Strict Linear Matrix Inequalities Research Articles

H∞ Control for 2D Singular Continuous Systems

This paper considers the problem of admissibility and admissibilization of 2D singular continuous systems described by the Roesser model. A necessary and sufficient admissibility condition is first proposed for 2D singular continuous systems in terms of a strict Linear Matrix Inequality (LMI). Then, a necessary and sufficient condition is established for the closed-loop system to be admissible (i.e., stable, regular, and impulse-free). Moreover, the stability condition is completed to give a sufficient condition to ensure a specified H∞ disturbance attenuation level for the state feedback closed loop. To illustrate the effectiveness of the proposed methodology, a numerical example is given to illustrate an admissibilization of a state feedback closed-loop system.

Admissibility LMI criteria for descriptor fractional-order systems with a changeable number of decision variables

This paper studies the admissibility problem of descriptor fractional-order systems (DFOSs) with order in (0,1). Firstly, an approach to stability for FOSs is derived which is effective for the case of at least two complex matrix variables. Secondly, based on the generalized linear matrix inequality (GLMI) region, a new admissibility criterion for DFOSs with multiple choices for the number of decision variables is presented, which does not involve complex variables and equality constraints. Moreover, all the theorems are necessary and sufficient conditions, and can be described by strict linear matrix inequalities (LMIs). Finally, two numerical examples are provided to verify the validity of theoretical conclusions.

New Robust Model for Stability and H∞ Analysis for Interconnected Embedded Systems

This paper presents a novel approach to analyzing the robust stability of interconnected embedded systems. The paper starts by discussing the challenges associated with designing stable and robust embedded systems, particularly in the context of interconnected systems. The proposed approach combines the H∞ control theory with a new model for interconnected embedded systems, which takes into account the effects of communication delays and data losses. The paper provides a detailed mathematical analysis of the new model and presents several theorems and proofs related to its stability. The effectiveness of the proposed approach is demonstrated through several practical examples, including a networked control system and a distributed sensor network. The paper also discusses the limitations of the proposed approach and suggests several directions for future research. The proposed filter design method establishes a sufficient condition for the asymptotic stability of the error system and the satisfaction of a predefined H∞ performance index for time-invariant bounded uncertain parameters. This is achieved through the use of the strict linear matrix inequalities (LMI) approach and projection lemma. The design is formulated in terms of linear matrix inequalities (LMI). Numerical examples are provided to demonstrate the effectiveness of the proposed filter design methods.

Quantized guaranteed cost dynamic output feedback control for uncertain nonlinear networked systems with external disturbance

In this article, the guaranteed cost control problem for a class of nonlinear networked control systems (NCSs) with dynamic quantization, external bounded disturbance, and data dropout is investigated via the dynamic output feedback. The Takagi–Sugeno (T-S) fuzzy model is used to represent the discrete-time nonlinear system discussed in this paper. Under the dynamic quantization strategy, two quantizers with dynamic parameters are used to quantize the measured output and control input, and the data dropout process is described by introducing the Bernoulli stochastic variable. By using the quadratic boundedness technique, the quadratic stability of the quantized NCS with external disturbance is described. Furthermore, the sufficient condition for the global design approach are proposed, that is, not only the dynamic output feedback controller but parameters of the quantizers are designed synchronously with strict linear matrix inequalities. Eventually, a simulation of nonlinear mass-spring-damper mechanical system is carried out to verify the effectiveness of the provided algorithm.

Robust fault estimation and proportional derivative fault tolerant control for a class of singular systems with interval time‐varying delay and disturbance

SummaryThis paper deals with observer‐based fault estimation (FE) and proportional derivative state feedback fault tolerant control (FTC) schemes for singular systems affected by actuator fault, interval time‐varying delay and external disturbance. Firstly, the system transformation method of constructing the derivative matrix into a full‐rank matrix is adopted to transform the singular system into a normal system, which effectively eliminates the impulse characteristic of the singular system. Secondly, in the design of the dynamic observer, fault estimation utilizes the real‐time output error and its derivative simultaneously, which can achieve better estimation performance when the fault change frequently. Most importantly, the observer‐based FTC adopts a proportional derivative state feedback control, considers disturbance and interval time‐varying delay, and the introduction of state estimation derivative term can eliminate the impulse property of the closed‐loop system, which optimizes the design scheme of the fault‐tolerant controller. The obtained sufficient conditions for stability can be solved by the strict linear matrix inequality (LMI), and the superiority of the conclusions will be further verified by the simulations.

