A link between input-output stability and Lyapunov stability

It is shown that incremental boundedness of an input-output operator ensures global asymptotic stability of any motion. A reciprocal statement is set.     

Equivalence of input-output stability and exponential stability for infinite-dimensional systems

For a wide class of infinite-dimensional linear systems it is shown that if the state-space realization is exponentially stabilizable and detectable then exponential stability is equivalent to input-output stability of the transfer function. Exponential stabilizability (or detectability) is a strong assumption as it implies that the system operator satisfies the spectrum decomposition assumption and has finitely many unstable eigenvalues of finite multiplicity. In addition, the finite-dimensional unstable projection of the system is controllable (or observable).     

Functional expansion and stability for non-linear input-output maps

The authors employ the global bilinearization of a linear analytic system using the tensor operator approach previously introduced by them in 1988. They generalize their 1988 result on finite dimensional bilinear systems to this class of systems and use the idea of diagonal dominance for tensor operators to derive an exponential stability result. A few examples arc presented to illustrate the theory.     

Sensitivity and robust stability of general input-output systems

The author considers a general model of an input-output system that is governed by nonlinear operator equations which relate the input, the state, and the output of the system. This model encompasses feedback systems as a special case. Assuming that the governing equations depend on a parameter A which is allowed to vary in a neighborhood of a nominal value A₀ in a linear space, the author studies the dependence of the system behavior on A. A system is considered insensitive if, for any fixed input, the output depends continuously on A. Similarly, the system is robust if it is stable for each A in a neighborhood of A₀. Stability is defined as an appropriate continuity of the input-output operator. The results give various sufficient conditions for insensitivity and robustness. Applications of the theory are discussed, including the estimation of the difference of operator inverses, and the insensitivity and robust stability of a Hilbert network, a feedback-feedforward system, a traditional feedback system, and a time-varying dynamical system described by a linear vector differential equation on (0, ∞)     

On graphs, conic relations, and input-output stability of nonlinearfeedback systems

This paper presents necessary and sufficient conditions for the stability of general interconnected dynamical systems. The results are expressed in terms of the graphs of the interconnection's subsystems. Conditions for robustness of stability for an interconnection with additive disturbances are also provided. Finally, the paper gives sufficient conditions for stability using conic sector conditions that generalize earlier conditions and that incorporate a recent nonlinear small gain idea     

Interconnected linear multivariable systems with delays: System properties and input-output stability

We consider an arbitrary interconnection of linear, time-invariant, distributed, multivariable subsystems; each subsystem is represented either by a strictly proper rational transfer function or by a finite sum of pure delay terms. Such models occur in transportation systems, in chemical processes, in production systems, in communication systems, and in models of boilers in nuclear power plants. Allowing inputs to be applied to any interconnection point, we show that, except for a special case, the input-output transfer function of such a system is in a class extensively stuided in the literature. Next we obtain necessary and sufficient conditions for input-output stability. An example illustrates the application of the test.     

Dwell-time approach to input-output stability properties for a class of discrete-time dynamical systems

We consider a class of discrete-time dynamical systems that consist of switching of many linear time-invariant systems, and it can be used to cover a sampled-data version of Switched Linear Systems, which have been widely studied in the literature. For this class of dynamical systems, we establish the LMI (Linear Matrix Inequality) formulation of analyzing their input-output stability properties, including both ?-2 stability and passivity, by constraining switching signals via the concept of dwell time. The LMI formulation of ?-2 stability is applied to evaluating the closed-loop performance of an industrial refrigeration process that is regulated by several proportional-integral (PI) controllers in a switching structure.     

Further results concerning finite-time stability

Weiss and Infante [1], [2] developed sufficient conditions for finite-time stability of a vector differential equation. It is shown here that some of their theorems actually imply the stronger property of uniform finite-time stability. A theorem is presented which gives sufficient conditions for stability but not necessarily uniform stability. It is shown that this theorem can in certain cases yield better results than those of Weiss and Infante's corresponding theorem.     

Further results on stability of

Our aim in this communication is to improve the existing results concerning the stability of . It is shown that the unstable characteristic roots could be more accurately located in a bounded and restrictive region of the complex plane. An iterative scheme is presented for numerical implementation of the results. Numerical computations are presented for illustration and comparison with previous results.     

Balanced input-output assignment

A method of actuator/ sensor location is presented. The state space representation obtained has assigned controllability and observability properties. The assignability conditions are defined, and algorithms for determining sensor and/or actuator locations are derived. The results are specified for systems with properties of flexible structures.     

