Stability in feedback systems with tapered and other special input spaces

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 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 study introduces a concept of mappings realized in presence of information granules and offers a design framework supporting the formation of such mappings. Information granules are conceptually meaningful entities formed on a basis of a large number of experimental input–output numeric data available for the construction of the model. We develop a conceptually and algorithmically sound way of forming information granules. Considering the directional nature of the mapping to be formed, this directionality aspect needs to be taken into account when developing information granules. The property of directionality implies that while the information granules in the input space could be constructed with a great deal of flexibility, the information granules formed in the output space have to inherently relate to those built in the input space. The input space is granulated by running a clustering algorithm; for illustrative purposes, the focus here is on fuzzy clustering realized with the aid of the fuzzy C-means algorithm. The information granules in the output space are constructed with the aid of the principle of justifiable granularity (being one of the underlying fundamental conceptual pursuits of Granular Computing). The construct exhibits two important features. First, the constructed information granules are formed in the presence of information granules already constructed in the input space (and this realization is reflective of the direction of the mapping from the input to the output space). Second, the principle of justifiable granularity does not confine the realization of information granules to a single formalism such as fuzzy sets but helps form the granules expressed any required formalism of information granulation. The quality of the granular mapping (viz. the mapping realized for the information granules formed in the input and output spaces) is expressed in terms of the coverage criterion (articulating how well the experimental data are “covered” by information granules produced by the granular mapping for any input experimental data). Some parametric studies are reported by quantifying the performance of the granular mapping (expressed in terms of the coverage and specificity criteria) versus the values of a certain parameters utilized in the construction of output information granules through the principle of justifiable granularity. The plots of coverage–specificity dependency help determine a knee point and reach a sound compromise between these two conflicting requirements imposed on the quality of the granular mapping. Furthermore, quantified is the quality of the mapping with regard to the number of information granules (implying a certain granularity of the mapping). A series of experiments is reported as well.

Towards simultaneous reduction of both input and output spaces for interactive simulation-based structural design

This paper studies robust stabilization of linear feedback systems. The special features of this study are: (1) the input and output signal spaces of systems are assumed to be any ℓ p spaces; (2) system perturbations are measured by the gap function.

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.

This paper discusses nonnegativity and positivity concepts and related properties for the state and output trajectory solutions of dynamic linear time-invariant systems described by functional differential equations subject to point time delays. The various nonnegativities and positivities are introduced hierarchically from the weakest one to the strongest one while separating the corresponding properties when applied to the state space or to the output space as well as for the zero-initial state or zero-input responses. The formulation is first developed by defining cones for the input, state and output spaces of the dynamic system, and then extended, in particular, to cones being the three first orthants each being of the corresponding dimension of the input, state, and output spaces.

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.

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.

It is shown that the form of multiplier Z(s) developed by Narendra and Yo Sung Cho for establishing the stability of feedback systems containing a single odd monotonic non-linearity is equally applicable to systems with a monotonic nonlinear feedback. The constraints on the coefficients of Z(s) are further relaxed.

While most mobile platforms offer motion sensing as input for creating tactile feedback, it is still hard to design such feedback patterns while the screen becomes larger, e.g. tabletop surfaces. This demonstration presents Ubitile, a finger-worn concept offering both motion sensing and vibration feedback for authoring of vibrotactile feedback on tabletops. We suggest the mid-air motion input space made accessible using Ubitile outperforms current GUI-based visual input techniques for designing tactile feedback. Additionally Ubitile offers a hands-free input space for the tactile output. Ubitile integrates both input and output spaces within a single wearable interface, jointly affording spatial authoring/editing and active tactile feedback on- and above- tabletops.

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.

A competitive exception learning algorithm (CELA) is proposed which derives exceptional patterns from a given set of data pairs. In the first step of the algorithm, the output space is fuzzily partitioned (using a competitive learning algorithm) and then characterized by a so-called unconditional cluster membership distribution. In the second step, the input space is fuzzily partitioned as well by again using competitive learning. After learning, the input clusters represent locations in the input space which correspond to the most exceptional patterns in the output space. These patterns are characterized by so-termed conditional cluster membership distributions. In the third step, a fuzzy rule base is set up describing the exceptions in linguistic terms. In addition, the complete fuzzy input-output map can be estimated using a mathematical function. Besides describing the various steps of the new algorithm, a theoretical framework is introduced using an information-theoretical point of view: all output data points are supposed to be generated by a non-stationary stochastic, fuzzy information source. A summary concerning the results of certain simulations completes the description of CELA. The article is concluded with a discussion and outlook.

This paper deals with singular value decomposition (SVD) for a class of linear time-varying systems. The class considered here describes switched linear systems with periodic switching. Based on an appropriate input-output description, the calculation method of singular values and singular vectors is derived. The SVD enables us to characterize the dominant input-output signals using singular vectors, which form orthogonal systems in input and output spaces. Then SVD is applied to switched linear systems to improve the transient response. A numerical example is provided to demonstrate the proposed method.