Solution Of Linear Matrix Inequalities Research Articles

Robust State Estimation for Fractional-Order Nonlinear Uncertain Systems via Adaptive Event-Triggered Communication Scheme

This paper is concerned with the adaptive event-triggered robust state estimation for a class of fractional-order nonlinear uncertain systems, where the fractional order satisfies 0 <; α <; 1. To save bandwidth resources effectively, an adaptive event-triggered scheme (AETS) is proposed to judge whether the measurement output of the sensor needs to be released to the estimator or not. Firstly, by taking the AETS into consideration, a novel fractional-order estimation error model is established. Then, based on this system model, a sufficient condition that can ensure the robust mean-square stability of the estimation error system has been derived by utilizing the Lyapunov functional approach and some properties of Mittag-Leffler functions. Meanwhile, by applying matrix's singular value decomposition (SVD) technique, the design of the state estimator has been solved in terms of solving the solution of linear matrix inequality (LMI). Finally, one provides a numerical example to verify the feasibility of the proposed method.

Finite‐Time Nonfragile Dissipative Control for Discrete‐Time Neural Networks with Markovian Jumps and Mixed Time‐Delays

This paper considers the stochastic finite‐time dissipative (SFTD) control problem based on nonfragile controller for discrete‐time neural networks (NNS) with Markovian jumps and mixed delays, in which the mode switching phenomenon, is described as Markov chain, and the mixed delays are composed of discrete time‐varying delay and distributed delays. First, by selecting an appropriate Lyapunov‐Krasovskii functional and applying stochastic analysis methods, some parameters‐dependent sufficient conditions for solvability of stochastic finite‐time boundedness are derived. Then, the main results are extended to SFTD control. Furthermore, existence condition of nonfragile controller is derived based on solution of linear matrix inequalities (LMIs). Finally, two numerical examples are employed to show the effectiveness of the obtained methods.

Event‐triggered sliding mode control for discrete‐time singular system

This study focuses on the discrete-time sliding mode control (SMC) based on event-triggered approach for discrete-time singular system. The purpose is to design a SMC law under event-triggered to ensure that the discrete-time singular system is admissible. A novel event-triggered sliding mode surface is constructed, and a sufficient condition, in the form of linear matrix inequality (LMI), is investigated to guarantee the admissibility of the discrete-time singular system. One can verify this condition by utilising the toolbox YALMIP of MATLAB. In addition, the expected controller gain matrix can be represented by a feasible solution of LMI. Then, based on the above solved sliding mode controller gain, a new discrete-time event-triggered SMC law is constructed to ensure that the discrete-time singular system can be driven onto the sliding surface in finite time and then is maintained there for all subsequent time. Finally, three examples are presented to illustrate the effectiveness of the theoretical results.

Collaborative operational fault tolerant control for stochastic distribution control system

Based on a class of industrial processes, a new distributed fault diagnosis approach and a collaborative operational fault tolerant control law are proposed for an irreversible interconnected stochastic distribution control (SDC) system with boundary conditions. This control method is different from the existing collaborative fault tolerant controllers which enable the output probability density function (PDF) to track a desired PDF as close as possible. When fault occurs, a setpoint redesigned fault tolerant approach is adopted to accommodate the fault instead of reconstructing the controller. An augmented PID nominal controller and a setpoint compensation item with linear structure are used to obtain a collaborative operational fault tolerant controller via solution of linear matrix inequalities (LMIs). Simulations are included to show the effectiveness of the proposed algorithms where encouraging results have been obtained.

Synthesis of strictly positive real ℋ2 controllers using dilated LMIs

ABSTRACTThis paper considers strictly positive real (SPR) controller design via direct minimisation of an upper bound on the closed-loop norm. To realise a tractable design problem, an observer-based control structure is assumed and a state-feedback gain that renders the controller SPR while minimising an upper bound on the closed-loop norm is sought. A convex optimisation problem subject to linear matrix inequality (LMI) constraints is realised by using a common Lyapunov variable. By using dilated LMIs, a partial separation of design variables is possible thereby enhancing design freedom. Numerical results comparing the common Lyapunov variable solution, the dilated LMI solution and method presented in the literature are included.

