Solution Of Linear Matrix Inequalities Research Articles

LMI solutions for H-two and H-infinity decentralized controllers applied to an aerothermic process

In this paper, we present a linear matrix inequality (LMI)-based solution to implement H-two and Hinfinity decentralized robust control strategies. Appropriate parametrization of optimal H-two and H-infinity controllers is used. The general formulation of the decentralized control design leads to the optimal determination of both the state feedback gains and the observer gains of the decentralized controllers. This formulation is two folds: first, a centralized controller is obtained, and then, a simplified decentralized solution is derived by optimizing only the observer gains. The mathematical determination of these gains is formulated as an LMI optimization problem that can be easily solved using LMI solvers. As an experimental evaluation of these controllers, a real time application to an aerothermic process is carried out. A continuous-time model of the process obtained with a suitable direct continuous-time identification approach is elaborated. Results illustrating the real performance obtained from the H-two and H-infinity decentralized controllers are discussed and compared with the centralized ones.

Integrated Design of Feedback Controller and UIO: An LMI Solution

Fault diagnosis in a closed-loop gas turbine engine control system poses a great challenge since that the involved controller is to keep system stability, which desensitizes fault diagnostic performance. In this paper, a design algorithm that integrates various feedback controllers and unknown input observer (UIO) has been proposed. In this method, the controller has been considered as the state of an augmented model by combining the controller and the gas turbine model. Then the UIOs for fault detecting and isolating has been designed for this augmented model to obtain the parameters of the concerned controller and UIOs simultaneously. Moreover, this integrated design process has been formulated as linear matrix inequalities (LMIs). One notable feature of this algorithm is that the integrated design can weaken the effect of a feedback controller on output residual generator and maximize the robustness of UIO against unknown disturbances. Experimental results on a gas turbine engine illustrate the effectiveness and applicability of the proposed algorithm.

Dissipative Output Tracking Control of Linear Systems with Time Delay

The problem of dissipative output tracking control of linear systems with time delay is investigated. Firstly, an augmented system is constructed to describe dissipative output tracking control error, and the concept of dissipative output tracking is defined. Based on this, some sufficient conditions are derived in terms of linear matrix inequalities (LMIs) technique, which ensure that the augmented system is dissipative and stable; then design methods of dissipative output tracking state-feedback controller are provided, and the desired controller gain can be expressed through the solutions of LMIs. Finally, a numerical simulation example is given to illustrate the validity of the proposed results.

Delay-range-dependent robust stabilisation and H ∞ control for nonlinear uncertain stochastic fuzzy systems with mode-dependent time delays and Markovian jump parameters

This article focuses on the problems of robust stabilisation and H ∞ control for nonlinear uncertain stochastic systems with mode-dependent time delay and Markovian jump parameters represented by the Takagi–Sugeno (T-S) fuzzy model approach. The system under consideration involves parameter uncertainties, Itô-type stochastic disturbances, Markovian jump parameters and unknown nonlinear disturbances. The purpose is to design a state feedback controller such that the closed-loop system is robustly exponentially stable in the mean square and satisfies a prescribed H ∞ performance level. Novel delay-range-dependent conditions in the form of linear matrix inequalities (LMIs) are derived for the solvability of robust stabilisation and H ∞ control problem. A desired fuzzy controller can be constructed by solving a set solutions of LMIs and can be easily calculated by Matlab LMI control toolbox. Finally, a numerical example is presented to illustrate the proposed method.

New sufficient conditions for robust stability analysis of interval matrices

This letter presents new sufficient conditions for robust Hurwitz stability of interval matrices. The proposed conditions are based on two approaches: (i) finding a common Lyapunov matrix for the interval family and (ii) converting the robust stability problem into a robust non-singularity problem using Kronecker operations. The main contribution of the letter is to derive accurate and computationally simple optimal estimates of the robustness margin and spectral bound of general interval matrices. The evaluation of the condition relies on the solutions of linear matrix inequalities (LMIs) and eigenvalue problems, both of which are solved very efficiently. The improvements gained by using the proposed conditions are demonstrated through application to previous examples in the literature.

Nonquadratic Stabilization of Continuous T–S Fuzzy Models: LMI Solution for a Local Approach

This paper is concerned with nonquadratic stabilization design problem for continuous-time nonlinear models in the Takagi-Sugeno (T-S) form obtained by sector nonlinearity approach. Most of the previous results found in the literature intended to establish global nonquadratic stabilization conditions which are hard to uphold due to the difficulty of handling time derivatives of the membership function. By changing the paradigm of global stabilization for something less restrictive, a local solution to overcome infeasible quadratic stabilization conditions is offered in this paper. It is shown that the derived local nonquadratic conditions actually lead to reasonable advantages over the existing quadratic approach, as well as some previous nonquadratic attempts. Moreover, conditions for the solvability of state feedback controller design given here are written in the form of linear matrix inequalities (LMIs) which can be efficiently solved by convex optimization techniques. Simulation examples are given to demonstrate the validity and applicability of the proposed approaches.