A new bounded real lemma for discrete‐time linear descriptor systems

AbstractThis paper presents a new bounded real lemma for discrete‐time linear descriptor systems. We first present a necessary and sufficient condition for the admissibility of discrete‐time descriptor systems expressed in terms of strict linear matrix inequalities (LMIs). We then prove that a discrete‐time descriptor system is ‐stable if and only if a symmetric matrix satisfying two strict LMIs can be found. Two illustrative examples are provided for verification.

Robust [formula omitted] dynamic output feedback control for nonlinear time-varying delay rectangular descriptor Markov jump systems

The robust H∞ dynamic output feedback (DOF) control for a class of nonlinear time-varying delay rectangular descriptor Markov jump systems (DMJSs) is discussed in this paper, there are no restrictions on the rows and columns of the differential matrix E. The rectangular DOF controller is designed to make the closed-loop system as square DMJSs. Firstly, by utilizing a mode-dependent and delay-dependent Lyapunov-Krasovskii (L-K) functional and implicit function theorem, sufficient conditions are given to guarantee that the augmented closed-loop systems are robust stochastically admissible with an H∞ performance and have unique solution simultaneously. Then based on certain matrix decoupling techniques, a new strict linear matrix inequality (LMI) sufficient condition for the existence of a rectangular DOF controller is obtained, and the controller is given. Finally, numerical examples are presented to verify the validity of the results.

Alternate Admissibility LMI Criteria for Descriptor Fractional Order Systems with 0 < α < 2

The paper focuses on the admissibility problem of descriptor fractional-order systems (DFOSs). The alternate admissibility criteria are addressed for DFOSs with order in (0,2) which involve a non-strict linear matrix inequality (LMI) method and a strict LMI method, respectively. The forms of non-strict and strict LMIs are brand new and distinguished with the existing literature, which fills the gaps of studies for admissibility. These necessary and sufficient conditions of admissibility are available to the order in (0,2) without separating the order ranges into (0,1) and [1,2). Based on the special position of singular matrix, the non-strict LMI criterion has an advantage in handling the DFOSs with uncertain derivative matrices. For the strict LMI form, a method involving least real decision variables is derived which is more convenient to process the practical solution. Three numerical examples are given to illustrate the validity of the proposed results.

Admissibility and robust stabilization of fractional-order singular discrete systems with interval uncertainties

ABSTRACT This paper investigates the admissibility and robust stabilization of fractional-order singular discrete systems with interval uncertainties. Firstly, based on the analysis of the regularity, causality and stability, novel admissibility conditions for nominal fractional-order singular discrete systems are derived including a necessary and sufficient condition in terms of spectral radius and a sufficient condition in terms of non-strict linear matrix inequalities. In order to eliminate the coupling terms and propose strict linear matrix inequality results, another novel admissibility condition is obtained, which is more tractable and reliable with the available linear matrix inequality software solver and more suitable for the controller design compared with the existing results. Secondly, the state feedback controller synthesis for the fractional-order singular discrete systems with interval uncertainties is addressed and the state feedback controller is designed. Finally, the efficiency of the proposed method is demonstrated by two numerical simulation examples.

Performance analysis for 2D discrete singular stochastic systems

This paper addresses the problem of H∞ performance analysis of 2D discrete singular stochastic system described by Roesser model which is challenging since it involves 2D random variables and the disturbance simultaneously. Sufficient conditions are established for the regularity, causality and stability of the system. The proposed results are expressed in terms of strict linear matrix inequalities. Furthermore, a mean square asymptotic stability with an disturbance level is developed. Simulation example is provided in order to illustrate the relevance of our approach.

Finite-time dissipative control design for one-sided Lipschitz nonlinear singular Caputo fractional order systems

ABSTRACT This paper presents an efficient analytical approach for solving the problem of dissipative control in one-sided Lipschitz singular Caputo fractional-order systems subject to nonlinear perturbations. The approach is based on mathematical transformations, and fractional calculus. A state feedback controller is designed to ensure that the closed-loop system is regular, impulse-free, and finite-time bounded with a dissipativity performance index. The results are established using tractable strict linear matrix inequalities (LMIs) without any equality constraint. Two numerical examples are provided to demonstrate the effectiveness of the proposed approach.

