Slack Matrices Research Articles

Robust State Feedback Control with D-Admissible Assurance for Uncertain Discrete Singular Systems

This study addresses the state feedback control associated with D-admissible assurance for discrete singular systems subjected to parameter uncertainties in both the difference term and system matrices. Firstly, a refined analysis criterion of D-admissible assurance is presented, where the distinct form embraces multiple slack matrices and has lessened linear matrix inequalities (LMIs) constraints, which may be beneficial for reducing the conservatism of admissibility analysis. In consequence, by hiring the state feedback control, controller design issues with pole locations, which directly dominate the system performance, are mainly treated. For all the presented criteria can be formulated by the strict LMIs, they are thus suitably solved via current LMI solvers to conduct a state feedback controller with specific poles’ locations of system’s performance requirements. Finally, two numerical examples illustrate that the presented results are efficient and practicable.

Asynchronous event‐triggered control for switched T–S fuzzy systems under dual‐channel hybrid cyber attacks

AbstractThis article presents an asynchronous event‐triggered scheme for switched Takagi–Sugeno (T–S) fuzzy systems against dual‐channel hybrid cyber attacks. Different from existing results, both sensor and controller channels are subjected to aperiodic denial‐of‐service attacks and random false data injection attacks. To efficiently utilize dual‐channel network communication resources while resisting hybrid attacks, two resilient event‐triggered mechanisms (ETMs) are constructed. Considering asynchronous ETMs and hybrid cyber attacks, the time‐delay switched T–S fuzzy system is derived by utilizing model transformation methods. Thereby, the stability conditions are derived by utilizing multiple Lyapunov functions technique, and slack matrices are introduced to further relax the conditions. Finally, two examples are given to demonstrate the effectiveness of the developed event‐based security control strategy.

Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices

This note aims to reduce the conservatism of stability criteria for neural networks with time-varying delay. To this goal, on the one hand, we construct an augmented Lyapunov–Krasovskii functional (LKF), incorporating some delay-product terms that capture more information about neural states. On the other hand, when dealing with the derivative of the LKF, we introduce several parameter-dependent slack matrices into an affine integral inequality, zero equations, and the S-procedure. As a result, more relaxed stability criteria are obtained by employing the so-called Lyapunov–Krasovskii Theorem. Two numerical examples show that the proposed stability criteria are of less conservatism compared with some existing methods.

A switched system approach to exponential stabilization of interval type-2 fuzzy nonlinear systems with packet dropouts

This paper investigates the exponential stabilization problem for interval type-2 (IT2) fuzzy system with parameter uncertainties and packet dropouts. The sample-and-hold characteristic is simulated by employing an input delay methodology, and we present a model for the packet dropout phenomenon using switched approach. Some novel two-side delay-dependent looped-functionals are proposed by considering state information in the intervals from tr to t and t to tr+1 which relaxes the monotonic constraint of the Lyapunov–Krasovskii functional (LKF) and enhances the utilization of the actual sampling pattern. Additionally, some free-weighting matrices, slack matrices, the bound information in membership functions (MFs) and the signs of the first-order time derivative of fuzzy MFs are used to obtain a novel relaxed sufficient conditions for exponential stabilization of interval type-2 fuzzy systems (IT2 FS). Based on the LKF theory, new relaxed sufficient conditions with linear matrix inequalities (LMIs) constraints are derived. According to the criterion, sampled-data controller is devised to ensure that the closed-loop system is exponential stabilization. Finally, an illustrative example is presented to demonstrate the practicality and effectiveness of the method proposed. Comparative analysis highlights the simplicity and practicality of the suggested algorithm.

Delay-dependent stability analysis for load frequency control systems with time-varying delays

This paper introduces a delay-dependent criterion that addresses both computational accuracy and efficiency in dealing with delay-dependent stability challenges within multi-area load frequency control (LFC) systems. Firstly, the traditional power system LFC model is decomposed into delay-related and delay-independent parts using a model reconstruction technique. Subsequently, an augmented Lyapunov–Krasovskii functional (LKF) is formulated, focusing on the delay-related part of the model. The double integral component within the derivative of the LKF is constrained using the parameter-dependent slack matrices approach, resulting in a less conservative stability criterion. Finally, the proposed result is validated through case studies. Simulation results demonstrate that the proposed stability criterion significantly improves computational accuracy while maintaining better validity compared to existing approaches. Moreover, compared to the traditional LFC model, the reconstructed model offers substantial computational efficiency improvements, albeit with a slight decrease in computational precision.

