LMI Stability Conditions Research Articles

LMI Stability Conditions for Nabla Fractional Order Systems With Order α ∈ (0, 2)

LMI Stability Conditions for Nabla Fractional Order Systems With Order α ∈ (0, 2)

Extending the TPTool MATLAB toolbox with LMI based observer and disturbance rejection design

Present paper demonstrates the extensions of MATLAB TPtool toolbox. The toolbox does not include the LMI based observer and disturbance rejection design. Thus, the suggested extensions include the observer and H∞ based disturbance rejection LMI conditions for TP models. Furthermore this toolbox contains obsolete algorithms. Therefore, in this paper, the recommended algorithms are presented. Showing the proposed design method is based on a nonlinear dynamic system; modified TP model of the TORA example is used. Thus, the detailed observer and controller design, the disturbance rejection design are presented with LMI stability conditions through the HOSVD based method in the PDC framework.

LMI Conditions for Stability and State-Feedback H∞ Control of Discrete-Time Multimode Multidimensional Systems

This paper deals with stability and state feedback control of discrete-time Multi-Mode Multi-Dimensional (M3D) linear systems. The M3D switch dynamics are modeled through a state mapping describing the mode transitions. This M3D model framework then allows to consider poly-quadratic Lyapunov functions to obtain Linear Matrix Inequalities conditions for stability proof and for the synthesis of state-feedback controllers under H∞ performance. A numerical example illustrates the improvement of the controller synthesis conditions here introduced for discrete-time M3D systems over independent point-wise mode solutions.

Sub-Predictors and Classical Predictors for Finite-Dimensional Observer-Based Control of Parabolic PDEs

We study constant input delay compensation by using finite-dimensional observer-based controllers in the case of the 1D heat equation. We consider Neumann actuation with nonlocal measurement and employ modal decomposition with N+1 modes in the observer. We introduce a chain of M sub-predictors that leads to a closed-loop ODE system coupled with infinite-dimensional tail. Given an input delay r, we present LMI stability conditions for finding M and N and the resulting exponential decay rate and prove that the LMIs are always feasible for any r. We also consider a classical observer-based predictor and show that the corresponding LMI stability conditions are feasible for any r provided N is large enough. A numerical example demonstrates that the classical predictor leads to a lower-dimensional observer. However, it is known to be hard for implementation due to the distributed input signal.

Stability analysis of a class of uncertain switched time‐delay systems with sliding modes

SummaryThis paper presents sufficient conditions to ensure the exponential stability of uncertain switched time‐delay systems with time‐delayed state‐dependent switching conditions, which can model high speed supercavitation vehicles (HSSVs). Based on the intrinsic properties of HSSVs, cavity can be represented by a relation between the states and time‐delayed states of the system. Moreover, a sliding behavior can be occurred at the cavity boundary in HSSVs. Therefore, the main contribution of this paper is to develop a set of LMI stability conditions in the presence of time‐delayed states in the linear state‐dependent switching signal and absorbing sliding modes in the state space, simultaneously. In addition, because of the possible existing uncertainties in HSSVs, another set of LMI‐based sufficient conditions is derived in order to analyze the stability of switched time‐delay systems with additive norm bounded uncertainties. The application of the proposed theorems are also validated through numerical examples. Finally, the stability of a two‐degrees‐of‐freedom longitudinal description of an HSSV is shown using the proposed stability theorems.

LMI stability conditions and stabilization of fractional-order systems with poly-topic and two-norm bounded uncertainties for fractional-order α: the 1 < α < 2 case

This article addresses the problem of robust stability and stabilization for linear fractional-order system with poly-topic and two-norm bounded uncertainties, and focuses particularly on the case of a fractional order α such that 1 < α < 2. First, the robust asymptotical stable condition is presented. Second, the design method of the state feedback controller for asymptotically stabilizing such uncertain fractional order systems is derived. In the proposed approach, linear matrix inequalities formalism is used to check and design. Lastly, two simulation examples are given to validate the proposed theoretical results.

LMI Stability Conditions for 2D Roesser Models

This note is devoted to the stability analysis of linear two-dimensional systems described by continuous, discrete or mixed Roesser models. Commonly used polynomial-based tests of stability are reduced to that of solving a set of linear matrix inequalities. The main contribution is that the approach is very general and the criteria, under certain hypothesis, are not conservative.

A Less Conservative LMI Condition for Stability of Discrete-Time Systems With Slope-Restricted Nonlinearities

Many conditions have been found for the absolute stability of discrete-time Lur'e systems in the literature. It is advantageous to find convex searches via LMIs where possible. In this technical note, we construct two less conservative LMI conditions for discrete-time systems with slope-restricted nonlinearities. The first condition is derived via Lyapunov theory while the second is derived via the theory of integral quadratic constraints (IQCs) and noncausal Zames-Falb multipliers. Both conditions are related to the Jury-Lee criterion most appropriate for systems with such nonlinearities, and the second generalizes it. Numerical examples demonstrate a significant reduction in conservatism over competing approaches.

Sufficient LMI stability conditions for Lur’e type systems governed by a control law designed on their Euler approximate model

In this paper, the stability issue of Lur’e systems governed by a control law stabilising their forward Euler approximate model is investigated. More specifically, the considered control law is obtained by exploiting the advantages of a new Lur’e type Lyapunov function with disconnected level sets. This Lyapunov function is adapted to discrete-time Lur’e systems and to the structure of the forward Euler approximate model. The main result consists of linear matrix inequality conditions allowing to guarantee that the continuous-time Lur’e system associated with the proposed digital control law is globally asymptotically stable. The relevance of this approach is illustrated using a numerical example.

LMI stability conditions for uncertain rational nonlinear systems

SUMMARYThis paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non‐convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S‐Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. Copyright © 2013 John Wiley &amp; Sons, Ltd.

A Nonconservative LMI Condition for Stability of Switched Systems With Guaranteed Dwell Time

Ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems. Several methods have been proposed in the literature to address this problem, but unfortunately they provide sufficient conditions only. This technical note proposes the use of homogeneous polynomial Lyapunov functions in the non-restrictive case where all the subsystems are Hurwitz, showing that a sufficient condition can be provided in terms of an LMI feasibility test by exploiting a key representation of polynomials. Several properties are proved for this condition, in particular that it is also necessary for a sufficiently large degree of these functions. As a result, the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well. Some examples illustrate the proposed approach.

Spherical domain of attraction estimation for nonlinear systems

This work proposes an algorithm to obtain a predefined-shape zone as large as possible, belonging to the local domain of attraction of a nonlinear system. In order to do it, fuzzy polynomial models will be used which analysis can be done by convex optimization (sum of squares). Most fuzzy control papers check LMI or SOS stability conditions for Takagi-Sugeno fuzzy models and, if feasibility is reached, forget to study the obtained region and its relationship with the original nonlinear problem. Having into account the shape of the membership functions in a particular region of interest, less conservative stability and stabilization conditions can be easily set up for doing iterations with the fuzzy modelling region in order to maximize the proved basin of attraction.

Decentralized stabilization of interconnected switched descriptors via a multiple switched Lyapunov functional

The decentralized stabilization of large scale switched descriptor systems under arbitrary switching laws is proposed in this paper. A large scale switched descriptor can be split into a set of smaller interconnected switched descriptors. Then, a set of switched controllers is employed to ensure the stabilization of the considered class of large scale descriptors. The proposed LMI stability conditions are obtained from a multiple switched Lyapunov-like candidate function. A numerical example is given to illustrate the effectiveness of the designed approach.

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>

STABILITY AND STABILIZATION OF A CLASS OF UNCERTAIN NONLINEAR DISCRETE-TIME SYSTEMS WITH SATURATING ACTUATORS

This paper presents some results on stability and stabilization of a class of uncertain nonlinear discrete-time systems under control saturations. The studied control law consists of the feedback of both the states and of the nonlinearity present in the dynamics of the controlled system. Saturations are taken into account by modeling the nonlinear saturated system through a deadzone nonlinearity satisfying a modified sector condition. Thus, as for precisely known systems, LMI stability and stabilization conditions are proposed, which can be cast into convex programming problems. Some relations of the proposed results with the poly-quadratic stability concept are included.

Stability region enlargement through anti-windup strategy for linear systems with dynamics restricted actuator

This paper addresses the problem of determination of stability regions for linear systems with amplitude and successive dynamics restricted actuator through anti-windup strategies. Considering a linear dynamic output feedback designed with respect to the linear system (without saturation), an anti-windup design method is investigated to guarantee both the stability of the closed-loop system and the respect of the controlled output constraints for a region of admissible initial states as large as possible. Based on the modelling of the closed-loop system resulting from the controller plus the anti-windup loop as a linear system with dead-zone and dynamics restricted nonlinearities, LMI stability conditions are formulated. Numerical optimization procedures are discussed.

Delay-dependent LMI conditions for stability and stabilization of T–S fuzzy systems with bounded time-delay

This paper studies the stability and stabilization problems for time-delay nonlinear systems through Takagi–Sugeno (T–S) fuzzy model approach. The time delays are assumed to be bounded but perhaps non-smooth. Delay-dependent conditions are presented in terms of LMIs. Illustrative examples are given to show the effectiveness and merits of the present results.

Anti-windup strategy with guaranteed stability for linear systems with amplitude and dynamics restricted actuator

This paper addresses the problem of the determination of regions of stability for linear systems with amplitude and successive dynamics restricted actuator through anti-windup strategies. Considering a linear dynamic output feedback designed with respect to the linear system (without saturation), an anti-windup design method is investigated to guarantee both the stability of the closed-loop system and the respect of the controlled output constraints for a region of admissible initial states as large as possible. Based on the modeling of the closed-loop system resulting from the controller plus the anti-windup loop as a linear system with dead-zone and dynamics restricted nonlinearities, LMI stability conditions are formulated. Numerical optimization procedures are discussed.

Positive polynomial matrices and improved LMI robustness conditions

Recently, several new LMI conditions for stability of linear systems have been proposed, introducing additional slack variables to reduce the gap between conservative convex quadratic stability conditions and intractable non-convex robust stability conditions. In this paper, we show that these improved LMI conditions can be derived with the help of some basic results on positive polynomial matrices. The approach allows us to derive in a unifying way results in the state-space and polynomial frameworks. Applications to robust stability analysis and robust stabilization of systems with multi-linear parametric uncertainty are fully described.

Rational multiplier IQCs for uncertain time-delays and LMI stability conditions

Describes a set of delay-dependent integral quadratic constraint (IQC) stability conditions for time-delay uncertainty. The IQCs are linearly parameterized in terms of a pair of rational stability multipliers, each active over one of a pair of complementary frequency intervals. Using the finite-frequency positive real lemma, each of these finite-frequency IQC conditions are shown to be equivalent to a frequency-independent linear matrix inequality condition, thereby dispensing with the need for frequency-sweeping.