Generalized dynamic observer design for quasi-LPV systems

Abstract This paper presents a new generalized dynamic observer (GDO) for quasi-linear parameter varying (LPV) systems. It generalises the structures of the proportional observer (PO) and proportional integral observer (PIO). The design of the GDO is derived from the solution of linear matrix inequalities (LMIs) and the solution of the algebraic constraints obtained from the estimation error analysis. The efficiency of the proposed approach is illustrated by a numerical example.

Interval Observer Design for Nonlinear Systems: Stability Radii Approach

This paper presents a new approach to design preserving order and interval observers for a family of nonlinear systems in absence and in presence of parametric uncertainties and exogenous disturbances. A preserving order observer provides an upper/lower estimation that is always above/below the state trajectory, depending on the partial ordering of the initial conditions, and asymptotically converges to its true values in the nominal case. An interval observer is then constituted by means of an upper and a lower preserving order observer. In the uncertain/disturbed case, the estimations preserve the partial ordering with respect to the state trajectory, and practically converge to the true values, despite of the uncertainties/perturbations. The design approach relies on the cooperativity property and the stability radii mathematical tools, both applied to the estimation error systems. The objective is to exploit the stability radii analysis for the family of linear positive systems under the time-varying nonlinear perturbations in order to guarantee the exponential convergence property of the observers, while the cooperativity condition determines the partial ordering between the trajectories of the state and the estimations. The proposed approach, defined for Lipschitz nonlinearities, depends only on two observer matrix gains. The design is reduced to the solution of linear matrix inequalities, which are given by the cooperative condition and convergence constraints. An illustrative example is presented to show the effectiveness of the theoretical results.

Robust H∞ Proportional-Integral Observer for Fault Diagnosis: Application to Vehicle Suspension

The main contribution of this paper is the design of robust H∞ proportional-integral (PI) observer for fault detection, isolation and magnitude estimation, applied for uncertain linear time-invariant (LTI) system. It is based on a combination of PI observer and H∞ norm in order to deal with system uncertainty and disturbance attenuation. This multi-objective problem is tackled by the solution of linear matrix inequality (LMI) feasibility problem, obtained from the analysis of estimation errors and the use of majoration lemma. The application to a real suspension system is then presented to highlight the performances of the proposed observer.

Sliding Mode Observer-Based Current Sensor Fault Reconstruction and Unknown Load Disturbance Estimation for PMSM Driven System

This paper proposes a new scheme of reconstructing current sensor faults and estimating unknown load disturbance for a permanent magnet synchronous motor (PMSM)-driven system. First, the original PMSM system is transformed into two subsystems; the first subsystem has unknown system load disturbances, which are unrelated to sensor faults, and the second subsystem has sensor faults, but is free from unknown load disturbances. Introducing a new state variable, the augmented subsystem that has sensor faults can be transformed into having actuator faults. Second, two sliding mode observers (SMOs) are designed: the unknown load disturbance is estimated by the first SMO in the subsystem, which has unknown load disturbance, and the sensor faults can be reconstructed using the second SMO in the augmented subsystem, which has sensor faults. The gains of the proposed SMOs and their stability analysis are developed via the solution of linear matrix inequality (LMI). Finally, the effectiveness of the proposed scheme was verified by simulations and experiments. The results demonstrate that the proposed scheme can reconstruct current sensor faults and estimate unknown load disturbance for the PMSM-driven system.

L1 synthesis of a static output controller for positive systems by LMI iteration

SummaryAn approach to find a static output feedback gain that makes the feedback system positive and minimizes the L1 gain is proposed. The problem of finding a static output feedback gain has 3 aspects: stabilizing the system, making the system positive, and then minimizing the L1 gain. Each subproblem is described by bilinear matrix inequality with respect to the feedback gain and the Lyapunov matrix or vector. Linear matrix inequality (LMI) that is sufficient to satisfy bilinear matrix inequality is derived using a convex‐concave decomposition, and the feedback gain sequence is calculated by an iterative solution of LMI. The sequence of the upper bounds on the design parameter is guaranteed to be monotonically nonincreasing for each algorithm. Similarly, 2 other LMIs are derived for each subproblem using another convex‐concave decomposition and PK iteration. The effectiveness of these algorithms is illustrated via several numerical examples.

