Linear Matrix Inequality Problem Research Articles

Control and Stability Analyzing of the Time-Delay Systems with Time-Varying Delays

Systems with delays are a specific group of dynamic systems. The fact that some delays can be short and some can be long makes system analysis and control design much more complex. In this paper we present a proposal to solve the problem of stability and control design of continuous time systems with time-varying delays. Based on the Lyapunov-Krasovskii functional the stability condition is derived using the linear matrix inequality (LMI) techniques, and convex optimization principle to find LMI variables which simultaneously satisfies the limitations given by the theory of Lyapunov-Krasovskii functionals. Obtained solution is the feasible convex LMI problem for the static controller design. Finally the design method is demonstrated using a system model example.

Robust Adaptive Attitude Tracking Control With L2-Gain Performance and Vibration Reduction of an Orbiting Flexible Spacecraft

This paper presents a dual-stage control system design method for flexible spacecraft attitude tracking control and active vibration suppression by an embedded smart material as sensors/actuators. More specifically, a conventional sliding mode controller with the assumption of knowing system parameters is first designed that ensures asymptotical convergence of attitude tracking error described by error quaternion and its derivative in the presence of bounded parameter variation/disturbance. Then it is redesigned, such that the need for knowing the system parameters in advance is eliminated by using an adaptive updating law. For the synthesis of the controller, to achieve the prescribed L2-gain performance criterion, the control gains are designed by solving a linear matrix inequality problem. Indeed, external torque disturbances/parametric error attenuations with respect to the performance measurement along with the control input penalty are ensured in the L2-gain sense. Even if this controller has the ability to reject the disturbance and deal with actuator constraint, it excites the elastic modes of flexible appendages, which will deteriorate the pointing performance. Then the undesirable vibration is actively suppressed by applying feedback control voltages to the piezoceramic actuator, in which the modal velocity feedback control method is employed for determining the control voltages. Numerical simulations are performed to show that attitude tracking and vibration suppression are accomplished, in spite of the presence of disturbances/parameter uncertainties and even control input constraint.

Stability analysis and robustness design of nonlinear systems: An NN-based approach

In this study a neural-network (NN) based approach is developed which combines H ∞ control performance with Tagagi–Sugeno (T–S) fuzzy control for the purpose of stabilization and stability analysis of nonlinear systems. A Takagi–Sugeno (T–S) fuzzy model and parallel-distributed compensation (PDC) scheme are first employed to design a nonlinear fuzzy controller for the stabilization of nonlinear systems. The neural-network model is adopted to overcome the modeling error problems found with nonlinear systems. A novel stability condition based on an NN-based controller design is derived to ensure the stability of the nonlinear system. The control problem can now be reformulated as a linear matrix inequality (LMI) problem. A simulation is provided in order to explore the feasibility of the proposed fuzzy controller design method.

Observers for descriptor systems with slope-restricted nonlinearities

This paper addresses the observer design problem for a class of nonlinear descriptor systems whose nonlinear terms are slope-restricted. The full-order observer is formulated as a nonlinear descriptor system. A linear matrix inequality (LMI) condition is derived to construct the full-order observer. The existence and uniqueness of the solution to the obtained observer system are guaranteed. Furthermore, under the same LMI condition and a common assumption, a reduced-order observer is designed. Finally, the design methods are reduced to a strict LMI problem and illustrated by a numerical example.

Fuzzy controller design for Takagi–Sugeno fuzzy models with multiplicative noises via relaxed non-quadratic stability analysis

In this paper, a relaxed non-quadratic stabilization analysis is developed to design fuzzy controllers for non-linear stochastic systems that are constructed by the Takagi–Sugeno (T–S) fuzzy models with multiplicative noises. The multiplicative noise term is introduced in the consequent part of the T–S fuzzy system for representing the stochastic behaviour. According to the proposed T–S fuzzy models, a relaxed non-quadratic technique is employed to derive the stability conditions for closed-loop fuzzy control systems. The relaxed non-quadratic stability conditions developed in this paper lead to the linear matrix inequality (LMI) problem. By solving the proposed LMI problem, the feasible solutions of the fuzzy control gains can be obtained. Finally, two numerical simulations are given to demonstrate the usefulness and applicability of the proposed fuzzy controller design approach.

