Linear Matrix Inequality Conditions Research Articles

Hierarchical stability conditions for linear systems with interval time-varying delay

This paper proposes the hierarchical stability conditions for linear systems with an interval time-varying delay. In the matter of the delay, the upper and lower bounds are restricted, but there is no constraint on its derivative. Firstly, based on the state vectors and multiple integral state vectors involved in the delay-related generalized free-matrix-based integral inequalities (GFIIs), the hierarchical Lyapunov-Krasovskii functional (LKF) candidates are constructed. Then, the GFIIs are applied for the approximation of the integral quadratic terms, which will introduce the delay related nonlinear terms in the LKF differential. Next, the novel adjusted matrix-valued high and odd degree polynomial negative definite conditions (NDCs) are provided to acquire the hierarchical linear matrix inequality (LMI) conditions and cope with the nonlinear terms introduced by the GFIIs. Eventually, the advantages of the obtained stability conditions are checked through some numeric examples.

Dynamic observer‐based fault estimation and fault‐tolerant control for switched stochastic systems with multiple faults

AbstractThe fault estimation (FE) and fault‐tolerant control (FTC) problems are studied for the switched system with sensor fault, actuator fault, and stochastic noise. The dynamic FE observers are used to reconstruct the system states and faults at the same time. Differently from the classical observer design method, in the process of observers design, both the output information and its derivative are introduced to improve the estimation performance, especially in the case of the systems subject time‐varying fault. Based on the coordinate transformation, the output derivative is not appeared in the observers, which means that the proposed dynamic observers can be used for the cased that the output derivative is unmeasured. Utilizing the estimate information, the observer‐based FTC scheme is given. For any switching signal, the linear matrix inequality (LMI) conditions are provided, which can ensure that the entire system is mean‐square exponentially stable (MSES). Finally, simulation examples show the effectiveness of the observer and controller.

Output Feedback Control Design for Switched Systems with Unmatched Uncertainties Based on the Switched Robust Integral Sliding Mode

This paper proposes an output feedback sliding-mode control design based on a switched robust integral sliding mode for switched systems with unmatched uncertainties. First, the control task based on the observer is given while the system state information cannot be measured directly. Then, the switched robust integral sliding mode is constructed on the space of the estimated state, while the parameters of the switched robust integral sliding mode are selected ensuring that the system state in the sliding mode is robustly and exponentially stable. Linear matrix inequality conditions for the stabilization switching rule and the common Lyapunov function criterion are achieved. Consequently, the corresponding sliding-mode controller is designed based on the state estimation and the switched robust integral sliding mode. Finally, the application simulation results in a one-link manipulator with the load change validate the effectiveness and feasibility.

Stability and Stabilization for Delay Delta Fractional Order Systems: An LMI Approach

The well-known delta fractional calculus is widely used in system analysis and synthesis. However, the irregular stable region brings many unexpected difficulties and challenges. The simple linear matrix inequality (LMI) condition, especially α(1,2), is lacking. This brief constructs a circular region to approximate the true stable region. Afterwards, the sufficient conditions in terms of LMIs are derived. It is shown that the elaborated results are valid and effective.

On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov–Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel–Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems.

Adaptive event‐triggered consensus of singular multi‐agent systems based on asynchronous data sampling

AbstractThis paper is devoted to the problem of event‐triggered consensus for a class of singular multi‐agent systems with nonlinear dynamics. Firstly, in consideration of the impossibility to reach all agents in some case, an observer is designed to access these agents' states. Secondly, a novel adaptive event‐triggered scheme is proposed based on asynchronous data sampling. In this scheme, the trigger parameter can be adjusted dynamically in order to reduce information transmissions among agents. Thirdly, by employing Lyapunov stability theory, a sufficient condition on the consensus of singular multi‐agent systems is obtained. Then, the control protocol and event‐triggering scheme are designed by solving a linear matrix inequality condition. Finally, an example is given to illustrate the proposed results.

Linear Quadratic Optimal Control of Discrete-Time Stochastic Systems Driven by Homogeneous Markov Processes

Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory.

