Quadratic Stability Condition Research Articles

On the continuous-time Takagi–Sugeno fuzzy systems stability analysis

In this paper, a new approach for the stability analysis of continuous-time Takagi–Sugeno (T-S) fuzzy system is proposed. The universe set is divided to subregions, and piecewise quadratic Lyapunov function is then found for each of them. This class of Lyapunov function candidates is much richer than the common quadratic Lyapunov function. By exploiting the piecewise continuous Lyapunov function, we derive stability conditions that can be verified via convex optimization over linear matrix inequalities (LMIs) or bilinear matrix inequalities (BMIs). These conditions are shown to be less conservative than some quadratic stabilization conditions published recently in the literature. Since this method uses low numbers of LMIs or BMIs and less computation Lyapunov functions, it is highly applicable and has less computation. This approach is not dependent on the shape of fuzzy sets and also, stability of the system is guaranteed in the presence of state feedbacks. At first, in order to decrease length of computation (amount of LMIs (BMIs)), an approach is introduced based on properties of the T-S system with two-overlapped fuzzy sets. Some criterions are obtained for stability analysis, stability analysis in the presence of parametric uncertainties and a stability criterion is presented to provide a reasonable performance for the system. To demonstrate the new approach, an illustrative example is presented.

Explicit use of probabilistic distributions in linear predictive control

The guarantee of feasibility given feasibility at initial time is an issue that has been overlooked by many of the recent papers on stochastic model predictive control. Effective solutions have recently been proposed, but these carry considerable online computational load and a degree of conservativism. For the case that the elements of the random additive disturbance vector are independent, the current paper ensures that probabilistic constraints are met and that a quadratic stability condition is satisfied. A numerical example illustrates the efficacy of the proposed algorithm, which achieves tight satisfaction of constraints and thereby attains near-optimal performance.

Diagonal stability of interval matrices and applications

Let A I = { A ∈ R n × n | A - ⩽ A ⩽ A + } be an interval matrix and 1 ⩽ p ⩽ ∞ . We introduce the concept of Schur and Hurwitz diagonal stability, relative to the Hölder p-norm, of A I , abbreviated as SDS p and HDS p , respectively. This concept is formulated in terms of a matrix inequality using the p-norm, which must be satisfied by the same positive definite diagonal matrix for all A ∈ A I . The inequality form is different for SDS p and HDS p . The particular case of p = 2 is equivalent to the condition of quadratic stability of A I . The SDS 2 inequality is equivalent to the Stein inequality ∀ A ∈ A I : A T PA - P ≺ 0 , and the HDS 2 inequality is equivalent to the Lyapunov inequality ∀ A ∈ A I : A T P + PA ≺ 0 ; in both cases P is a positive definite diagonal matrix and the notation “ ≺ 0 ” means negative definite. The first part of the paper • provides SDS p and HDS p criteria, • presents methods for finding the positive definite diagonal matrix requested by the definition of SDS p and HDS p , • analyzes the robustness of SDS p and HDS p and • explores the connection with the Schur and Hurwitz stability of A I . The second part shows that the SDS p or HDS p of A I is equivalent to the following properties of a discrete- or continuous-time dynamical interval system whose motion is described by A I : • the existence of a strong Lyapunov function defined by the p-norm and • the existence of exponentially decreasing sets defined by the p-norm that are invariant with respect to system’s trajectories.

Robust Reliability as a Measure of Stability of Controlled Dynamic Systems with Bounded Uncertain Parameters

Based on an appropriate uncertainty description, a new robust reliability method for stability issue of dynamic systems is developed to deal with bounded uncertainties. The presented methods provide necessary and sufficient conditions for quadratic stability and stabilization of uncertain systems and are suitable both for cases where the bounds of uncertain parameters are known and where they are unknown. Using this method, a robust reliability measure of the stability of parametric uncertain systems can be provided, and the maximum robustness bounds of uncertain parameters such that the system can be stable can be obtained. The design of a controller for stabilizing uncertain systems is carried out by solving a robust-reliability-based convex optimization problem. This makes it possible to take both the robustness with respect to uncertainties and the control cost into account simultaneously in the controller design. Moreover, the presented procedures are within the framework of linear matrix inequality and can be implemented conveniently. Two examples are provided to demonstrate the effectiveness and feasibility of the proposed methods. By numerical simulations and compared with existing results, it is shown that increasing conservatism in controller design by traditional methods does not mean increasing reliability, and so it is significant to take the robust reliability into account in the controller design of uncertain systems.

Stabilization of Discrete-time 2-D T-S Fuzzy Systems Based on New Relaxed Conditions

This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications.First,a discrete-time 2-D T-S fuzzy model is proposed to represent the underlying nonlinear 2-D system.Second,new quadratic stabilization conditions are proposed by applying relaxed quadratic stabilization technique for 2-D case.Third,for sake of further reducing conservatism,new non-quadratic stabilization conditions are also proposed by applying a new parameter-dependent Lyapunov function,matrix transformation technique,and relaxed technique for the underlying discrete-time 2-D T-S fuzzy system.Finally,a numerical example is provided to illustrate the effectiveness of the proposed results.