Cellular Neural Networks for Gray Image Noise Cancellation Based on a Hybrid Linear Matrix Inequality and Particle Swarm Optimization Approach

This paper describes a technique for gray image noise cancellation. This method employs linear matrix inequality (LMI) and particle swarm optimization (PSO) based on cellular neural networks (CNN).We use two images that one is desired image and the other is corrupted to find the CNN template. The Lyapunov stability theorem is employed to derive the criterion for uniqueness and global asymptotic stability of the CNN equilibrium point. The current study characterizes the template design problem as a standard LMI problem and the optimization parameters of the templates are carried out by PSO. Finally, the examples are given to illustrate the effectiveness of the proposed method.

Linear Matrix Inequality Conditions of Nonlinear Systems by Genetic Algorithm-based H ∞ Adaptive Fuzzy Sliding Mode Controller

In this paper, the stability analysis of a genetic algorithm-based (GA-based) H∞ adaptive fuzzy sliding model controller (AFSMC) for a nonlinear system is discussed. First, a nonlinear plant is well-approximated and described with a reference model and a fuzzy model, both involving fuzzy logic control (FLC) rules. Then, FLC rules and the consequent parameter are decided on via a genetic algorithm. After this, we guarantee a new H∞ tracking performance inequality for the control system. The H∞ tracking problem is characterized to solve an eigenvalue problem. Next, an AFSMC is proposed to stabilize the system so as to achieve good H∞ control performance. Lyapunov’s direct method can be used to ensure the stability of the nonlinear system. It is shown that the stability analysis can reduce nonlinear systems into a linear matrix inequality problem. Finally, a numerical simulation is provided to demonstrate the control methodology.

RBF neural network based H ∞ synchronization for unknown chaotic systems

In this paper, we propose a new H ∞ synchronization strategy, called a Radial Basis Function Neural NetworkH ∞ synchronization (RBFNNHS) strategy, for unknown chaotic systems in the presence of external disturbance. In the proposed framework, a radial basis function neural network (RBFNN) is constructed as an alternative to approximate the unknown nonlinear function of the chaotic system. Based on this neural network and linear matrix inequality (LMI) formulation, the RBFNNHS controller and the learning laws are presented to reduce the effect of disturbance to an H ∞ norm constraint. It is shown that finding the RBFNNHS controller and the learning laws can be transformed into the LMI problem and solved using the convex optimization method. A numerical example is presented to demonstrate the validity of the proposed RBFNNHS scheme.

Constrained multivariable stable predictive control based on Toeplitz equation

A stable constrained multivariable predictive control algorithm based on the Toeplitz predictive equation is presented. An analytical method for transformation between left and right coprime factorization of multivariable system is put forward, and then an unconstrained control law and its stable condition are given. Furthermore, for the constrained multivariable controlled system, a predictive controller and its asymptotically stable condition are presented and proved. Finally, the constrained receding horizon optimization is transformed into a typical linear matrix inequality (LMI) problem and solved. The detailed control algorithm is depicted in a numerical example, and the simulation results reveal the effectiveness and practicability of the presented algorithm. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Application of Fuzzy-model-based Control to Nonlinear Structural Systems with Time Delay: an LMI Method

In this study we present a fuzzy robust control design which combines H ∞ control performance with Tagagi-Sugeno (T-S) fuzzy control for the control of delayed nonlinear structural systems under external excitations. We design a nonlinear fuzzy controller based on parallel distributed compensation schemes. The controller design problem is reformulated as a linear matrix inequality (LMI) problem as derived from the Lyapunov theory. This robust method is designed to overcome the modeling error that can occur between delayed nonlinear structural systems with T-S fuzzy models. Given the fuzzy-model-based H ∞ control and the stability conditions, the stability of a delayed nonlinear structural system under external excitation is ensured. Furthermore, the delayed nonlinear structural system is equipped with a tuned mass damper designed according to the first mode of frequency. The feedback gain of the fuzzy controller is found via the Matlab LMI toolbox. The proposed method is then applied to a delayed nonlinearly tuned mass damper system. The simulation results show that not only is the proposed method able to stabilize delayed nonlinear structural systems, but also has strong robustness in terms of preventing modeling errors and external excitations.

