Sampling‐interval‐dependent stability for sampled‐data systems with state quantization

This paper is concerned with the stability of sampled‐data systems with state quantization. A new piecewise differentiable Lyapunov functional is first constructed by fully utilizing information about sampling instants. This functional has two features: one is that it is of the second order in time t and of every term being dependent on time t explicitly and the other is that it is discontinuous and is only required to be definite positive at sampling instants. Then, on the basis of this piecewise differentiable Lyapunov functional, a sampling‐interval‐dependent exponential stability criterion is derived by applying the technique of a convex quadratic function with respect to the time t to check the negative definiteness for the derivative of the piecewise differentiable Lyapunov functional. In the case of no quantization, a new sampling‐interval‐dependent stability criterion is also obtained. It is shown that the new stability criterion is less conservative than some existing one in the literature. Finally, two examples are given to illustrate the effectiveness of the stability criterion. Copyright © 2013 John Wiley & Sons, Ltd.     

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.     

Stability of periodically time-varying systems: Periodic Lyapunov functions

This paper proposes a novel approach to stability analysis of discrete-time nonlinear periodically time-varying systems. The contributions are as follows. Firstly, a relaxation of standard Lyapunov conditions is derived. This leads to a less conservative Lyapunov function that is required to decrease at each period rather than at each time instant. Secondly, for linear periodic systems with constraints, it is shown that compared to standard Lyapunov theory, the novel concept of periodic Lyapunov functions allows for the calculation of a larger estimate of the region of attraction. An example illustrates the effectiveness of the developed theory.     

Novel Stability Criteria for Sampled-Data Systems With Variable Sampling Periods

This paper is concerned with a novel Lyapunovlike functional approach to the stability of sampled-data systems with variable sampling periods. The Lyapunov-like functional has four striking characters compared to usual ones. First, it is time-dependent. Second, it may be discontinuous. Third, not every term of it is required to be positive definite. Fourth, the Lyapunov functional includes not only the state and the sampled state but also the integral of the state. By using a recently reported inequality to estimate the derivative of this Lyapunov functional, a sampled-interval-dependent stability criterion with reduced conservatism is obtained. The stability criterion is further extended to sampled-data systems with polytopic uncertainties. Finally, three examples are given to illustrate the reduced conservatism of the stability criteria.     

Improved stability criteria for linear time-varying systems on time scales

ABSTRACT

In this paper, asymptotic stability problems of linear time-varying (LTV) systems on time scales are considered based on a less conservative Lyapunov inequality, whose right side is not required to be necessarily negative. It is shown that the Lyapunov inequality covers not only the corresponding trivial (continuous and discrete) ones but also nontrivial ones. Based on this inequality, some necessary and sufficient conditions for asymptotic stability, exponential stability, uniformly exponential stability of LTV systems on time scales are obtained. An example about nontrivial systems is given for illustrating the effectiveness of the proposed results.     

Finite-time stability and instability of stochastic nonlinear systems

This paper presents a new definition of finite-time stability for stochastic nonlinear systems. This definition involves stability in probability and finite-time attractiveness in probability. An important Lyapunov theorem on finite-time stability for stochastic nonlinear systems is established. A theorem extending the stochastic Lyapunov theorem is also proved. Moreover, an example and a lemma are presented to illustrate the scope of extension. A useful inequality, extended from Bihari’s inequality, is derived, which plays an important role in showing the Lyapunov theorem. Finally, a Lyapunov theorem on finite-time instability is proved, which states that almost surely globally asymptotical stability is not equivalent to finite-time stability for some stochastic systems. Two simulation examples are given to illustrate the theoretical analysis.     

Regional stability of two‐dimensional nonlinear polynomial Fornasini‐Marchesini systems

This paper addresses the problem of regional stability analysis of 2‐dimensional nonlinear polynomial systems represented by the Fornasini‐Marchesini second state‐space model. A method based on a polynomial Lyapunov function is proposed to ensure local asymptotic stability and provide an estimate of the domain of attraction of the system zero equilibrium point. The proposed results that build on recursive algebraic representations of the polynomial vector function of the system dynamics and Lyapunov function are tailored via linear matrix inequalities that are required to be satisfied at the vertices of a given bounded convex polyhedral region of the state space. Numerical examples demonstrate the effectiveness of the proposed method.     

