Sufficient Linear Matrix Inequality Conditions Research Articles

Observer‐based networked predictive controller with bounded and unknown time‐varying delay

AbstractThis paper investigates a networked control system with bounded and time‐varying delays. Also, an observer‐based predictive controller is developed for the active compensation of the network communication delay since it may lead to poor performances or even unstable dynamics for the systems. The existence of the state observer can be established by choosing an appropriate Lyapunov function, and to do this, Luenberger observer gain is computed with Linear Matrix Inequality (LMI) based on the stability conditions of the system. The practical aspect of this research, different from previous works, is the accessibility of the states that are not always available for the system. Controller and observer gains and other essential variables are derived through LMI. In the closed‐loop system, which is modeled as a switching system, the switches are based on communication delays. Using the Lyapunov stability method for switching systems, sufficient LMI conditions are derived to guarantee the stability of the closed‐loop system. Finally, the results of the simulation have demonstrated the performance of the methodology presented.

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.

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.

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.

Evaluation of Ride Performance of Active Inerter-Based Vehicle Suspension System with Parameter Uncertainties and Input Constraint via Robust H∞ Control

In this study, we investigate a robust H∞ controller for a quarter-car model of an active inerter-based suspension system under parameter uncertainties and road disturbance. Its main objective is to improve the inherent compromises between ride quality, handling performance, suspension stroke, and energy consumption. Inerters have been extensively used to suppress unwanted vibrations from various kinds of mechanical structures. The advantage of inerter is that the realized ratio of equivalent mass (inertance relative to the mass of the primary structure) is greater than its actual mass ratio, resulting in higher performance for the same effective mass. First, the dynamics and state space of the active inerter-based suspension system were achieved for the quarter-car model with parameter uncertainties. In order to attain the defined objectives, and ensure that the closed-loop system achieves the prescribed disturbance attenuation level, the Lyapunov stability function, and linear matrix inequality (LMI) techniques have been utilized to satisfy the robust H∞ criterion. Furthermore, to limit the gain of the controller, some LMIs have been added. In the case of feasibility, sufficient LMI conditions by solving a convex optimization problem afford the stabilizing gain of the robust state-feedback controller. According to numerical simulations, the active inerter-based suspension system in the presence of parameter uncertainties and external disturbance performs much better than both a passive suspension with inerter and active suspension without inerter.

Control of Switched Affine Systems with Bounded Sampling Time Rate on the Switching Function: Application to Buck DC-DC Converter

The paper addresses the problem of designing a stabilizing control for switched affine and its experimental verification based on Linear Matrix Inequalities (LMIs). The main contribution is on the determination of a switching function that exploits the potential of LMI control approaches and which assures global stability and minimizes a guaranteed quadratic cost. Slack variables are introduced to reduce the design conservatism and new sufficient LMI conditions for the synthesis of the controllers are presented. Thus, it is showed that the performance of the control system is superior with a smaller guaranteed cost upper bound of that afforded by recent results. In addition, a theorem with sufficient conditions for the control of switched affine systems that allows a way to garantee a bounded sample time rate on the switching function is proposed. The implementation of the switching function taking into account a bounded sampling time is analysed and discussed with particular interest. Finally, the theoretical results are applied to Buck DC-DC converter. Several simulations show the usefulness of the methodology and experimental results obtained from a prototype validate this approach.

On the finite-time stability and stabilisation of LTI systems via an S-variable approach

This paper deals with the finite-time boundedness (FTB) of linear time-invariant systems that are affected by a disturbance generated by an exosystem. By using a time-invariant quadratic Lyapunov function, Finsler's lemma and a dual system approach, new sufficient linear matrix inequalities (LMIs) conditions are given for analysis and synthesis for FTB. These results are extended: first, to finite-time stability (FTS) of linear time-invariant systems, second, to their input–output FTS associated, or not, to FTB. All the conditions are given in terms of LMIs, which make them easily tractable. Finally, the computational efficiency and the conservatism reduction of our approach are illustrated through an illustrative example.