An auxiliary approach to interval observer design for linear systems with disturbances

An auxiliary filtering system approach is adopted in this paper to treat the interval observer design for linear systems with state and output disturbances. By augmenting the filtering system along with the disturbances boundaries, interval observer is built under the unknown input observer framework. Relying on the Lyapunov stability method and H∞ technique, the existence of the interval observer is formulated into strict linear matrix inequalities instead of the combination of linear programming and linear matrix inequalities. Compared with the existing result, the proposed method could render tighter state intervals from simulation studies.

Adaptive sliding mode control of Markov jump systems with completely unknown mode information

AbstractThis paper aims to investigate the problem of adaptive sliding mode control for Markov jump systems with deficient transition probability. Different from the existing literature, we propose a novel method to design a mode‐dependent sliding mode controller when the mode information is completely unknown, while the controller gain parameters can be designed by solving feasible conditions. Firstly, by designing an integral‐type sliding surface, on which an ideal sliding mode dynamics is obtained. Then, for different types of mode transition information, a set of feasible easy‐checking stochastic stability criteria are proposed for the sliding mode dynamics in terms of strict linear matrix inequalities. Further, relying on the parameters obtained from the stability criteria, an adaptive sliding mode controller is successfully designed with respect to different types of system mode transition information. Finally, a numerical example is provided to illustrate the advantage of the developed strategy.

Sliding mode control for discrete-time singular semi-Markovian jumping models

In this paper, the sliding mode control (SMC) is studied for discrete-time uncertain singular semi-Markovian jumping models. The aim is to construct an appropriate SMC strategy such that the stochastic admissibility conditions in the form of strict linear matrix inequalities are established for the corresponding closed-loop system under the framework of the semi-Markovian jumping process. Firstly, considering the particularity of singular systems, a common sliding surface with the singular matrix is designed, which can effectively reduce the influence of repetitive jumping among subsystems on the sliding surface and the numerical problem caused by the existence of the singular matrix. Then, in line with an improved reaching condition, the desired SMC mechanism is synthesized to drive the system state onto the pre-designed sliding surface. Finally, the DC motor model is taken as an example to demonstrate the effectiveness of the proposed SMC approach.

Vehicle Actuator Fault Detection With Finite-Frequency Specifications via Takagi-Sugeno Fuzzy Observers: Theory and Experiments

This article presents a new nonlinear observer-based method to detect the faults of both steering and torque actuators of autonomous ground vehicles. To this end, the nonlinear vehicle system is reformulated in a Takagi-Sugeno (TS) fuzzy model with both measured and unmeasured nonlinear consequents, called N-TS fuzzy model. Differently from the classical TS fuzzy technique with linear consequents, this N-TS fuzzy reformulation enables an effective use of differential mean value theorem to deal with unmeasured nonlinearities, which is known as a major challenge in TS fuzzy observer design. Moreover, the N-TS fuzzy form also allows reducing not only the design conservatism but also the numerical complexity of the design conditions as well as the observer structure, which is crucial for real-time vehicle application. To minimize the disturbance effect with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_\infty$</tex-math></inline-formula> performance and maximize the fault sensibility with an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_-$</tex-math></inline-formula> performance, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> information on the disturbance/fault frequency ranges is taken into account in the N-TS fuzzy observer design via the generalized Kalman–Yakubovich–Popov (KYP) lemma. Based on Lyapunov stability theory, the design of the proposed multiobjective <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_-/\mathscr {H}_\infty$</tex-math></inline-formula> N-TS fuzzy fault detector is recast as an optimization problem with strict linear matrix inequality (LMI) constraints, which can be effectively solved with numerical solvers. Both numerical and experiments are performed under realistic driving conditions to demonstrate the theoretical and practical interests of the new finite-frequency fuzzy fault detection method.

Novel non-fragile extended dissipative synchronization of T-S fuzzy complex dynamical networks with interval hybrid coupling delays

&lt;abstract&gt;&lt;p&gt;This paper presents the first investigation of extended dissipative synchronization in a specific type of Takagi-Sugeno (T-S) fuzzy complex dynamical networks with interval hybrid coupling delays. First, the decoupling method is employed to reorganize the multiple communication dynamical system, which comprises discrete-time, partial and distributed coupling delays. Second, the non-fragile control, which allows for uncertainty management within predefined norm bounds, has been applied to networks. Moreover, it becomes possible to derive a less conservative condition by utilizing multiple integral Lyapunov functionals, a decoupling strategy, Jensen's inequality, Wirtinger's inequality, and mathematical inequality techniques. This condition ensures that the T-S fuzzy complex dynamical networks, with interval hybrid coupling delays, can attain asymptotic synchronization with the assistance of a non-fragile feedback controller. Additionally, we extended this system to the extended dissipativity analysis, including passivity, $ L_2-L_\infty, H_{\infty} $ and dissipativity performance in a unified formulation. A set of strict linear matrix inequalities (LMIs) conditions is a sufficient criterion. Finally, two simulation examples are proposed to verify the merit of the obtained results.&lt;/p&gt;&lt;/abstract&gt;