Polytopic energy-to-peak filter design for uncertain continuous systems over finite frequency ranges

A new approach for the synthesis of polytopic L 2 − L ∞ filters is proposed for uncertain systems over Finite Frequency (FF) ranges. The novelty derives from the use of FF specifications combined with the L 2 − L ∞ performance index: First, conditions are provided for the asymptotic stability of the polytopic augmented systems, by trace calculation of matrices and the use of Finsler's Lemma. Then, synthesis conditions are proposed as LMIs for the polytopic filter parameters over several FF intervals. Furthermore, the use of slack matrices through Finsler's Lemma provides design conditions with less conservative outcomes. A numerical example is solved and simulated to demonstrate the effectiveness of the ongoing approach when compared with previous results in the literature.

Finite-frequency fuzzy fault-tolerant static output feedback H∞ control for Diesel engine air-path system

This paper focuses on the design of H∞ finite-frequency (FF) fault-tolerant static output feedback (FTSOFC) problem of Diesel engine air-path system with consideration of external disturbances and actuator faults. Initially, a Diesel engine air-path nonlinear model is described by Takagi-Sugeno (T-S) fuzzy model based descriptor approach. The aim is to regulate intake and exhaust manifold pressures to the desired reference pressures by controlling the Geometry Turbine (VGT) and Exhaust Gas Recirculation (EGR) valves. Then, the robust integrator-based control strategy is developed to track the desired reference signals despite the presence of disturbances and actuator faults. By using the extended Generalized Kalman Yakubovich Popov (GKYP) lemma, Lyapunov functions and independent slack matrices, sufficient conditions are established to ensure both the good tracking of reference pressures and the prescribed H∞ performance with FF domain of the fuel flow variation and actuator faults. Finally, simulation results are given to demonstrate the effectiveness of the proposed approach.

A novel nonzero functional method to extended dissipativity analysis for neural networks with Markovian jumps

<abstract><p>This paper explored the topic of extended dissipativity analysis for Markovian jump neural networks (MJNNs) that were influenced by time-varying delays. A distinctive Lyapunov functional, distinguished by a non-zero delay-product types, was presented. This was achieved by combining a Wirtinger-based double integral inequality with a flexible matrix set. This novel methodology addressed the limitations of the slack matrices found in earlier research. As a result, a fresh condition for extended dissipativity in MJNNs was formulated, utilizing an exponential type reciprocally convex inequality in conjunction with the newly introduced nonzero delay-product types. A numerical example was included to demonstrate the effectiveness of the proposed methodology.</p></abstract>

Robust tracking control for nonlinear fractional order delayed systems: Modified repetitive control approach

AbstractThis study focuses on the problem of perturbation suppression and tracking control of nonlinear fractional order systems with time‐varying delay. To simplify the control process for such systems, the Takagi–Sugeno fuzzy model is used to describe the nonlinear fractional order delayed systems. Additionally, a modified repetitive controller is applied to develop an output feedback technique for the proposed systems. To ensure system stability, the study establishes new delay‐dependent stabilization conditions using a novel Lyapunov–Krasovskii functional (LKF). These stability conditions are expressed in formation of linear matrix inequalities (LMIs) by adopting a frequency‐distributed model transformation and some slack matrices. The proposed method demonstrates that the system output can track the desired input with ignorable error, while effectively rejecting disturbances. Finally, the efficacy of the proposed method is evaluated through simulations of the modified Chua circuit and the hydraulic turbine governing system (HTGS).

A flexible membership-dependent functional method to stability and stabilization for T-S fuzzy systems with interval time-varying delay

In this paper, the stability and stabilization problems for T-S fuzzy delayed systems is studied. A flexible membership-dependent Lyapunov functional (FMDLF) is constructed to fully utilize the information of time-varying delay and fuzzy membership functions (FMFs). Different from some exiting ones, the upper and lower limits of the integral terms in the FMDLF are adjustable. By first introducing exponential gains in reciprocally convex inequality, an exponential-type reciprocally convex inequality (ETRCI) is developed to estimate the FMDLF, which generates some existing results. Based on the FMDLF and ETRCI, a novel stability criterion is obtained. By using a decoupling technique, the corresponding controller design problem is solved. Importantly, the dimensions of the solution space increase since the constraints on slack matrices in some previous studies are eliminated. The effectiveness of the proposed results can be demonstrated by some examples.

