LMI Conditions Research Articles

Delay‐dependent LMI condition for global asymptotic stability of discrete‐time uncertain state‐delayed systems using quantization/overflow nonlinearities

AbstractA delay‐dependent criterion for the global asymptotic stability of a class of uncertain discrete‐time state‐delayed systems using various combinations of quantization and overflow nonlinearities is presented. The proposed criterion is in the form of a linear matrix inequality and, hence, computationally tractable. A numerical example highlighting the usefulness of the proposed criterion is given. Copyright © 2010 John Wiley & Sons, Ltd.

Guaranteed Cost Output Feedback Control of Fuzzy Systems via LMI Approach

This paper is concerned with the stability with guaranteed cost for a fuzzy system with immeasurable premise variables via output feedback. It is well known that Takagi-Sugeno fuzzy model describes a wide class of nonlinear systems especially when its premise variables include immeasurable functions. However, when it comes to output feedback control design of such a fuzzy system, a conventional Parallel Distributed Compensator (PDC) is not feasible because the PDC shares the same immeasurable premise variables as those of a fuzzy system. In this paper, we introduce an output feedback controller with the estimate of the premise variables of an original fuzzy system. We then formulate the stabilization problem with guaranteed cost for a fuzzy system with immeasurable premise variables. Our control design method is based on a set of strict LMI conditions. No tuning parameter is necessary a priori to solve them. The stability with guaranteed cost takes care of not only stabilization but also control performance. Our proposed method attempts to minimize the upper bound of the performance index, which results in the satisfactory trajectories of the system. Finally, numerical examples are given to illustrate our control design method.

Stability and [formula omitted] objective characterizations in control: A comparative study

This paper is concerned with stability and H 2 objective characterizations in control. New forms of alternative LMI characterizations are presented for continuous-time stability and H 2 objective. A key relationship between the alternative characterizations for continuous-time systems and the existing discrete-time versions is illustrated. The existing enhanced LMI characterization for continuous-time systems, called the reciprocal projection lemma, is revisited and an alternative proof is provided with a new interpretation. An ϵ -embedded LMI characterization is presented for a comparative study with the reciprocal projection lemma and for numerical analysis on the conservativeness of synthesis conditions. Through an application in control, it is discussed that, when applied to multi-objective problems in control, the alternative characterizations in this paper lead to an LMI solution with further significant conservativeness reduction, compared to the reciprocal projection lemma. In addition, through the comparison with the existing dilated LMI condition, it is shown that the dilated condition is an equivalent continuous-time counterpart of the existing discrete-time versions.

영구자석 동기전동기의 비선형 속도 제어기 및 퍼지토크관측기 설계에 대한 연구

This paper proposes a new nonlinear speed controller with a fuzzy load torque observer based on the Takagi-Sugeno fuzzy method for a permanent magnet synchronous motor(PMSM). The LMI conditions are derived for the existence of the proposed nonlinear speed controller and fuzzy load torque observer, and the LMI parameterization to obtain the gain matrices of the controller and observer is given. In this paper, to verify the performance of the proposed nonlinear speed controller and fuzzy load torque observer, and the simulation and experimental results are demonstrated under motor parameter and load torque variations.

Stability analysis for linear systems with input backlash through sufficient LMI conditions

This paper addresses the problem of stability analysis for a given class of nonlinear systems resulting from the connection of a linear system with an isolated backlash operator. Constructive conditions based on LMIs to ensure closed-loop stability are proposed by using some suitable Lyapunov functionals with quadratic terms and Lure type terms, and generalized sector conditions. Additionally, the boundary of the associated set of all the admissible equilibrium points is precisely defined.

Parameter-dependent Lyapunov function approach to robust [formula omitted] filter design for uncertain time-delay systems

An LMI-based method for robust L 2 – L ∞ filter design is proposed for polytopic uncertain time-delay systems using a parameter-dependent Lyapunov function approach. Based on a new characterization, a strict LMI condition for L 2 – L ∞ filter design is obtained, which does not involve any iterations for design parameter search, any couplings between the system matrices and the Lyapunov function matrix, nor any system-dependent filter parametrization. Therefore, the proposed condition enables us to easily employ, with the help of efficient numerical solvers, a parameter-dependent Lyapunov function approach to robust L 2 – L ∞ filter design for uncertain time-delay systems. A numerical example is presented to show the effectiveness of the proposed method.

Controller synthesis for networked control systems

This paper presents a discrete-time model for networked control systems (NCSs) that incorporates all network phenomena: time-varying sampling intervals, packet dropouts and time-varying delays that may be both smaller and larger than the sampling interval. Based on this model, constructive LMI conditions for controller synthesis are derived, such that stabilizing state-feedback controllers can be designed. Besides the proposed controller synthesis conditions a comparison is made between the use of parameter-dependent Lyapunov functions and Lyapunov–Krasovskii functions for stability analysis. Several examples illustrate the effectiveness of the developed theory.

PDC Control Design for Non-holonomic Wheeled Mobile Robots with Delayed Outputs

This paper presents a new technique for tracking-error model-based Parallel Distributed Compensation (PDC) control for non-holonomic vehicles where the outputs (measurements) of the system are delayed and the delay is constant. Briefly, this technique consists of rewriting the kinematic error model of the mobile robot tracking problem into a TS fuzzy representation and finding a stabilizing controller by solving LMI conditions for the tracking-error model. The state variables are estimated by nonlinear predictor observer where the outputs are delayed by a constant delay. To illustrate the efficiency of the proposed approach a comparison between the TS fuzzy observer and the nonlinear predictor observer is shown. For this study the reference trajectory is built by taking into account the acceleration limits of the mobile robot. All experiments are implemented on simulation and the real-time platform.

Robust Track-Following Controller Design in Hard Disk Drives Based on Parameter Dependent Lyapunov Functions

This paper presents a novel technique for designing robust track-following output-feedback controllers in hard disk drives (HDDs). In this paper, the manufacturing variations of HDDs are modeled as polytopic parametric uncertainties in linear time-invariant discrete-time systems. For this model, the robust track-following control problem is formulated as the worst-case H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> performance optimization. The optimization problem reduces to the one with bilinear matrix inequalities (BMIs), using the parameter dependent Lyapunov functions and the extended LMI condition introduced by de Oliveira. Although the formulated problem is nonconvex, and thus it is difficult to ensure global optimality, a numerical technique called ¿G- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> iteration¿ is applied for optimization to guarantee monotonic non-increase of the worst-case performance during iterations. The proposed design technique will be useful in improving the track-following performance, and thus increasing the storage capacity of HDDs.

Improved LMI conditions for H ∞ output feedback stabilization of linear discrete-time systems

The problem of H∞ output feedback control for discrete-time systems is investigated in this paper. The main contribution is to provide a uniform framework for generating sufficient linear matrix inequality conditions. Those conditions are classified into two parallel parts based on the way of slack variable selection. Moreover, it is shown that the existing result is a special case of the new conditions by taking few of the additional free matrix parameters to be zero. This directly leads to more flexibility and less conservativeness in the H∞ output feedback control design. Numerical examples are carried out for illustration.

Design of a Fuzzy Speed Controller for a Permanent Magnet Synchronous Motor

This paper proposes a new fuzzy speed controller based on the Takagi-Sugeno fuzzy method to achieve a robust speed control of a permanent magnet synchronous motor(PMSM). The proposed controller requires the information of the load torque, so the second-order load torque observer is used to estimate it. The LMI condition is derived for the existence of the proposed fuzzy speed controller, and the LMI parameterization to calculate the gain matrices of the controller is provided. It is proven that the augmented control system including the fuzzy speed controller and the load torque observer is exponentially stable. To evaluate the performance of the proposed fuzzy speed controller, the simulation and experimental results are presented under motor parameter and load torque variations. Finally, it is clearly verified that the proposed control method can be used to accurately control the speed of a permanent magnet synchronous motor.

LMI Approach to Output Feedback Stabilization for Fuzzy Systems with Immeasurable Premise Variables

This paper is concerned with output feedback control design for a fuzzy system with immeasurable premise variables. It is well known that Takagi-Sugeno fuzzy model describes a wide class of nonlinear systems especially when its premise variables include immeasurable functions. However, when it comes to control design of such a fuzzy system with immeasurable premise variables, a conventional parallel distributed compensator (PDC) is not feasible because it shares the same premise variables as those of a fuzzy system. In this paper, we introduce an output feedback controller with the estimate of the premise variables of an original fuzzy system. We then formulate the stabilization problem for a fuzzy system with immeasurable premise variables. Our control design method is based on a set of strict LMI conditions. No tuning parameter is necessary a priori to solve LMI conditions. Our method includes tuning matrices for control gains in a controller and hence they can be chosen to optimize the control performance of the system. Numerical examples are finally given to illustrate our control design method.

New Dilated LMI Characterization for the Multiobjective Full-Order Dynamic Output Feedback Synthesis Problem

This paper introduces new dilated LMI conditions for continuous-time linear systems which not only characterize stability and performance specifications, but also, performance specifications. These new conditions offer, in addition to new analysis tools, synthesis procedures that have the advantages of keeping the controller parameters independent of the Lyapunov matrix and offering supplementary degrees of freedom. The impact of such advantages is great on the multiobjective full-order dynamic output feedback control problem as the obtained dilated LMI conditions always encompass the standard ones. It follows that much less conservatism is possible in comparison to the currently used standard LMI based synthesis procedures. A numerical simulation, based on an empirically abridged search procedure, is presented and shows the advantage of the proposed synthesis methods.

Decentralized Robust Control of Linear Uncertain Systems with Reduced Conservatism

In this paper a novel decentralized robust control design scheme is proposed, considering the structured uncertain system matrix with a constant part (typical case for a system with dynamic controller as PID); the resulting robust control design is based on developed robust stability LMI condition which is less conservative than those from literature. In the following we use parameter dependent Lyapunov function and structured auxiliary matrices to receive “structured robust stability condition” which is the base for robust controller design. All developments are performed in general formulation including both discrete-time and continuous-time systems.

Delay-interval dependent robust stability criteria for stochastic neural networks with linear fractional uncertainties

In this paper, we study the delay-interval dependent robust stability criteria for stochastic neural networks with linear fractional uncertainties. The time-varying delay is assumed to belong to an interval and is a fast time-varying function. The uncertainty under consideration includes linear fractional norm-bounded uncertainty. Based on the new Lyapunov–Krasovskii functional, some inequality techniques and stochastic stability theory, delay-interval dependent stability criteria are obtained in terms of linear matrix inequalities. Finally, some numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed LMI conditions.

Stability of Networked Control Systems With Uncertain Time-Varying Delays

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this technical note, a new approach for the stability analysis and controller synthesis of networked control systems (NCSs) with uncertain, time-varying, network delays is presented. Based on the Jordan form of the continuous-time plant, a discrete-time representation of the NCS is derived. Using this model for delays that can be both smaller and larger than the sampling interval, sufficient LMI conditions for stability and feedback stabilization are proposed. The results are illustrated by a typical motion control example. </para>

Delay-scheduled state-feedback design for time-delay systems with time-varying delays—A LPV approach

This paper is concerned with the synthesis of delay-scheduled state-feedback controllers which stabilize linear systems with time-varying delays. In this framework, it is assumed that the delay is approximately known in real-time and used in the controller in a scheduling fashion. First, a new model transformation turning a time-delay system into an uncertain LPV system is introduced. Using this transformation, a new delay-dependent stability test based on the so-called full block S -procedure is developed and from this result, a new delay-dependent stabilization result is derived. Since the resulting LMI conditions depend polynomially on the parameters, a relaxation result is then applied in order to obtain a tractable finite set of finite-dimensional LMIs. The interest of the approach resides in (1) the synthesis of a new type of controller scheduled by the delay value which has a lower memory consumption than controllers with memory (since it is not necessary to store past values of the state), and (2) an easy consideration of uncertainties on the delay knowledge.

Lag synchronization for fuzzy chaotic system based on fuzzy observer

A new fuzzy observer for lag synchronization is given in this paper. By investigating synchronization of chaotic systems, the structure of drive-response lag synchronization for fuzzy chaos system based on fuzzy observer is proposed. A new lag synchronization criterion is derived using the Lyapunov stability theorem, in which control gains are obtained under the LMI condition. The proposed approach is applied to the well-known Chen’s systems. A simulation example is presented to illustrate its effectiveness.

逐次 LMI 化手法と双対反復法の関係について

逐次 LMI 化手法と双対反復法の関係について

Static Output Feedback Control for Discrete-time Piecewise Linear Systems: an LMI Approach

This paper investigates the problem of static output feedback (SOF) control for discrete-time piecewise linear systems. Based on piecewise quadratic Lyapunov functions, new sufficient LMI conditions for the synthesis of SOF stabilization controllers are presented. Meanwhile, by using Finsler's lemma, a set of slack variables with special structure are introduced to reduce design conservatism. Compared to the existing methods, the proposed method has a good performance and can work successfully in situations where the existing methods fail. An extension of this method is also given in order to incorporate H ∞ performance. Three examples are given to illustrate the effectiveness of our method.