Controller tuning freedom under plant identification uncertainty: double Youla beats gap in robust stability

In iterative schemes of identification and control one of the particular and important choices to make is the choice for a model uncertainty structure, capturing the uncertainty concerning the estimated plant model. Structures that are used in the recent literature encompass e.g. gap metric uncertainty, coprime factor uncertainty, and the Vinnicombe gap metric uncertainty. In this paper, we study the effect of these choices by comparing the sets of controllers that guarantee robust stability for the different model uncertainty bounds. In general these controller sets intersect. However in particular cases the controller sets are embedded, leading to uncertainty structures that are favourable over others. In particular, when restricting the controller set to be constructed as metric-bounded perturbations around the present controller, the so-called double Youla parametrization provides a set of robustly stabilizing controllers that is larger than corresponding sets that are achieved by using any of the other uncertainty structures. This is particularly of interest in controller tuning problems.     

鲁棒稳定后推控制设计

研究严格反馈形式系统的鲁棒稳定控制问题.考虑输入噪声、测量噪声以及控制对象扰动的情况,通过非线性距离度量框架结构,建立了鲁棒稳定后推控制设计方法.作为应用,考虑一个时滞的二维系统,得到了闭环系统对时滞的鲁棒稳定性.     

The graph metric for unstable plants and robustness estimates for feedback stability

In this paper, a "graph metric" is defined that provides a measure of the distance between unstable multivariable plants. The graph metric induces a "graph topology" on unstable plants, which is the weakest possible topology in which feedback stability is robust. Using the graph metric, it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of RHP poles. If the perturbed and unperturbed systems have the same RHP poles, then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations. As an application of these results, the design of stabilizing controllers for unstable singularly perturbed systems is studied. Finally, the relationship of the graph metric to the "gap metric" introduced by Zames and El-Sakkary is studied in detail. In particular, it is shown that the robustness results of Zames and El-Sakkary do not enable one to conclude the causality, of the perturbed system, whereas the present results do.     

Optimal robustness in the gap metric

The application of the gap metric to robust stabilization of feedback systems is considered. In particular, a solution to the problem of robustness optimization in the gap metric is presented. The problem of robust stabilization under simultaneous plant-controller perturbations is addressed, and the least amount of combined plant-controller uncertainty, measured by the gap metric, that can cause instability of a nominally stable feedback system is determined. Included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the T-gap metric. A key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations. This settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well-defined metric     

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     

Robust output regulation and the preservation of polynomial closed‐loop stability

In this paper, we study the robust output regulation problem for distributed parameter systems with infinite‐dimensional exosystems. The main purpose of this paper is to demonstrate the several advantages of using a controller that achieves polynomial closed‐loop stability, instead of a one stabilizing the closed‐loop system strongly. In particular, the most serious unresolved issue related to strongly stabilizing controllers is that they do not possess any known robustness properties. In this paper, we apply recent results on the robustness of polynomial stability of semigroups to show that, on the other hand, many controllers achieving polynomial closed‐loop stability are robust with respect to large and easily identifiable classes of perturbations to the parameters of the plant. We construct an observer based feedback controller that stabilizes the closed‐loop system polynomially and solves the robust output regulation problem. Subsequently, we derive concrete conditions for finite rank perturbations of the plant's parameters to preserve the closed‐loop stability and the output regulation property. The theoretical results are illustrated with an example where we consider the problem of robust output tracking for a one‐dimensional heat equation.Copyright © 2013 John Wiley & Sons, Ltd.     