Input and Output Quantized Feedback Linear Systems

Although there has been a lot of research on analysis and synthesis of quantized feedback control systems, most results are developed for the case of a single quantizer (either measurement quantization or control signal quantization). In this technical note, we investigate the case of feedback control systems subject to both input and output quantization. This is motivated by the fact that it is common in remotely controlled systems that measurement and control signals are shared over a single digital network. More specifically, we consider a single-input single-output linear system with memoryless logarithmic quantizers. We firstly show that the output feedback quadratic stabilization problem in this setting can be addressed with no conservatism by means of a sector bound approach. Secondly, we provide a sufficient condition for quadratic stabilization via the solution of a scaled H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> control problem. Finally, we analyze a problem of bandwidth allocation in the communication channel for finite-level input and output quantizers.

Stabilization of Discrete-time 2-D T-S Fuzzy Systems Based on New Relaxed Conditions

This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications. First, a discrete-time 2-D T-S fuzzy model is proposed to represent the underlying nonlinear 2-D system. Second, new quadratic stabilization conditions are proposed by applying relaxed quadratic stabilization technique for 2-D case. Third, for sake of further reducing conservatism, new non-quadratic stabilization conditions are also proposed by applying a new parameter-dependent Lyapunov function, matrix transformation technique, and relaxed technique for the underlying discrete-time 2-D T-S fuzzy system. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.

Composite Systems with Uncertain Couplings of Fixed Structure: Scaled Riccati Equations and the Problem of Quadratic Stability

We consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure. The couplings are viewed as structured perturbations of the block-diagonal system representing the collection of the separate nominal subsystems (the “nominal system”). We define spectral value sets and stability radii for these time-invariant structured perturbations and derive formulas for their computation. Scaled Riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time-varying and possibly nonlinear perturbations of the given structure. From these we derive necessary and sufficient conditions under which the stability radii with respect to time-invariant and time-varying perturbations are equal. These results are obtained by constructing joint quadratic Liapunov functions of optimal robustness. With their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized Aizerman conjecture.

Robust stabilization of uncertain chaotic systems via interval fuzzy control

This paper presents a robust interval fuzzy control scheme to stabilize a class of uncertain chaotic systems with H∞ performance. First, we apply the Takagi-Sugeno (T-S) fuzzy modelling scheme to the uncertain chaotic systems to obtain the corresponding T-S interval fuzzy model. Next, the parallel distributed compensation (PDC) method is employed to derive the fuzzy controller. The quadratic stabilizability and performance problem of the interval fuzzy control system with disturbances are then solved. Both the quadratic stability condition and performance are represented by linear matrix inequality (LMI) problems. Finally, an uncertain chaotic system is utilized to demonstrate the feasibility and validity of the proposed interval fuzzy control scheme.

New LMI-Based Conditions for Quadratic Stabilization of LPV Systems

This paper is concerned with quadratic stabilization problem of linear parameter varying (LPV) systems, where arbitrary time-varying dependent parameters are belonging to a polytope. It provides improved linear matrix inequality- (LMI-) based conditions to compute a gain-scheduling state-feedback gain that makes closed-loop system quadratically stable. The proposed conditions, based on the philosophy of Pólya's theorem, are written as a sequence of progressively less and less conservative LMI. More importantly, by adding an additional decision variable, at each step, these new conditions provide less conservative or at least the same results than previous methods in the literature.

A KYP Lemma for LMI Regions

In this technical note, a Kalman-Yakubovich-Popov (KYP) lemma is discussed for linear matrix inequality (LMI) regions. Sufficient quadratic stability conditions are developed for an uncertain linear system subject to time varying uncertainty satisfying a quadratic inequality. Furthermore, the quadratic stability conditions are shown to guarantee the satisfaction of a frequency domain inequality.

Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions

In this paper, we present several conditions which are both necessary and sufficient for quadratic stability of an uncertain/nonlinear system. These conditions involve multiplier matrices which characterize the uncertain/nonlinear terms in the system description. It is known that some of these conditions are sufficient for quadratic stability. One of the main contributions of this paper is to demonstrate that these conditions are also necessary conditions, hence, they are not conservative conditions for quadratic stability. By presenting multiplier matrices for many common types of uncertain/nonlinear terms, the paper also demonstrates the usefulness of the multiplier matrix approach in the analysis and control of nonlinear/time-varying/uncertain systems.

