Sufficient Linear Matrix Inequality Research Articles

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.

Bounded real lemma for singular linear continuous-time fractional-order systems

This paper proposes novel necessary and sufficient strict linear matrix inequalities (LMIs), to characterize admissibility of singular fractional-order linear continuous-time systems with the fractional derivative of order α belonging to 1≤α<2. Then, the problem of the bounded real lemma corresponding to the H∞ norm computation is addressed involving additional variables. Necessary and sufficient conditions are established via a set of LMIs that can be effectively used to design H∞ controllers. Based on the corresponding bounded real lemma, a state feedback control with a prescribed H∞ performance index for the underlying systems is proposed. Finally, numerical examples are provided to show effectiveness of the given results.

Cluster synchronization of fractional-order complex networks via uncertainty and disturbance estimator-based modified repetitive control

The combined problems of cluster synchronization and disturbance rejection for a family of fractional-order complex networks subject to coupling delay, unknown uncertainty and disturbances (UDs) are examined in this study. In particular, the existence of coupling delay is taken into the account with both known and unknown cases. First, a new uncertainty and disturbance estimator (UDE)-based control protocol is formulated for the concerned system to estimate and compensate for the effects of UD. Although the UDE strategy has proven to be a viable tool to deal with slowly changing UDs in control design, the presence of rapidly changing UDs or sinusoidal disturbances is not an effective tool. A well-known modified iterative control (MRC) block is built internally in a closed feedback control loop to solve this problem. After implementing UDE and MRC blocks into the feedback loop, the resulting system becomes almost UD-free. Moreover, a set of sufficient linear matrix inequality constraints are established to ensure the cluster synchronization of the resulting system. Lastly, the benefits, feasibility and robustness of the established UDE-based MRC scheme are confirmed by two illustrative examples.

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.

H model reduction design in finite frequency ranges for discrete-time Takagi-Sugeno fuzzy systems

This paper addresses the problem of H∞ model reduction design in finite frequency ranges for discrete-time nonlinear systems. With the aid of the generalised Kalman-Yakubovich-Popov (gKYP) lemma, the Lyapunov theory, and some matrix inequality techniques, sufficient conditions for the finite frequency model reduction problem are derived. The design conditions are given in terms of sufficient parameter-dependent linear matrix inequalities that can be solved through relaxations based on semi-definite programming. The advantages of the proposed approach are illustrated through numerical examples and comparisons with other available techniques.

H&lt;SUB align="right"&gt;∞ model reduction design in finite frequency ranges for discrete-time Takagi-Sugeno fuzzy systems

This paper addresses the problem of H∞ model reduction design in finite frequency ranges for discrete-time nonlinear systems. With the aid of the generalised Kalman-Yakubovich-Popov (gKYP) lemma, the Lyapunov theory, and some matrix inequality techniques, sufficient conditions for the finite frequency model reduction problem are derived. The design conditions are given in terms of sufficient parameter-dependent linear matrix inequalities that can be solved through relaxations based on semi-definite programming. The advantages of the proposed approach are illustrated through numerical examples and comparisons with other available techniques.

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.

T–S Fuzzy Sampled-Data Control for Nonlinear Systems With Actuator Faults and Its Application to Wind Energy System

This article concerns the problem of stability analysis of nonlinear systems under the sampled-data control (SDC) with actuator faults. The addressed nonlinear systems can be expressed by a number of linear arrangements, based on the Takagi–Sugeno (T–S) fuzzy technique. The time-dependent SDC with actuator faults of the addressed nonlinear systems delineates based on the input model of Markovian variability. The time-scheduled Lyapunov functional and the looped-functional are taken together in the production of a Lyapunov functional candidate. Based on a free matrix-based integral inequality and the Lyapunov functional, some sufficient linear matrix inequality conditions are delivered to guarantee the mean-square asymptotic stability under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance of the considered sampled-data T–S fuzzy systems with actuator faults. Finally, we demonstrate the feasibility and strength of our proposed method through some examples.

Strong Stability Analysis of Linear Delay-Difference Equations With Multiple Time Delays

This article is concerned with the strong stability problem of linear continuous-time delay-difference equations with multiple time delays. A family of linear matrix inequalities (LMIs), indexed by a positive integer k, is derived to assess strong stability. A time-domain interpretation of the proposed LMI-based condition is given in terms of a quadratic integral Lyapunov functional, which allows us to reveal relations with an existing result. The LMI condition can easily be reformulated in a form where the dependence on the coefficients of the delay-difference equation is affine, which is instrumental to establishing a sufficient LMI condition for robust strong stability of delay-difference equations with norm-bounded uncertainty. A necessary and sufficient condition for robust strong stability is also given, in the form of a structured singular value characterization. Examples are given to illustrate the effectiveness of the presented results.

Fault detection for T-S fuzzy singular systems via integral sliding modes

This paper investigates the problem of sliding mode observer based fault detection for a class of T-S fuzzy singular systems. A H−/H∞ residual generator, which is based on the integral sliding modes, is developed. Firstly, an integral-type switching function is designed to deal with the unknown nonlinearity. Then, on the basis of H∞ disturbance attenuation performance and H− fault sensitivity performance, a sufficient linear matrix inequality condition is presented to guarantee that the residual signal is sensitive to fault while robust against disturbance. Moreover, a neural network based adaptive sliding mode controller is designed to estimate the nonlinearity, as well as guarantee the reachability condition. Finally, two examples are presented to verify the effectiveness of the proposed method.

Induced [formula omitted]-gain computation for rational LPV systems using Finsler’s lemma and minimal generators

This paper proposes a novel method to compute an upper bound on the induced L2-gain for a linear parameter varying (LPV) system with rational parameter dependence. The proposed method relies on a standard dissipation inequality condition. The storage function is a quadratic function of the state and a rational function of the parameters. The specific parameter dependence is restricted to involve (fixed) rational functions and an affine term with free decision variables. Finsler’s lemma and affine annihilators are used to formulate sufficient linear matrix inequality (LMI) conditions for the dissipativity relation. The dimension and conservatism of the resulting LMI problem are reduced by the joint application of minimal generators and maximal annihilators. An LPV model of a pendulum–cart system is used to demonstrate the proposed method and compare it to existing techniques in the literature.

$H_{\infty }$ Weighted Integral Event-Triggered Synchronization of Neural Networks With Mixed Delays

This article considers an event-triggered H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> synchronization of neural networks (NNs) with mixed (discrete and distributed) delays. To release the communication burden, a novel weighted integral event-triggered scheme (IETS) is proposed based on the past information of the system dynamics. In this scheme, for the first time, a weighting function is proposed to weight the system state over a given period, which can be viewed as a forgetting factor. Moreover, a waiting time interval is added in the proposed IETS to exclude Zeno phenomenon. By constructing a novel Lyapunov-Krasovskii functional with the distributed delay kernel and the weighting function, sufficient linear matrix inequality conditions for the existence of an event-triggered controller that guarantees an exponential synchronization of the delayed NNs with an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance are derived. Finally, an illustrative example and an application to image encryption are used to demonstrate the advantage of the proposed approach.

Design of Robust PID Controller using Static Output Feedback framework

This paper considers the design of a robust PID controller for higher order MIMO plants. The design problem is first recast into a Static Output Feedback (SOF) controller design problem and then the transformed SOF problem is solved within the framework of Linear Matrix Inequalities (LMIs) through a decomposition of the Lyapunov matrix variable. Sufficient LMI criteria are derived that ensure H∞ performance of the underlying system. By means of a numerical example, it is shown that the designed controller yields less conservative results. Also, a comparative study is done with the existing techniques to demonstrate the efficacy of the proposed method.

Co-Design of Event-Triggered Scheme and H∞ Output Control for Markov Jump Systems Against Deception Attacks

This paper is concerned with the co-design problem of event-triggered scheme and $H_{\infty }$ static output control of linear Markov jump systems with deception attacks. To save the previous communication resources, a mode-dependent event-triggered scheme is utilized based on system output. To describe the deception attacks, a random variable satisfying Bernoulli distribution is employed. By using a separation approach, sufficient linear matrix inequality conditions for the existence of event-triggered output controllers that ensure the stochastic stability with prescribed $H_{\infty }$ are obtained. Then, a co-design algorithm is proposed to obtain the trade-off between the communication cost and $H_{\infty }$ performance. Lastly, the validity of the developed method is verified by a numerical example and a practical single-link robot arm system.

Formula omitted] non-minimal filter design in finite frequency ranges for discrete-time Takagi–Sugeno fuzzy systems with time-varying delays

This paper addresses the problem of H∞ fuzzy filter design in finite frequency intervals for discrete-time nonlinear systems subject to time-varying delays. Fuzzy filters whose order is greater (non-minimal) than the order of the system are synthesized by transforming the original time-delayed system into an equivalent switched nonlinear system. The design conditions are given in terms of sufficient parameter-dependent linear matrix inequalities that can be solved through relaxations based on semi-definite programming. The advantages of the proposed approach are illustrated through numerical examples and comparisons with other available techniques.

Algorithm 998

The ROLMIP (Robust LMI Parser) is a toolbox specialized in control theory for uncertain linear systems, built to work under MATLAB jointly with YALMIP, to ease the programming of sufficient Linear Matrix Inequality (LMI) conditions that, if feasible, assure the validity of parameter-dependent LMIs in the entire set of uncertainty considered. This article presents the new version of the ROLMIP toolbox, which was completely remodeled to provide a high-level user-friendly interface to cope with distinct uncertain domains (hypercube and multi-simplex) and to treat time-varying parameters in discrete- and continuous-time. By means of simple commands, the user is able to define polynomial matrices as well as to describe the desired parameter-dependent LMIs in an easy way, considerably reducing the programming time to end up with implementable LMI conditions. Therefore, ROLMIP helps the popularization of the state-of-the-art robust control methods for uncertain systems based on LMIs among graduate students, researchers, and engineers in control systems.

Robust H∞ Consensus Control for Linear Discrete-Time Swarm Systems with Parameter Uncertainties and Time-Varying Delays

Robust H∞ consensus control problems of linear swarm systems with parameter uncertainties and time-varying delays are investigated. In this literature, a linear consensus protocol for high-order discrete-time swarm systems is proposed. Firstly, the robust H∞ consensus control problem of discrete-time swarm systems is transformed into a robust H∞ control problem of a set of independent uncertain systems. Secondly, sufficient linear matrix inequality conditions for robust H∞ consensus analysis of discrete-time swarm systems are given by the stability theory, and a H∞ performance level γ is determined meanwhile. Thirdly, the convergence result is derived as a final consensus value of swarm systems. Finally, numerical examples are presented to demonstrate theoretical results.