Pointwise gap metrics on transfer matrices

A family of metrics, pointwise gap metrics, in the space of real rational matrices of fixed size is developed and used to study open-loop and closed-loop stability robustness of linear time-invariant finite-dimensional continuous-time systems. It is shown that pointwise gap metrics have the desired qualitative properties for the study of stability robustness. Necessary and sufficient conditions on the robustness are obtained in terms of the radii of the pointwise gap metric balls centered at the nominal plant and/or the nominal controller. Comparison with the gap metric and the graph metric is made. All of these metrics induce the same topology. Many of the quantitative properties of pointwise gap metrics are the same as those of the gap metric, although they differ in value. Pointwise gap metrics, in the scalar case, have a very simple expression which is potentially useful for accessing the relationship between the uncertainty of physical parameters and uncertainty measured by pointwise gap metrics     

Graph topology and gap topology for unstable systems

A reformation is provided of the graph topology and the gap topology for a general setting (including lumped linear time-invariant systems and distributed linear time-invariant systems) in the frequency domain. Some essential properties and their comparisons are clearly presented in the reformulation. It is shown that the gap topology is suitable for general systems rather than square systems with unity feedback. It is shown that whenever an unstable plant can be stabilized by feedback, it is a closed operator, mapping a subspace of the input space to the output space. Hence, the gap topology can always be applied whenever the unstable plant can be stabilized. The graph topology and the gap topology are suitable for different subsets of systems and have many similar characteristics. If one confines them to the same subset, they will be identical. The definitions of the graph metric and the gap metric are discussed     

Performance robustness of discrete-time systems with structureduncertainty

Given an interconnection of a nominal discrete-time plant and a stabilizing controller together with structured, norm-bounded, nonlinear/time-varying perturbations, necessary and sufficient conditions for robust stability and performance of the system are provided. It is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem. The resulting stability robustness problem is shown to be equivalent to a simple algebraic one, the solution of which provides the desired necessary and sufficient conditions for performance/stability robustness. These conditions provide an effective tool for robustness analysis and can be applied to a large class of problems. In particular, it is shown that some known results can be obtained immediately as special cases of these conditions     

BIBO stability robustness in the presence of coprime factorperturbations

A necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given. The computation of the largest stability margin is discussed via solving a nonsquare l¹ optimal control problem. An algorithm for obtaining lower approximations of the minimum value of the optimization problem μ₀ is proposed. This, together with the standard algorithm which provides upper approximations, allows μ₀ to be computed within any degree of accuracy     

The gap metric: Robustness of stabilization of feedback systems

In this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open-loop systems. We elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply, By admissible uncertainties we mean those which preserve closed-loop stability and a specified small tolerance on the I/O behavior of a feedback system. We show that admissible uncertainties are precisely those which are constrained in the gap. Finally, we conclude that any metric which preserves a continuous relationship between open-loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric.     

Feedback stability under simultaneous gap metric uncertainties in plant and controller

The stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric. A fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller.     

A test for stability robustness of linear time‐varying systems utilizing the linear time‐invariant ν‐gap metric

A stability robustness test is developed for internally stable, nominal, linear time‐invariant (LTI) feedback systems subject to structured, linear time‐varying uncertainty. There exists (in the literature) a necessary and sufficient structured small gain condition that determines robust stability in such cases. In this paper, the structured small gain theorem is utilized to formulate a (sufficient) stability robustness condition in a scaled LTI ν‐gap metric framework. The scaled LTI ν‐gap metric stability condition is shown to be computable via linear matrix inequality techniques, similar to the structured small gain condition. Apart from a comparison with a generalized robust stability margin as the final part of the stability test, however, the solution algorithm implemented to test the scaled LTI ν‐gap metric stability robustness condition is shown to be independent of knowledge about the controller transfer function (as opposed to the LMI feasibility problem associated with the scaled small gain condition which is dependent on knowledge about the controller). Thus, given a nominal plant and a structured uncertainty set, the stability robustness condition presented in this paper provides a single constraint on a controller (in terms of a large enough generalized robust stability margin) that (sufficiently) guarantees to stabilize all plants in the uncertainty set. Copyright © 2008 John Wiley & Sons, Ltd.     

