Singularly Perturbed Systems 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.

Dissipative control for nonlinear singularly perturbed systems with dynamic quantization and actuator failure

The paper discusses the dissipative control problem of nonlinear singularly perturbed systems (SPSs) with dynamic quantization and actuator failure. The nonlinear singularly perturbed plant is described by Takagi–Sugeno (T–S) fuzzy model. The dynamic quantizer is considered to realize the effective utilization of the limited network channel bandwidth resources, and the possible transmission failure between the controller and the actuator is also considered. Based on linear matrix inequalities, sufficient design conditions for the ϵ-independent state feedback controller are given such that the constructed quantized closed-loop system is asymptotically stable and satisfies a predefined dissipative performance. Furthermore, a short search algorithm is given to search the maximum stability bound ϵ̄. Finally, two examples are provided in order to prove the feasibility and effectiveness of the proposed method.

Distributed Formation Control for Multiple Quadrotor UAVs Based on Distributed Estimator and Singular Perturbation System

Distributed Formation Control for Multiple Quadrotor UAVs Based on Distributed Estimator and Singular Perturbation System

ADP-Based Optimal Control of Linear Singularly Perturbed Systems With Uncertain Dynamics: A Two-Stage Value Iteration Method

We study the problem of adaptive dynamic programming (ADP) based on optimal control of linear singularly perturbed systems (SPSs) subject to completely unknown dynamics. Previous works on ADP-based optimal control of SPSs require that the fast dynamics of the system are either known a priori or unknown but asymptotically stable, and such results are limited to standard cases, i.e., the internal dynamics of their fast subsystems are nonsingular. In this brief, these conditions can be removed through a sequential procedure to design feedback gains of both fast and slow states in the framework of ADP. In this procedure, two optimal control problems are formulated for the prefeedback fast subsystem and the modified slow subsystem. Then, a data-driven two-stage value iteration algorithm is imposed to learn the optimal controller without using any system dynamics. Fundamentally different from existing works, an initial admissible control policy is no longer needed during learning. Finally, the effectiveness of the learning algorithm is proved by a simulation example.

Output feedback H∞ control for discrete time singularly perturbed systems with Markov lossy network: The round-robin-like protocol case

In this paper, the output feedback H∞ control is investigated for the networked singularly perturbed systems (SPSs) with Markov lossy network, in which the packet loss probability varies in different modes. A new network mode-dependent round-robin-like protocol (RRLP) is introduced to the networked SPSs, which can dynamically adjust the number of the selected sensor nodes according to the network environment condition. Based on this RRLP and Markov lossy network, a Markov mode and scheduling order co-dependent output feedback controller is constructed to ensure the system control performance. By constructing a novel Lyapunov-Krasovskii functional dependent on the singular perturbation parameter, Markov mode, and scheduling order simultaneously, sufficient conditions of mean square stability for SPSs are obtained, and numerical stiffness is avoided. Finally, a direct current motor-controlled inverted pendulum example is proposed to verify the effectiveness of our main results.

Nonfragile observer-based event-triggered fuzzy tracking control for fast-sampling singularly perturbed systems with dual-layer switching mechanism and cyber-attacks

This paper investigates the problem of nonfragile observer-based tracking control for a class of fuzzy fast-sampling singularly perturbed systems (SPSs) with sensor saturation, which subject to event-triggered scheme and random cyber-attacks. The switching topology of the fast-sampling SPSs is dominated by a dual-layer switching mechanism, in this case, the Markov process in the underlying system is nonhomogeneous and its time-varying transition probabilities (TPs) are governed by a persistent dwell-time switching signal. First, in order to avoid asynchronous phenomenon of premise variable between the fuzzy fast-sampling SPSs and the fuzzy controller under network environments, the non-parallel distributed compensation (non-PDC) technique is applied to improve the design flexibility and reduce the conservativeness of the results. Furthermore, a novel fuzzy tracking control design scheme is proposed, in which integrating the fuzzy nonfragile observer and reference model signal into the synthetic design of the tracking controller can be expected to reduce the tracking error. Second, by constructing a well-designed time-delay and mode-dependent Lyapunov function, some sufficient conditions are derived such that the closed-loop system is mean-square exponential stable and with extended dissipative tracking performance. Finally, the effectiveness and applicability of the proposed control protocol are illustrated by two simulation examples.