Robust state and sensor fault estimation for switched nonlinear systems based on asynchronous switched fuzzy observers

SummaryThis article aims to address state and sensor fault estimation issue for a class of continuous‐time nonlinear switched systems, modelled as Takagi‐Sugeno (T‐S) models with nonlinear consequent parts and subject to norm bounded disturbances. This modelling approach helps to prevent Unmeasured Premise Variables (UPVs) problem. The primary contribution of this study consists of proposing robust asynchronous switched observers to simultaneously estimate state and sensor faults under state‐dependent switching. Based on a candidate multiple Lyapunov function, the design of the proposed observer is formulated in terms of Linear Matrix Inequalities (LMIs) conditions. These conditions are dwell‐time‐independent and less conservative compared with similar previous studies, thanks to some usual relaxation techniques and the incremental quadratic constraints applied on unmeasured nonlinear consequent parts. Another contribution consists to perform the estimation of the attraction domain of the estimation error using an optimization procedure. In order to illustrate the effectiveness of the proposed design approach, two simulation examples are considered. The first one concerns the conservatism reduction brought by our proposal compared with previous studies, while the second one is dedicated to show through an illustrative example the performance of the proposed switched T‐S observers under mismatching switching laws.

Linear Matrix Inequality Design of Exponentially Stabilizing Observer-Based State Feedback Port-Hamiltonian Controllers

The design of an observer-based state feedback (OBSF) controller with guaranteed passivity properties for port-Hamiltonian systems (PHS) is addressed using linear matrix inequalities (LMIs). The observer gain is freely chosen and the LMIs conditions such that the state feedback is equivalent to control by interconnection with an input strictly passive (ISP) and/or an output strictly passive (OSP) and zero state detectable (ZSD) port-Hamiltonian controller are established. It is shown that the proposed controller exponentially stabilizes a class of infinite-dimensional PHS and asymptotically stabilizes a class of finite-dimensional non-linear PHS. A Timoshenko beam model and a microelectromechanical system are used to illustrate the proposed approach.

Finding the LQR Weights to Ensure the Associated Riccati Equations Admit a Common Solution

This paper addresses the problem of finding the linear quadratic regulator (LQR) weights such that the associated discrete-time algebraic Riccati equations admit a common optimal stabilising solution. Solving such a problem is key to designing LQR controllers to stabilise discrete-time switched linear systems under arbitrary switching, or stabilise polytopic systems (e.g., Takagi-Sugeno fuzzy systems and linear parameter varying systems) in the entire operating region. To ensure problem tractability and reduce the searching space, this paper proposes an efficient framework of finding only the state weights based on the given input weights. Linear matrix inequality conditions are derived to conveniently check feasibility of the problem. An iterative algorithm with quadratic convergence and low computational complexity is developed to solve the problem. Efficacy of the proposed method is illustrated through numerical simulations of systems with various sizes.

Trajectory‐based robustness analysis for nonlinear systems

AbstractThis article considers the robustness of an uncertain nonlinear system along a finite‐horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients. First, the nonlinear system is approximated by a linear time‐varying (LTV) system via linearization along a trajectory. This linearization introduces an additional forcing input due to the nominal trajectory. Second, the input/output behavior of the perturbation is described by time‐domain, integral quadratic constraints (IQCs). Third, a dissipation inequality is formulated to bound the worst‐case deviation of an output signal due to the uncertainty. These steps yield a differential linear matrix inequality (DLMI) condition to bound the worst‐case performance. The robustness condition is then converted to an equivalent condition in terms of a Riccati differential equation. This yields a computational method that avoids heuristics often used to solve DLMIs, for example, time gridding. The approach is demonstrated by a two‐link robotic arm example.

Fault detection based on adaptive interval observer and its application in aeroengine

This paper is concerned with the problem of sensor fault detection (FD) for discrete-time linear systems. First, considering the influence of disturbance, an adaptive law is designed based on the interval observer theory, which can effectively compress the error interval of output estimation, and improve the sensitivity to sudden faults. Then, based on adaptive interval observer (AIO), a flexible event-triggering fault detection mechanism is designed. Different from the existing methods, the event-triggering condition has greater flexibility. At the same time, the adaptive law can ensure that the zero signal belongs to the upper and lower bound of residual. Based on the robustness theory, the linear matrix inequality conditions (LMIs) satisfying the H∞ performance index are derived. Finally, the effectiveness of the proposed method is demonstrated by an aeroengine simulation example.

Hybrid‐triggered based reliable filter design for interval type‐2 fuzzy system under deception attacks and missing measurements

AbstractThis study inspects the problem of fault‐tolerant control based filter design for interval type‐2 (IT‐2) fuzzy systems with deception attacks and missing measurements, under hybrid‐triggered mechanism. At first, hybrid‐triggered mechanism is commenced to preserve the network bandwidth and also to maintain the competent system performance. Next, the existence of deception attack in the filter infuses unwanted data in the measurement output in order to limit the transmitted signal in the communication networks. The behaviors of triggering mechanism and attack signals are governed by stochastic Bernoulli distribution. Moreover, the missing measurements phenomenon is also characterized by stochastic random variables, which observes the data loss between system and the filter during the transference of information. The sensor failure, deception attack and missing measurement are the key factors that will destruct the network security completely. Further, by endowing Lyapunov stability theory and linear matrix inequality (LMI), a set of adequate conditions is fabricated to ensure the mean‐square asymptotic filtering of the proposed IT‐2 fuzzy system. Subsequently, the filter gain matrices are exhibited with desired LMI conditions. Eventually, the capability and usefulness of the proposed fault‐tolerant filter scheme is substantiated via two numerical examples which includes a tunnel diode application.

Robust fuzzy predictive switching control for nonlinear multi-phase batch processes with synchronous vs asynchronous cases

Multi-phase batch processes (MPBP) have the characteristics of nonlinearity and various cases of switching, where the switching between two adjacent phases can be divided into synchronous vs asynchronous cases. The present study introduces a robust fuzzy predictive method for switching control to ensure that an MPBP can still operate stably in the above two cases. First, considering the multi-phase characteristics of MPBP, multiple stable sub-models are established under the synchronous case. On this basis, unstable sub-models are established between adjacent phases under the asynchronous case. Second, in order to deal with the inherent nonlinearity of MPBP, each sub-model is locally linearized and a mixed Takagi-Sugeno (T-S) model is established. Third, using a mapping relationship between controller and local subsystem including direct mapping and fully associative mapping, a global output control law is designed. By combining the Lyapunov stability theory, linear matrix inequality (LMI) theory, mode-dependent average dwell time method, etc., corresponding LMI conditions are derived by the mapping relationship of different cases, making each phase asymptotically stable and each batch exponentially stable. Then, the distribution compensation gains of the corresponding controllers, the shortest operating time of the stable subsystem and the longest operating time of the unstable subsystem can be obtained by solving corresponding LMI conditions in a synchronous or an asynchronous case. Finally, if the MPBP is in an asynchronous case, according to the longest operating time of the unstable subsystem, a time compensation method based on the obtained operating time is proposed to avoid the occurrence of an unstable subsystem. The simulation results demonstrate that the proposed method has greater advantages than the reference methods.

LMI based stability condition for delta fractional order system with sector approximation

This paper concerns the stability of linear time invariant delay delta fractional order systems. Due to the complexity of the stable region, it is difficult or even impossible to develop a sufficient and necessary linear matrix inequality (LMI) condition. After analyzing the feature of the suggested stable region, a sector region is constructed to be the proper subset of true stable region. To control the angle of the sector region, a variable θ is introduced which improves the degree of freedom. Afterwards, the equivalent LMI condition corresponding to the sector region is developed. Notably, both the cases of α∈(0,1) and α∈(1,2) are considered. Besides the stability analysis, the developed method also be applied in controller design. Finally, the validity and efficacy of the elaborated method are illustrated by simulation study.

Hierarchical conditions for exponential stabilisation of time-varying delay systems using Bessel–Legendre inequalities and Finsler's lemma

This paper addresses the exponential stabilisation of time-varying delay systems by means of Lyapunov–Krasovskii approaches, Bessel–Legendre inequalities and Linear Matrix Inequalities (LMIs). The key aspect of the proposed control synthesis method is that the LMI conditions are proved for the first time to be hierarchical under time-varying delays, that is to say, a progressive conservatism reduction is achieved as long as the degree of Legendre polynomials is increased. To this end, Finsler's lemma has been properly applied to decouple the Lyapunov matrices from the controller gain. Finally, simulation results are provided to illustrate the effectiveness of the proposed method.

Consensus Congestion Control for Ad Hoc Networks: Time-Delay and Saturation

This work focuses on the congestion control design for ad hoc networks affected by time-delay and asymmetrical input saturation. First, an innovative general TCP/AQM model for mobile and wireless ad hoc networks (MANET/WANET) is proposed using graph theory and TCP/AQM system dynamics in a dumbbell topology. Next, a new characteristic condition for sector linearity is developed, which plays the role of a specification to be checked and allows us to assume that the dynamics of the nodes are not saturated. Finally, we propose a new consensus-based congestion control protocol while satisfying the input constraints based on the Lyapunov method, the new sector of linearity condition, the Bessel Legendre-based integral inequality, and the reciprocal convex combination. The proposed results have the advantage of being given as a set of linear matrix inequality (LMI) conditions, which can be solved using convex optimization toolboxes. The merits of the theory are evaluated with the help of various real and challenging scenarios with the use of MATLAB and NS3.

Observer‐based sliding mode fault tolerant control for nonlinear descriptor systems

AbstractIn this article, the design and analysis of fault tolerant control (FTC) are investigated for stochastic nonlinear descriptor systems by using the sliding mode approach. First, an integral sliding mode‐based descriptor observer (ISM‐DO) is constructed to estimate the state and fault. The proposed ISM‐DO retains the basic characteristics of descriptor systems, meanwhile, it is not only suitable for dual normalized descriptor systems, but also for non‐dual normalized descriptor systems. It is more universal than the existing sliding mode fault tolerant control (SMFTC) methods. In the sequel, a novel restricted equivalent form based algorithm is developed to solve the sliding parameters. The proposed method provides a general algorithm, and it does not impose additional rank constraints. Further, a new adaptive sliding mode fault tolerant controller is constructed to stabilize the fault system, the assumption related to the diffusion matrix is removed. Based on these ingredients, the asymptotically mean square admissibility of the closed‐loop fault system and the observer gain matrix are got by a sufficient linear matrix inequality condition. The effectiveness of the proposed method is verified by simulations.

Synchronization of Packet Coupled Low-Accuracy RC Oscillator Clocks for Wireless Networks

Time-sensitive wireless applications have strict requirements on real-time data transmission and control operation. Even though time synchronisation has been extensively studied for providing a common timing among distributed wireless nodes, there still exists a lack of research for low-accuracy and large-drifting clocks, such as internal Resistor-Capacitor (RC) oscillator clocks with around 4 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> parts per million (ppm), which are widely used in wearable sensor systems. This paper proposes a Proportional Packet-Coupled Oscillators (P-PkCOs) protocol for synchronising poor-performing internal RC oscillator clocks with high disturbances in the single-cluster wireless network. The behaviour of such a drifting clock is described as a non-identical and time-varying model. To achieve time synchronisation on low-accuracy internal RC oscillator clocks, a packet-coupled synchronisation scheme is proposed for adjusting drifting clocks via the proportional control-based correction scheme. The RC oscillator frequency in an embedded system cannot be corrected, and this work utilises the clock threshold adjustment as a substitute for frequency correction. The stability region of controller parameters is given to guarantee that the clock threshold approaches a value, which is jointly determined by the nominal threshold and the corresponding clock frequency. We also propose a linear matrix inequality condition to prove that the P-PkCOs performance is robust against the large clock disturbances. We demonstrate the implementation of P-PkCOs. The experimental results show that P-PkCOs can achieve and maintain robust time synchronisation on the internal RC oscillator clocks.

Dynamic event‐triggered dissipative control of nonlinear networked control systems with deception attacks

AbstractFor nonlinear networked control systems with deception attacks, the problem of dynamic event‐triggered dissipative control is researched in this article. First, on the basis of nonlinear networked control systems model, a dynamic event triggered scheme is proposed to save impact of network bandwidth. Second, considering the impact of time delay, a networked time‐delay system model under deception attacks is established. Third, by constructing a new and less conservative Lyapunov function, using linear matrix inequality and unilateral Lipschitz condition, we derive some new sufficient conditions to guarantee the asymptotically stability of nonlinear networked control systems and strictly ‐dissipative performance. Besides, a cooperative design strategy of dynamic event triggered scheme parameters and controller gain is put forward. Finally, a simulation example is given to prove feasibility and rationality of the proposed strategy.