Relaxed LMI conditions for closed-loop fuzzy systems with tensor-product structure

Current fuzzy control research tries to obtain the less conservative conditions to prove stability and performance of fuzzy control systems. In many fuzzy models, membership functions with multiple arguments are defined as the product of simpler ones, where all possible combinations of such products conform a fuzzy partition. In particular, such situation arises with widely used fuzzy modelling techniques for nonlinear systems. These type of fuzzy models will be denoted as tensor-product (TP) fuzzy systems, because its expressions can be understood as operations on multi-dimensional arrays. This paper discusses the generalisation to TP fuzzy systems of the results in Kim and Lee [2000. New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Transactions on Fuzzy Systems 2, 1571–1582] and Xiaodong and Qinling [2003. New approaches to H ∞ controller designs based on fuzzy observers for T–S fuzzy systems via LMI. Automatica 39, 1571–1582]. The procedures here will allow to set up linear matrix inequality conditions which are less conservative than the cited ones, by exploiting the TP structure of the membership functions. A numerical example illustrates the achieved improvement.

Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design

We derive conditions for quadratic stabilizability of linear networked control systems by dynamic output feedback and communication protocols. These conditions are used to develop a simultaneous design of controllers and protocols in terms of matrix inequalities. The obtained protocols do not require knowledge of controller and plant states but only of the discrepancies between current and the most recently transmitted values of nodes’ signals, and are implementable on controller area networks. We demonstrate on a batch reactor example that our design guarantees quadratic stability with a significantly smaller network throughput than previously available designs.

Correction to "Quadratic Stability and Stabilization of Dynamic Interval Systems"

It is pointed out that the counterexample in a previous comment is not a counterexample. A new counterexample is presented to show that the necessity part of Theorem 1 of the above paper is not valid in some cases. Furthermore, based on the M-matrix theory, a new quadratic stability condition for dynamic interval systems is proposed to correct the corresponding result in the above paper

A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems

This paper proposes a new quadratic stabilization condition for Takagi-Sugeno (T-S) fuzzy control systems. The condition is represented in the form of linear matrix inequalities (LMIs) and is shown to be less conservative than some relaxed quadratic stabilization conditions published recently in the literature. A rigorous theoretic proof is given to show that the proposed condition can include previous results as special cases. In comparison with conventional conditions, the proposed condition is not only suitable for designing fuzzy state feedback controllers but also convenient for fuzzy static output feedback controller design. The latter design work is quite hard for T-S fuzzy control systems. Based on the LMI-based conditions derived, one can easily synthesize controllers for stabilizing T-S fuzzy control systems. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, the validity and applicability of the proposed approach are successfully demonstrated in the control of a continuous-time nonlinear system.

Some properties of networked reset control systems

The dynamics and stability of reset control systems in a networked environment are studied considering the impact of network-induced delays. Lyapunov-like exponential stability and robust stability criteria for such systems are obtained. A quadratic stability condition, which is, from a practical point of view and thus more easily testable, is derived using the Kalman-Yakubovich-Popov (KYP) lemma. This result is demonstrated by examples to be applicable to the asymptotic tracking and disturbance rejection problems of such systems.

State feedback control of switched linear systems: An LMI approach

This paper addresses the problem of state feedback control of continuous-time switched linear systems with arbitrary switching rules. A quadratic Lyapunov function with a common matrix is used to derive a stabilizing switching control strategy that guarantees: (i) the assignment of all the eigenvalues of each linear subsystem inside a chosen circle in the left-hand half of the complex plane; (ii) a minimum disturbance attenuation level for the closed-loop switched system. The proposed design conditions are given in terms of linear matrix inequalities that encompass previous results based on quadratic stability conditions with fixed control gains. Although the quadratic stability based on a fixed Lyapunov matrix has been widely used in robust control design, the use of this condition to provide a convex design method for switching feedback gains has not been fully investigated. Numerical examples show that the switching control strategy can cope with more stringent design specifications than the fixed gain strategy, being useful to improve the performance of this class of systems.

Quadratic stabilization for uncertain stochastic systems

This paper discusses the robust quadratic stabilization control problem for stochastic uncertain systems, where the uncertain matrix is norm bounded, and the external disturbance is a stochastic process. Two kinds of controllers are designed, which include state feedback case and output feedback case. The conditions for the robust quadratic stabilization of stochastic uncertain systems are given via linear matrix inequalities. The detailed design methods are presented. Numerical examples show the effectiveness of our results.

New results on robust control design of discrete-time uncertain systems

The paper considers the problem of robust control redesign of a class of linear discrete-time systems with norm-bounded parametric uncertainties and controller-gain perturbations. In both cases of additive and multiplicative perturbations, the control objective is to design a resilient feedback-stabilisation schemes using guaranteed cost control and H∞ control approaches. The problem under consideration is cast as convex optimisation over linear matrix inequalities which led to the derivation of necessary and sufficient conditions for closed-loop robust quadratic stability. Three system examples are provided to illustrate the theoretical developments.