HMM-based asynchronous synthesis for Interval type-2 fuzzy semi-Markovian jump systems with its applications: A bilateral looped functional methodology

This article mainly addresses the hidden-Markovian model-based asynchronous sampled-data control (SDC) synthesis problem for interval type-2 (IT2) fuzzy semi-Markovian jump systems (S-MJSs) with stochastic actuator failures (SAFs). Herein, the asynchronous factor resulted from twofold: (1) hidden Markovian model (HMM) and (2) imperfectly matched membership functions (IMMFs) between the original plant and the controller. Aiming at approximating the nonlinearity along with randomness with high accuracy and broadening practical application background, the IT2 fuzzy S-MJSs associated with IMMFs is established. At the beginning, a HMM is borrowed to portray the discrepancy between the corresponding plant and the desired controller. Additionally, in comparison with majority of some previous sampled-data control mechanisms, a intrinsic lag signal and gain fluctuations are purposefully introduced in the double asynchronous control strategy, which exploited initially to settle the control synthesis issue. By considering the information of sampling intervals x(tk)→x(t)→x(tk+1) along with x(tk−τ)→x(t−τ)→x(tk+1−τ) adequately, a fresh looped Lyapunov functional called bilateral looped Lyapunov functional is innovatively constructed. Whereafter, by suitably incorporating with the skills of stochastic analysis and relaxed integral inequality (including generalized free matrix-based integral inequality and canonical Bessel–Legendre integral inequality) together with symmetric slack matrices, numerically testable sufficient conditions are established in the form of a group of feasible multi-parameterized linear matrix inequalities (M-PLMIs). The theoretical developments are implies the hierarchy character, and the conservatism decreases with the increase of the degree of hierarchy. Conclusively, the design methodology brought forward herein is demonstrated by two practical examples, which show the potency and practicability.

Security-Based Control for Networked Interval Type-2 Fuzzy Systems With Multiple Cyber-Attacks: An Improved Dynamic Event-Triggered Scheme

This paper addresses the problem of dynamic event-triggered (DET) control for networked interval type-2 (IT2) fuzzy systems subject to multiple cyber-attacks, which contain deception attacks and aperiodic denial-of-service attacks simultaneously. Firstly, a multiple cyber-attack model is established for the IT2 fuzzy systems with multiple cyber-attacks and external disturbances. Secondly, an improved DET mechanism (DETM) is proposed to reduce the communication burden. Contrasting with existing works, by introducing two adjustable parameters into the novel DETM, the presented DETM has more flexibility in adjusting the data packet transmissions to conserve more limited network resources. Meanwhile, the Zeno behavior is probably excluded under the proposed DETM while preserving the system performance. Thirdly, imperfectly matched membership functions (MFs) are considered between fuzzy controller and IT2 fuzzy model, and the global boundary information of MFs and slack matrices are introduced to relax stability conditions. Besides, by constructing a proper Lyapunov-Krasovskii functional, sufficient conditions are obtained to ensure the exponentially mean-square stable of the IT2 fuzzy systems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> performance. Finally, two simulation examples are provided to illustrate the effectiveness of the theoretical results.

Self-Dual Polyhedral Cones and Their Slack Matrices

Self-Dual Polyhedral Cones and Their Slack Matrices

Sliding-Mode Control for Interval Type-2 Fuzzy Systems: Event-Triggering WTOD Scheme.

This article investigates the sliding-mode control issue for interval type-2 (IT2) T-S fuzzy systems under limited communication resources. An event-triggering weight try-once-discard (ET-WTOD) protocol is formulated via two thresholds to determine the transmission of the state signal. The proposed ET-WTOD protocol can dynamically adjust the transmitted nodes and permits only partial components with larger error to be sent at each triggering instant, which is just the key distinction from the existing protocols. Under the imperfect premise matching framework, the controller's membership functions are reconstructed via the received state and the known upper and lower bounds, and then, a new scheduling signal set is established to design the scheduling-signal-dependent fuzzy sliding-mode controller. With the aid of the membership-function-dependent approach, the mismatching premise variables of the fuzzy model and the controller are effectively handled by introducing some slack matrices, while the relaxed stability conditions are derived to ensure the stability of the closed-loop system and the reachability of the specified sliding surface. Moreover, an optimized sliding domain is further obtained via the genetic algorithm (GA). Finally, the proposed control strategy is verified via the mass-spring-damper system.

Event-triggered non-fragile state estimator design for interval type-2 Takagi–Sugeno fuzzy systems with bounded disturbances

In this paper, the state estimator design problem of interval type-2 Takagi–Sugeno fuzzy systems suffering from bounded disturbances is studied. To enhance the resilience of the estimator, a non-fragile design scheme is proposed to tackle the estimator gain variations. Meanwhile, an event-triggered communication mechanism is introduced for relieving the transmission burden over networks. To settle down the non-fragile estimator design issue subject to bounded disturbances and event-induced error, we propose a new definition of quadratic boundedness via the multiple Lyapunov functions. Based on this definition, a novel co-design method of estimator and event generator for fuzzy system models in the presence of both measurable and immeasurable premise variables is presented. In virtue of quadratic boundedness framework, less conservative conditions of the existence and quadratic stability of the fuzzy estimators are obtained, and the upper bound of estimation error is given explicitly. The desired estimator gains are determined by convex optimization technique using slack matrices. Two illustrative examples are exploited to validate the availability and superiority of the addressed design approach.