DC Power Grids With Constant-Power Loads—Part II: Nonnegative Power Demands, Conditions for Feasibility, and High-Voltage Solutions

In this two-part article, we develop a unifying framework for the analysis of the feasibility of the power flow equations for dc power grids with constant-power loads. Part II of this article, explores further implications of the results in Part I. We present a necessary and sufficient linear matrix inequality (LMI) condition for the feasibility of a vector of power demands (under small perturbation), which extends a necessary condition in the literature. The alternatives of these LMI conditions are also included. In addition, we refine these LMI conditions to obtain a necessary and sufficient condition for the feasibility of nonnegative power demands, which allows for an alternative approach to determine power flow feasibility. Moreover, we prove two novel sufficient conditions, which generalize known sufficient conditions for power flow feasibility in the literature. Finally, we prove that the unique long-term voltage semistable operating point associated to a feasible vector of power demands is a strict high-voltage solution. A parametrization of such operating points, which is dual to the parametrization in Part I, is obtained, as well as a parametrization of the boundary of the set of feasible power demands.

Necessary and sufficient admissibility conditions of dynamic output feedback for singular linear fractional‐order systems

AbstractThis paper studies the stabilization problem for linear continuous‐time singular fractional‐order systems with the fractional‐order satisfying . New formulations of the admissibility for the closed‐loop system by dynamic output feedback controller are obtained using necessary and sufficient (linear matrix inequality) conditions, to generalize existing results in the literature for integer‐order systems. The derived results are validated by numerical examples demonstrating applicability and efficacy of the presented approach.

Non-Fragile Observer-Based Adaptive Integral Sliding Mode Control for a Class of T-S Fuzzy Descriptor Systems With Unmeasurable Premise Variables.

The issue of non-fragile observer-based adaptive integral sliding mode control for a class of Takagi–Sugeno (T-S) fuzzy descriptor systems with uncertainties and unmeasurable premise variables is investigated. General nonlinear systems are represented by nonlinear T-S fuzzy descriptor models, because premise variables depend on unmeasurable system states and fuzzy models have different derivative matrices. By introducing the system state derivative as an auxiliary state vector, original fuzzy descriptor systems are transformed into augmented systems for which system properties remain the same. First, a novel integral sliding surface, which includes estimated states of the sliding mode observer and controller gain matrices, is designed to obtain estimation error dynamics and sliding mode dynamics. Then, some sufficient linear matrix inequality (LMI) conditions for designing the observer and the controller gains are derived using the appropriate fuzzy Lyapunov functions and Lyapunov theory. This approach yields asymptotically stable sliding mode dynamics. Corresponding auxiliary variables are introduced, and the Finsler's lemma is employed to eliminate coupling of controller gain matrices, observer gain matrices, Lyapunov function matrices, and/or observer gain perturbations. An observer-based integral sliding mode control strategy is obtained to assure that reachability conditions are satisfied. Moreover, a non-fragile observer and a non-fragile adaptive controller are developed to compensate for system uncertainties and perturbations in both the observer and the controller. Finally, example results are presented to illustrate the effectiveness and merits of the proposed method.

Stabilization and disturbance rejection with decay rate bounding in discrete‐time linear parameter‐varying systems via ℋ ∞ gain‐scheduling static output feedback control

The stabilization of discrete-time linear parameter-varying (LPV) systems via gain-scheduling static output feedback (SOF) control is addressed in this article. We propose new sufficient linear matrix inequalities (LMI) conditions for synthesizing a gain-scheduled SOF controller that ensures asymptotic stability and also imposes a lower bound on the closed-loop decay rate. The SOF controller design is based on a two-step method: a state-feedback controller is obtained in a first-stage design, which is then used as input information in the second stage for computing the desired gain-scheduled SOF controller. The proposed LMI constraints are given in terms of the existence of affine parameter-dependent Lyapunov functions, considering arbitrarily fast variation of the time-varying parameters. An extension for coping with disturbance rejection is also proposed, in terms of the H ∞ $$ {\mathcal{H}}_{\infty } $$ guaranteed cost optimization. Some numerical experiments are presented to illustrate the control synthesis procedure and its efficacy. Also, feasibility analyses are presented to compare and show the advantages of our results over other available strategies present in literature.