Stability analysis for fractional-order neural networks with time-varying delay

This paper focuses on the stability analysis for fractional-order neural networks with time-varying delay. A novel Lyapunov's asymptotic stability determination theorem is proved, which can be used for fractional-order systems directly. Different from the classical Lyapunov stability theorem, constraint condition on the derivative of Lyapunov function is revised as an uniformly continuous class-K function in the fractional-order case. Based on this novel Lyapunov stability theorem and free weight matrix method, a new sufficient condition on Lyapunov asymptotic stability of fractional-order Hopfield neural networks is derived by constructing a suitable Lyapunov function. Moreover, two numerical examples are provided to illustrate the effectiveness of these criteria.     

Stability analysis in identification of interval type‐2 adaptive neuro‐fuzzy inference system: Contribution to a novel Lyapunov function

In recent years, control research has strongly highlighted the issue of training stability in the identification of non‐linear systems. This paper investigates the stability analysis of an interval type‐2 adaptive neuro‐fuzzy inference system (IT2ANFIS) as an identifier through a novel Lyapunov function. In so doing, stability analysis is initially conducted on the IT2ANFIS identifier, while performing the online training of both the antecedent and the consequent parameters by the gradient descent (GD) algorithm. In addition, the same stability analysis is carried out when the antecedent and the consequent parameters are trained by GD and forgetting factor recursive least square (FRLS) algorithms, respectively (GD + FRLS). A novel Lyapunov function is proposed in this study in order for the identifier stability to attain the required conditions. These conditions determine the permissible boundaries for the covariance matrix and the learning rates at every iteration of the identification procedure. Stability analysis reveals that wide range of learning rates is obtained. Furthermore, simulation results indicate that when the permissible boundaries are selected according to the proposed stability analysis, a stable identification process with appropriate performance is achieved.     

Stability of a class of switched positive linear time‐delay systems

This paper addresses the problem of stability for a class of switched positive linear time‐delay systems. As first attempt, the Lyapunov–Krasovskii functional is extended to the multiple co‐positive type Lyapunov–Krasovskii functional for the stability analysis of the switched positive linear systems with constant time delay. A sufficient stability criterion is proposed for the underlying system under average dwell time switching. Subsequently, the stability result for system under arbitrary switching is presented by reducing multiple co‐positive type Lyapunov–Krasovskii functional to the common co‐positive type Lyapunov–Krasovskii functional. A numerical example is given to show the potential of the proposed techniques. Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive controller which provides Lyapunov stability

An adaptive controller that can provide exponential Lyapunov stability for an unknown linear time-invariant (LTI) system is presented. The only required a priori information about the plant is that the order of an LTI stabilizing compensator be known, although this can be reduced to assuming only that the plant is stabilizable and detectable at the expense of using a more complicated controller. This result extends the work of M. Fu and B.R. Barmish (see ibid., vol.AC-31, p.1097-1103, Dec. 1986) in which it is shown that there exists an adaptive controller which provides exponential Lyapunov stability if it is assumed that an upper bound on the plant order is known and that the plant lies in a known compact set; it shows that adaptive stabilization is possible under very mild assumptions without large state deviations     

Input-to-state stability and interconnections of discontinuous dynamical systems

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov’s solution concept, that is appropriate for ‘open’ systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.     

Parameter-dependent Lyapunov functions and the Popov criterion inrobust analysis and synthesis

Many practical applications of robust feedback control involve constant real parameter uncertainty, whereas small gain or norm-bounding techniques guarantee robust stability against complex, frequency-dependent uncertainty, thus entailing undue conservatism. Since conventional Lyapunov bounding techniques guarantee stability with respect to time-varying perturbations, they possess a similar drawback. In this paper we develop a framework for parameter-dependent Lyapunov functions, a less conservative refinement of “fixed” Lyapunov functions. An immediate application of this framework is a reinterpretation of the classical Popov criterion as a parameter-dependent Lyapunov function. This result is then used for robust controller synthesis with full-order and reduced-order controllers     

Stability studies of control systems using a non-Luré type Lyapunov function

Stability of control systems with multiple nonlinearities is discussed. A non-Luré type Lyapunov function is presented, which surpasses the Luré-type function from the point of view of the stability region guaranteed. This function is used to establish a stability criterion for the system. The superiority of the function proposed is indicated by a numerical example, comparing the stability boundary to that obtained by a Luré-type Lyapunov function.     