Event-Triggered Finite-Time Dissipative Filtering for Interval Type-2 Fuzzy Complex Dynamical Networks With Cyber Attacks

This article aims to resolve the event-based finite-time dissipative filtering issue for interval type-2 fuzzy complex dynamical networks with coupling delays and cyber attacks. An event-triggered scheme introducing the estimation error is designed, which is a tradeoff between state estimator performance and network communication bandwidth in accordance with the practical requirements. Moreover, the cyber attack model is established, which consists of randomly occurring deception attacks. The stochastic variable obeying the Bernoulli distribution is utilized to describe the attack condition. Through handling the linear matrix inequalities (LMIs) with some slack matrices, the event-triggered finite-time filter is designed to guarantee that the resulting systems are finite-time bounded and strictly dissipative. The proposed filter design parameters are obtained by solving LMIs. Finally, a numerical example and a practical example of a tunnel diode circuit system are used to show the effectiveness and applicability of the proposed event-based dissipative filtering scheme.

Relaxed stability criteria of time-varying delay systems via delay-derivative-dependent slack matrices

This paper is focused on delay-dependent stability problem of time-varying delay systems. By introducing delay-derivative-dependent slack matrices, relaxed stability conditions are derived based on Lyapunov-Krasovskii functional approach. As the delay-derivative-dependent slack matrices provide extra freedom to optimize the Lyapunov matrices, less conservative results are obtained. Two benchmark examples are provided to verify the effectiveness of the proposed approach.

Stability and stabilisation for time-varying delay systems based on flexible augmented Lyapunov–Krasovskii functional

This paper studies the problems of stability and stabilisation for time-varying delay systems. A flexible augmented Lyapunov–Krasovskii functional (FALKF) including some triple integral terms and delay-product-type quadratic terms is first constructed, in which the upper and lower limits of integral terms are flexible. Compared with some existing ones, the information of time-varying delay can be fully utilised. A parameter-dependent reciprocally convex inequality (PDRCI) is proposed, which covers some existing ones as its special cases. Based on the FALKF and PDRCI, a less conservative stability condition is obtained to ensure the time-varying delay systems to be asymptotically stable. By using a matrix inequality decoupling technique, the corresponding controller for the closed-loop systems is derived. Compared with some existing works, the constraints on introduced slack matrices are avoided. It directly provides extra free dimensions in the solution space. Two examples are employed to illustrate the effectiveness of the proposed methods.

Refined Admissible Analysis and Design Conditions for Discrete Fuzzy Singular Systems with Multiple Difference Matrices

ABSTRACT This study mainly discusses extended admissibility and fuzzy parallel distributed compensation (PDC) control issues for discrete singular fuzzy systems with multiple difference matrices existing in the rules. By the overall system associated with the discrete singular models with multiple difference matrices, we first propose an extended admissibility analysis criteria, where the new results not only involve some slack matrices but also have a less number of linear matrix inequalities’ (LMIs) constraints. Furthermore, by hiring the fuzzy PDC, explicit design criteria are further developed for the regarded system. Noticeably, the new design method can cope with controller synthesis of the admissibility and D-admissibility issues. Due to all the presented criteria are formed by the strict LMIs, we can readily evaluate them via some existing LMI solvers. Finally, two numerical examples are involved to demonstrate the applicability and the feasibility of the developed results.

Stochastic Stability of Markovian Neural Networks With Generally Hybrid Transition Rates.

This article studies the problem of the stability for Markovian neural networks (MNNs) with time delay. The transition rate is considered to be generally hybrid, which treats those existing ones as its special cases. The introduced generally hybrid transition rates (GHTRs) make these systems more general and practical. Apropos of the GHTRs, a double-boundary approach rather than the traditional estimation method is introduced to make full use of the error information in GHTRs. In order to fully capture system information, a parameter-type-delay-dependent-matrix (PTDDM) approach is proposed, in which the PTDDM approach removes some zero components on slack matrices in previous works. Thus, the PTDDM approach can fully link the relationship among time delay and state-related vectors. Based on these ingredients, a novel stochastic stability condition is proposed for MNNs with GHTRs. A numerical example is illustrated to demonstrate the effectiveness of the proposed approaches.