Computation of the robustness margin with the skewed μ tool

Methods for the direct computation of the maximal structured singular value (s.s.v.) over the frequency range require a recursive application of μ analysis. Introducing the v measure as a skewed s.s.v., we point out the problem can be solved by a single application of the v tool. A mixed v upper bound is then proposed, which provides a direct solution to the problem. The relationship between the mixed μ and v upper bounds is moreover clarified. The nonnegativity of the sensitivity of the mixed μ upper bound is finally obtained as a corollary.     

Reliable computation of the robustness margin for a flexible aircraft

In this paper, an efficient method for directly computing a reliable estimate of the maximal structured singular value μ is proposed. A method, which directly computes a μ upper bound over a frequency interval, is first extracted from Ferreres and Biannic (Proc. ACC 4 (1998) 2294–2298). The problem is then to evaluate the conservatism of this upper bound by computing an accurate μ lower bound which is the central issue of the paper. The proposed algorithm is successfully applied—with a reasonable computational effort—to a challenging aeronautical application, namely the evaluation of the robustness properties of the flight control system of a flexible transport aircraft.     

On the robustness of uncertain time-delay systems with structured uncertainties

This paper considers the robust stability and robust performance of uncertain delay systems subject to possibly structured uncertainties using structured singular values with phase information. Computationally efficient sufficient conditions for robust stability and performance problem are derived.     

A numerical method for computing the structured singular value

This article suggests a new approach to computing Doyle's structured singular value (SSV) of a matrix. The SSV is a notion important in robust control and several iteration schemes exist for approximation a solution [1,2,4,-8,10,11].Our idea is to pick a special case of the general problem, which we believe to be natural and give an algorithm for studying it based on ‘off the shelf’ packages. Once we are committed to this ‘special case’ we discuss a very general plant uncertainty problem; it embraces real as well as complex plant perturbations of many kinds. The idea is simple and we believe very natural to the problem.     

Analysis of robust performance with multiple objectives

Performance robustness with multiple objectives of linear control systems having structured norm-bounded uncertainty is considered. In order to deal with any two objective functions Ψ₁ and Ψ₂ associated with a single feedback control system, the combined criterion Ψ[Ψ₁ ψ₂]Ψ S 1 is often used. The paper, however, considers Ψ ψ₁ Ψ 1 and Ψ W, 11 1 directly. It is shown that it is possible to assess robust performance of two objectives by a single μ test if repeated nonscalar blocks are adequately introduced in the structure of μ. In other words, a performance robustness problem with multiple objectives is proved to be equivalent to a stability robustness problem with extra repeated uncertainty blocks. This equivalence theorem is applicable to various system and norm setups including sampled-data systems.     

A detailed comparative analysis of all practical algorithms to compute lower bounds on the structured singular value

μ analysis is one of the most efficient techniques to evaluate the stability margins and the performance levels of linear time-invariant systems in the presence of structured time-invariant uncertainties. The exact computation of the structured singular value μ is known to be NP hard in the general case, but several methods have been developed in the last 30 years to compute accurate and reliable bounds. In this paper, all existing μ lower bound algorithms are reviewed and the most relevant ones are evaluated on a wide set of real-world benchmarks, corresponding to various fields of application, system dimensions and structures of the uncertainties. The results are thoroughly analyzed and simple improvements to the existing algorithms are proposed to approach the exact value of μ with a reasonable computation cost. Conclusions show that non-conservative values can be obtained in almost all cases. A brief extension to skew-μ analysis confirms the good results obtained in the classical μ case.     

Robustness of the stability of feedback systems with respect to small time delays

It is shown that a feedback system is robustly stable with respect to small time delays if and only if it is stable for zero time delay and a structured singular value is less than one.     

A new upper bound for the skewed structured singular value

The structured singular value (s.s.v)μ enables the study of robust stability and performance of a controller in the presence of real parametric uncertainties and complex uncertainties corresponding to neglected dynamics. In spite of the NP-hard characteristic of the problem, it is now possible to compute an interval for the s.s.v. μ using polynomial-time algorithms. The skewed s.s.v. ν was introduced by Fan and Tits in the context of robust performance analysis. The primary aim of this paper is to propose a new mixed ν upper bound, which is applicable to problems with a special, but practically important, structure. We then illustrate through a realistic missile example that certain problems naturally require the ν tool rather than the μ tool. Copyright © 1999 John Wiley & Sons, Ltd.     

MIXED μ PROBLEMS AND BRANCH AND BOUND TECHNIQUES

The computation of the general structural singular value (μ) is NP hard, so quick solutions to medium sized problems must often be approximate. In many of the cases where the current approximate methods are unsatisfactory, improved solutions are highly desirable. It is shown that, despite their worst-case combinatorial nature, branch and bound techniques can give substantially improved solutions with only moderate computational cost. © 1997 by John Wiley & Sons, Ltd.     