Neural-Network-Based Adaptive Control of Uncertain MIMO Singularly Perturbed Systems With Full-State Constraints.

This article investigates the tracking control problem for a class of nonlinear multi-input-multi-output (MIMO) uncertain singularly perturbed systems (SPSs) with full-state constraints. The underlying issues become more challenging because two-time-scale characteristics and full state constraints are involved. To this end, first, the adaptive neural network (NN) control method is designed to handle system uncertainties in the design process. Second, the nonlinear state-dependent coordinate transformation functions are employed to avoid the violation of full-state constraints and feasibility conditions for intermediate controllers. Furthermore, by introducing an appropriate ϵ-dependent Lyapunov function, the potential ill-conditioned numerical problems in the design process of SPSs are avoided, and the stability of the nonlinear SPSs is proven. Finally, two examples are presented to illustrate the validity of the proposed adaptive NN control scheme.

Learning-based near-optimal tracking control for industrial processes with slow and fast modes

This paper devotes to solving the optimal tracking control (OTC) problem of singular perturbation systems in industrial processes under the framework of reinforcement learning (RL) technology. The encountered challenges include the different time scales in system operations and an unknown slow process. The immeasurability of slow process states especially increases the difficulty of finding the optimal tracking controller. To overcome these challenges, a novel off-policy ridge RL method is developed after decomposing the singular perturbed systems using the singular perturbation (SP) theory and replacing unmeasured states using important mathematical manipulations. Theoretical analysis of approximate equivalence of the sum of solutions of subproblems to the solution of the OTC problem is presented. Finally, a mixed separation thickening process (MSTP) and a numerical example are used to verify the effectiveness.

Canard Mechanism and Rhythm Dynamics of Neuron Models

Canards are a type of transient dynamics that occur in singularly perturbed systems, and they are specific types of solutions with varied dynamic behaviours at the boundary region. This paper introduces the emergence and development of canard phenomena in a neuron model. The singular perturbation system of a general neuron model is investigated, and the link between the transient transition from a neuron model to a canard is summarised. First, the relationship between the folded saddle-type canard and the parabolic burster, as well as the firing-threshold manifold, is established. Moreover, the association between the mixed-mode oscillation and the folded node type is unique. Furthermore, the connection between the mixed-mode oscillation and the limit-cycle canard (singular Hopf bifurcation) is stated. In addition, the link between the torus canard and the transition from tonic spiking to bursting is illustrated. Finally, the specific manifestations of these canard phenomena in the neuron model are demonstrated, such as the singular Hopf bifurcation, the folded-node canard, the torus canard, and the “blue sky catastrophe”. The summary and outlook of this paper point to the realistic possibility of canards, which have not yet been discovered in the neuron model.

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.

Designing Distributed Impulsive Controller for Networked Singularly Perturbed Systems

This paper develops a novel synthesis approach for synchronization of a network of singularly perturbed systems (SPSs) with a small singular perturbation parameter (SPP) <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> via distributed impulsive control. First, a decoupling method in the setting of directed networks is employed to decompose networked SPSs related to complex eigenvalues of the Laplacian matrix. Then, based on an improved piecewise continuous Lyapunov function, an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent synchronization criterion is established. The relationship among the impulse interval, the impulse gain matrix and <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> is revealed. By employing the newly-obtained synchronization criterion, sufficient conditions on the existence of an <inline-formula><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula>-dependent impulse gain matrix are derived. Finally, an example is simulated to verify the effectiveness of the theoretical results.

