $ into account properly. With the aid of some appropriate Lyapunov functionals, the sufficient conditions are derived to ensure the asymptotic stability of the sliding mode dynamics and the reachability of the specified sliding surface. Besides, it is shown that the quasi-sliding motion is dependent on the dynamic event-triggered parameters and its bound converges to a constant. The solving algorithm for the dynamic event-triggered SMC law is formulated in a convex optimization framework. Finally, an numerical example is provided to illustrate the effectiveness of the proposed results.

Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension

We introduce and analyse the so-calledReference Solution IMplicit-EXplicit schemeas a flux-splitting method for singularly-perturbed systems of balance laws. RS-IMEX scheme’s bottom-line is to use the Taylor expansion of the flux function and the source term around a reference solution (typically the asymptotic limit or an equilibrium solution) to decompose the flux and the source into stiff and non-stiff parts so that the resulting IMEX scheme is Asymptotic Preserving (AP) w.r.t. the singular parameter tending to zero. We prove the asymptotic consistency, asymptotic stability, solvability and well-balancing of the scheme for the case of the one-dimensional shallow water equations when the singular parameter is the Froude number. We will also study several test cases to illustrate the quality of the computed solutions and to confirm the analysis.