Robust H∞$H_\infty$ dynamic output feedback control of the DC–AC converter with a neutral leg

This paper investigates the problem of robust H ∞ $H_\infty$ dynamic output feedback control of 3-phase 4-wire converter with neutral bridge arm in the application of smart grid and distributed generation. Based on Lyapunov functions, a novel sufficient Linear Matrix Inequality condition analysis was performed on the dynamic output feedback stabilisation controllers. Meanwhile, some auxiliary slack variables with special structure and well-established performance conditions are employed to reduce design conservatism. This topology along with the H ∞ $H_\infty$ controller can solve the DC link balancing problem and maintain the deviation between the neutral point and the midpoint of DC supply very small. The achievable performance is analysed quantitatively and the controller design method was conducted and verified both in MATLAB simulation and the real-time simulation platform built by StarSim and MT, respectively.

Lyapunov function computation for autonomous systems with complex dynamic behavior

A computational approach is presented in this paper to construct local Lyapunov functions for autonomous dynamical systems with multiple isolated locally asymptotically stable (such as point-like, periodic, or strange) attractors. We consider systems of nonlinear ODEs, where the right-hand-side of the dynamic equations is given in the form of rational functions (i.e., fraction of polynomials). The Lyapunov function is searched in a parameterized quadratic form of rational terms of the state variables. The quadratic decomposition of the rational state-dependent inequalities is performed using the linear fractional transformation (LFT) and further algebraic/numeric simplification steps. Unlike the sum of squares (SOS) approach, the sufficient linear matrix inequality (LMI) conditions for the Lyapunov function are formulated only locally on a compact polytopic subset of the state space, which allows indefinite matrix solutions for the quadratic decomposition. The local solution is enforced using affine annihilators with matrix Lagrange multipliers. Alongside the typical Lyapunov conditions, further boundary LMI constraints are prescribed using Finsler’s lemma to ensure the required geometric properties of the Lyapunov function. The results are illustrated on four planar benchmark models having either multiple locally stable equlibria or a limit cycle, and on the Lorenz system.

Performance analysis for interconnected time-delay systems with networked communication

This paper studies the network-based performance analysis for interconnected time-delay system (ITDS), where each subsystem experiences delays in the state and connects arbitrarily via the packet-based communication network. The ITDS with networked communication is modeled as hybrid systems with memory to incorporate the delay and network-induced imperfections into a unified framework. Based on this model, a general hybrid Lyapunov–Krasovskii functional is constructed. Sufficient linear matrix inequality conditions for the ITDS to be asymptotically stable and L2 stable are proposed, respectively. These conditions are only related to the interconnected structure, the parameters of each subsystem and local network, leading them convenient for feasibility computation. Two examples, including a numerical example and a three-area time-delay power system, are given to show the feasibility of the derived results.

Comments on ‘Less conservative conditions for robust LQR-state-derivative controller design: an LMI approach’ and new sufficient LMI conditions for invertibility of a convex combination of matrices

This note is concerned with conditions on a set of non-singular matrices , , so that any convex combination of these matrices is also non-singular. The first part of the note points out that Theorem 2.3 in a previous paper [Beteto et al. (2021). Less conservative conditions for robust LQR-state derivative controller design: An LMI approach. International Journal of Systems Science] provides only necessary conditions, which are not sufficient in the general case. In the second part, some stability results based on Linear Matrix Inequalities (LMIs) for a class of fractional order systems are used to establish new sufficient conditions. Numerical examples are presented for illustration. The results suggest that the new LMI conditions may be less conservative compared to a test proposed in the literature on P-matrices, and also to a positive-definiteness test based on matrix cross-products.