Optimal design of periodic test input signals for multivariable impulse response models

AbstractThis paper considers an optimal test input design problem occurring in a control‐relevant, iterative system identification scheme. The optimization is based on minimizing the expected a posteriori closed‐loop error based on the a priori information given by the data already collected. Instead of directly solving the resulting non‐linear optimization to obtain the test input sequences, two different computationally efficient approaches are proposed. The first approach is cast in the frequency domain and we solve for optimal discrete spectra of the periodic signals under some reasonable approximation. The solution can be realized as an input sequence by performing a spectral factorization and implementing the spectral factor as a digital Finite Impulse Response (FIR) filter. A second approach is suggested by reformulating the optimal design problem as a linear matrix inequalities (LMI) problem. The efficacy of these approaches is demonstrated through a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.

Low-order robust power system stabilizer for single-machine systems: An LMI approach

This paper describes a linear matrix inequality (LMI) approach to the design of robust loworder power system stabilizers (PSSs), which are used to damp out local-mode oscillations of synchronous generators. The performance of a PSS is expressed as the location of the closed-loop poles, and a single fixed-gain pole-placement controller is synthesized for a wide range of operating conditions. The synthesis results in simultaneous regional pole-placement stabilization through static output feedback, and is formulated as a novel LMI feasibility problem with a rank condition. A penalty method is applied to solve the rank-constrained LMI problem. Numerical experiments with a single machine connected to an infinite bus system were performed to demonstrate the proposed LMI method, and the results were compared with those of previous work.

Admissible Model Matching using 𝒟R‐regions: Fault accommodation and robustness against FDD inaccuracies

AbstractThis paper proposes a reformulation of the Admissible Model Matching (AMM) method for Fault‐Tolerant Control (FTC) using Linear Matrix Inequality (LMI) regional pole placement. The essential feature of the AMM approach is that the FTC design is based on the specification of a set of admissible closed‐loop behaviors whose limits represent the acceptable control performance degradation in the presence of faults. In the original formulation, this set is characterized by inequalities relating the coefficients of the closed‐loop system matrix, derived ‘ad hoc’ for the particular system and the provided control specification, and the FTC design problem is formulated as a non‐linear constrained optimization problem. In this paper, the set of admissible behaviors is specified by restricting the closed‐loop spectrum in a union of particular convex regions called 𝒟R‐regions. On one hand, this allows a straightforward representation for a wide variety of practical control specifications while preserving the convexity of the problem. On the other hand, 𝒟R‐regions are characterized by LMIs and the fault accommodation can be formulated in terms of several LMI (feasibility) problems. Moreover, the robustness of the solutions against errors in the fault estimations provided by the Fault Detection and Diagnosis module is characterized as well as a LMI‐constrained optimization problem. The versatility and the effectiveness of the proposed method are illustrated using an example. Copyright © 2010 John Wiley & Sons, Ltd.

DELAYED FEEDBACK $\mathcal{H}_{\infty}$ SYNCHRONIZATION FOR TIME-DELAYED CHAOTIC SYSTEMS BASED ON T–S FUZZY MODEL

In this letter, we propose a new [Formula: see text] synchronization method, called a delayed feedback fuzzy [Formula: see text] synchronization (DFFHS) method, for time-delayed chaotic systems with external disturbance. Based on Lyapunov–Krasovskii theory, T–S fuzzy model, and delayed feedback control scheme, the DFFHS controller is presented to not only guarantee aymptotical synchronization but also reduce the effect of external disturbance to an [Formula: see text] norm constraint. The proposed controller can be obtained by solving the linear matrix inequality (LMI) problem. A numerical example for time-delayed Lorenz system is presented to demonstrate the validity of the proposed DFFHS method.

A T-S Fuzzy Model-Based Adaptive Exponential Synchronization Method for Uncertain Delayed Chaotic Systems: An LMI Approach

This paper proposes a new fuzzy adaptive exponential synchronization controller for uncertain time-delayed chaotic systems based on Takagi-Sugeno (T-S) fuzzy model. This synchronization controller is designed based on Lyapunov-Krasovskii stability theory, linear matrix inequality (LMI), and Jesen's inequality. An analytic expression of the controller with its adaptive laws of parameters is shown. The proposed controller can be obtained by solving the LMI problem. A numerical example for time-delayed Lorenz system is presented to demonstrate the validity of the proposed method.

Adaptive H Anti-Synchronization for Time-Delayed Chaotic Neural Networks

In this paper, an adaptive H∞ control scheme is developed to study the antisynchronization behavior of time-delayed chaotic neural networks with unknown parameters. This adaptive H∞ anti-synchronization controller is designed based on LyapunovKrasovskii theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The proposed synchronization method guarantees the asymptotical anti-synchronization of drive and response systems. Furthermore, this method reduces the effect of external disturbance to an H∞ norm constraint. The proposed controller can be obtained by solving a linear matrix inequality (LMI) problem. An illustrative example is given to demonstrate the effectiveness of the proposed method. Subject Index: 044, 055

Anti-synchronization of Time-delayed Chaotic Neural Networks Based on Adaptive Control

This paper investigates the adaptive anti-synchronization problem for time-delayed chaotic neural networks with unknown parameters. Based on Lyapunov-Krasovskii stability theory and linear matrix inequality (LMI) approach, the adaptive anti-synchronization controller is designed and an analytic expression of the controller with its adaptive laws of unknown parameters is shown. The proposed controller can be obtained by solving the LMI problem. An illustrative example is given to demonstrate the effectiveness of the proposed method.

Fuzzy Static Output Feedback May Be Possible in LMI Framework

Takagi-Sugeno (T--S) fuzzy-model-based static output-feedback controller design is a challenging issue due to the nonconvexity in the Lyapunov stability condition. Recently, a convexified linear matrix inequality (LMI) condition was proposed for the problem by additionally importing as many equality constraints as the number of the fuzzy rules with a variable in the paper from the study of Fang <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . However, numerical experiences seem to imply the infeasibility of the proposed LMI. This letter shows that their LMI problem may be feasible.

Neural network ℋ∞ chaos synchronization

This paper proposes a new neural network ℋ∞ synchronization (NNHS) scheme for unknown chaotic systems. In the proposed framework, a dynamic neural network is constructed as an alternative to approximate the chaotic system. Based on this neural network and linear matrix inequality (LMI) formulation, the NNHS controller and the learning law are presented to reduce the effect of disturbance to an ℋ∞ norm constraint. It is shown that finding the NNHS controller and the learning law can be transformed into the LMI problem and solved using the convex optimization method. A numerical example is presented to demonstrate the validity of the proposed NNHS scheme.

Adaptive Neural Control Design for Nonlinear Distributed Parameter Systems With Persistent Bounded Disturbances

In this paper, an adaptive neural network (NN) control with a guaranteed L(infinity)-gain performance is proposed for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities and persistent bounded disturbances. Initially, Galerkin method is applied to the PDE system to derive a low-order ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, based on the low-order slow model and the Lyapunov technique, an adaptive modal feedback controller is developed such that the closed-loop slow system is semiglobally input-to-state practically stable (ISpS) with an L(infinity)-gain performance. In the proposed control scheme, a radial basis function (RBF) NN is employed to approximate the unknown term in the derivative of the Lyapunov function due to the unknown system nonlinearities. The outcome of the adaptive L(infinity)-gain control problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal controller is obtained in the sense of minimizing an upper bound of the L(infinity)-gain, while control constraints are respected. Furthermore, it is shown that the proposed controller can ensure the semiglobal input-to-state practical stability and L(infinity)-gain performance of the closed-loop PDE system. Finally, by applying the developed design method to the temperature profile control of a catalytic rod, the achieved simulation results show the effectiveness of the proposed controller.