A relaxed stability criterion for T-S fuzzy discrete systems

Stability conditions for Tanaka-Sugeno (T-S) fuzzy discrete systems based on a single quadratic Lyapunov function or a weighting dependent Lyapunov function have been mentioned in lots of literature. The existence of a common matrix P is required in the former, and r (rules' number) positive matrices satisfying r2 Lyapunov inequalities are necessary in the latter. In this paper, the weighting dependent Lyapunov function is used again. Moreover according to the idea of the firing rule group and the distance estimation between two successive states of the system, the relaxed stability criterion of T-S fuzzy discrete system is proposed.     

Robust absolute stability for general neutral type Lurie indirect control systems

This note concerns the robust absolute stability analysis for a class of general neutral type Lurie indirect control systems with nonlinearity located in an infinite sector or finite one. By using Lyapunov functional of quadratic form with integral term and introducing some free‐weighting matrices, some delay‐dependent robust absolute stability criteria are presented in terms of strict linear matrix inequalities. Neither model transformation nor bounding technique is required here. The obtained criteria are less conservative than previous ones, which are illustrated by numerical examples. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stability criteria for large-scale systems

Recent research into large-scale system stability has proceeded via two apparently unrelated approaches. For Lyapunov stability, it is assumed that the system can be broken down into a number of subsystems, and that for each subsystem one can find a Lyapunov function (or something akin to a Lyapunov function). The alternative approach is an input-output approach; stability criteria are derived by assuming that each subsystem has finite gain. The input-output method has also been applied to interconnections of passive and of conic subsystems. This paper attempts to unify many of the previous results, by studying linear interconnections of so-called "dissipative" subsystems. A single matrix condition is given which ensures both input-output stability and Lyapunov stability. The result is then specialized to cover interconnections of some special types of dissipative systems, namely finite gain systems, passive systems, and conic systems.     

Finite‐time stability of switched positive linear systems

This brief paper addresses the finite‐time stability problem of switched positive linear systems. First, the concept of finite‐time stability is extended to positive linear systems and switched positive linear systems. Then, by using the state transition matrix of the system and copositive Lyapunov function, we present a necessary and sufficient condition and a sufficient condition for finite‐time stability of positive linear systems. Furthermore, two sufficient conditions for finite‐time stability of switched positive linear systems are given by using the common copositive Lyapunov function and multiple copositive Lyapunov functions, a class of switching signals with average dwell time is designed to stabilize the system, and a computational method for vector functions used to construct the Lyapunov function of systems is proposed. Finally, a concrete application is provided to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison of two novel MRAS based strategies for identifying parameters in permanent magnet synchronous motors

Two model reference adaptive system (MRAS) estimators are developed for identifying the parameters of permanent magnet synchronous motors (PMSM) based on the Lyapunov stability theorem and the Popov stability criterion, respectively. The proposed estimators only need online measurement of currents, voltages, and rotor speed to effectively estimate stator resistance, inductance, and rotor flux-linkage simultaneously. The performance of the estimators is compared and verified through simulations and experiments, which show that the two estimators are simple, have good robustness against parameter variation, and are accurate in parameter tracking. However, the estimator based on the Popov stability criterion, which can overcome parameter variation in a practical system, is superior in terms of response speed and convergence speed since there are both proportional and integral units in the estimator, in contrast to only one integral unit in the estimator based on the Lyapunov stability theorem. In addition, the estimator based on the Popov stability criterion does not need the expertise that is required in designing a Lyapunov function.     

External stability of Caputo fractional‐order nonlinear control systems

This paper investigates external stability of Caputo fractional‐order nonlinear control systems. Following the idea of a traditional Lyapunov function method, we point out the problems that would appear when applying it for fractional external stability. These problems are shown to be solvable by employing results on smoothness of solutions, but this method generalized for Caputo fractional‐order nonlinear control systems requires strong conditions to be imposed on vector field functions and inputs. To further explore the fractional external stability, diffusive realizations and Lyapunov‐like functions are taken into consideration. Specifically, a Caputo fractional‐order nonlinear control system with certain assumptions proves to be equivalent to its diffusive realization; a Lyapunov‐like function based on the realization exhibits properties useful to prove the external stability. As expected, this Lyapunov‐like method has weaker requirements. Finally, it is applied to the external stabilization of a Caputo fractional‐order Chua's circuits with inputs.