Observability of Discrete-Time Two-Time-Scale Multi-Agent Systems with Heterogeneous Features under Leader-Based Architecture

This paper investigates the observability of discrete-time two-time-scale multi-agent systems with heterogeneous features under leader–follower architecture. First, a singular perturbation difference model for the discussed system is established based on consensus agreement. Second, to eliminate the numerical ill-posed problem that may arise from the singularly perturbed small parameter that distinguishes different time scales in the observability analysis, the order of the system model is reduced using the boundary layer theory of the singular perturbation system to obtain a slow-time-scale subsystem and a fast-time-scale subsystem. Then, based on the matrix theory, some algebraic and graphical features that guarantee the observability of the system are obtained. Finally, the validity of the theoretical results is verified by a numerical example.

Synchronization of discontinuous competitive networks modeled by Filippov singular perturbation system: time-scales dependent settling-time

Synchronization of discontinuous competitive networks modeled by Filippov singular perturbation system: time-scales dependent settling-time

Horizontal Trajectory Tracking Control for Underactuated Autonomous Underwater Vehicles Based on Contraction Theory

In this paper, contraction theory is applied to design a control law to address the horizontal trajectory tracking problem of an underactuated autonomous underwater vehicle. Suppose that the vehicle faces challenges such as model uncertainties, external environmental disturbances, and actuator saturation. Firstly, a coordinate transformation is introduced to solve the problem of underactuation. Then, a disturbance observer is designed to estimate the total disturbances, which are composed of model uncertainties and external environmental disturbances. Next, a saturated controller is designed based on singular perturbation theory and contraction theory. Meanwhile, contraction theory is used to analyse the convergence properties of the observer and the full singular perturbation system, and make quantitative analysis of the estimation error and the tracking error. Finally, the results of numerical simulations prove that the method in this paper enables the vehicle to track the desired trajectory with relatively high accuracy, while the control inputs do not exceed the limitations of the actuators.

New results on [formula omitted] control for interval type-2 fuzzy singularly perturbed systems with fading channel: The weighted try-once-discard protocol case

This article deals with the interval type-2 (IT2) fuzzy model-based H∞ control problem for networked singularly perturbed systems (SPSs) under fading channel. To prevent data collision, a weighted try-once-discard (WTOD) communication scheme is adopted to adjust the communication sequence of sensor nodes. Based on this transmission strategy, only one sensor with more considerable difference is allowed to be transmitted at each time instant. On the other hand, a time-varying stochastic process model is introduced to describe the behavior of the fading channels in unreliable networks. Based on fading state signal and known bounds, the imperfectly matched membership functions (MFs) are reconstructed in the controller. Accordingly, under the effect of fading channels and WTOD protocol, a new IT2 fuzzy controller is constructed for the networked IT2 fuzzy SPSs. Meanwhile, by constructing a singular perturbation parameter (SPP) dependent Lyapunov function, sufficient conditions are established to guarantee the mean square stability of networked IT2 fuzzy SPSs, and the numerical stiffness issue is avoided. Finally, the practicability of the proposed methods is proved through a tunnel diode circuit simulation.

A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs

Motivated by the fact that electrical transients are rather fast compared with mechanical response, the traditional cascade control structure constituted by the inner current and outer speed loops is usually employed in the permanent magnet synchronous motors (PMSMs) servo control community. According to the above-mentioned time-scale characteristic of the PMSMs drive systems, this technique addresses the problems of the non-cascade sliding mode control (SMC) strategy for the surface-mounted PMSMs. Firstly, by appropriately introducing the singular perturbation theory, the corresponding mathematical equations are modeled as a singular perturbation system. Meanwhile, a composite sliding mode surface is constructed based on the Lyapunov equation, such that the system stability can be also guaranteed. Then, according to the exponential reaching law, a standard non-cascade SMC law is designed. Furthermore, an optimal nonlinear function-based tracking differentiator (TD) is presented to smooth the reference velocity value, while providing differential signals. As a result, a novel TD-based SMC strategy is synthesized by incorporating a nonlinear function, thus improving the inherent chattering phenomenon. Finally, a surface-mounted PMSM servo system is performed to illustrate the advantages and effectiveness of the proposed approaches. The main contribution of this paper is to present an alternative non-cascade SMC framework based on the singular perturbation approach, which provides a novel control structure for a PMSM speed regulation system.

Trajectory tracking control for autonomous underwater vehicle with disturbances and input saturation based on contraction theory

Autonomous underwater vehicle (AUV) is a complex nonlinear system and its control is accompanied by various challenges. This paper focuses on the three dimensional (3D) trajectory tracking control of a fully-actuated AUV in the presence of model uncertainties, unmeasured velocity, time-varying external disturbance and input saturation. First, taking the model uncertainties and external disturbances as the total disturbances, an extended state observer (ESO) is designed to estimate the unmeasured velocity and total disturbances. Then, the saturated controller based on contraction theory and its application in singular perturbation system (SPS) is obtained so that the AUV tracks the desired trajectory and avoids exceeding the limit of the actuator. The estimation error, tracking error and the error between the ideal controller and the actual controller are analyzed by contraction theory, and the explicit bounds of these errors are given. At last, comparative numerical simulations are provided to show the effectiveness of the ESO and the advantages of the saturated controller.

HMM-Based Fuzzy Control for Nonlinear Markov Jump Singularly Perturbed Systems With General Transition and Mode Detection Information.

In this article, the hidden Markov model (HMM)-based fuzzy control problem is addressed for slow sampling model nonlinear Markov jump singularly perturbed systems (SPSs), in which the general transition and mode detection information issue is considered. The general information issue is formulated as the one with not only the transition probabilities (TPs) and the mode detection probabilities (MDPs) being partly known but also with the certain estimation errors existing in the known elements of them. This formulation covers the cases with both the TPs and the MDPs being fully known, or one of them being fully known but another being partly known, or both them being partly known but without the certain estimation errors, which were considered in some previous literature. By utilizing the HMM with general information, some strictly stochastic dissipativity analysis criteria are derived for the slow sampling model nonlinear Markov jump SPSs. In addition, a unified HMM-based fuzzy controller design methodology is established for slow sampling model nonlinear Markov jump SPSs such that a fuzzy controller can be designed depending on whether the fast dynamics of the systems are available or not. A numerical example and a tunnel diode circuit are finally used to illustrate the validity of the obtained results.

Robust multiobjective control of singularly perturbed systems using linear matrix inequality

SUMMARYThe article investigates the issue of designing H∞ output feedback controller for a fuzzy nonlinear singularly perturbed systems (SPSs). From overall perspective, we design a series of linear matrix inequalities conditions, which provides the optional stability bounds Ɛ0, verifies the existence of Ɛ0 and ensures that the T–S fuzzy SPSs are asymptotic stable for all Ɛ ∈ (0, Ɛ0]. According to the characteristics of SPSs, we design the asymmetric auxiliary matrix P(Ɛ), which reduces the conservatism and expands the application scope of Ɛ in contrast to the previous results. The approach gives the Ɛ‐dependent Lyapunov function and deduces the fuzzy Ɛ‐dependent robust controller. It is not only can be applied to the standard or nonstandard nonlinear systems but also eliminates the ill‐conditioned problems. Finally, a numerical example testifies the feasibility of conclusion.

Reinforcement Learning Based Optimal Control of Linear Singularly Perturbed Systems

This brief studies the optimal control problem of linear singularly perturbed systems (SPSs) via reinforcement learning (RL). We first present an offline model-based algorithm on the basis of Kleinman algorithm to solve the underlying algebraic Riccati equation with singular perturbation parameter and its convergence is proven. This revised version of Kleinman algorithm is the key to develop the subsequent learning algorithm. Then, by means of two time-scale characteristics of SPSs, we present a novel model-free learning algorithm relating to the online measurement of state and input to design the optimal controller. Compared with the existing learning approaches, the obtained RL algorithm is free of ill-conditioned numerical issues caused by the co-existence of fast and slow modes in SPSs and thus is more robust with respect to the computation and measurement errors. Finally, we verify the effectiveness of the developed results by a simulation example.