Stabilization Problem Of Systems Research Articles

Design of a Finite-Time Bounded Tracking Controller for Time-Delay Fractional-Order Systems Based on Output Feedback

This paper focuses on a class of fractional-order systems with state delays and studies the design problem of the finite-time bounded tracking controller. The error system method in preview control theory is first used. By taking fractional-order derivatives of the state equations and error signals, a fractional-order error system is constructed. This transforms the tracking problem of the original system into an input–output finite=time stability problem of the error system. Based on the output equation of the original system and the error signal, an output equation for the error system is constructed, and a memory-based output feedback controller is designed by means of this equation. Using the input–output finite-time stability theory and linear matrix inequality (LMI) techniques, the output feedback gain matrix of the error system is derived by constructing a fractional-order Lyapunov–Krasovskii function. Then, a fractional-order integral of the input to the error system is performed to derive a finite-time bounded tracking controller for the original system. Finally, numerical simulations demonstrate the effectiveness of the proposed method.

Event‐Triggered Impulsive Control of Nonlinear Stochastic Systems With Exogenous Disturbances

ABSTRACTIn this work, the stability problem of nonlinear stochastic systems is investigated under exogenous disturbances through event‐triggered impulsive control (ETIC). The ETIC strategy proposed in this paper incorporates three levels of events, taking into account four indicators: threshold, control‐free index, inspection interval, and waiting time for stochastic systems, which is more practically significant and can effectively eliminate the Zeno behavior. By utilizing Lyapunov stability theory, stochastic analysis techniques, and some fundamental inequalities, sufficient conditions for the moment input‐to‐state stability (‐ISS) and exponentially ‐ISS of the considered system can be achieved through the implementation of the designed ETIC. Then, the theoretical results are employed in actual nonlinear stochastic systems, leading to the establishment of LMI‐based criteria for exponential ‐ISS. Ultimately, the feasibility of ETIC strategy is confirmed through two instances.

Leader-Following Sampled-Data Consensus of Multiagent Systems With Successive Packet Losses and Stochastic Sampling.

In practical applications, sampled-data systems are often affected by unforeseen physical constraints that cause the sampling interval to deviate from the expected value and fluctuate according to a certain probability distribution. This probability distribution can be determined in advance through statistical analysis. Taking into account this stochastic sampling interval, this article focuses on addressing the leader-following sampled-data consensus problem for linear multiagent systems (MASs) with successive packet losses. First, the relationship of the equivalent sampling interval between two successive update instants is established, taking into account the double randomness introduced by both SPLs and stochastic sampling. Then, the equivalent discrete-time MAS is obtained, and the overall leader-following consensus problem is formulated as a stochastic stability problem of the equivalent system by incorporating the sampled-data consensus protocol and properties of the Laplacian matrix. Based on the equivalent the discrete-time system, a consensus criterion is derived under a directed graph by using the Lyapunov theory and leveraging a vectorization technique. By the introduction of a matrix reconstruction approach, the mathematical expectation of a product of three matrices, including the system matrix and its transpose, can be determined. Then, the consensus protocol gain is designed. Finally, an example is provided to validate our theoretical results.

Optimal Periodic Impulsive Control for Orbital Stabilization of Underactuated Systems

Abstract The orbital stabilization problem for underactuated systems with a single passive degree-of-freedom is revisited. The impulse controlled Poincaré map (ICPM) approach, in which stabilizing impulsive inputs are applied on a Poincaré section, has distinct advantages over existing methods but feedback compensation once every oscillation limits the rate of convergence to the desired orbit. To overcome these limitations, we propose stabilization through application of multiple impulsive inputs during each oscillation. An optimal control problem is formulated to minimize a quadratic cost functional and the optimal inputs are obtained by solving a discrete periodic Riccati equation. Simulation results for a Pendubot are presented, highlighting the advantages of the control design over the ICPM method in terms of convergence rate and robustness to parameter uncertainty.

Static Output Feedback Controller Design for Switching Polynomial Fuzzy Time-Varying Delay System

This study focuses on the stability and stabilization problems of a switching polynomial fuzzy system with a time-varying delay. The switching method is based on the operating sub-domains, and the switching polynomial Lyapunov function with a time-varying delay is used to design the switching polynomial static output feedback controller. The switching polynomial Lyapunov function contains a double-integral term for analyzing the upper bound of the time-varying delay. The stabilization of the polynomial system is investigated, using the boundary information of the membership functions and introducing slack polynomial matrices, which can reduce the conservatism of the stability conditions. Subsequently, the sum of squares conditions are obtained, which are convex and can be solved using SOSTOOLS. Finally, the viability and validity of the proposed approach are demonstrated using a numerical example.

Decentralized Robust Tracking Control of Interconnected Nonlinear‐Constrained Systems: A Dynamic Event‐Sampled Method

ABSTRACTThis paper presents a decentralized robust dynamic event‐sampled tracking (EST) control law for interconnected nonlinear‐constrained systems. The core of developing such a control law is to convert the original EST control problem into the event‐sampled decentralized stabilization problem of augmented interconnected systems. To address the transformed decentralized stabilization problem, an indirect approach relying on the optimal control methodology is proposed. Initially, a group of cost functions are constructed for the nominal subsystems related to the augmented interconnected systems. Then, the dynamic event‐sampling mechanisms are introduced for lessening the computational burden. Meanwhile, the event‐sampled Hamilton–Jacobi–Bellman equations (ES‐HJBEs) are proposed for the augmented interconnected systems. To approximately solve the ES‐HJBEs, the critic approximators are used with their parameters tuned under the reinforcement learning framework. After that, the uniform ultimate boundedness of the tracking errors and the approximators' parameter estimation errors are assured based on the Lyapunov theorem. Finally, a nonlinear plant is provided to validate the decentralized robust dynamic EST control law.

Fast finite‐time stabilization of stochastic nonlinear systems with output constraints and an application to single‐link robot

AbstractThis article studies the fast finite‐time stabilization problem for stochastic nonlinear systems with asymmetric output constraints. To address the issue of slow convergence encountered in previous finite‐time control designs of stochastic nonlinear constrained systems, this paper extends the fast finite‐time stabilization mechanism to such systems. First, a novel asymmetric barrier Lyapunov function is employed to tackle the output constraint. Then, by adding a power integrator technique and the barrier Lyapunov function, a state‐feedback controller for stochastic nonlinear systems is explicitly designed using the backstepping method. The designed controller achieves fast finite‐time stability in probability of the stochastic nonlinear system while ensuring no violation of the output constraints. Finally, two examples are presented to demonstrate the effectiveness of the controller scheme.

Modified preview repetitive control with equivalent-input-disturbance based on two-dimensional model

This paper deals with the problem of two-dimensional (2D) system-based modified preview repetitive control (MPRC) with equivalent-input-disturbance (EID) for continuous-time systems. First, an EID-based MPRC law is used to improve the tracking performance and disturbance attenuation. Next, by using a new equality constraint, a continuous-discrete 2D system is established and the design problem of the EID-based MPRC law is transformed into the stability problem of the 2D system. Then, we employ the 2D system theory together with the linear matrix inequality (LMI) approach, a sufficient condition is derived to ensure the stability of the closed-loop system. By solving an LMI, the gains of the controller and state observer can be obtained. Finally, two numerical examples are provided to illustrate the effectiveness and superiority of the developed controller design technique.

Collaborative Optimization of Multi-regional Integrated Energy System Based on Improved Beluga Whale Optimization Algorithm

With the installed capacity of the renewable energy power generation is growing at a high speed, a large number of the renewable energy is connected to the grid, the energy problem has been solved to a large extent, but the ensuing problems can not be ignored, the first is the consumption of the renewable energy, which is followed by the stability problem of the power system, and a multi-regional integrated energy system (MRIES) is constructed to solve the problem. In view of the wind power uncertainty, the absorption treatment is carried out by the equipment layer and the optimization layer respectively. A hybrid energy storage system (HESS) is introduced in the equipment layer to suppress the influence of the wind power uncertainty on the system stability. A combination of the convolutional filtering algorithm, the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise algorithm and the fuzzy control algorithm is introduced in the optimization layer to develop the charging and discharging strategy of the HESS. With the strategy, the impact of the electric power fluctuations on the power system during the optimization process can be reduced largely. Then, on the basis of fully considering the energy coupling in different equipments, a collaborative optimization model of the MRIES is constructed. And the integrated demand response model considering the time-of-use electric price is introduced in the load side. Finally, the improved Beluga optimization(IBWO) algorithm is developed to optimize the model. The optimization results show that the IBWO algorithm plays a good optimization effect both in the participation of energy supply equipments and the economy, plays a collaborative optimization role in the MRIES and ensures the stability of the whole power system.

