Stability of feedback systems containing a single monotonic nonlinearity

Some extensions on recent work in passive stability theory by Desoer and the author are presented. The stability of feedback systems is considered where the forward-loop and return-loop subsystems each have been subdivided into several systems operating in parallel. Results are obtained in the areas of Lyapunov stability, L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> input L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> output stability, and bounded-input bounded-state stability. Before applying these results to a specific multiloop feedback system, a pseudoenergy analysis must be done on each sub sub-system. Two specific types of systems are so analyzed. The first is a general linear first-order time-varying system. The second is a linear time-varying infinite-dimensional system; the analysis of this system takes the form of a Kalman-Yacubovich-type lemma. By using these two new systems, along with several others that have been analyzed previously, stability theorems for many specific multiloop feedback systems can be proven. One such example is given.

The global stability of positive continuous-time standard and fractional order nonlinear feedback systems is investigated. New sufficient conditions for the global stability of these classes of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear systems.

The global stability of positive discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

The effect of weighted input and output spaces on the stability of feedback systems is examined. For a weighted space, the recent past of a signal is emphasized and the remote past is deemphasized. Tapered spaces are a subclass of weighted spaces. Results concerning weighted spaces used on linear systems are given. The circle conditions are recovered for feedback systems with exponentially tapered spaces.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Using the parameter-dependent Lyapunov function framework of Haddad and Bernstein this paper extends and connects some of the classical absolute stability results on discrete-time systems with monotonic and odd monotonic feedback nonlinearities to modern robust stability and performance problems for constant real parameter uncertainty. As in the continuous-time case, it is shown that an immediate application of this framework provides a generalization and reformulation of discrete-time mixed-/spl mu/ theory in terms of Lyapunov functions and Riccati equations while blending H/sub 2/ robust performance within the /spl mu/ framework.

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

The object of this paper is to outline a stability theory for input-output problems using functional methods. More particularly, the aim is to derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invariant feedback system, stability can be assessed using Nyquist's criterion; roughly speaking, stability depends on the mounts by which signals are amplified and delayed in flowing around the loop. An attempt is made here to show that similar considerations govern the behavior of feedback systems in general-that stability of nonlinear time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related to amplification and delay. This paper is divided into two parts: Part I contains general theorems, free of restrictions on linearity or time invariance; Part II, which will appear in a later issue, contains applications to a loop with one nonlinear element. There are three main results in Part I, which follow the introduction of concepts of gain, conicity, positivity, and strong positivity: THEOREM 1: If the open loop gain is less than one, then the closed loop is bounded. THEOREM 2: If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. Results analogous to Theorems I-3, but with boundedness replaced by continuity, are also obtained.

THE PURPOSE of this paper is to present a technique which may be applied to improve the stability of sampled-data feedback control systems having more than one sampler without the use of additional compensating elements, and to illustrate by means of some examples the type of analysis procedure which may be used and the kind of results which may be expected. The technique which is referred to deals with the phasing of samplers in a multiple sampler feedback system to obtain an optimum condition of stability for a given configuration of the system. The technique is principally applicable either to existing systems where it is desired to improve upon a marginally satisfactory degree of stability in the most economical fashion, or to systems where the sampling results from inherent characteristics of the system data measuring or processing elements and is not introduced purposely to utilize the advantages of digital compensation. For the purposes of this paper the discussion will be limited to consideration of multiple sampler feedback systems containing only two samplers operating at the same sampling frequency and it will be assumed that a zero-order hold circuit follows each sampler. The results can be readily extended to systems with more than two samplers. These conditions are not exceedingly restrictive and many practical systems can be conceived which meet them in an engineering sense.

In adaptive dynamical networks, the dynamics of the nodes and the edges influence each other. We show that we can treat such systems as a closed feedback loop between edge and node dynamics. Using recent advances on the stability of feedback systems from control theory, we derive local, sufficient conditions for steady states of such systems to be linearly stable. These conditions are local in the sense that they are written entirely in terms of the (linearized) behavior of the edges and nodes. We apply these conditions to the Kuramoto model with inertia written in an adaptive form and the adaptive Kuramoto model. For the former, we recover a classic result, and for the latter, we show that our sufficient conditions match necessary conditions where the latter are available, thus completely settling the question of linear stability in this setting. The method we introduce can be readily applied to a vast class of systems. It enables straightforward evaluation of stability in highly heterogeneous systems.

In this paper, we study the problems of robustness and disturbance rejection in a general setting, whereby the signal space in which the inputs and outputs reside is not necessarily L 2 . It is well known that, if the signal space is taken as L 2 , then both optimal robustness design and optimal disturbance rejection can be formulated as H ∞ -norm minimization problems. Three distinct 'Aizerman' type of conjectures regarding the stability of nonlinear feedback systems are formulated, each of which happens to be true in the special case when the underlying signal space is L 2 . It is shown that, in a general setting, only one of the three Aizerman-type conjectures is true, namely: If a feedback system is stable for all linear, possibly time-varying feedback elements belonging to a specified sector, then the feedback system remains stable for all nonlinear, possibly time-varying feedback elements belonging to the same sector. It is shown that the remaining two conjectures are equivalent to each other, and necessary and sufficient conditions for each of the two conjectures to hold are derived. Next, it is shown that, in general, the problems of optimal disturbance rejection and optimal robustness design are quite distinct. In the special case where the signal space is L 2 and the corresponding Banach algebra of causal stable LTI systems is H ∞ , both problems coincide. But in general, the problem of optimal disturbance rejection is that of minimizing the norm of the weighted sensitivity matrix, whereas the problem of optimal robustness design is that of minimizing something like the spectral radius of the weighted complementary sensitivity matrix.

The author considers stability and robustness of feedback systems, where plant and compensator need not be strictly proper. In his earlier paper (2001) he described a functional R/sub /spl infin// which, when negative, guarantees closed-loop instability as a result of parasitic interactions in the feedback loop. In his main result, Theorem 5, he proves that, when R/sub /spl infin//>0. there exist perturbations of plant and compensator from a narrow class which result in closed-loop stability and convergence. Hence, one may view R/sub /spl infin//>0 as a necessary and sufficient condition for closed-loop robustness in non-strictly-proper feedback loops.

A new criterion for absolute stability in the sector [a, η], a ≠ 0, is presented.

The problem of deriving frequency-domain conditions that ensure asymptotic stability with probability one (ASWP1) of discrete time non-linear feedback systems with state-dependent noise is considered. It is shown that if the noise is linearly related with the state of the system, a relatively simple frequency-domain inequality that guarantees ASWPl of the system, exists. The discrete stochastic version of the second method of Liapunov, along with Szego-Kalman typo lemmae are used to derive such conditions, assuming that only sector information or both sector and slope information are available. The case of a difference system with white-noise parameter perturbation is also considered and an illustrative example is presented.

The global stability of continuous-time fractional orders multi–input multi-output nonlinear feedback systems with interval matrices of positive linear parts is investigated. New sufficient conditions for the global stability of these class of positive nonlinear systems are established. A procedure for computation of gain matrix characterizing the class of nonlinear elements is given and illustrated on simple example.

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions.