Linear Matrix Inequalities Research Articles

Preview control for linear parameter‐varying discrete‐time systems using a linear fractional representation

AbstractIn order to create a mixed tracking controller for discrete‐time linear parameter‐varying (LPV) systems, this work introduces a unique control strategy employing a linear fractional representation (LFR). Under the premise that the reference signal can be previewable, a design strategy for a controller with preview compensations is suggested. First, a preview tracking control issue is changed into a regulator problem by adding two additional linked auxiliary parameters into the initial system state/input, which further creates more enhanced error systems incorporating future information of reference signals. The suggested conditions then depend on utilizing slack variables with decision matrices related to the LFR approach in order to give innovative preview controller designs. We also construct reduced conservative synthesis situations by taking into account parameter‐dependent Lyapunov functions and full‐block multipliers. In order to create state‐feedback and output‐feedback preview tracking controllers, design requirements are characterized as linear matrix inequalities. Two simulation studies are completed as the last step to highlight how successful the suggested control strategy is.

Reachable Set Estimation for Singular Discrete‐Time Nonlinear Systems With Time‐Varying Delays

ABSTRACTIn this article, the reachable set estimation problems for a class of singular discrete‐time nonlinear systems (SDNS) with time‐varying delay (TVD) are presented. First, we employ a Lyapunov function scheme and utilize zero‐type free matrix equations. Based on this, a criterion is derived in the form of linear matrix inequalities (LMIs) to guarantee that the considered system to be regular, causal, and the state trajectory of the system is bounded. Second, by developing the singular systems decomposition technique into slow and fast subsystems and using the introduced auxiliary lemma, we establish that the state trajectory of the fast subsystem is also bounded, and the reachable set of singular nonlinear system is bounded within the ellipsoid. Finally, Numerical examples are presented to illustrate the effectiveness of the proposed method.

H∞ Filtering of Mean Field Stochastic Differential Systems

This paper addresses the H∞ filtering problem for mean field stochastic differential systems that involve both state-dependent and disturbance-dependent noise. We assume that the state as well as the measurement output is distracted by an uncertain exogenous disturbance. Firstly, a sufficient condition for the stochastic-bounded real lemma is given. Next, H∞ filtering, which is built upon a stochastic-bounded real lemma, is put forward by two linear matrix inequalities. Furthermore, the validation of the theoretical analysis is demonstrated with two examples.

Asymptotic synchronisation for coupled heterogeneous complex networks via L' Hoptial' rule

This paper considers the global asymptotic synchronisation (GAS) for the drive-response coupled heterogeneous complex networks (DRCHCNS). By using the different inequality techniques (DIT), L' Hoptial' rule and limit algorithms, three criteria to assure the GAS for the DRCHCNS are derived. Up to now, the results on GAS for DRCHCNS have been obtained usually by applying linear matrix inequality (LMI) and matrix measure approach (MMA). But, in this paper, without applying above approaches, we establish the GAS criteria for the DRCHCNS by using L' Hoptial' rule and limit algorithms. So, our study approach for the complex networks (CNS) extends the study fields of synchronisation of the CNS.

Gain-scheduled filter design with different time delays for T-S fuzzy systems

In this article, the authors study a problem of parameter-varying systems that gain-scheduled [Formula: see text] filtering T-S fuzzy problems by appearing of time-varying delays. Our paper also aims to establish filters that deal with communication delays. Traditional studies often study delay-free filters, but these filters have a more general form. In addition, a nonlinear fractional transformation structure has been incorporated into this filter, which allows its parameters to vary. The authors implement fuzzy Lyapunov–Krasovskii functionals used to intend the delayed fuzzy filters of this study, and delay-dependent circumstances are obtained for stability and performance analysis. The solution of the presented linear matrix inequalities (LMIs) provides the filter parameters. This method is finally justified with an example that illustrates its effectiveness.

New delay‐dependent finite‐time stabilization of Takagi–Sugeno fuzzy approaches for delayed nonlinear systems

AbstractIn this paper, the finite‐time stability and stabilization of nonlinear systems with delays is studied, via a Takagi–Sugeno approach. By using a novel Lyapunov–Krasovskii functional and introducing some fuzzy free‐weighting matrices, sufficient conditions are derived, for bounded and differentiable time‐varying delays in terms of an upper bound of the delay derivatives. Then, we achieve closed‐loop stabilization in finite time through an efficient parallel distributed compensation design. The sufficient conditions are formulated as linear matrix inequalities to achieve the desired performance. Finally, the proposed methodology is applied to various case studies, highlighting its significance.

Event‐Triggered Generalized Extended State Observer‐Based Control for Nonlinear Networked Systems Under Gain Variation and Multi‐Channel Attacks

ABSTRACTThis paper investigates the event‐triggered generalized extended state observer‐based estimation issue for a class of nonlinear networked control systems with unmodeled dynamics, external disturbances and multi‐channel attacks. In order to resist different forms of attack threats on multiple communication channels from sensors to the observer, Markov chain is introduced to describe the stochastic switching or jumping behavior between different attack modes. Considering the limited network resources, the measured outputs are transmitted to the observer only when the triggering conditions are met. Moreover, the parameters modeled by the Bernoulli process are adopted to help analyze potential random gain variations of the generalized extended state observer. By employing Lyapunov stability theory and stochastic systems analysis, sufficient conditions are derived to ensure that the augmented system is exponentially bounded in mean square, and the expected observer gains are further determined through linear matrix inequalities. Finally, a numerical example and a simulation related to the RLC series circuit system are conducted, illustrating the proposed method's effectiveness.

Performance Optimization with LPV Synthesis for Disturbance Attenuation in Planar Motors

Optimizing the performance of motion control systems with variations in nonlinear parameters is not an easy task. To accomplish this task, it is important to design the controller using the linear system approach. In this study, a linear parameter varying (LPV) control method is proposed in which nonlinearities are treated as parameter variations for planar motors. The proposed control method consists of the force and torque modulation with the commutation scheme and the nonlinear current controller with H∞ state feedback control in the form of LPV synthesis to improve the position-tracking performance. An interpolated gain-scheduling controller based on LPV synthesis is determined by applying H∞ control to a linear matrix inequality technique. An interpolated gain-scheduling controller can attenuate disturbance without disturbance estimation. The effectiveness of the proposed control method is evaluated using simulation results and compared with the conventional proportional–integral–derivative control to verify both improved position-tracking performance and disturbance attenuation.

Model predictive control for space robot manipulator modeled by switched systems: A data‐driven approach

AbstractThis article investigates the data‐driven based model predictive control (MPC) problem for a class of space robot manipulator (SRM). First, the SRM system is modeled as a class of discrete‐time switched system to reflect more system characteristics. Second, only the input‐state data are used for end‐to‐end controller design, and the system identification process is avoided. In order to meet the real‐time requirements of the manipulator arm angles, a kind of constrained predictive control method is designed to prevent the overheating or damage of SRM motor, optimizing the cost function for each decision cycle under the safe control input constraints. Then, based on the average dwell time method, the asymptotic stability is assured for the presented SRM system. The design conditions for data‐driven based MPC are provided in the form of linear matrix inequalities. Finally, the effectiveness and advantage for the developed data‐driven based MPC strategy are validated via a case study of SRM system.

Q‐learning‐based H∞ control for LPV systems

AbstractIn this paper, a model‐free H∞ control method is developed for discrete‐time linear parameter‐varying (LPV) systems by leveraging Q‐learning, which employs the data encompassing system states, control inputs, exogenous disturbance, and time‐varying convex hull rather than the dynamical model known a priori. A policy iteration algorithm presented via linear matrix inequality formed by such collected data is developed with convergence guarantee. Our analysis demonstrates that, in the presence of sufficiently rich disturbances, the attenuation level converges to a lower value than the traditional H∞ control solution derived from an exact LPV model. A numerical example is demonstrated to verify the stabilization, disturbance attenuation, adaptivity, and convergence of the H∞ attenuation level. Moreover, a continuous stirred tank reactor (CSTR) approximated by a discrete‐time LPV system is employed to show the practical potential of the developed model‐free approach.

Discrete event-triggered security control for Markovian CVNNs with additive time-varying delays under random deception attacks

This paper is concerned with the security stabilization problem for a class of Complex-valued Neural Networks (CVNNs) with Markov Jump Parameters (MJPs) and Additive Time-varying Delays (ATVDs) under Random Deception Attacks (RDAs). Different from the existing literature, the instant and strength of RDAs considered in this paper is both random, which is more in line with the real situation. Secondly, a general Lyapunov–Krasovskii Functional (LKF) contains more information about MJPs and ATVDs is constructed, and a new Complex-valued Reciprocally Convex Inequality (CVRCI) containing more free matrices and ATVDs parameters is proposed, which play a key role in reducing the conservativeness of security stabilization criteria. Thirdly, a Discrete Event-triggered Mechanism (DETM) is introduced to mitigate the transmission burden of communication networks, in which the triggering condition of DETM mainly relies on the current sampled state and the last triggered state. Then, by combining with the LKF, CVRCI, DETM, and other analysis techniques, some less conservative security stabilization criteria for the underlying systems are provided in terms of Linear Matrix Inequalities (LMIs). Finally, the effectiveness of our results are verified by two numerical examples and a practical example.

Distributed Bipartite Consensus of Multi-Agent Systems via Disturbance Rejection Control Strategy

This work aims to focus on analyzing the consensus control problem in cooperative–competitive networks in the occurrence of external disturbances. The primary motive of this work is to employ the equivalent input-disturbance estimation technique to compensate for the impact of external disturbances in the considered multi-agent system. In particular, a suitable low-pass filter is implemented to enhance the accuracy of disturbance estimation performance. In addition, a specific signed, connected, and structurally balanced undirected communication graph with positive and negative edge weights is considered to express the cooperation–competition communication among neighboring agents. The cooperative–competitive multi-agent system reaches its final state with same magnitude and in opposite direction under the considered structurally balanced graph. By utilizing the properties of Lyapunov stability theory and graph theory, the adequate conditions assuring the bipartite consensus of the examined multi-agent system are established as linear matrix inequalities. An illustrative example is delivered at the end to check the efficacy of the designed control scheme.

Design of extended dissipative-based composite anti-disturbance reliable tracking control for stochastic Takagi–Sugeno fuzzy systems

This study focuses on examining the issues of extended dissipative-based composite anti-disturbance reliable tracking control for stochastic Takagi–Sugeno fuzzy systems afflicted by sensor faults and multiple disturbances. Explicitly, the multiple disturbances include the autonomous Wiener process and non random noises with partly known information that are generated by an exogenous plant. To address the difficulty of estimating the non random disturbance while considering the partially known information, the study incorporates a fuzzy stochastic disturbance observer. Furthermore, to solve the output tracking issue of the system under consideration, a modified repetitive tracking control is implemented. Then, combining the methodologies of disturbance observer-based control technique with modified repetitive control, a composite anti-disturbance reliable tracking control strategy is developed for achieving the intended tracking goal despite the multiple disturbances and sensor faults encountered. Additionally, the proposed approach involves extended dissipativity scheme, which includes H ∞ , passivity, dissipativity, and L 2 − L ∞ performances in a unified framework. In particular, by employing Lyapunov–Krasovskii functional approach, the essential stability criteria for the proposed closed-loop system are articulated in the form of linear matrix inequalities. Additionally, numerical illustrations are presented to exemplify the application and efficiency of the developed control protocol.

Reliable Finite‐Time Control for Nonlinear Chaotic Semi‐Markov Jump Systems With Incomplete Transition Rates

ABSTRACTThis article studies the finite‐time control problem for a class of nonlinear chaotic semi‐Markov jump systems with incomplete transition rates described by the T‐S fuzzy model approach. As a means to depict the dynamical properties pertaining to the examined system, parametric uncertainties, faults, external disturbances and input saturation are taken into consideration. The foremost objective of this study is to come up with a composite control mechanism that effectively rejects and attenuates the repercussions of faults and disturbances. In particular, we primarily built a disturbance observer in order to obtain a precise estimation of the external disturbances. After which, the fault diagnosis observer is designated to effectively estimate the faults. Specifically, a composite reliable control mechanism is developed by fusing the output of the constructed observers with the mode‐dependent fuzzy‐rule based state feedback controller. By employing suitable Lyapunov functions in conjunction with the linear matrix inequality technique, a set of mode‐dependent conditions is derived to ensure the finite‐time stochastic boundedness of the underlying closed‐loop and estimation error systems. Following this, the anticipated controller and observer gain matrices are elicited by resolving the established linear matrix inequalities. Thereafter, intending to ascertain the efficacy and usefulness of accrued theoretical findings, simulation results performed on Chua's circuit system is endowed.

H ∞ prescribed tracking performance-based fractional-order sliding mode control for disturbed discrete-time systems: an LMI approach

This paper presents a novel approach to the design of a discrete-time Fractional-Order Sliding Mode Control (FSMC) system for Single Input Single Output (SISO) applications, utilising the Autoregressive with exogenous input (ARX) method for system modelling. The proposed methodology emphasises the Prescribed Performance Control (PPC) to ensure minimal tracking errors through a transformation of tracking errors based on a specified performance function. A Fractional-Order Quasi-Sliding Mode Control (FQSMC) is subsequently developed to maintain the tracking error within predetermined bounds, effectively addressing the chattering phenomenon commonly associated with the traditional sliding mode control. The integration of fractional calculus enhances the smoothness of the output while ensuring accurate tracking, making it ideal for real-world applications. Additionally, a novel Linear Matrix Inequality (LMI) observer is introduced to bolster the system's resilience against uncertainties and disturbances without complicating the computational process. The effectiveness of the proposed Prescribed Performance Fractional-Order Quasi-Sliding Mode Control (PP-FQSMC) is validated through extensive MATLAB simulations and experimental studies conducted on a Tank system. Results demonstrate that the PP-FQSMC significantly outperforms traditional methods, with maximum tracking errors of 1.6 × 10−4 compared to 0.2 for the PP-QSMC and a Root Mean Square Tracking Error (RMSTE) of 1.3 × 10−3 compared to 1.2 × 10−4 for the PP-QSMC, ensuring strong tracking performance and stability. The findings indicate that the proposed controller effectively mitigates chattering in control signals while maintaining a smooth output. It showcases its potential for practical applications in environments characterised by model inaccuracies, time delays, and external disturbances.

Non-Fragile Sampled Control Design for an Interconnected Large-Scale System via Wirtinger Inequality

A control design for a linear large-scale interconnected system composed of identical subsystems is presented in this paper. The control signal of all subsystems is sampled. For different subsystems, the sampling times are not identical. Nonetheless, it is assumed that a bound exists for the maximal sampling time. The control algorithm is designed using the Wirtinger inequality, and the non-fragile control law is proposed. The size of the linear matrix inequalities to be solved by the proposed control algorithm is independent of the number of subsystems composing the overall system. Hence, the algorithm is computationally effective. The results are illustrated by two examples. The first example graphically illustrates the function of the proposed algorithm while the second one compares with a method for stabilizing a large-scale system obtained earlier, thus illustrating the improved capabilities of the presented algorithm.

Finite Time Stability Analysis and Feedback Control for Takagi–Sugeno Fuzzy Time Delay Fractional-Order Systems

This study treats the problem of Finite Time Stability Analysis (FTSA) and Finite Time Feedback Control (FTFC), using a Linear Matrix Inequalities Approach (LMIA). It specifically focuses on Takagi–Sugeno fuzzy Time Delay Fractional-Order Systems (TDFOS) that include nonlinear perturbations and interval Time Varying Delays (ITVDs). We consider the case of the Caputo Tempered Fractional Derivative (CTFD), which generalizes the Caputo Fractional Derivative (CFD). Two main results are presented: a two-step procedure is provided, followed by a more relaxed single-step procedure. Two examples are presented to show the reduction in conservatism achieved by the proposed methods. The first is a numerical example, while the second involves the FTFC of an inverted pendulum, which exhibits both symmetrical dynamics for small angular displacements and asymmetrical dynamics for larger deviations.

Relaxed conditions for parameterized linear matrix inequality in the form of nested fuzzy summations

The aim of this study is to investigate less conservative conditions for parameterized linear matrix inequalities (PLMIs) that are formulated as nested fuzzy summations. Such PLMIs are commonly encountered in stability analysis and control design problems for Takagi-Sugeno (T-S) fuzzy systems. Utilizing the weighted inequality of arithmetic and geometric means (AM-GM inequality), we develop new, less conservative linear matrix inequalities for the PLMIs. This methodology enables us to efficiently handle the product of membership functions that have intersecting indices. Through empirical case studies, we demonstrate that our proposed conditions produce less conservative results compared to existing approaches in the literature.

Adaptive event-triggered stochastic estimator-based sampled-data fuzzy control for fractional-order permanent magnet synchronous generator-based wind energy systems

This study aims to develop an adaptive event-triggered estimation sampled-data control method for a fractional-order permanent magnet synchronous generator (PMSG)-based wind energy system (WES) using the Takagi–Sugeno (T–S) fuzzy approach. Unlike existing PMSG-based WES control schemes, we propose an aperiodic event-triggered communication scheme with an adaptive mechanism to reduce data transmission. To address the challenge of limited communication resources, we introduce an adaptive event-triggered (AET) estimation method with aperiodic sampling, significantly reducing communication overhead while maintaining stable WES performance. This method employs transmission techniques based on absolute errors, resulting in high data transmission efficiency. First, the fractional-order model for the PMSG-based WES is represented using linear sub-models through the T–S fuzzy approach. Next, a fuzzy aperiodic AET mechanism is proposed for the PMSG model with augmented estimation error systems. Then, the fractional Lyapunov function theory is employed to derive linear matrix inequalities (LMIs), ensuring bounded mean-square stability for the PMSG-based WES with the augmented estimation error systems. Additionally, the desired estimator control gains are determined through solvable LMIs. Finally, simulation studies are presented to demonstrate the superiority and feasibility of the proposed control scheme.

Incremental robust predictive control for anti-pitching in catamarans with appendage constraints

To solve the problem that high-speed catamarans have excessive amplitude of vertical motion in rough sea conditions, an anti-pitching method based on incremental robust predictive control of catamarans is proposed, which takes into account the model uncertainty and input rate of appendage requirements. Thus, a high-speed catamaran vertical control model is established with T-foils and flaps serving as anti-pitching appendages, and the hydrodynamic coefficient uncertainties is analyzed, which is converted into a state space model with norm-bounded uncertainty to facilitate control system design; moreover, transforming the norm-bounded uncertainty model of a high-speed catamaran into an incremental model, which can increase the integral action to improve the anti-pitching performance. On this basis, the hybrid H2/H∞ robust predictive control method is proposed to achieve wave disturbance resistance capability and robust stability of the catamaran system, taking the rate of input of the anti-pitching appendages as the input constraints, and the linear matrix inequality (LMI) receding-horizon optimization is used to solve the incremental control input of appendages. Ultimately, the effectiveness of the proposed algorithm is verified by digital simulation.