H∞ dynamic observer design for linear discrete-time systems

The objective of this paper is to propose a new form of H∞ dynamic observer for linear discrete-time systems. The widely used proportional observer (PO) and proportional integral observer (PIO) can be considered as particular cases of the proposed observer. The observer design is derived from the solution of linear matrix inequalities (LMIs), based on the solutions of algebraic constraints obtained from the unbiasedness conditions of estimation error. A numerical example is provided to show the performance of the proposed observer, compared with PO and PIO.

Frequency-domain tools for stability analysis of reset control systems

The potential of reset controllers to improve the transient performance of linear (motion) systems has been extensively demonstrated in the literature. The design and stability analysis of these reset controllers generally rely on the availability of parametric models and on the numerical solution of linear matrix inequalities. Both these aspects may hamper the application of reset control in industrial settings. To remove these hurdles and stimulate broader application of reset control techniques in practice, we present new sufficient conditions, based on measured frequency response data of the system to be controlled, to guarantee the stability of closed-loop reset control systems. The effectiveness of these conditions is demonstrated through experiments on an industrial piezo-actuated motion system.

State Estimation for Static Neural Networks With Time-Varying Delays Based on an Improved Reciprocally Convex Inequality.

This brief is concerned with the problem of neural state estimation for static neural networks with time-varying delays. Notice that a Luenberger estimator can produce an estimation error irrespective of the neuron state trajectory. This brief provides a method for designing such an estimator for static neural networks with time-varying delays. First, in-depth analysis on a well-used reciprocally convex approach is made, leading to an improved reciprocally convex inequality. Second, the improved reciprocally convex inequality and some integral inequalities are employed to provide a tight upper bound on the time-derivative of some Lyapunov-Krasovskii functional. As a result, a novel bounded real lemma (BRL) for the resultant error system is derived. Third, the BRL is applied to present a method for designing suitable Luenberger estimators in terms of solutions of linear matrix inequalities with two tuning parameters. Finally, it is shown through a numerical example that the proposed method can derive less conservative results than some existing ones.

Tracking control for a cushion robot based on fuzzy path planning with safe angular velocity

This study proposes a new nonlinear tracking control method with safe angular velocity constraints for a cushion robot. A fuzzy path planning algorithm is investigated and a realtime desired motion path of obstacle avoidance is obtained. The angular velocity is constrained by the controller, so the planned path guarantees the safety of users. According to Lyapunov theory, the controller is designed to maintain stability in terms of solutions of linear matrix inequalities and the controller U+02BC s performance with safe angular velocity constraints is derived. The simulation and experiment results confirm the effectiveness of the proposed method and verify that the angular velocity of the cushion robot provided safe motion with obstacle avoidance.

T-S Fuzzy Model Based H-Infinity Control for 7-DoF Automobile Electrohydraulic Active Suspension System

This paper presents a double loop controller for a 7-DoF automobile electrohydraulic active suspension via T-S fuzzy modelling technique. The outer loop controller employs a modified H-infinity feedback control based on a T-S fuzzy model to provide the actuation force needed to ensure better riding comfort and handling stability. The resulting optimizing problem is transformed into a linear matrix inequalities solution issue associated with stability analysis, suspension stroke limit, and force constraints. Integrating these via parallel distributed compensation method, the feedback gains are derived to render the suspension performance dependent on the perturbation size and improve the efficiency of active suspensions. Adaptive Robust Control (ARC) is then adopted in the inner loop design to deal with uncertain nonlinearities and improve tracking accuracy. The validity of improvements attained from this controller is demonstrated by comparing with conventional Backstepping control and a passive suspension on a 7-DoF simulation example. It is shown that the T-S fuzzy model based controller can achieve favourable suspension performance and energy conservation under both mild and malevolent road inputs.

