LMI Region Research Articles

Characterizing matrices with eigenvalues in an LMI region: a dissipative-Hamiltonian approach

In this paper, we provide a dissipative Hamiltonian (DH) characterization for the set of matrices whose eigenvalues belong to a given LMI region. This characterization is a generalization of that of Choudhary et al. (Numer. Linear Algebra Appl, 2020) to any LMI region. It can be used in various contexts, which we illustrate on the nearest Ω-stable matrix problem: given an LMI region Ω ⊆ C and a matrix A ∈ R n × n , find the nearest matrix to A whose eigenvalues belong to Ω. Finally, we generalize our characterization to more general regions that can be expressed using LMIs involving complex matrices.

Generalized Criteria for Admissibility of Singular Fractional Order Systems

Unified frameworks for fractional order systems with fractional order 0<α<2 are worth investigating. The aim of this paper is to provide a unified framework for stability and admissibility for fractional order systems and singular fractional order systems with 0<α<2, respectively. By virtue of the LMI region and GLMI region, five stability theorems are presented. Two admissibility theorems for singular fractional order systems are extended from Theorem 5, and, in particular, a strict LMI stability criterion involving the least real decision variables without equality constraint by isomorphic mapping and congruent transform. The equivalence between the admissibility Theorems 6 and 7 is derived. The proposed framework contains some other existing results in the case of 1≤α<2 or 0<α<1. Compared with published unified frameworks, the proposed framework is truly unified and does not require additional conditional assignment. Finally, without loss of generality, a unified control law is designed to make the singular feedback system admissible based on the criterion in a strict LMI framework and demonstrated by two numerical examples.

Generalization of the concept of diagonal dominance with applications to matrix D-stability

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D⊂C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D. Moreover, we show that in some cases, diagonal D-dominance implies (D,D)-stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

The Problem of Generalized D-Stability in Unbounded LMI Regions and Its Computational Aspects

We generalize the concepts of D-stability and additive D-stability of matrices. For this, we consider a family of unbounded regions defined in terms of Linear Matrix Inequalities (so-called LMI regions). We study the problem when the localization of a matrix spectrum in an unbounded LMI region is preserved under specific multiplicative and additive perturbations of the initial matrix. The most well-known particular cases of unbounded LMI regions (namely, conic sectors and shifted halfplanes) are considered. A new D-stability criterion as well as sufficient conditions for generalized D-stability are analyzed. Several applications of the developed theory to dynamical systems are shown.

Robust Stabilization in LMI Regions for Constrained Input Systems

Stabilization of constrained uncertain linear systems in LMI regions is studied. Robustness is in demand and exact pole assignment is no longer suitable for uncertain systems. Therefore, LMI region pole assignment encompasses the exact one preserving time domain performances in the presence of uncertainties. Robust constrained state feedback controllers are designed such that asymptotic stability and desired performances are achieved. Constraints are assumed symmetric and the LMI regions are closed under complex conjugacy. Hence, LMI’s are deduced such that robust pole assignment is obtained and constraints on the input are respected. Namely, two design strategies are treated: the unsaturating and the saturating case. Optimization problem for enlarging the initial conditions set is also achieved. Simulation examples are given to show the application and the ease of the proposed approach. Comparison with different approaches for both given examples is also presented.

H-infinity controller with graphical LMI region profile for liquid slosh suppression

This paper presents a H-infinity synthesis with pole clustering based on LMI region schemes for liquid slosh control. Using LMI approach, the regional pole placement known as LMI region combined with design objective in H-infinity controller guarantee a fast input tracking capability and very minimal liquid slosh. A graphical profile of the transient response of liquid slosh suppression system with respect to pole placement is very useful in giving more flexibility to the researcher in choosing a specific LMI region. With the purpose to confirm the design of control scheme, a liquid slosh model is considered to represent the lateral slosh movement. Supremacy of the proposed approach is shown by comparing the results with hybrid model-free fuzzy-PID controller with derivative filter. The performance of the control schemes is examined in terms of time response specifications of lateral tank tracking capability and level of liquid slosh reduction.

