Gap Metric Research Articles

Gap Metric Concept and Implications for Multilinear Model-Based Controller Design

The gap metric concept is used within the context of multilinear model-based control. The concept of distance between dynamic systems is used as a criterion for selecting a set of models that can explain the nonlinear plant behavior in a given operating range. The case studies presented include a CSTR and a pH neutralization reactor. The gap metric is used to analyze the relationships among candidate models, resulting in a reduced model set that provides enough information to design multilinear controllers. The simulation and experimental results indicate good performance and stability features.

The Vinnicombe metric for nonlinear operators

Describes an extension of the Vinnicombe metric on linear operators to a pseudometric on nonlinear operators. A metric for finite-dimensional time-varying operators is shown to be capable of guaranteeing stability and performance robustness and reduces to the standard Vinnicombe metric for the time-invariant operator case, which is known to be less conservative than the gap metric. The analysis exploits the time-varying operator equivalents of unstable poles and normalized coprime fractional descriptions. In addition, a time-varying operator equivalent of the winding number is defined.

Linear feedback systems and the graph topology

It is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed-loop stability and closed-loop performance are robust properties. In earlier works, similar topological results were obtained by exploiting the existence of particular coprime-factor system representations, not known to exist in general. By contrast, the development here does not rely on any specific system representations. Systems are simply characterized as subspaces of norm bounded input-output pairs, and the analysis hinges on the underlying geometric structure of the feedback stabilization problem. Unlike other work developed within such a framework, fundamental issues concerning the causality of feedback interconnections are discussed explicitly. The key result of this paper concerns the difference between linear feedback interconnections, with identical controllers, in terms of a performance/robustness related closed-loop mapping. Upper and lower bounds on the induced norm of this difference are derived, allowing for possibly infinite-dimensional input-output spaces and time-varying behavior. The bounds are both proportional to the gap metric distance between the plants, which clearly demonstrates the gap to be an appropriate measure of the difference between open-loop systems from the perspective of closed-loop behavior. To conclude, an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting.

Robustness analysis of an integrated flight and propulsion control system using μ and the ν-gap metric

Abstract This paper describes the application of two frequency domain analysis tools, the structured singular value μ and the ν -gap metric, to the problem of evaluating the robustness properties of a multivariable integrated flight and propulsion control (IFPC) system for an experimental V/STOL aircraft configuration. Variations in aircraft parameters and dynamics over the V/STOL flight envelope are represented as linear fractional transformation (LFT) based parametric uncertainty descriptions, using a simple ‘black-box’ type modelling approach. Tight bounds on μ for the resulting high-order real parametric uncertainty model are computed by introducing small amounts of additional complex uncertainty, in order to analyse the stability robustness properties of the IFPC system. Stability and performance robustness of the IFPC system is also evaluated using the ν -gap metric. Results generated by both analysis tools are seen to be consistent, and to agree well with evaluation results obtained for the IFPC system in piloted simulation.

Gap metrics for handwritten Korean word segmentation

A gap-based method for the segmentation of handwritten Korean text lines into separate words is described. Three kinds of gap metrics have been evaluated, to see how much the gap information contributes to the task of handwritten Korean word segmentation.

Some Alternate New Gap Metric

The paper introduces some new alternate gap metrics for measuring the distance between two transfer functions. These metrics amplify the principal frequency domain where the sensitivity equals to one and/or where the closed-loop frequency gain equals to one. Some examples are shown when the new metrics are more sensitive to parameter variations, than the well known v-gap metric

Remarks on "Robustness analysis of nonlinear feedback systems: an input-output approach"

In our previous paper (1999) Sec. VII contains two examples of adaptive controllers (Examples 8 and 9) for which the nonlinear gap robustness margin is zero and which are destabilized by arbitrarily small perturbations in the gap metric. The proof that the robustness margin is zero makes use of the fact that the parameter estimate /spl Theta/(t) tends to infinity as t/spl rarr//spl infin/ when there is a constant input disturbance of size /spl epsi/>0. The purpose of this present note is: a) to complete the argument given in the above paper to prove the unboundedness of /spl Theta/(t) and b) to give an alternative proof that the convergence of the state x(t)/spl rarr/0 is a consequence of the unboundedness of /spl Theta/(t) in Example 9.

補償器の低次元化による変動を考慮したロバスト制御

This paper presents a designing method of a robust controller which can be reduced within a prescribed degradation of a control performance. The method uses p-synthesis to take each variation of a plant and a controller into account. A numerical example is presented and its result is analyzed by using the Gap Metric.

Gap metric on the subgraphs of systems and the robustness problem

The gap metric between the shift invariant subspaces of the graphs, or subgraphs, of systems is investigated. Under certain index conditions, it is shown that the gap metric on the subgraphs shares the same fundamental property of robust stability as those well known metrics such as the gap metric and the /spl nu/-gap metric. It is also shown that the /spl nu/-gap metric between two systems is the distance, measured by the gap metric, between their respective sets of all subgraphs under certain index conditions.

Existence of Right and Left Representations of the Graph for Linear Periodically Time-Varying Systems

The graph representation of a system (the set of all input-output pairs) has gained considerable attention in the control literature in view of its usefulness for the analysis of feedback systems. In this paper it is shown that the graph of any stabilizable, linear, periodically time-varying (LPTV), continuous-time system can be expressed as the range and kernel of bounded, causal, LPTV systems that are, respectively, left and right invertible by bounded, causal, LPTV systems. These so-called strong-right and strong-left representations are closely related to the perhaps more common notion of coprime factor representations. As an example of their usefulness, a neat characterization of closed-loop stability is obtained in terms of strong-right and strong-left representations of the plant and controller graphs. This in turn leads to a Youla-style parametrization of stabilizing controllers. All of the results obtained accommodate possibly infinite-dimensional input and output spaces and apply, as a special case, to sampled-data control-systems. Furthermore, they are particularly useful for robustness analysis in terms of the gap metric.

Robust stabilization of discrete-time multirate systems

In this paper we present a state space solution to the robust stabilization problem of general discrete-time multirate systems where the uncertainty is described in terms of the gap metric. This robust stabilization problem is converted to a constrained H ∞ optimal control problem by using the lifting technique. Then the optimal robust stabilization margin is explicitly computed and a method is provided to get the controllers satisfying the optimal robust stabilization margin. The solution amounts to solving two discrete-time algebraic Riccati equations and an extended Parrot problem.

Computation of the singular values of Toeplitz operators and the gap metric

It is widely recognized that the computation of gap metric is equivalent to a certain two-block H ∞ problem, i.e., the gap is equal to the norm of a certain two-block operator. However, it can also be characterized as the smallest singular value of a certain Toeplitz operator. This paper derives a simple computational method for finding such singular values and the gap between two plants by using a state space approach.

Robustness and Trade-offs in Repetitive Control

We investigate some robustness compromises involved in achieving disturbance rejection for periodic disturbances. We consider asymptotic rejection or attenuation of disturbance harmonics and transient response. Robustness is measured with respect to uncertainty in the gap metric.

Robust detection with the gap metric

In a multipath communication channel, the optimal receiver is matched to the maximum likelihood (ML) estimate of the multipath signal. In general, this leads to a computationally intensive multidimensional nonlinear optimization problem that is not feasible in most applications. We develop a detection algorithm that avoids finding the ML estimates of the channel parameters while still achieving good performance. Our approach is based on a geometric interpretation of the multipath detection problem. The ML estimate of the multipath signal is the orthogonal projection of the received signal on a suitable signal subspace S. We design a second subspace G, which is the representation subspace, that is close to S but whose orthogonal projection is easily computed. The closeness is measured by the gap metric. The subspace G is designed by using wavelet analysis tools coupled with a reshaping algorithm in the Zak transform domain. We show examples where our approach significantly outperforms the conventional correlator receiver (CR) and other alternative suboptimal detectors.

Explicit formulas for optimally robust controllers for delay systems

This paper considers single-input/single-output systems whose transfer functions take the form of a strictly proper rational function times a delay. A closed-form expression is presented for the controller which is optimally robust with respect to perturbations measured in the gap metric. The formula allows the H/sub /spl infin// loop-shaping procedure of Glover-McFarlane to be carried out explicitly for this class of systems without the need to first find a rational approximation of the plant. The form of the controller involves a certain algebra of "pseudo-derivation" operators. These operators, and their matrix generalizations, play a central role in the derivation of the controller. A discussion of the main properties of these operators is given. An example is presented of a controller design to achieve disturbance attenuation and robust set-point following for a plant with two lightly damped poles and a nontrivial time delay. The performance is compared, and shown to be superior, to that of a Smith predictor.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Continuity of optimal robustness and robust stabilization in slowly varying systems

Continuity properties of the optimal design of robust stabilization in the gap metric are investigated. In general the optimal design in the gap metric lacks continuity properties required for certain H ∞ adaptation schemes for slowly varying systems. On the other hand, the δ-suboptimal central design is shown to be Lipschitz continuous. Applications of the continuity properties and frozen-time stability analysis lead to a suboptimal design for robust stabilization in slowly time-varying plants which are represented by local normalized coprime factorizations.

Differentiable families of subspaces

We show how the differentiable families of subspaces can be studied, from a geometrical point of view, by means of their representation as a differentiable mapping from the manifold of parameters M into the Grassmann manifold Gr k,n of all k -dimensional subspaces of K n , and the consideration of principal bundles over Gr k,n and M . In order to achieve this, first we present a survey of the topology, the differentiable structure and the bundle structure of Gr k,n . In particular, we show that the topology induced by the gap metric is equivalent to the usual quotient topology. Second, we use these structures to prove two local characterizations of the differentiability of a family of subspaces, and we state that these characterizations can be globalized if M is contractible. Finally, we apply these properties to study the existence of global Jordan basis of class C r .

Robust Stability of Feedback Systems: A Geometric Approach Using the Gap Metric

A geometric framework for robust stabilization of infinite-dimensional time-varying linear systems is presented. The uncertainty of a system is described by perturbations of its graph and is measured in the gap metric. Necessary and sufficient conditions for robust stability are generalized from the time-invariant case. An example is given to highlight an important difference between the obstructions, which limit the size of a stabilizable gap ball, in the time-varying and time-invariant cases. Several results on the gap metric and the gap topology are established that are central in a geometric treatment of the robust stabilizability problem in the gap. In particular, the concept of a “graphable” subspace is introduced in the paper. Subspaces that fail to be graphable are characterized by an index condition on a certain semi-Fredholm operator.

Upper and lower bounds for approximation in the gap metric

Upper and lower bounds for the closest approximant of degree k<n in the gap metric to a plant of degree n are obtained. The bounds are expressed in terms of the singular values of two Hankel operators defined in terms of the symbol of the graph of the plant. The question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric.

On the gap metric and coprime factor perturbations

New conditions are derived for when the distance between two linear systems in the gap metric is less than one. By including a coprimeness assumption in the coprime factor uncertainty description it is shown that an open ball in the gap metric is equivalent to an open ball phrased in terms of coprime factor perturbations. A new criterion is given for robust stabilization.