Neutral-type of delayed inertial neural networks and their stability analysis using the LMI Approach

A theoretical investigation of neutral-type of delayed inertial neural networks using the Lyapunov stability theory and Linear Matrix Inequality (LMI) approach is presented. Based on a suitable variable transformation, an inertial neural network consisting of second-order differential equations can be converted into a first-order differential model. The sufficient conditions of the delayed inertial neural network are derived by constructing suitable Lyapunov functional candidates, introducing new free weighting matrices, and utilizing the Writinger integral inequality. Through the LMI solution, we analyse the global asymptotic stability condition of the resulting delayed inertial neural network. Simulation examples are presented to demonstrate the effectiveness of the derived analytical results.

Computational Stability Analysis of Lotka-Volterra Systems

Abstract This paper concerns the computational stability analysis of locally stable Lotka-Volterra (LV) systems by searching for appropriate Lyapunov functions in a general quadratic form composed of higher order monomial terms. The Lyapunov conditions are ensured through the solution of linear matrix inequalities. The stability region is estimated by determining the level set of the Lyapunov function within a suitable convex domain. The paper includes interesting computational results and discussion on the stability regions of higher (3,4) dimensional LV models as well as on the monomial selection for constructing the Lyapunov functions. Finally, the stability region is estimated of an uncertain 2D LV system with an uncertain interior locally stable equilibrium point.

Non-fragile sampled-data robust synchronization of uncertain delayed chaotic Lurie systems with randomly occurring controller gain fluctuation

This paper proposes a new non-fragile stochastic control method to investigate the robust sampled-data synchronization problem for uncertain chaotic Lurie systems (CLSs) with time-varying delays. The controller gain fluctuation and time-varying uncertain parameters are supposed to be random and satisfy certain Bernoulli distributed white noise sequences. Moreover, by choosing an appropriate Lyapunov-Krasovskii functional (LKF), which takes full advantage of the available information about the actual sampling pattern and the nonlinear condition, a novel synchronization criterion is developed for analyzing the corresponding synchronization error system. Furthermore, based on the most powerful free-matrix-based integral inequality (FMBII), the desired non-fragile sampled-data estimator controller is obtained in terms of the solution of linear matrix inequalities. Finally, three numerical simulation examples of Chua's circuit and neural network are provided to show the effectiveness and superiorities of the proposed theoretical results.

Global asymptotic stabilisation of rational dynamical systems based on solving BMI

ABSTRACTIn this paper, the global asymptotic stabiliser design of rational systems is studied in detail. To develop the idea, the state equations of the system are transformed to a new coordinate via polynomial transformation and the state feedback control law. This in turn is followed by the satisfaction of the linear growth condition (i.e. Lipschitz at zero). Based on a linear matrix inequality solution, the system in the new coordinate is globally asymptotically stabilised and then, leading to the global asymptotic stabilisation of the primary system. The polynomial transformation coefficients are derived by solving the bilinear matrix inequality problem. To confirm the capability of this method, three examples are highlighted.

Some novel approaches on state estimation of delayed neural networks

This paper studies the issue of state estimation for a class of neural networks (NNs) with time-varying delay. A novel Lyapunov-Krasovskii functional (LKF) is constructed, where triple integral terms are used and a secondary delay-partition approach (SDPA) is employed. Compared with the existing delay-partition approaches, the proposed approach can exploit more information on the time-delay intervals. By taking full advantage of a modified Wirtinger’s integral inequality (MWII), improved delay-dependent stability criteria are derived, which guarantee the existence of desired state estimator for delayed neural networks (DNNs). A better estimator gain matrix is obtained in terms of the solution of linear matrix inequalities (LMIs). In addition, a new activation function dividing method is developed by bringing in some adjustable parameters. Three numerical examples with simulations are presented to demonstrate the effectiveness and merits of the proposed methods.