Robust PI, gain scheduled and PSS controller design using LMI regions: Comparative studies

Abstract This paper considers development of a new design procedure for PI and Gain scheduled robust controller with PSS for turbogenerator using LMI regional regions. The obtained PI+PSS and GSC+PSS robust controllers with output and state derivative feedback for uncertain polytopic turbogenerator model ensure that all closed -loop eigenvalues lay in the prescribed LMI region. The proposed method is to guarantee the less conservativeness with respect to previous methods due to introducing a design procedure based on new auxiliary matrices. The effectiveness of this procedure is illustrated by the examples of two controllers for turbogenerator.

DR-Admissibility With Static Output Feedback For Uncertain Descriptor Fuzzy System

In this paper, the issue of DR-admissibility analysis and synthesis for a class of uncertain continuous fuzzy descriptor systems is investigated. The descriptor Takagi-sugeno T-S fuzzy model with parameter uncertainties is adopted for admissibility analysis and control design. Using linear matrix inequality approach and introducing some free matrices, sufficient criterion is established to guarantee the admissibility and DR-stability of the considered class of nonlinear systems. From this new characterization, fuzzy static output feedback controller is designed not only to ensure the admissibility of the closed loop descriptor systems but also the transient performances in a specified LMI region. An example is introduced to illustrate the effectiveness of the proposed condition.

Fuzzy regional pole placement based on fuzzy Lyapunov functions

This paper studies the regional pole placement problem for a class of T–S fuzzy systems. Firstly, a definition for the fuzzy systems to be D stable is given from the viewpoint of Lyapunov functions. Then a new sufficient condition is proposed to guarantee all the poles of the fuzzy systems located within a prescribed LMI region by using the fuzzy Lyapunov functions method and introducing some free matrices. And then the controller design approach is given by solving a set of LMIs. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

Pole placement with LMI constraint of fuzzy descriptor system

This paper investigates the pole placement problem for a class of fuzzy descriptor system. Firstly, based on the fuzzy Lyapunov function approach and with the introduction of some free matrices, a new sufficient condition for the fuzzy descriptor system to be D stable is presented. Then based on this new condition, a controller design method is proposed to guarantee the poles of the closed-loop system lying in a given LMI region. At last, numerical examples are given to show the efficiency of the proposed approach.

Quadratic Design of D-stabilizing Non-PDC Controllers for Quasi-LPV/T-S Models

This paper deals with the D-stabilization of quasi-LPV/Takagi-Sugeno (T-S) models. Based on a Quadratic Lyapunov Function (QLF) candidate, basic conditions are presented for the design of D-stabilizing Parallel Distributed Compensation (PDC) control, i.e with LMI region constraints. Then, relaxed quadratic LMI conditions are proposed in order to design a D-stabilizing non-PDC controller. The proposed results are illustrated in simulation through an academic example, then through the benchmark of the stabilization of a flexible robot with single joint.

Robust State Feedback Controller Design for DC-Motor System

This paper proposes an approach to a robust state feedback controller design for real uncertain DC-motor system in time domain. Because only output vector of this real system is measurable, it is necessary to observe state vector. Polytopic model of uncertain system is considered as a plant model to be controlled. Stability and performance of the closed loop system is determined through the LMI region, where the closed loop poles of whole uncertainty domain are placed. To reduce the conservativeness of quadratic stability approach the parameter-dependent Lyapunov function (PDLF) is used.

H-Infinity Controller Based on LMI Region for Flexible Robot Manipulator

This study presents investigations into the development of H-infinity controller with LMI region schemes for trajectory tracking and vibration control of a Flexible Robot Manipulator System. A constrained planar single-link flexible manipulator is considered and the dynamic model of the system is derived using the assume mode method. H-infinity synthesis with pole clustering schemes is used to control the hub angle with very minimal vibration at the end-point of the manipulator. The performance of the flexible robot manipulator system is examined in terms of time response specifications of hub angle, minimal vibration at the end-point of the manipulator and minimal control input torque. To examine the effectiveness of the proposed controller, it is compared with hybrid input shaping with collocated PD controller schemes. The implementation results show that H-infinity controller with confined closed-loop poles based on LMI region produce a fast tracking capability with very minimal end-point vibration and smooth control input torque.

