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In this paper, a new linear matrix inequality-based model predictive control (MPC) problem is studied for discrete-time nonlinear systems described as Takagi–Sugeno fuzzy systems. A recent local stability approach is applied to improve the performance of the proposed MPC scheme. At each time ${k}$ , an optimal state-feedback gain that minimizes an objective function is obtained by solving a semidefinite programming problem. The local stability analysis, the estimation of the domain of attraction, and feasibility of the proposed MPC are proved. Examples are given to demonstrate the advantages of the suggested MPC over existing approaches.

This paper studies the problem of stabilizing discrete-time switched linear control systems (SLCSs) using continuous input by a user against adversarial switching by an adversary. It is assumed that at each time the adversary knows the user's decision on the continuous input but not vice versa. A quantitative metric of stabilizability is proposed. Systems at the margin of stabilizability are further classified and studied via the notions of defectiveness and reducibility. Analytical bounds on the stabilizability metric are derived using (semi)norms, with tight bounds provided by extremal norms. Numerical algorithms are also developed for computing this metric. An application example in networked control systems is presented.

The goal of this paper is to study the exponential stabilization problem for autonomous discrete-time switched linear systems (SLSs), where only the discrete mode can be controlled. Our approach is based on periodic control Lyapunov functions whose value decreases periodically instead of at each time step as in the classical control Lyapunov functions. Using periodic control Lyapunov functions, we develop stabilizability analysis and controller synthesis conditions that are less conservative than existing results in that they apply to a larger class of SLSs. Utilizing recent results on the switched optimal control problems, a constructive way to find periodic control Lyapunov functions is presented.

Recently, the local stability and local stabilization problems have been investigated by several authors for nonlinear systems described by Takagi-Sugeno (T-S) fuzzy systems. In this paper, we study the local stabilization problem for discrete-time T-S fuzzy systems with magnitude- and energy-bounded disturbances. The control design problem is cast as a linear matrix inequality (LMI) optimization. The state-feedback controller is designed in such a way that, if the initial state is within a certain domain, then the state trajectories under bounded disturbances will not escape the fuzzy modelling region of the state space. Finally, examples are given to demonstrate the validity of the proposed method.

Recently, the so-called FIR-type (FIR, finite impulse response) state-feedback memory controller and periodic memory controller have been developed, and it was revealed that this control design approach can reduce the conservatism of stability and stabilization conditions for periodic or uncertain systems. In these work, the system duality and a descriptor representation of the original system are used. In this paper, we focus only on the FIRtype state-feedback memory controller design problem, which is a special case of the general memory controller structure previously developed. Although the structure of the controller considered in this paper is a particular case of the existing one, this special consideration gives some benefits, which are the main purpose of this paper. Finally, examples demonstrate the effectiveness of the FIR-type state-feedback memory controller in terms of less conservatism.

In this paper, we propose an iterative linear matrix inequality (LMI) approach to design static output feedback (SOF) controllers for continuous-time linear time-invariant (LTI) systems. The method suggested in this paper can be viewed as a version of some previous iterative LMI methods. Although the proposed method cannot always provide improved results in comparison with the previous one, it can be used as a less conservative alternative in some cases. In addition, it can be combined with the previous algorithm to improve the results. Finally, an example is gSiven to demonstrate the validity of the proposed methods.

The goal of this paper is to propose linear matrix inequality (LMI) formulations to analyze the local stability and local stabilization of discrete-time nonlinear systems represented by Takagi–Sugeno (T–S) fuzzy systems with time-delay. In order to estimate the domain of attraction (DA) of the time-delay systems, the so-called fuzzy Lyapunov-Krasovskii functionals are used to characterize the subsets of the DA as sublevel sets of the Lyapunov- Krasovskii functionals. We use the information of the variation rates of the membership functions to obtain less conservative results. Finally, examples are given to illustrate the proposed methods.

The problem of designing continuous control input to stabilize switched linear control systems against adversarial switching is studied. It is assumed that the continuous controller has access to the current switching mode and can be of the form of an ensemble of mode-dependent state feedback controllers. The fastest stabilizing rate under the given information structure is proposed as a quantitative metric of the system’s stabilizability, and its bounds are derived using seminorms. Computation algorithms for the stabilizing rate are developed and illustrated through examples.

In this technical note, we are concerned with linear matrix inequality (LMI) approaches for solving sampled-data state-feedback control design problems of linear time-invariant systems. This short contribution discusses a simple alternative to a family of LMI approaches based on the polytopic linear difference inclusion representation of the original problem. With the aid of the so-called matrix-dilation technique, the proposed condition is capable of being directly derived from the discrete-time Lyapunov inequality without the polytopic modeling. Finally, numerical experiments are given to illustrate the results.

This paper addresses local H ∞ controller design problems for continuous-time Takagi–Sugeno (T–S) systems with magnitude- and energy-bounded disturbances. The design procedure is formulated as optimizations subject to linear matrix inequalities (LMIs) which can be solved by means of convex optimization techniques. The designed controllers not only guarantee the H ∞ performance but also ensure the state not to escape an invariant set that is included by the region where the T–S fuzzy model is defined. Finally, an example is given to illustrate the proposed method.

In this paper, robust H 2 and H ∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.

In this paper, we propose linear matrix inequality (LMI) formulations to analyze local stability and design controllers that locally stabilize continuous-time nonlinear systems represented by Takagi-Sugeno (T-S) fuzzy systems. In order to estimate the domain of attraction (DA), the so-called fuzzy Lyapunov function is used to characterize the subsets of the DA as sublevel sets of the Lyapunov function. Quadratic bounds on the time-derivative of the membership functions are employed to derive the main results. Finally, examples are given to illustrate the proposed methods.

In the previous work, the authors studied the problem of robust discretization of linear time-invariant systems with polytopic uncertainties, where linear matrix inequality (LMI) conditions were developed to find an approximate discrete-time (DT) model of a continuous-time (CT) system with uncertainties in polytopic domain. The system matrices of obtained DT model preserved the polytopic structures of the original CT system. In this paper, we extend the previous approach to solve the problem of robust discretization of polytopic uncertain systems with aperiodic sampling. In contrast with the previous work, the sampling period is assumed to be unknown, time-varying, but contained within a known interval. The solution procedures are presented in terms of unidimensional optimizations subject to LMI constraints which are numerically tractable via LMI solvers. Finally, an example is given to show the validity of the proposed techniques.

In order to guide the robots move along a collision-free path efficiently and reach the goal position quickly in the unknown multi-obstacle environment, this paper presented the navigation problem of a wheel mobile robot based on proximity sensors by fuzzy logic controller. Then a genetic algorithm was applied to optimize the membership function of input and output variables and the rule base of the fuzzy controller. Here the environment is unknown for the robot and contains various types of obstacles. The robot should detect the surrounding information by its own sensors only. For the special condition of path deadlock problem, a wall following method named angle compensation method was also developed here. The simulation results showed a good performance for navigation problem of mobile robots.

The so-called digital redesign (DR) is a sampled-data (SD) controller design method where an analogue controller is designed firstly, and then transformed to an approximately equivalent digital controller in the sense of state-matching. In this approach, the SD controller is designed by reducing the discrepancy between the discrete-time (DT) counterpart of the closed-loop SD control system and the continuous-time (CT) closed-loop system. In this paper, we develop a DR strategy for CT linear time-invariant systems. More specifically, H∞ norm of the error dynamic system between the CT and DT plants is minimized for the optimal state-matching performance at every sampling point. The design problem is formulated as linear matrix inequalities which can be efficiently solved by using convex optimization techniques. Finally, an example is given to illustrate the effectiveness of the proposed method.

For continuous-time linear time-invariant (LTI) systems with polytopic uncertainties, we develop a robust sampled-data state-feedback control design scheme in terms of linear matrix inequalities (LMIs). Truncated power series expansions are used to approximate a discretized model of the original continuous-time system. The system matrices obtained by using the power series approximations are then expressed as homogeneous polynomial parameter-dependent (HPPD) matrices of finite degrees, and conditions for designing the controller are formulated as a HPPD matrix inequality, which can be solved by means of a recent LMI relaxation technique to test the positivity of HPPD matrices with variables in the simplex. To take care of the errors induced by the remainder terms of the truncated power series, the terms are considered as norm bounded uncertainties and then incorporated into the proposed LMI conditions. Finally, examples are used to illustrate the approach.

The goal of this paper is to study the state-feedback stabilization of controlled discrete-time switched linear systems (SLSs), where the discrete switching control input and the continuous control input coexist. We propose a periodic hybrid-control policies, periodic control Lyapunov functions (PCLFs), and prove that the existence of a quadratic PCLF is a necessary and sufficient condition for the stabilizability of SLSs. Moreover, computational algorithms are developed to find quadratic PCLFs and test the stabilizability.

The goal of this paper is to study switching stabilization problems for discrete-time switched linear systems (SLSs). To analyze the switching stabilizability, we introduce the notion of graph control Lyapunov functions (GCLFs). The GCLF is a set of Lyapunov functions which satisfy several Lyapunov inequalities associated with a weighted directed graph (digraph). Each Lyapunov function represents each node in the digraph, and each Lyapunov inequality represents edges connecting a node and its out-neighbors. The weight of each directed edge indicates the decent rate of the Lyapunov functions from the tail to the head of the edge. The GCLF is an extension of the graph Lyapunov function for stability analysis of SLSs to switching stabilization problems, and unifies several control Lyapunov functions in a single framework. In this paper, we proposed a GCLF theorem, which provides a necessary and sufficient condition for switching stabilizability and develop computational algorithms to the stabilizability.

In this paper, local stability and local stabilization problems are considered for discrete-time nonlinear systems represented by Takagi–Sugeno (T–S) fuzzy systems. Improved methods to assess the local stability, design locally stabilizing control laws, and estimate the domain of attraction are developed in terms of single-parameter minimization problems subject to linear matrix inequality (LMI) constraints. The improvement is achieved by applying a convergent LMI relaxation technique to prove positivity of homogeneous polynomial parameter-dependent matrices of arbitrary degree with variables in the simplex. We also take advantage of the recently developed parameter variation modeling technique to deal with the case that the variation rate of the membership functions of the T–S fuzzy systems is bounded and less than one. Finally, several illustrative examples demonstrate the validity and efficiency of the proposed method.

Summary This paper suggests a simple convex optimization approach to state-feedback adaptive stabilization problem for a class of discrete-time LTI systems subject to polytopic uncertainties. The proposed method relies on estimating the uncertain parameters by solving an online optimization at each time step, such as a linear or quadratic programming, and then, on tuning the control law with that information, which can be conceptually viewed as a kind of gain-scheduling or indirect adaptive control. Specifically, an admissible domain of stabilizing state-feedback gain matrices is designed offline by means of linear matrix inequality problems, and based on the online estimation of the uncertain parameters, the state-feedback gain matrix is calculated over the set of stabilizing feedback gains. The proposed stabilization algorithm guarantees the asymptotic stability of the overall closed-loop control system. An example is given to show the effectiveness of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.

This technical note focuses on the development of a general periodically time-varying memory filter (PTVMF) to improve the robust H∞ filtering performance for discrete-time linear time-invariant (LTI) systems with polytopic parameter uncertainties. A sufficient condition to design the H∞-PTVMF is given in terms of convex optimizations expressed as linear matrix inequalities (LMIs). Finally, examples demonstrate its potential less conservatism.

This paper proposes a strategy to estimate invariant subsets of the domain of attraction (DA) for asymptotically stable zero equilibrium points of continuous-time Takagi-Sugeno (T-S) fuzzy systems. Specifically, by using Lyapunov's stability theory and the linear matrix inequality (LMI) technique, sufficient conditions for proving the local stability are provided in terms of single-parameter minimization problems subject to LMI constraints or eigenvalue problems, which are solvable via convex optimizations. The fuzzy Lyapunov functions (FLFs), expressed by the so-called multi-dimensional fuzzy summations, are employed to characterize invariant subsets of the DA as sublevel sets of the FLFs. To compute a larger inner estimate of the DA, an iterative LMI algorithm is also developed. Finally, illustrative examples show the efficacy of the approach.

