L/sub 2/ robust stability analysis for the fuzzy feedback linearization regulator

The robust stability analysis problem of a linear time invariant feedback system is investigated. Based on the stability radii, a set of perturbed open-loop plants and/or controllers can be found such that the closed-loop system is still stable. Also, the derived robust stability conditions are used on a linear time invariant singularly perturbed system such that a stability bound /spl epsiv/~ is obtained and conditions are determined to ensure the stability property subjected to real or complex unstructured perturbations.

The robust stability of linear continuous-time uncertain systems in polytopic domains is investigated in this paper. The uncertain parameters are assumed as time-varying with bounded rates of variation. The robust stability conditions are obtained from the definition of a Lyapunov function with a particular structure, depending on integer powers κ of the dynamic uncertain matrix of the system. As a consequence, parametrized linear matrix inequalities conditions are obtained in terms of κ. As κ grows, the robust stability conditions can take into account bounds on the successive time-derivatives of the uncertain parameters, reducing the conservativeness of the evaluations. Numerical examples illustrate the effectiveness of the proposed methodology.

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

ABSTRACTThis paper addresses the problem of robust stabilisation of nD hybrid systems described by Roesser model. The model is in a descriptor form where all the system matrices are corrupted by uncertainty in a general form as implicit linear fractional representations . Based on the S-procedure technique, robust stability and robust stabilisation conditions are established for hybrid systems. Simulation example is provided in order to illustrate the relevance of our approach.

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.

This article is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power κ of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by κ. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of κ needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be introduced, yielding linear matrix inequalities optimisation problems of improved convergence.

This paper concerns the problems of robust stability analysis and stabilization for uncertain linear system with time-varying delay. In order to obtain a robust stability criterion with less conservative for uncertain linear delay system, a quadratic integral inequality is improved by matrix decomposition. Further, two different kinds of controller, i.e., linear memoryless state feedback controller and linear state feedback controller with state delay, are derived in terms of the robust stability criteria. Two numeral examples are provided to demonstrate the validity of the proposed methods.

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

In this paper, we have studied a numerical stability analysis method for the robust fuzzy feedback linearization regulator using Takagi-Sugeno fuzzy model. To analyze the robust stability, we assume that uncertainty is included in the model structure with known bounds. For these structured uncertainty, the robust stability of the closed system is analyzed by applying Linear Matrix Inequalities theory following a transformation of the closed loop systems into Lur'e systems.

Given a time-delay system, we are interested in finding new robust stability conditions for the system. We apply the integral quadratic constraint approach to obtain two results which allow us to test the robust stability using the linear matrix inequality technique. Both results give an estimate of the maximum time-delay which preserves robust stability. The first result is simpler to apply while the second one gives a less conservative robust stability condition.

In this paper, a new method for robust Lagrange stability analysis of a class of nonlinear discrete-time systems is proposed. Both the linear part structured and unstructured uncertainties are considered in a unified way. Sufficient, conditions for robust Lagrange stability are established in terms of linear matrix inequalities (LMIs). With this LMI approach, the largest allowable magnitude of the admissible perturbation was given explicitly by solving a, generalized eigenvalue minimization problem which is essentially a convex optimization problem and numerically efficient, illustrative example confirmed the efficiency and accuracy of the proposed approach

For a linear system affected by real parametric uncertainty, this paper focuses on robust stability analysis via quadratic-in-the-state Lyapunov functions polynomially dependent on the parameters. The contribution is twofold. First, if n denotes the system order and m the number of parameters, it is shown that it is enough to seek a parameter-dependent Lyapunov function of given degree 2nm in the parameters. Second, it is shown that robust stability can be assessed by globally minimizing a multivariate scalar polynomial related with this Lyapunov matrix. A hierarchy of LMI relaxations is proposed to solve this problem numerically, yielding simultaneously upper and lower bounds on the global minimum with guarantee of asymptotic convergence.

In this paper, the Takagi-Sugeno (TS) fuzzy modelling and the fuzzy regulation theory are considered in order to design a fuzzy controller capable of taking the output of a fuzzy plant to the reference signal generated by an external system. It is well-known that TS fuzzy modelling allows the stabilizer to be obtained by means of numerical techniques. As a result, stability region defined by the fuzzy stabilizer is larger than the one obtained when a simple linear stabilizer is considered. On the other hand, a fuzzy regulator constructed from linear controllers does not solve the exact regulation problem in general [6], [12], [13], [15]. Therefore, a fully nonlinear regulator, defined on the overall TS fuzzy model, is considered in this work. The results are applied on a simple fuzzy model in order to illustrate the improvement achieved with the proposed approach.

Because the inter-sample behavior can be considered, the lifting technique now becomes the unique method for design and analysis of sampled-data systems. However, when we consider the robust stability problem by using the small gain theorem, the state transition operator is zero, and the current lifting algorithms can not be used directly. The lifting computation formulas for the case of the robust stability analysis are derived in detail. Further study shows that when the lifting technique is used for robust stability analysis, its result is conservative. A new discretization method for robust stability analysis is proposed in the paper, and it is not conservative when the condition of the small gain theorem is used. Two examples are given here to illustrate the above ideas.

Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties