Linear Matrix Inequality Framework Research Articles

Truncated predictive tracking control design for semi-Markovian jump systems with time-varying input delays

A state tracking control problem for a class of semi-Markovian jump systems (SMJSs) with input delay and disturbances is addressed in this study. In particular, an improved-equivalent-input-disturbance estimator approach (IEIDEA)-based truncated predictive controller is designed to synchronously compensate the effect of unknown external disturbances and time-varying delay. Specifically, by making use of IEIDEA, the unknown disturbances are estimated with high precision and incorporated into the control input channel, which eases the way to achieve the desired tracking performances. Specifically, the state predictor satisfies the delay-free state-space equation and is used in controller design along with prediction. In accordance with linear matrix inequality framework and Lyapunov stability theory, a set of adequate conditions are established to guarantee the state tracking performance of SMJSs. Finally, the numerical examples with simulation results are provided to validate the efficacy and practicability of the established control technique.

Extended dissipative criteria for delayed semi-discretized competitive neural networks

This brief investigates the extended dissipativity performance of semi-discretized competitive neural networks (CNNs) with time-varying delays. Inspired by the computational efficiency and feasibility of implementing the networks, we formulate a discrete counterpart to the continuous-time CNNs. By employing an appropriate Lyapunov–Krasovskii functional (LKF) and a relaxed summation inequality, sufficient conditions ensure the extended dissipative criteria of discretized CNNs are obtained in the linear matrix inequality framework. Finally, to refine our prediction, two numerical examples are provided to demonstrate the sustainability and merits of the theoretical results.

Guaranteed cost event-triggered H∞ control of uncertain linear system via output disturbance observer

This paper is concerned with anti-disturbance output feedback control of uncertain systems via dynamic guaranteed cost triggering mechanism. An output disturbance observer is constructed to attenuate matched disturbances and the mismatched one is attenuated by the H∞ approach. A new dynamic event-triggered mechanism is established in terms of guaranteed cost triggering condition and triggering threshold. A composite controller is built on an observer based state feedback controller and an output disturbance compensator. By employing the Lyapunov stability method, a sufficient condition is presented in the framework of linear matrix inequality to ensure the asymptotic stability of the closed-loop system with the prescribed cost. Comparative simulation studies are supplied to verify the effectiveness of the proposed composite control scheme.

A robust damping control for battery energy storage integrated power systems to mitigate inter-area oscillations

This paper presents the effect of a Battery Energy Storage System (BESS) on the power system inter-area oscillations under changing load conditions. The dynamic interaction between the BESS control modes and synchronous generators, along with BESS control parameters, influence the damping of prominent inter-area oscillations under different loading conditions. Furthermore, H∞ mixed sensitivity scheme-based Wide-Area Damping Controller (WADC) is proposed for BESS. This controller is designed using the Linear Matrix Inequality (LMI) framework to mitigate the prominent inter-area oscillation of the power system. Sensitivity to control parameter variation and the geometric measure of the observability are utilized to select the location and feedback signal for the WADC. The performance of the proposed WADC is compared with an Energy Storage System-based stabilizer and WADC proposed in an existing work on a modified New England ten-machine benchmark system. The Real-Time Digital Simulator (RTDS) platform is used to assess the real-time viability of the proposed WADC. The simulation results demonstrate that the proposed WADC can effectively diminish several inter-area modes. Additionally, it offers robust performance under varying load conditions, contingencies and with communication delays in feedback signals. The proposed controller is also effective under the system integrated with Doubly Fed Induction Generator-based Wind Turbine System.

Set‐membership estimation of switched LPV systems: Application to fault/disturbance estimation

SummaryThis paper proposes a set‐membership state estimation method for Switched Linear Parameter Varying (SLPV) systems subject to unknown but bounded parametric uncertainties, disturbances and noises. A zonotopic outer approximation of the state estimation domain is computed at every time iteration. The size of this zonotope is designed to be convergent and bounded by satisfying ‐radius‐based and Average Dwell Time (ADT) conditions that are formulated in the Linear Matrix Inequality (LMI) framework. An extension of the state estimation method is presented to address the fault/disturbance estimation problem for SLPV systems. By using the state augmentation technique, the fault/disturbance estimation problem is transformed into a state estimation problem of the generated augmented descriptor switched LPV system. An application to vehicle lateral dynamics fault estimation is used for assessment purposes. Simulation results demonstrate the effectiveness of the proposed algorithm and highlight its advantages over the existing methods.

Some novel results for DNNs via relaxed Lyapunov functionals

<abstract><p>The focus of this paper was to explore the stability issues associated with delayed neural networks (DNNs). We introduced a novel approach that departs from the existing methods of using quadratic functions to determine the negative definite of the Lyapunov-Krasovskii functional's (LKFs) derivative $ \dot{V}(t) $. Instead, we proposed a new method that utilizes the conditions of positive definite quadratic function to establish the positive definiteness of LKFs. Based on this approach, we constructed a novel the relaxed LKF that contains delay information. In addition, some combinations of inequalities were extended and used to reduce the conservatism of the results obtained. The criteria for achieving delay-dependent asymptotic stability were subsequently presented in the framework of linear matrix inequalities (LMIs). Finally, a numerical example confirmed the effectiveness of the theoretical result.</p></abstract>

Integrated Fault-Tolerant Control Design With Sampled-Output Measurements for Interval Type-2 Takagi-Sugeno Fuzzy Systems.

This article is concerned with the integrated design of fault estimation (FE) and fault-tolerant control (FTC) for uncertain nonlinear systems suffering from actuator faults and external disturbance. The uncertain nonlinear systems are characterized as the interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy model, and IT2 membership functions are employed to effectively handle uncertainties. A fuzzy observer, utilizing only sampled-output measurements, is applied to simultaneously estimate actuator faults and system states. Based on the estimation, the fault-tolerant controller is designed to ensure the system stability under a predefined H∞ performance. The sampling behavior complicates the system dynamics and makes the integrated FTC design more challenging. To confront this issue, the discontinuous Lyapunov functional technique is exploited to enhance stability results by considering the sampling characteristic, upon which FE and FTC units are co-designed in the linear matrix inequality (LMI) framework. To further relax stability criteria, the analysis process incorporates the bound information of membership functions through the membership-function-dependent (MFD) method. Additionally, the relationship of mismatched premise variables resulting from the sampling scheme is also taken into account. Moreover, considering the imperfect premise matching (IPM) framework, the proposed fault-tolerant controller provides greater flexibility in selecting the shapes of membership functions and number of fuzzy rules that can vary from the counterpart of the fuzzy system. Finally, the efficacy of the proposed FTC technique is validated through a detailed numerical example.

Reachable set estimation and H∞$H_\infty$ performance for delayed fuzzy multi‐agent systems under false data injection attacks

AbstractAddressed in this paper is the reachable set estimation (RSE) problem for fuzzy‐model‐based leader‐follower multi‐agent systems with time‐varying delays and false data injection attacks. First, the aperiodic sampled‐data control is designed for the follower agents with randomly occurring false data injection attacks. Then, using the Kronecker product, the error system between the leader and the follower is obtained in a compact general form. Next, a novel Lyapunov‐Krasovskii functional is constructed with the knowledge of sampling patterns and time‐varying delays. In the framework of linear matrix inequalities, sufficient consensus conditions are determined from the performance index and Lyapunov theory to guarantee that its reachable set is enclosed by an ellipsoid in the existence of bounded perturbations. In the end, the Duffing Van der Pol oscillator and the single‐link robot arm models are employed to validate the derived theoretical results.

Disturbance suppression based quantized tracking control for periodic piecewise polynomial systems

The investigation presented here deals with the issue of disturbance suppression and the design of robust tracking control with quantization mechanism for periodic piecewise polynomial systems prone to parameter uncertainties, time delays and external disturbances by means of a proportional integral observer (PIO)-based parallel equivalent input disturbance (PEID) approach. Primarily, in the PEID technique, in order to reduce estimate errors, the PEID notion has been put forward, wherein these errors are perceived as artificial disturbances and a chain of EID compensators has been employed to make amends for them. Meanwhile, PIO’s integrating part aids in blending a relaxing variable into the system’s layout, which allows for greater flexibility in the system’s framework while rendering the system more robust. Subsequently, with the information of estimates from PEID and PIO, disturbance suppression-based quantized tracking control is designed, which simultaneously makes the system states follow the reference states and mitigates the disturbances from the system. Further, the input signals are quantized before being sent as a result of the limited capacity of the channel via which the data is transmitted. Subsequently, through configuring a periodic piecewise polynomial matrix and blending Lyapunov stability theory with the matrix polynomial lemma, adequate requirements are derived in the framework of linear matrix inequalities which ensure the desired outcomes. After which, the requisite controller and observer gain matrices are generated by solving the stated linear matrix inequality-based relations. Ultimately, the simulation portion offers a numerical example that verifies the potential of the acquired findings.

Observer‐based non‐fragile event‐triggered control of extra‐corporeal blood circulation process in finite‐time interval

AbstractAn observer‐based event‐triggered non‐fragile control problem for regulating the levels of oxygen and carbon dioxide (O and CO) in patients blood during the extra‐corporeal blood circulation (ECC) process with random communication packet losses is explored in this work. The main intention of this work is to design an observer‐based event‐triggered and non‐fragile controller for ensuring the finite‐time boundedness criterion for the resulting system with mixed and passivity performance index over a finite interval of time. In addition to the limited bandwidth of the channels, such random packet losses might occur simultaneously between the sensors and the controllers, and between the controllers and the actuators. Unlike most studies in the literature, we have found that, since the controller‐actuator link has packet dropouts, the observer's control input differs from that of the plant. The sufficient conditions derived are presented in a linear matrix inequalities frame work. Finally, simulation results are used to validate the efficacy of the proposed control method. The results show that the proposed observer‐based event‐triggered non‐fragile controller keeps the blood gases within physiological limits during a stable ECC process.

Adaptive Threshold-based Fault Detection for Systems Exposed to Model Uncertainty and Deterministic Disturbance

The fault detection problem is investigated for discrete-time linear uncertain systems. Instead of designing a fault detection system from the viewpoint of observer design for robust residual generation, an adaptive threshold approach is proposed to attain robustness against disturbance and norm-bounded model uncertainty. The main goal of the research is to develop a threshold design method that could establish an appropriate trade-off between false alarms and missed fault detection in the presence of model uncertainty. For this purpose, the H∞ optimization technique is adopted in the linear matrix inequality framework to compute the unknown parameters of an adaptive threshold. It is shown that the proposed fault detection system based on an adaptive threshold depends only on the system parameters and the control input of the monitored system. It is independent of robust residual generator designs in traditional observer-based fault detection systems. The effectiveness of the proposed approach is verified on two well-known benchmark systems: a direct-current motor and three tank systems. Several types of faults are successfully detected in both applications.

Observer based stabilization of linear time delay systems using new augmented LKF

This paper addresses the stabilization problem of linear time-varying delay systems with unmeasurable states. A novel augmented Lyapunov–Krasovskii functional (LKF) is proposed that effectively accounts for the impact of time delays, and an observer based stabilization controller is developed employing linear matrix inequality (LMI) based optimization technique. The utilization of extended reciprocally convex matrix inequality (ERCMI) is employed in this work to establish less conservative stabilization conditions within the framework of linear matrix inequalities (LMIs). By formulating a convex optimization problem, the observer gain and controller gains are determined. Simulation results are used to validate the design, and two numerical examples are considered to prove the usefulness of the proposed method over existing methods.