Stabilization of Unknown Nonlinear Discrete-Time Delay Systems Based on Neural Network

This paper discusses about the stabilization of unknown nonlinear discrete-time fixed state delay systems. The unknown system nonlinearity is approximated by Chebyshev neural network (CNN), and weight update law is presented for approximating the system nonlinearity. Using appropriate Lyapunov-Krasovskii functional the stability of the nonlinear system is ensured by the solution of linear matrix inequalities. Finally, a relevant example is given to illustrate the effectiveness of the proposed control scheme.

Design of Robust Stochastic Controller for Stabilizing Queues of TCP/AQM System

In this article, a robust stochastic controller is designed analytically using the linearized fluid flow model of TCP/AQM behavior. The stability criteria found is delay dependent. The control law is obtained as a solution of linear matrix inequalities (LMIs) and not only asymptotically stabilizes the queue size to the desired operating point, but also guarantees the robust stability of network against the linearization error in estimating round-trip time delay. The performance of the proposed controller is verified and compared with the PI and ARED AQM schemes using the NS-2 simulator. The simulation results show that the proposed scheme has better performance and robustness than PI and ARED controllers with faster response, smaller overshoot and less bias.

Design of Robust Power System Stabilizer Based on Particle Swarm Optimization

In this paper, we examine the problem of designing power system stabilizer (PSS). A new technique is developed using particle swarm optimization (PSO) combined with linear matrix inequality (LMI). The main feature of PSO, not sticking into a local minimum, is used to eliminate the conservativeness of designing a static output feedback (SOF) stabilizer within an iterative solution of LMIs. The technique is further extended to guarantee robustness against uncertainties wherein power systems operation is changing continuously due to load changes. Numerical simulation ahs illustrated the utility of the developed technique.

Robust H<SUB align="right">∞ synchronisation for a class of complex dynamical networks with time-varying delay

This paper deals with the problem of robust H ∞ synchronisation for a class of complex dynamical networks with time-varying delay. Each node of the network is a general Lur’e system subject to an energy bounded input noise. A synchronisation criterion formulated in the form of linear matrix inequalities (LMIs) is obtained under which the controlled network can be robustly stabilised onto an expected homogeneous state with a guaranteed performance. The controller designed can be constructed via feasible solutions of LMIs. Finally, a dynamical network composed of identical Chua’s oscillators is adopted as a numerical example to demonstrate the effectiveness of the proposed results.

Motion control for a vertical rigid rotor rotating in electromagnetic bearings

The problems of control of a motion of a rigid rotor in electromagnetic bearings are considered. The main ides of synthesis of the control laws is the approach based on feedback linearization of the original nonlinear mathematical model of the system. The method of Lyapunov functions and methods connected with solution of linear matrix inequalities are used in synthesis of control laws.

State Feedback Guaranteed Cost Control of Parameter Uncertain Nonlinear System

In an actual system, the effects of nonlinear factors are inevitable. So in real practice, when a model for complex system is being built, all the features in it will be linearized. Though simplifying the designing and analyzing process, the model being built in this way is thought to be incapable of revealing the true characteristics of the system. In order to solve this problem, the paper analyzes a model combining both linear and nonlinear features while taking the parameter perturbation of the linear part into consideration, which enables the model to retain as many characteristics of the actual system as possible. Provided that the nonlinear function satisfies the Lipschitz constraint conditions, the robust guaranteed cost state feedback control law of nonlinear system is deduced using the Lyapunov function and then converted into the feasible solutions of linear matrix inequality (LMI). The proposed method optimizes the design of controller by modifying the previous oversimplified models that fail to reveal the real characteristics of the actual system, and the effectiveness of the proposed method is being verified through an algorithm simulation example.

An Approach to H ∞ Control of a Class of Nonlinear Stochastic Systems

This paper studies the problems of the stochastic stabilization and H∞ control of a class of nonlinear stochastic systems that can be represented by a stochastic heterogeneous model. Based on this model, the problem of stabilization with state feedback controller is first considered, and a new H∞ controller design method is then developed. Synthesis of both stabilization and H∞ controller is based on the solutions of linear matrix inequalities (LMIs). Finally, an example is given to illustrate the effectiveness of the approach proposed in this paper.