H-Infinity Controller with LMI Region Schemes for a Lab-Scale Rotary Pendulum Crane System

This study presents an H-infinity controller with LMI region schemes for a lab-scale rotary pendulum crane system. H-infinity synthesis with pole clustering schemes is used to control the hub angle of the arm of rotary pendulum crane system with very minimal sway angle. The performance of the lab-scale rotary pendulum crane system is examined in terms of time response specifications of hub angle of the arm, minimal sway angle of pendulum and minimal control input. To examine the effectiveness of the proposed controller, it is compared with LQR controller schemes. The implementation results show that H-infinity controller with confined closed-loop poles based on LMI region produce a fast tracking capability with very minimal sway and control input.

Vehicle spacing control using robust fuzzy control with pole placement in LMI region

This work presents a robust pole placement in an LMI region for Takagi–Sugeno (TS) (Takagi T., Sugeno M., 1985. Fuzzy identification of systems and its application to modelling and control. IEEE Transactions on SMC 15 (1), 116–132.) fuzzy models. The objective is to find a set of linear matrix inequalities in order to ensure that the linear model poles of the nonlinear TS model remain in a specified region of the complex plane, even in presence of model uncertainties. As an illustration, the obtained conditions are applied to the spacing policy control of an automated electric vehicle. Simulation and real-time results are presented.

MISSILE AUTOPILOT DESIGN USING OBSERVER-BASED GAIN SCHEDULING

This paper deals with the problem of output tracking for a missile axis autopilot. For a small number of plant linearizations across the missile's operating domain, linear output feedback controllers are first designed using H∞ optimization with pole placement constraints: these controllers incorporate integral action while maintaining the eigenvalues of the closed loop system in a convex LMI region. The controllers are afterwards converted to observer based ones using Youla parameterization and their gains are scheduled, providing a nonlinear controller to cover the entire operating domain. To complete such a nonlinear control scheme, an open loop control using only output reference information is added, along with a feedforward controller to reduce output overshoot while tracking reference signals.

ROBUST POLE PLACEMENT IN A LMI REGION FOR TAKAGI-SUGENO MODELS: APPLICATION TO VEHICLE SPACING CONTROL

Robust pole placement in a LMI region for Takagi-Sugeno (Takagi & Sugeno 1985) fuzzy models is presented. The objective is to find a set of Linear Matrix Inequalities in order to ensure that the linear models poles remain in a specified region of the complex plane, even in presence of model uncertainties. As an illustration, the obtained conditions are applied to the spacing policy control of an automated electric vehicle. Simulation and real time results are presented.

Pole placement in lmi region with takagi-sugeno fuzzy systems

A fuzzy state feedback control law that places the poles of the closedloop subsystems in Takagi-Sugeno (TS) fuzzy systems in a convex region is proposed in this paper. This convex region is defined by the intersection of three elementary regions: an α-stability region, a disk, and a conic sector, mathematically characterized by linear matrix inequalities (LMIs). Sufficient conditions are expressed in terms of LMls, and the resulting formulation is therefore numerically tractable via LMl optimization. The validity and application of this approach are illustrated on the inverted pendulum and the mass-spring-damper examples.

LPV techniques for control of an inverted pendulum

The authors present a comprehensive application of linear fractional transformation and polytopic control techniques to the control of an arm-driven inverted pendulum, The particular interest of this application lies in the fact that all ingredients of the design problem have to be taken into account; from the specifications up to the constraints inherent to real-world implementations. In this context, it has been shown that currently available synthesis methodologies, such as /spl mu/ and LPV techniques, may fail to provide acceptable answers, A major obstacle is undoubtedly the implementation constraint that puts hard limitations on the controller dynamics, These limitations are generally difficult to handle within the usual formulation of LPV control techniques. It has been shown that a suitable extension of these techniques including LMI region constraints on the closed-loop dynamics can overcome this difficulty. When implementable, it has been observed that LPV controllers outperform fixed /spl mu/ controllers both in robustness and performance. These observations were confirmed by simulations but more importantly by a number of records on the physical experiment.