Observer and fault reconstruction-based control design for interval type-2 fuzzy systems

In this study, the finite-time stabilization and multiple faults reconstruction issues are scrutinized for nonlinear control systems in the presence of time-varying state delay, actuator faults and sensor faults in which the underlying system is approximated by dint of fuzzy approach. To be specific, the interval type-2 (IT2) fuzzy technique is deployed in tandem with a non-parallel distribution compensation scheme to linearize such a faulty system. Further, the existence of unknown inputs in both the system and its output are effectively reconstructed with the aid of IT2 fuzzy-based observer, namely the IT2 fuzzy unknown input (UI) observer. Simultaneously, the designed observer also ensures the reconstruction of system state vectors that are not able to be measured. With the help of this reconstruction information, a fault-tolerant control scheme is established to maintain the performance of the addressed system even in the occurrence of unknown faults. Particularly, in this work, the sampling strategy is incorporated with the developed controller design. Overall, the proposed IT2 fuzzy UI observer-based sampled fault-tolerant controller guarantees the stability of the resulting system within a finite-time duration. Furthermore, the Lyapunov stability theory enables to formulate the delay-dependent adequate constraints in the framework of linear matrix inequalities. On the grounds of established criteria, the relations are also constructed for obtaining the controller and observer gain matrices. Eventually, the simulation study is conducted with two numerical examples to validate the theoretical and practical significance of the developed control procedure.

Actuator fault and Bernoulli random delay-based H∞ reliable fuzzy control of singular hybrid vehicle suspension systems with impulsive perturbations

The problem of actuator fault and Bernoulli random delay-based H∞ reliable fuzzy state feedback control of singular hybrid vehicle suspension systems (SHVSSs) with impulsive perturbations is studied in this paper. By taking the acceleration of each part of the suspension as the component of the system state, the vehicle suspension system is successfully extended to a singular vehicle suspension system. The regularity and non-impulsiveness of the time-varying delays SHVSSs with impulsive perturbations are proved via singular value decomposition (SVD) method. By means of constructing a novel Lyapunov-Krasovskii (L-K) functional embodied with Markovian jump modes and impulse times information, neoteric conditions are smoothly concluded under linear matrix inequalities (LMIs) framework, which guarantee that the closed-loop SHVSS acquires H∞ stochastic admissibilization. By exploiting Takagi-Sugeno (T-S) fuzzy technique and parallel-distributed compensation (PDC) method, a reliable Bernoulli random delay-based fuzzy mode-dependent feedback (FMDF) controller is proposed to handle the change of sprung mass and actuator fault of the controlled time-varying delays SHVSSs. Simulation results verify the validity of the method and riding comfort level of the vehicle.

Optimal guaranteed cost control of discrete-time uncertain systems subjected to state saturation nonlinearity

This paper investigates the optimal guaranteed cost (GC) control problem for uncertain discrete-time (DT) systems with state saturation nonlinearities via static-state feedback. The parameter uncertainties in the system state and input matrix are considered to be norm bounded. Utilizing Lyapunov theory and sector-based characterization of saturation nonlinearities, a novel criterion for the existence of GC controllers is proposed. The criterion is expressed in the framework of linear matrix inequalities (LMIs) and is thus computationally tractable. The proposed GC controllers make the closed-loop (CL) system asymptotically stable and ensure an adequate level of performance for all admissible parameter uncertainties. To design an optimum GC controller, a convex optimization problem with LMI constraints is formulated. The effectiveness of the proposed approach is illustrated by examples.

A Robust Model Predictive Control for a Photovoltaic Pumping System Subject to Actuator Saturation Nonlinearity

In this paper, a new robust model predictive control (RMPC) for uncertain nonlinear systems subject to actuator saturation is designed to regulate the terminal voltage of a photovoltaic generator (PVG) that feeds a DC motor-pump via a buck DC–DC converter. The considered system is a combination of a PVG-converter and DC motor-pump, which possesses nonlinear behavior along with under a saturating control signal highly dependent on the operation point and climate conditions of solar radiation and temperature. As a result, the control task is complex due to the nonlinearity of the system and its dependence on climate conditions. Based on the dead-zone property, the presented paper introduces a new RMPC technique to provide an innovative and efficient solution to ensure the closed-loop system’s robust stability in the presence of actuator nonlinearity. In this paper, the nonlinear system is described in polytypic form, and an appropriate linear feedback control law is designed and used to minimize an infinite horizon cost function under the framework of linear matrix inequalities (LMIs). Furthermore, sufficient state-feedback control law conditions are synthesized to guarantee the robust stability of the closed-loop system in the presence of polytypic uncertainties. Simulation results are provided, in which the results illustrate the effectiveness of the proposed method.