Sliding mode observer based control for T-S fuzzy descriptor systems

<p style='text-indent:20px;'>In this paper, the problem of sliding mode observer (SMO) based sliding mode control (SMC) for nonlinear descriptor delay systems is studied. First, based on the T-S fuzzy dynamic modeling technique, the nonlinear descriptor system is transformed into a combination of local linear models. Then, a integral-type sliding surface (ITSS) based SMO is constructed for the error system. In the sequel, sufficient linear matrix inequality (LMI) conditions are established to ensure the admissibility of the sliding motions and obtain the observer gain matrix. Furthermore, two novel SMC laws are developed to ensure the reachability conditions and stabilize the descriptor systems. Finally, simulations are provided to show the effectiveness of the method.</p>

Robust Structural Stability, Stability Margins, and Maximum Uncertainty Amplification for 2D Uncertain Systems via Structured Lyapunov Functions and Matrix Annihilators

This paper considers 2D uncertain systems with homogeneous or mixed dynamics, and addresses three problems: establishing robust structural stability, computing robust structural stability margins, and determining the maximum uncertainty amplification that preserves robust structural stability. Two sufficient linear matrix inequality (LMI) conditions are proposed for establishing robust structural stability obtained by introducing an equivalent closed-loop complex system and by searching for real or complex structured Lyapunov functions and matrix annihilators parameterized by the uncertainties and auxiliary quantities. Moreover, it is shown that lower bounds of the introduced robust structural stability margins and maximum uncertainty amplification can be obtained by solving quasi-convex optimization problems under some restrictions on the sought Lyapunov functions or on the set of uncertainties. Lastly, the nonconservatism of the proposed results is analyzed, showing that the proposed LMI condition based on the use of complex Lyapunov functions is not only sufficient but also necessary under some assumptions.

Sliding-Mode Observer-Based Fault Reconstruction for T-S Fuzzy Descriptor Systems

This article discusses the problem of sliding-mode observer (SMO)-based fault reconstruction for T-S fuzzy descriptor systems by using the RBF neural network. First, a descriptor learning SMO, without the observability condition of fast subsystem, is developed such that the reconstruction of fault and states is achieved, simultaneously. Then, a sufficient linear matrix inequality (LMI) condition is presented to guarantee the admissibility of the sliding motion. Meanwhile, the LMI condition ensures the boundedness of the state and fault estimation errors. By utilizing robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> strategy, the impact of disturbance on the system can be reduced. Moreover, an RBF neural network-based sliding-mode control (SMC) strategy is adopted to estimate the nonlinearity, unknown positive constants, and ensure the reachability condition, simultaneously. Finally, an example is presented to verify the efficacy of our approach.

Exact LMI Conditions for Stability and $\mathcal {L}_2$ Gain Analysis of 2-D Mixed Continuous–Discrete Time Systems via Quadratically Frequency-Dependent Lyapunov Functions

This article addresses the problems of establishing structural stability and the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> gain of 2-D mixed continuous&#x2013;discrete time systems. The first contribution is to show that Lyapunov functions quadratically dependent on the frequency are exact for establishing structural stability. This is particularly important since the existing works that exploit Lyapunov functions provide a much larger upper bound on the dependence on the frequency or other parameters. The second contribution is to propose a novel linear matrix inequality (LMI) necessary and sufficient condition for establishing the existence of such Lyapunov functions. It is shown, analytically and through several examples, for both best and worst cases, that the numerical complexity of this novel condition is significantly smaller than that of the existing methods. The third contribution is to show that the proposed methodology can be used to establish upper bounds on the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> gain, in particular, deriving a novel necessary and sufficient LMI condition based on Lyapunov functions quadratically dependent on the frequency. Finally, the article presents the generalization of the proposed methodology to nonmixed 2-D systems.

Gain‐scheduled control design for discrete‐time nonlinear systems using difference‐algebraic representations

SummaryThis paper addresses the local stabilization problem of nonlinear systems described by Difference‐Algebraic Representations (DAR). A novel set of sufficient Linear Matrix Inequalities (LMI) conditions are developed to design gain‐scheduled state feedback controllers. The proposed approach uses parameter‐dependent Lyapunov functions and new auxiliary decision variables aiming to obtain less conservative results. Two optimization problems are proposed to either obtain the largest estimated Domain‐of‐Attraction (DOA) or minimize theℓ2‐gain from the energy‐bounded disturbance input to the performance output. Furthermore, this investigation considers control input saturation and system states constrained in a polyhedral region. Numerical examples are provided to illustrate the effectiveness of the proposed methodology, showing favorable comparisons with recently published similar approaches.