Stability analysis of aperiodic impulsive delay systems using sum of squares approach

AbstractThe stability problem of aperiodic impulsive delay systems with unstable continuous and discrete dynamics is addressed in this article. New delay‐dependent stability conditions are proposed by employing a clock‐dependent Lyapunov–Krasovskii‐like functional composed of a Lyapunov–Krasovskii functional and a high‐order polynomial matrix function. The stability conditions are then solved using sum of squares programming techniques, yielding less conservative results than the looped‐functionals approach. Finally, the simulation results demonstrate the effectiveness of the results.

Static output feedback control for positive 2D discrete systems in Roesser model

Abstract This paper investigates the output stabilization problem for multi-input multi-output positive 2D systems described by a linear discrete-time Roesser system. This kind of systems have the property that the horizontal and vertical states take non-negative values whenever the initial boundaries are non-negative. Conditions for the existence of desired static output feedback controllers guaranteeing the resultant closed-loop system to be positive and asymptotically stable are obtained. Also, the synthesis of static output feedback controllers, including the requirement of positivity of the controllers, is solved. These results are then extended to the case of uncertain plants. All of the proposed conditions are expressed in terms of Linear Programming, which can be easily verified by using standard numerical software. Finally, several numerical examples are included to illustrate the validity and the effectiveness of the developed results.

Stability Analysis of Matrices and Single-Parameter Matrix Families on Special Regions

This paper proposes an efficient method for determining the D-stability of matrices using Kelley’s cutting-plane method. The study examines the D-stability of matrices in symmetric regions of the complex plane, defined by quadratic matrix inequalities (QMI) and polynomial functions. Using Kelley’s cutting-plane method, we present an iterative algorithm for determining the D-stability of a given matrix. Further, the robustness of single-parameter matrix families is analyzed, and a method is proposed for considering their D-stability. Through examples, we indicate the possible applicability of the proposed approach in addressing stability problems in linear systems.

Disturbance observer based adaptive predefined‐time sliding mode control for robot manipulators with uncertainties and disturbances

AbstractThis article develops a predefined‐time sliding mode control approach for systems with external disturbances and uncertainties through a nonlinear disturbance observer (DO). For addressing predefined‐time stabilization problem of robotic manipulator system, a predefined‐time sliding mode surface is proposed, ensuring system states converge to origin within a predefined‐time once sliding mode surface is attained. Compared to conventional fixed‐time and finite‐time control strategies, a distinctive advantage of this scheme is that system settling time can be explicitly chosen in advance and independent of system states. To achieve predefined‐time performance, a disturbance observer is introduced to generate the disturbance estimate, which can be incorporated into controller to counteract disturbance. To address the systems uncertainty, an adaptive law is employed to estimate the unknown upper boundary of system uncertainties. Finally, the effectiveness and performance of the proposed scheme are illustrated by simulation and experiment.

Input-to-state hybrid impulsive formation stabilization for multi-agent systems with impulse delays

This paper addresses the input-to-state formation stabilization problem of nonlinear multi-agent systems within a hybrid impulsive framework, considering delay-dependent impulses, strong nonlinearity, and deception attack signals. By leveraging Lyapunov functionals, impulsive comparison theory, average impulsive interval methods, and graph theory, we develop novel criteria for possessing locally input-to-state and integral input-to-state formation stabilization across different impulse sequence classes. These criteria are expressed in terms of continuous/impulsive feedback gains, time delay size, nonlinearity strength, uniform upper bound of impulsive interval, and length of average impulsive interval. Notably, the design of control impulses benefit the destabilizing continuous dynamics in the formation stabilization process. To demonstrate the effectiveness and validity of the analytical results, we provide numerical simulation examples involving various types of external attack signals.

STABILIZATION OF DISCRETE 2D LINEAR SYSTEMS IN FORNASINI-MARCHESINI MODEL

This paper is concerned with the stabilization problem of discrete 2D linear systems described by the second Fornasini-Marchesini model. A necessary and sufficient condition involving the characteristic polynomial is first quoted by which the unforced system is structurally or exponentially stable. On the basis of the derived stability condition, a tractable condition is formulated in the form of linear matrix inequality for obtaining the controller gain of a desired stabilizing state-feedback controller.