Structure-preserving numerical algorithm for solving discrete-time LMI and DARS

We present a numerical algorithm to solve a discrete-time linear matrix inequality (LMI) and discrete-time algebraic Riccati system (DARS). With a given system ( A , B , C , D ) we associate a para-hermitian matrix pencil. Then we transform it by an orthogonal transformation matrix into a block-triangular para-hermitian form. Under either of the two assumptions (1) matrix pair ( A , B ) is controllable or (2) matrix pair ( A , B ) is reachable and ( A , B , C , D ) is a left invertible system, we extract the solution of LMI and DARS by the entries of the orthogonal transformation matrix.

A robust model predictive control algorithm for incrementally conic uncertain/nonlinear systems

AbstractThis paper presents a robustly stabilizing model predictive control algorithm for systems with incrementally conic uncertain/nonlinear terms and bounded disturbances. The resulting control input consists of feedforward and feedback components. The feedforward control generates a nominal trajectory from online solution of a finite‐horizon constrained optimal control problem for a nominal system model. The feedback control policy is designed off‐line by utilizing a model of the uncertainty/nonlinearity and establishes invariant ‘state tubes’ around the nominal system trajectories. The entire controller is shown to be robustly stabilizing with a region of attraction composed of the initial states for which the finite‐horizon constrained optimal control problem is feasible for the nominal system. Synthesis of the feedback control policy involves solution of linear matrix inequalities. An illustrative numerical example is provided to demonstrate the control design and the resulting closed‐loop system performance. Copyright © 2010 John Wiley & Sons, Ltd.

Dissipative Control for Nonlinear Neutral Delay Systems

The problem of delay-dependent dissipative control for nonlinear neutral delay systems is dealt with. We develop the design method of dissipative static state feedback controller such that the closed-loop system is absolutely stable and strictly-dissipative. Sufficient conditions for the existence of the quadratic dissipative controller are obtained by using linear Matrix Inequality(LMI) approach. Furthermore, a procedure of constructing such a controller from the solution of LMI is given. It is shown that the solvability of a dissipative controller design is implied by the feasibility of LMIs.

LMI solutions to the mixed ℋ︁−/ℋ︁∞ fault detection observer design for linear parameter‐varying systems

AbstractThis paper addresses the mixed ℋ︁−/ℋ︁∞ fault detection observer design issue for a class of linear parameter‐varying (LPV) systems. Analogous to the definition of the quadratic ℋ︁∞ performance for LPV systems and the ℋ︁− index for linear time invariant (LTI) systems, the quadratic ℋ︁− index and the affine quadratic ℋ︁− index for LPV systems are defined in terms of linear matrix inequalities (LMIs). The first algorithm for designing the mixed ℋ︁−/H∞ observer is proposed, which aims at minimizing the quadratic ℋ︁∞ performance and maximizing the quadratic ℋ︁− index of the observer error dynamic systems. To reduce the conservativeness of this algorithm, the affine quadratic ℋ︁∞ performance and the affine ℋ︁− index for LPV systems are utilized. The robustness conditions and affine ℋ︁− index conditions for the underlying observer optimization issue are formulated as parameter‐dependent LMIs. The Gridding technique and multi‐convexity concept are applied, respectively, for reducing the parameter‐dependent LMIs to finite LMI constraints. Correspondingly, two iterative algorithms are proposed. Furthermore, the threshold design and the estimation of the worst undetectable fault size are investigated. An example is studied to demonstrate the effectiveness of the proposed algorithms. Copyright © 2010 John Wiley & Sons, Ltd.

Localization of eigenvalues of polynomial matrices

We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize the known method for the localization of the spectrum of polynomial matrices based on the solution of linear matrix inequalities. We also develop a method for the localization of eigenvalues of a parametric family of matrix polynomials in the form of a system of linear matrix inequalities.

Uncertainty Modeling and Robust Control in Urban Traffic

AbstractThe paper investigates the problem of uncertainty modeling and constrained robust control of urban traffic. Linear polytopic approach is used by state-space representations to describe the uncertain network system. In order to handle model mismatches, robust and infinite horizon model predictive control (MPC) method is proposed. The control strategy is an efficient method to reduce travel time and improve homogeneous traffic flow under changing model conditions. Centralized numerical solution has been carried out as a solution of Linear Matrix Inequalities (LMI) by using semidefinite programming (SDP). Closed-loop control results were tested in simulation environment taking alternative model uncertainty levels into account.

Non-fragile H∞ and Guaranteed Cost Control for a Class of UncertainDescriptor Systems

Based on linear matrix inequality (LMI), the problem of H∞ guaranteed cost control for a class of descriptor system is addressed, where the uncertainties exist in the systematic matrix and the controller gain. Under the additive perturbation, the sufficient condition of non-fragile H∞ guaranteed cost is presented, and corresponding controller design is given in terms of the feasible solutions of LMI.