Robust Fixed-Order Controller Design for Uncertain Systems With Generalized Common Lyapunov Strictly Positive Realness Characterization

This article investigates the design of a robust fixed-order controller for single-input–single-output (SISO) polytopic systems with interval uncertainties, with the aim that the closed-loop stability is appropriately ensured and the performance specifications on sensitivity shaping are conformed in a specific finite frequency range. Utilizing the notion of generalized common Lyapunov strictly positive realness (CL-SPRness), the equivalence between strictly positive realness (SPRness) and strictly bounded realness (SBRness) is established; and then, the specifications on robust stability and performance are transformed into the SPRness of newly constructed systems and further characterized in the framework of linear matrix inequality (LMI) conditions. The proposed methodology avoids the tedious yet mandatory evaluations of the specifications on all vertices of the uncertain polytopic system in an explicit form. Instead, solving five LMIs exclusively suffices for ensuring the robust stability and performance regardless of the number of vertices, and thus, the typically heavy computational burden is considerably alleviated. It is also noteworthy that the proposed methodology additionally provides the necessary and sufficient conditions for this robust controller design with the consideration of a prescribed finite frequency range, and therefore, significantly less conservatism is attained in the system performance.

An LMI framework for contraction-based nonlinear control design by derivatives of Gaussian process regression

Contraction theory formulates the analysis of nonlinear systems in terms of Jacobian matrices. Although this provides the potential to develop a linear matrix inequality (LMI) framework for nonlinear control design, conditions are imposed not on controllers but on their partial derivatives, which makes control design challenging. In this paper, we illustrate this so-called integrability problem can be solved by a non-standard use of Gaussian process regression (GPR) for parameterizing controllers and then establish an LMI framework of contraction-based control design for nonlinear discrete-time systems, as an easy-to-implement tool. Later on, we consider the case where the drift vector fields are unknown and employ GPR for functional fitting as its standard use. GPR describes learning errors in terms of probability, and thus we further discuss how to incorporate stochastic learning errors into the proposed LMI framework.

Design of a robust model predictive control for nonlinear switched systems: A low‐conservative and recursive feasibility strategy

AbstractThis article proposes a robust H∞ based model predictive control scheme for Lipschitz nonlinear switched systems with persistent dwell time. In the developed method, the constraints corresponding to the recursive feasibility are included and the recursive feasibility is guaranteed. In this scheme, the existence of unstabilizable sub‐systems and exogenous disturbances are included which makes it possible to avoid considering an ideal and simplistic assumption about switched systems. By simultaneously designing the controller and the switching signal and using multiple Lyapunov functions, the unacceptable conservatism in methods involving common Lyapunov functions and arbitrary switching signals is avoided. In the proposed solution, a time‐varying prediction horizon is used for finite horizon cost functions, which in turn will be effective in reducing conservatism. In the suggested design strategy, the non‐convex control scheme is formulated as an optimization problem in the framework of linear matrix inequality. Finally, a numerical example is developed and the performance of the proposed scheme is evaluated.

Secure Communication in Complex Dynamical Networks via Time-Delayed Feedback Control

In this article, a synchronization method of complex dynamical networks (CDNs) with time-varying delay feedback control is proposed to secure communication between the command system and each node of CDNs. To advance the security of the communication between the command system and each node of CDNs, the original information signal transmitted from the command system is encrypted with the techniques of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> -shift cipher and public key. Based on the Lyapunov stability sense and linear matrix inequality (LMI) framework, a new delay-dependent synchronization criterion is established to restore the original information signal on each node of the CDN as well as to ensure stable synchronization for secure communication of all nodes of the command system and CDN. With the support of numerical simulation, the validity of the proposed method is verified.