Quadratic Admissibility for a Class of LTI Uncertain Singular Fractional-Order Systems with 0 &lt; α &lt; 2

This paper provides a unified framework for the admissibility of a class of singular fractional-order systems with a given fractional order in the interval (0, 2). These necessary and sufficient conditions are derived in terms of linear matrix inequalities (LMIs). The considered fractional orders range from 0 to 2 without separating the ranges into (0, 1) and [1, 2) to discuss the admissibility. Moreover, the uncertain system with the fractional order in the interval (0, 2) is norm-bounded. The quadratic admissibility and general quadratic stability of the system are analyzed, and the equivalence between the two is proved. All the above can be expressed in terms of strict LMIs to avoid any singularity problem in the solution. Finally, the effectiveness of the method is illustrated by three numerical examples.

A New Method of Fault Estimation Observer Design for T-S Fuzzy Singular Systems With Disturbances Using Dissipativity Theory

This brief dedicates to addressing the robust fault estimation (FE) problem for a class of T-S fuzzy singular systems (TSFSSs) subject to faults and disturbances. A dissipativity-based FE observer design scheme is proposed to realize fault reconstruction, by which the FE performance can be improved in contrast to the existing one. With the help of the Lyapunov approach, dissipativity theory and matrix transformation technique, a brand-new condition is established to ensure observation error dynamics to be strictly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\mathscr {Q}, \mathscr {U}, \mathscr {R})-\iota -$ </tex-math></inline-formula> dissipative. It is worth mentioning that different from some previous studies, our method can solve observer gains simultaneously in a strict linear matrix inequality (LMI) framework rather than determine some of them artificially. Moreover, to bring extra degrees of freedom, slack scalars and matrices are added in FE observer design. The superiority of our method is fully illustrated via simulations.

Dissipativity‐based adaptive integral sliding mode control of Takagi‐Sugeno fuzzy descriptor systems

This paper focuses on the problem of adaptive integral sliding mode control (ISMC) with dissipativity for a type of complex non-linear system represented by Takagi-Sugeno (T-S) fuzzy descriptor models. First, T-S fuzzy descriptor systems with different derivative matrices are transformed into augmented systems that have the same properties as the original systems, and an integral sliding mode surface function is designed considering the descriptor matrices. Then, based on the dissipativity theory and the non-parallel distributed compensation (non-PDC) method, ISMC strategies and dissipativity-based ISMC strategies are proposed, and the system uncertainties and external disturbances with unknown upper bounds are addressed with an adaptive controller. According to the Lyapunov theory and the dissipativity theory, a fuzzy Lyapunov function is designed, and the obtained criteria are presented in the form of strict linear matrix inequalities (LMIs); this approach guarantees that the closed-loop system is asymptotically stable and strictly ( Q , R , S ) − α $(\mathcal{Q},\mathcal{R},\mathcal{S}) - {{\alpha}}$ dissipative. Moreover, the controller gain matrices and Lyapunov matrices are decoupled by introducing the corresponding auxiliary variables and employing Finsler's lemma. Finally, two examples are provided to illustrate the effectiveness and merit of the proposed method.

Tracking control and disturbance rejection for nonlinear switched control systems with saturation effects

AbstractThis article focuses on the concerns of a tracking control and active disturbance rejection for nonlinear switched systems described by Takagi–Sugeno fuzzy framework with state‐dependent nonlinear perturbations, actuator saturations and disturbances. Especially, a nonlinear equivalent‐input‐disturbance technique is employed to guarantee that only the external disturbances are rejected while retaining the beneficial nonlinearity of the system. Notably, with the aid of the lifting technique, the modified repetitive control are accurately described by a continuous‐discrete two‐dimensional model is designed to improve the tracking precision. The fuzzy‐membership‐function‐dependent piecewise Lyapunov–Krasovskii functional (LKF) by exploiting the knowledge of the membership functions are built to assure the exponential stability of the investigated system. Moreover, the controller and observer gains can be obtained as solutions to a set of strict linear matrix inequalities. Finally, an application example based on the cognitive radio model is given to verify the efficacy of the proposed control protocol.