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 ν-gap metric. It is also shown that the ν-gap metric between two systems is the distance, measured by the gap metric, between their respective sets of all subgraphs under certain index conditions     

Robust stability under structured and unstructured perturbations

The problem of robust stability for linear time-invariant single-output control systems subject to both structured (parametric) and unstructured (H_∞) perturbations is studied. A generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time-invariant dynamic system under perturbations of mixed type is presented. The solution involves calculating the H_∞ -norm of a finite number of extremal plants. The problem of calculating the exact structured and unstructured stability margins is then constructively solved. A feedback control system containing a linear time-invariant plant which is subject to both structured and unstructured perturbations is considered. The case where the system to be controlled is interval is treated, and a nonconservative, easily verifiable necessary and sufficient condition for robust stability is given. The solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered     

Perturbations of closed-loop systems in robust feedback stabilization

The largest robust stability radius γ(P₀) of a system P₀ is defined as the radius of the largest ball B_max in the gap metric centred at P₀ which can be stabilized by one single controller. Any controller stabilizing B_max is called an optimally robust controller of P₀ . A controller, regarded as a system, should have its own largest robust stability radius also. In this note it is first shown that the largest robust stability radius of any optimally robust controller of P₀ is larger than or equal to γ(Po)- The main result of this paper is the estimate of the variations (in the L∞-norm) of the closed-loop transfer matrix caused by the perturbations of the system or of the optimally robust controller. Finally, the schemes of designing finite-dimensional controllers are presented via the largest robust stability radius. These schemes guarantee that the designed finite-dimensional controllers will stabilize the original infinite-dimensional systems. Moreover, the closed-loop transfer matrices can be estimated.     

Robust stabilization of normalized coprime factor plantdescriptions with H∞-bounded uncertainty

The problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by H _∞ bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system. This problem can be reduced to a Nehari extension problem directly and gives an optimal stability margin. All controllers satisfying a suboptimal stability margin are characterized, and explicit state-space formulas are given     

The robust control of a servomechanism problem for linear time-invariant multivariable systems

Necessary and sufficient conditions are found for there to exist a robust controller for a linear, time-invariant, multivariable system (plant) so that asymptotic tracking/regulation occurs independent of input disturbances and arbitrary perturbations in the plant parameters of the system. In this problem, the class of plant parameter perturbations allowed is quite large; in particular, any perturbations in the plant data are allowed as long as the resultant closed-loop system remains stable. A complete characterization of all such robust controllers is made. It is shown that any robust controller must consist of two devices 1) a servocompensator and 2) a stabilizing compensator. The servocompensator is a feedback compensator with error input consisting of a number of unstable subsystems (equal to the number of outputs to be regulated) with identical dynamics which depend on the disturbances and reference inputs to the system. The sorvocompensator is a compensator in its own right, quite distinct from an observer and corresponds to a generalization of the integral controller of classical control theory. The sole purpose of the stabilizing compensator is to stabilize the resultant system obtained by applying the servocompensator to the plant. It is shown that there exists a robust controller for "almost all" systems provided that the number of independent plant inputs is not less than the number of independent plant outputs to be regulated, and that the outputs to be regulated are contained in the measurable outputs of the system; if either of these two conditions is not satisfied, there exists no robust controller for the system.     

Robustness analysis of nonlinear feedback systems: an input-outputapproach

This paper presents an approach to robustness analysis for nonlinear feedback systems. We pursue a notion of model uncertainty based on the closeness of input-output trajectories which is not tied to a particular uncertainty representation, such as additive, parametric, structured, etc. The basic viewpoint is to regard systems as operators on signal spaces. We present two versions of a global theory where stability is captured by induced norms or by gain functions. We also develop local approaches (over bounded signal sets) and give a treatment for systems with potential for finite-time escape. We compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations, e.g., small-time delays. We also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations. The paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection     

Stability and stabilization of equilibrium positions of nonlinear nonautonomous mechanical systems

Stability analysis methods and stabilizing controls for nonlinear nonautonomous mechanical systems are discussed. Theorems on stability of equilibrium positions in the case of essentially nonlinear positional forces are proved on the basis of decomposition of systems under study. Time estimates for transient processes are found, and the effect of nonstationary perturbations on stability of equilibrium positions is studied. The results obtained are used for solving stabilization problem for mechanical systems with regard to the structure of control forces. Situations where stabilizing controls can be constructed are determined, and the nonlinear forces that play the key role in these situations are identified.