Conditions for local almost sure asymptotic stability

In this paper, we investigate local asymptotic stability ensured by the addition of Gaussian white noise into dynamical systems. There are different stability notions for stochastic systems, such as asymptotic stability in probability (ASiP) and uniform almost sure asymptotic stability (UASAS). The local ASiP property is incapable of ensuring that sample paths converge to the origin with probability one, whereas the local UASAS property is capable of it. However, in general, the local UASAS property requires tight conditions. Here, we provide our notion of local almost sure asymptotic stability (local ASAS) to relax the conditions with both almost sure convergence of sample paths to the origin and the existence of bounded (weak) invariant sets. We find that the addition of Gaussian white noise always prevents the origin from being locally UASAS as long as we consider smooth Lyapunov functions; however, it is possible to make the origin locally ASAS. The result is confirmed by a simple example of elimination of unstable equilibria by deliberately adding Gaussian white noise.     

A priori stability results for PFC

Despite its popularity in industry and obvious efficacy, predictive functional control has few rigorous a priori stability results in the literature. In many cases, common sense and intuition with some trial and error are the main design tools. This paper seeks to tackle that gap by providing some analysis of the control law and showing what forms of stability assurances can be given and how these depend on the user choices of coincidence horizon and desired closed-loop pole. The conditions are separated into necessary, but not sufficient conditions for stability and, conversely, sufficient but not necessary conditions. Numerical examples demonstrate the efficacy of these conditions and the ease of use.     

Extreme-point robust stability results for discrete-timepolynomials

This paper provides some new extreme-point robust-stability results for discrete-time polynomials with special uncertainties in the coefficient space. The proofs, obtained using the bilinear transformation, are simple, and the results specialize to existing robust-stability results     

Connections between local stability in Lyapunov and input/outputsenses

The concept of gain over set was recently introduced as a tool for local input/output analysis of nonlinear systems. In this setting, the finiteness of the gain over set defines a local stability in the input/output sense. In this paper, we show that the finite gain over ball stability is related to the local stability in the sense of Lyapunov     

Robust local stability of multilayer recurrent neural networks

We derive a condition for robust local stability of multilayer recurrent neural networks with two hidden layers. The stability condition follows from linking theories about linearization, robustness analysis of linear systems under nonlinear perturbation, and matrix inequalities. A characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic Lyapunov function. Similar to the NL(q) theory, the local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion, while the proven basin of attraction is relatively small due to conservatism of the criterion. Modification of the dynamic backpropagation by the new stability condition is discussed and illustrated by simulation examples.     

Analysis of the local robustness of stability for flows

In this paper the problem of measuring the robustness of stability for a perturbed continuous-time nonlinear system at a singular fixed point is studied. Various stability radii are introduced and their values for the nonlinear system and its linearization are compared. It is shown that they generically coincide. This result may also be used to show generic continuity of linear real stability radii. Some examples are presented showing that it is sometimes necessary to consider the nonlinear system directly, and not simply to rely on the information provided by the linearization.     

Generalization and comparison of two recent frequency-domain stability robustness results

The generalization of two recent frequency-domain methods for evaluating the robust stability of uncertain systems to the complex polynomial case is presented. Both methods are based on Mikhailov criteria and have been established recently for real continuous-time systems. Besides generalization to the complex case, these two methods are related for both the real and complex cases, and an objective comparison is made between them. Some illustrative examples, are given.     

Modeling and input-output decoupling of hypersonic vehicles

This paper studies the problems of modeling and input-output decoupling of generic hypersonic vehicles. Dynamical equations of hypersonic vehicles are derived using Lagrangian approach, which capture the dominating characteristics and primary interactions. Then, based on the simplified model, the original decoupling problem is reformulated as an asymptotical stability problem of the corresponding error system. The popular dynamic inversion is employed to design the decoupling controller, which can achieve steady-state decoupling. However, external disturbance will greatly destroy the effect of decoupling before the system reaches steady state. To this end, based on the error system, robust H _∞ theorem can be easily used to address this issue by reducing the impact of disturbance on error system outputs, which ultimately results in approximate decoupling. Moreover, the degree of approximate decoupling can be enhanced by choosing a small performance index γ. Simulations verify the effectiveness of proposed controllers.     

New converse Lyapunov theorems and related results on exponential stability

We treat the problem of constructing Lyapunov functions for systems which are, by assumption, exponentially stable. The construction we present results in a larger set of functions than those obtainable by previously known methods. A useful property of the proposed Lyapunov functions is that they preserve information on the rate of exponential convergence of the system. Some useful applications are given. The first author was supported by the US National Science Foundation under Grants MSM-87-06927 and MSS-90-57079. The activities of the second author were performed during a stay at the School of Aeronautics and Astronautics, Purdue University, and were supported by Consiglio Nazionale delle Ricerche, under Grant 203.07.17.