An optimization‐based method for robust stabilization of linear delay systems

AbstractThis paper investigates the robust stabilization problem of linear delay systems with parametric uncertainty. First, by means of the argument principle, a delay‐dependent sufficient condition is derived for designing a feedback controller, with which the closed‐loop system can achieve robust stability. Then, based on the fundamental matrix of the linear delay systems, a constrained optimization problem is formulated for solving the feedback gain matrices. The effectiveness of the proposed method is verified by some simulation results.

Stochastic configuring networks based adaptive sliding mode control strategy to improving stability of SG-VSG paralleled microgrid

This article proposes an adaptive sliding mode control (SMC) strategy based on stochastic configuring networks (SCNs) to improve the stability of synchronous generator (SG) and virtual synchronous generator (VSG) paralleled microgrid and enhance the dynamic performance. First, derives an equivalent swing equation what coupling of inherent inertia/damping of SG and virtual inertia/damping of VSG, the parameters design problem is established to a stabilization problem of second-order uncertain nonlinear system with bounded uncertainties. Second, an adaptive sliding mode control based on variable-speed reaching law is designed to maintain the second-order nonlinear system stable, and the switch function of adaptive reaching law is improved to reduce system chattering. Then, the stochastic configuring networks is introduced to approximate the nonlinear term, and the approximation ability of SCNs is improved by designing the configuration range of input parameters of hidden layer nodes. The stability of proposed control strategy is proved by Lyapunov theory. Finally, the feasibility and effectiveness of the proposed control strategy are verified by the designed simulations and experiments.

Fixed-time Lyapunov criteria of stochastic nonlinear systems revisited and its applications

In all the references on stochastic fixed-time stability, the customary treatment of the stochastic noise in the worst-case sense is that it is treated as the unfavorable factor for system stability. Consequently, stochastic fixed-time Lyapunov-type conditions are rather restrictive. Realizing this limitation, we revisit stochastic fixed-time stability and present a generalized fixed-time stability theorem for stochastic differential equations. This theorem completely removes the assumption in these references that the differential operator of the Lyapunov function must be strictly negative, and reveals a positive role of the stochastic noise in stochastic fixed-time stability. As the application of this theorem and its corollary, we solve the problem of fixed-time stabilization for stochastic nonlinear systems with high-order and low-order nonlinearities.

Composite learning prescribed performance control of strict-feedback nonlinear systems with mismatched parametric uncertainties

A composite learning prescribed performance control (CLPPC) approach is presented for strict-feedback nonlinear systems with mismatched parametric uncertainties. A finite-time performance function based on polynomial is introduced to predefined a restriction region with respect to the tracking error. By introducing an error transformation function, the tracking error restriction problem of the original system is transformed into the stability problem of an equivalent transformation system. To guarantee the convergence of unknown parameters, online recording data and instantaneous are used to generate a prediction error, which is used together with filtered tracking errors to update parameter estimations under an interval excitation condition. All signals are proven to be ultimately uniformly stable under the proposed CLPPC strategy. Furthermore, the tracking error is limited within the predefined region after the predefined time. Simulation results verify the effectiveness of the proposed approach.

On robustification of digital event-based controllers for control-affine nonlinear systems

In this paper, the robust digital stabilization problem of nonlinear systems is investigated. In particular, a methodology for the design of robust quantized sampled-data stabilizers updated via an event-triggered mechanism is provided for time-varying control-affine nonlinear systems affected by actuation disturbances and measurement noises. The notion of time-varying steepest descent feedback (TSDF), continuous or not, and the Input-to-State Stability (ISS) redesign methodology are used for the development of the proposed robust event-based digital controller. Under the assumption that the actuation disturbances and measurement noises are bounded with a-priori known bounds and that the amplitude of the measurement noises satisfies a certain condition related to the new added robustification term, the following result is proved: there exist a suitably fast sampling and an accurate quantization of the input/output channels such that the digital implementation of robustified TSDF controllers, updated through a proposed event-triggered mechanism, ensures semi-global practical stability of the related closed-loop system regardless of the above uncertainties. In the methodology here proposed, time-varying sampling periods and the non-uniform quantization of the input/output channels are allowed. Moreover, the theory here developed includes the analysis of the intersampling system behaviour. Possible discontinuities in the function describing the TSDF at hand are also managed. The provided results are validated through a numerical example.