The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective and easily computable control laws. These two aspects are complementary, since the latter approaches are, typically, based upon the inductive verification of the validity of the former system properties under compositions (in the terminology used in [62], the “activation” of theoretical concepts leads to “constructive” control). This expository presentation addresses the first of these aspects, and in particular the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability. We provide an intuitive and informal presentation of the main concepts. More precise statements, especially about older results, are given in the cited papers, as well as in several previous surveys such as [103] and [105] (of which the present paper represents an update), but we provide a little more detail about relatively recent work. Regarding applications and extensions of the basic framework, we give some pointers to the literature, but we do not focus on feedback design and specific engineering problems; for the latter we refer the reader to textbooks such as [43], [60], [58], [96], [66], [27], [44].

Input to State Stability: Basic Concepts and Results

A Glimpse at Set Theory. The Real Numbers. The Topology of Cartesian Spaces. Convergence. Continuous Functions. Functions of One Variable. Infinite Series. Differentiation in RP Integration in RP.

The elements of real analysis (2nd edition), by Robert G. Bartle. Pp xv, 480. £10. 1976. SBM 0 471 05464 X (Wiley)

Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach

This paper derives some sufficient conditions for asymptotic stability of neural networks with constant or time-varying delays. The Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach are employed to investigate the problem. It shows how some well-known results can be refined and generalized in a straightforward manner. For the case of constant time delays, the stability criteria are delay-independent; for the case of time-varying delays, the stability criteria are delay-dependent. The results obtained in this paper are less conservative than the ones reported so far in the literature and provides one more set of criteria for determining the stability of delayed neural networks.

LMI-based approach for asymptotically stability analysis of delayed neural networks

This paper is concerned with the robust exponential synchronization problem of a class of chaotic delayed neural networks with different parametric uncertainties. A novel impulsive control scheme (so-called dual-stage impulsive control) is proposed. Based on the theory of impulsive functional differential equations, a global exponential synchronization error bound together with some new sufficient conditions expressed in the form of linear matrix inequalities (LMIs) is derived in order to guarantee that the synchronization error dynamics can converge to a predetermined level. Furthermore, to estimate the stable region, a novel optimization control algorithm is established, which can deal with the minimum problem with two nonlinear terms coexisting in LMIs effectively. The idea and approach developed in this paper can provide a more practical framework for the synchronization of multiperturbation delayed chaotic systems. Simulation results finally demonstrate the effectiveness of the proposed method.

Robust Global Exponential Synchronization of Uncertain Chaotic Delayed Neural Networks via Dual-Stage Impulsive Control

In this paper a novel approach to assess the stability of dynamic neural networks is presented. Using a Lyapunov function, we determine conditions to guarantee input-to-state stability (ISS) which also ensures global asymptotic stability (GAS). The applicability of these conditions is illustrated by two examples.

Input-to-state stability (ISS) analysis for dynamic neural networks

This paper deals with real-time adaptive tracking for discrete-time induction motors in the presence of bounded disturbances. A high-order neural-network structure is used to identify the plant model, and based on this model, a discrete-time control law is derived, which combines discrete-time block-control and sliding-mode techniques. This paper also includes the respective stability analysis for the whole system with a strategy to avoid adaptive weight zero-crossing. The scheme is implemented in real time using a three-phase induction motor.

Real-Time Discrete Neural Block Control Using Sliding Modes for Electric Induction Motors

Chaos stabilization: an inverse optimal control approach

Real-Time Integral Backstepping with Sliding Mode Control for a Quadrotor UAV

This paper presents the input-to-state (ISS) analysis for dynamic neural networks. We determine, using a Lyapunov function, conditions to guarantee ISS; they also guarantee globally asymptotically stability.

Edgar N. Sanchez

Papers

Input-to-state stability (ISS) analysis for dynamic neural networks

Real-Time Discrete Neural Block Control Using Sliding Modes for Electric Induction Motors

Chaos stabilization: an inverse optimal control approach

Real-Time Integral Backstepping with Sliding Mode Control for a Quadrotor UAV

Input-to-state stability (ISS) analysis for dynamic neural networks