In this paper, the problem of robust discretisation of linear time-invariant LTI systems with polytopic uncertainties is introduced. More specifically, the main objective is to provide a systematic way to find an approximate discrete-time DT model of a continuous-time CT plant with uncertainties in polytopic domain. The system matrices of polytopic DT model to be found are expressed as parameter-dependent matrices which are homogeneous polynomials of arbitrary degree with respect to the uncertain variables in the simplex, and is obtained in such a way that the norm between the system matrices and the truncated power series of the exact DT model is minimised while preserving the polytopic structure of the original CT plant. The solution procedures proposed are presented in terms of single-parameter minimisation problems subject to linear matrix inequality LMI constraints which are numerically tractable via LMI solvers. Finally, examples are given to show the validity and effectiveness of the proposed techniques.

In this paper, a sufficient linear matrix inequality (LMI) condition is presented for robust stability analysis of continuous-time linear time-invariant (LTI) systems in polytopic domains. The underlying idea behind the proposed approach is to introduce a family of complex functions which map the closed right-hand side of the complex plane into the inside of the closed unit circle centered at the origin. Then, the mapping properties are used to assure that all the eigenvalues of a system are located in the open left-hand side of the complex plane. Examples show the validity of the proposed condition.

This study deals with the problem of local stability analysis and the computation of invariant subsets of the domain of attraction (DA) for discrete-time Takagi–Sugeno fuzzy systems. Based on the fuzzy Lyapunov functions, new sufficient conditions and an iterative scheme are proposed in order to prove the local stability and to estimate the DA. The mean value theorem and polytopic type bounds on the gradient of the membership functions are used to consider the relation between the membership functions at samples k and k + 1. Each step of the iterative procedure consists of linear matrix inequalities (LMIs) or single-parameter minimisation problems subject to LMI constraints, which are solvable via convex optimisations. Finally, examples compare the proposed conditions with existing tests.

The goal of this paper is to study the switching stabilization of discrete-time autonomous switched linear systems (SLSs) based on the periodic Lyapunov functions. In recent papers, a convex switching stabilizability condition was presented. We investigate properties of the condition. In particular, it is proved that the condition is only a sufficient condition, and also provide a class of stabilizable SLSs which cannot be identified as stabilizable by the condition. In addition, for two-dimensional SLSs, another sufficient and necessary stabilizability condition is proposed by using the S-procedure and considering sufficiently fine partitions of the state space. Examples are given to show the validity of the proposed approaches.

SUMMARY In this paper, the finite frequency robust filtering problem (-FFRFP), design of a robust filter minimizing the norm from the disturbance input to the estimation error evaluated over a prescribed finite frequency domain, is considered for continuous-time and discrete-time linear time-invariant (LTI) systems with polytopic parameter uncertainties. By means of the generalized Kalman–Yakubovich–Popov (KYP) lemma in combination with a result known as Finsler's lemma, the -FFRFP is cast as a linear matrix inequality (LMI) optimization. Examples are given to demonstrate that the proposed condition can achieve improvement over the previous ones in the literature. Copyright © 2012 John Wiley & Sons, Ltd.

This paper deals with stability analysis and control design problems for continuous-time Takagi-Sugeno (T-S) fuzzy systems. The first aim is to present less conservative linear matrix inequality (LMI) conditions to design controllers and assess the stability. The second relevant contribution is to present a new strategy to find an inner estimate of the domain of attraction (DA) via LMIs. The results are based on the fuzzy Lyapunov functions (FLFs) and non-parallel distributed compensation (non-PDC) approaches. Finally, examples illustrate the effectiveness and merits of the proposed methods.

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.

This paper presents a new fuzzy Lyapunov function (FLF) for the stability analysis of continuous-time Takagi-Sugeno (T-S) fuzzy systems. Unlike conventional FLFs, the proposed one depends not only on the fuzzy weighting functions of the T-S fuzzy systems but on their first-order time derivatives as well. Based on the proposed FLF, a sufficient stability condition is derived in the form of linear matrix inequalities, depending on the upper bounds on the second-order time derivative of the fuzzy weighting functions, as well as the first-order ones. Finally, some examples demonstrate that the proposed condition can provide less conservative results than the previous ones in the literature.

This paper provides simple and effective linear matrix inequality (LMI) characterizations for the stability and stabilization conditions of discrete-time Takagi-Sugeno (T-S) fuzzy systems. To do this, more general classes of non-parallel distributed compensation (non-PDC) control laws and non-quadratic Lyapunov functions are presented. Unlike the conventional non-quadratic approaches using only current-time normalized fuzzy weighting functions, we consider not only the current-time fuzzy weighting functions but also the l-step-past (l>=0) and one-step-ahead ones when constructing the control laws and Lyapunov functions. Consequently, by introducing additional decision variables, it can be shown that the proposed conditions include the existing ones found in the literature as particular cases. Examples are given to demonstrate the effectiveness of the approaches.

In this paper, a systematic procedure to design the robust H ∞ fuzzy controller for a nonlinear active magnetic bearing (AMB) system affected by time-varying parametric uncertainties is presented. First, the continuous-time Takagi-Sugeno (T-S) fuzzy model is employed to represent the nonlinear AMB system. Next, based on the obtained fuzzy model, sufficient conditions are derived in terms of linear matrix inequalities (LMIs) for robust stability and H ∞ performance of the control system. The main feature of this paper is that some drawbacks existing in the previous approaches such as truncation errors and nonconvex bilinear matrix inequality (BMI) problem are eliminated by utilizing the homogeneous fuzzy model which includes no bias terms in the local state space models rather than the affine one which includes bias terms. Hence, the design method presented here will prove to be more tractable and accessible than the previous ones. Finally, numerical simulations demonstrate the effectiveness of the proposed approach.

This paper introduces a -gain state feedback fuzzy controller design for the time delay nonlinear system represented by Takagi-Sugeno(T-S) fuzzy model. First, the T-S fuzzy model is employed to represent the time delay nonlinear system. Next based on the fuzzy model, a fuzzy state feedback controller is developed to achieve -gain performance. Finally, sufficient conditions are derived for -gain performance. The sufficient conditions are formulated in the format of linear matrix inequalities (LMIs). The effectiveness of the proposed controller design methonology is finally demonstrated through numerical simulations.

The problem of stabilizing discrete-time switched linear control systems using continuous input by the user and against adversarial switching by an adversary is studied. It is assumed that the adversary has the advantage in that at each time it knows the user's decision on the continuous control input but not vice versa. Stabilizability conditions and bounds on the fastest stabilizing rates are derived. Examples are given to illustrate the results.

This paper addresses local H ∞ controller design problems for continuous-time Takagi-Sugeno (T-S) systems with magnitude- and energy-bounded disturbances. The design procedure is formulated as optimizations subject to linear matrix inequalities (LMIs) which can be solved by means of convex optimization techniques. The designed controllers not only guarantee the H ∞ performance but also ensure the state not to escape an invariant set that is included by the region where the T-S fuzzy model is defined. Finally, examples are given to illustrate the proposed method.

This paper addresses the problem of stabilizing discrete-time LTI systems with polytopic uncertainties via a static output feedback controller. We approach this problem using the recently developed periodically time-varying memory state-feedback controller design scheme, and a sufficient linear matrix inequality condition for the static output control design is proposed. Finally, numerical examples are given to demonstrate the usefulness of the proposed method.

This paper addresses the problem of static output feedback (SOF) stabilization for discrete-time LTI systems. We approach this problem using the recently developed periodically time-varying memory state-feedback controller (PTVMSFC) design scheme. A bilinear matrix inequality (BMI) condition which uses a pre-designed PTVMSFC is developed to design the periodically time-varying memory SOF controller (PTVMSOFC). The BMI condition can be solved by using BMI solvers. Alternatively, we can apply two-steps and iterative linear matrix inequality algorithms that alternate between the PTVMSFC and PTVMSOFC designs. Finally, an example is given to illustrate the proposed methods., Comment: This paper has been submitted as a supplemental document of "Periodically time-varying memory static output feedback control design for discrete-time LTI systems" submitted to Automatica

This paper addresses the problems of local stability analysis, local stabilization, and computation of invariant subsets of the domain of attraction for continuous-time Takagi-Sugeno fuzzy systems. Improvements on the existing stability and stabilization conditions are achieved through the reduction of conservatism. To release the conservatism further, the so-called multidimensional fuzzy summation approach is adopted. The design and analysis conditions are expressed as one-parameter minimization problems which can be solved via a sequence of linear matrix inequality optimizations or treated as eigenvalue problems. Examples are given to show the validity of the proposed method.

This paper proposes a strategy to estimate invariant subsets of the domain of attraction (DA) for continuous-time Takagi-Sugeno (T-S) fuzzy systems. The result is based on the so-called fuzzy Lyapunov function, which is used to characterize the subsets of the DA as sublevel sets of the Lyapunov function. The key idea relies on casting the Lyapunov inequality that includes the time-derivative of the membership functions into linear matrix inequalities (LMIs) by considering a bound on the nonlinear function dependent quadratically on the time-derivative of the membership functions. Finally, an example is given to demonstrate the effectiveness of the proposed approach.

This paper investigates the problem of robust D-stabilization for polynomial matrix polytopes. By employing proportional-derivative (PD) state-feedback controllers, a new sufficient condition is formulated in terms of a linear matrix inequality (LMI) feasibility problem. Example is given to demonstrate the validity and efficiency of the proposed approach.

In this paper, stability of continuous-time Takagi-Sugeno (T-S) fuzzy systems is investigated by means of a new fuzzy Lyapunov function which depends on the first-order timederivative of the fuzzy weighting functions (TDFWF). As a result, both the first and second-order TDFWF appear in the Lyapunov inequality and the stability condition. The additional dependence on the second-order TDFWF makes it possible to reduce the conservativeness of the previous result. The stability condition is formulated in terms of linear matrix inequalities (LMIs). The effectiveness of the proposed analysis approach is demonstrated by numerical examples.

This paper presents the relaxed nonquadratic stabilization conditions of discrete-time Takagi-Sugeno (T-S) fuzzy systems. To do this, we propose a new fuzzy controller and Lyapunov function by generalizing the nonparallel distributed compensation (non-PDC) control law and nonquadratic Lyapunov function, respectively. By exploiting Po?lya's theorem and algebraic properties of a homogeneous polynomials of normalized fuzzy weighting functions, an infinite family of sufficient conditions for the asymptotic stabilizability is derived. These conditions are formulated in the format of linear matrix inequalities (LMIs) and, hence, are numerically tractable via convex programming techniques. Finally, an example is given to illustrate advantages of the proposed method.

This paper is an attempt to pursue the relaxed stabilization conditions for discrete-time Takagi–Sugeno (T– S) fuzzy systems. The main results given here concern their stabilization using extended non-quadratic Lyapunov functions, a non-parallel distributed compensations (non-PDC). New sufficient conditions in the terms of linear matrix inequalities (LMIs) have been proposed. The conditions are shown to be less conservative than some previous results in the literature. Two examples are given to illustrate the advantage of the proposed methods.

This study aims at developing new linear matrix inequality (LMI) conditions for robust κ-stability and κ-stabilisation of linear time-invariant uncertain systems in polytopic domains. An existing LMI condition for robust κ-stability is lifted into those of larger dimensions that are given as functions of a certain integer κ. Based on the consequent conditions, new sufficient conditions for the existence of a state-feedback control law ensuring the robust κ-stability of the closed-loop system is proposed in terms of LMIs. Finally, examples are given to illustrate the benefit of the proposed methods.