Stability and performance of current harmonic controllers for multiphase PMSMs

Multiphase electric machines can provide significant benefits over conventional three-phase machines. A drawback to multiphase machines is that they are known to have problems with stator current harmonics. The harmonics can be eliminated by various current harmonic control methods. However, there appears to be no clear agreement on the most suitable method for multiphase machines. This paper aims to compare different current harmonic controllers in terms of stability and performance under model uncertainty. A detailed theoretical analysis of the harmonic controllers is given by taking a modern multi-input multi-output approach based on a structured singular value analysis. Further, the performance of the harmonic controllers is studied with experimental results from a dual three-phase permanent magnet synchronous machine. The analysis and results of this paper show how to design robust high-performance current harmonic controllers for multiphase machines.     

The rank one mixed μ problem and ‘kharitonov-type’ analysis

The general mixed μ problem has been shown to be NP hard, so that the exact solution of the general problem is computationally intractable, except for small problems. In this paper we consider not the general problem, but a particular special case of this problem, the rank one mixed μ problem. We show that for this case the mixed μ problem is equivalent to its upper bound (which is convex), and it can in fact be computed easily (and exactly). This special case is shown to be equivalent to the so-called ‘affine parameter variation’ problem (for a polynomial with perturbed coefficients) which has been examined in detail in the literature, and for which several celebrated ‘Kharitonov-type’ results have been proven.     

复结构奇异值的几何解释及其计算:两块情形

应用几何方法给出两块情形复结构奇异值的几何解释,并提出一种简单迭代计算方法.该方法求的是结构奇异值的真值而非其某一上界或下界.     

Development of a skew μ upper bound

Exploitation of the NP hard, mixed μ problem structure provides a polynomial time algorithm that approximates μ with usually reasonable answers. When the problem is extended to the skew μ problem, an extension of the existing method to the skew μ formulation is required. The focus of this paper is to extend the μ upper bound derivation to the skew μ upper bound and show its direct computation by way of an Osborne balancing algorithm and an linear matrix inequality (LMI) algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

IQC analysis and synthesis via nonsmooth optimization

Integral quadratic constraints (IQCs) are used in system theory to model nonlinear phenomena within the framework of linear feedback control. IQC theory addresses parametric robustness, saturation effects, sector nonlinearities, passivity, and much else. In IQC analysis specially structured linear matrix inequalities (LMIs) arise and are currently addressed by structure exploiting LMI solvers. Controller synthesis under IQC constraints is nonconvex and much harder and has been attempted sporadically by global optimization techniques such as branch and bound, cutting plane or D-K-type coordinate descent ideas. Here, we revisit IQC theory and propose a completely different algorithmic solution based on local and nonsmooth optimization methods. This is less ambitious than global methods, but is very promising in practice. Our approach, while aiming high at IQC synthesis, offers new answers even for IQC analysis, because we optimize without Lyapunov variables. For high-order systems this leads to a significant reduction of the number of unknowns.     

Measures of overshoot, speed of response and robustness

Performance measures for a linear multivariable system with nonlinear perturbations are defined in terms of the eigenvalues and the eigenvectors of the linearized system matrix, and an interactive computer-aided design of controller parameters is given. The overshoot and the speed of response are the parameters of an exponential envelope constraining the norm of the states under initial condition perturbations. The robustness is shown to be related to these measures, and is bounded by their ratio.     

Instability analysis and improvement of robustness of adaptive control

The effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed. Five possible types of instability mechanisms—parameter drift, ‘linear’ instability, ‘fast adaptation’ instability, ‘high frequency’ instability, and ‘throughput’ instability—are analyzed using simple examples. A procedure is used to construct Lyapunov-like functions for a modified adaptive controller applied to a dominant plant of relative degree one, in the presence of parasitics and disturbances, and obtain sufficient conditions under which none of the five types of instability can occur. The modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set. The size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast, the region of attraction becomes the whole space.     

Open-loop and closed-loop robust optimal control of batch processes using distributional and worst-case analysis

The recognition that optimal control trajectories for batch processes can be highly sensitive to model uncertainties has motivated the development of methods for explicitly addressing robustness during batch processes. This study explores the incorporation of robust performance analysis into open-loop and closed-loop optimal control design. Several types of robust performance objectives are investigated that incorporate worst-case or distributional robustness metrics for improving the robustness of batch control laws, where the distributional approach computes the distribution of the performance index caused by parameter uncertainty. The techniques are demonstrated on a batch crystallization process. A comprehensive comparison of the robust performance of the open-loop and closed-loop system is provided.     

Robust model predictive control of constrained linear systems with bounded disturbances

This paper provides a novel solution to the problem of robust model predictive control of constrained, linear, discrete-time systems in the presence of bounded disturbances. The optimal control problem that is solved online includes, uniquely, the initial state of the model employed in the problem as a decision variable. The associated value function is zero in a disturbance invariant set that serves as the ‘origin’ when bounded disturbances are present, and permits a strong stability result, namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances, to be obtained. The resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control.     

Improving the robustness of Nussbaum-type regulators by the use of σ-modification — local results

The bursting phenomenon in a Nussbaum-type regulator is discussed. Its cause is associated with some properties of the equilibrium manifold. A σ-modification factor is introduced, and the resulting equilibrium set is completely characterized using linearization techniques. Local robustness against bounded disturbances is proved by Malkin's Theorem, and verified by simulation.