Reliable state-space upper bounds for the peak structured singular value

A state-space method for computing upper bounds for the peak of the structured singular value over frequency for both real and complex uncertainties is presented. These bounds are based on the positivity and Popov criteria for one-sided, sector-bounded and for norm-bounded, block-structured linear uncertainty. These criteria are restated and used to derive upper bounds for the peak structured singular value by equating the feasibility of a linear matrix inequality which involves a plant state-space realization to the strict positive realness of a transfer function. Numerical examples are given to illustrate these upper bounds. © 1998 John Wiley & Sons, Ltd.     

An upper bound of structured singular value

An improved upper bound of structured singular value for mixed uncertainties with purely real uncertainty blocks is proposed.     

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.     

Lower bounds for the smallest singular value

Let A = (a_ij) be an n × n complex matrix. Suppose that G(A), the undirected graph of A, has no isolated vertex. Let E be the set of edges of G(A). We prove that the smallest singular value of A, σ_n, satisfies: σ_n ≥ min σ_ij | (i, j) ∈ E, where g_ij ≡ a_i + a_j − [(a_i − a_j)² + (r_i + c_i)(r_j + c_j)]^1/2/2 with a_i ≡ |a_ii| and r_i,c_i are the i^th deleted absolute row sum and column sum of A, respectively. The result simplifies and improves that of Johnson and Szulc: σ_n ≥ min_i≠j σ_ij. (See [1].)     

An implicit small gain condition and an upper bound for the real structured singular value

In this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem. The implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and, unlike prior bounds, does not require frequency-dependent scales or multipliers. Numerical results show that the implicit small gain bound compares favorably with real-μ bounds that involve frequency-dependent scales and multipliers.     

Improved upper and lower bounds on the optimization of mixed chordal ring networks

Recently, Chen, Hwang and Liu [S.K. Chen, F.K. Hwang, Y.C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 1 (2003) 3-16] introduced the mixed chordal ring network as a topology for interconnection networks. In particular, they showed that the amount of hardware and the network structure of the mixed chordal ring network are very comparable to the (directed) double-loop network, yet the mixed chordal ring network can achieve a better diameter than the double-loop network. More precisely, the mixed chordal ring network can achieve diameter about as compared to for the (directed) double-loop network, where N is the number of nodes in the network. One of the most important questions in interconnection networks is, for a given number of nodes, how to find an optimal network (a network with the smallest diameter) and give the construction of such a network. Chen et al. [S.K. Chen, F.K. Hwang, Y.C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 1 (2003) 3-16] gave upper and lower bounds for such an optimization problem on the mixed chordal ring network. In this paper, we improve the upper and lower bounds as and , respectively. In addition, we correct some deficient contexts in [S.K. Chen, F.K. Hwang, Y.C. Liu, Some combinatorial properties of mixed chordal rings, J. Interconnection Networks 1 (2003) 3-16].     

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.     

A study of the gap between the structured singular value and itsconvex upper bound for low-rank matrices

The size of the smallest structured destabilizing perturbation for a linear time-invariant system can be calculated via the structured singular value (μ). The function μ can be bounded above by the solution of a convex optimization problem, and in general there is a gap between μ and the convex bound. This paper gives an alternative characterization of μ which is used to study this gap for low-rank matrices. The low-rank characterization provides an easily computed bound which can potentially be significantly better than the standard convex bound. This is used to find new examples with larger gaps than previously known     

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.     

Polynomial methods for the structured singular value with real parameters

We consider the structured singular value problem with real parametric uncertainty only. Using techniques from algebraic geometry, we propose two algorithms that in principle can yield the precise value of the structured singular value at a fixed frequency. Their ability to do so depends upon their ability to find all common roots to a system of polynomial equations. The first algorithm is applicable to problems with two real parameters each of multiplicity two. The second algorithm is applicable to problems with n distinct real parameters. These algorithms have proved useful in applications to aerospace control law analysis.     

Some new properties of the structured singular value

J.C. Doyle et al. (1982) have shown that a necessary and sufficient condition for robust stability or robust performance in the H^∞-frame work may be formulated as a bound on the structured singular value (μ) of a specific matrix M which includes information on the system model, the controller, the model uncertainty, and the performance specifications. Often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices (T) which are of direct interest to the engineer, e.g. sensitivity or complementary sensitivity. The present paper shows how to derive bounds on σ(T) from bounds on μ(M)     

Real structured singular value conditions for the strong D-stability

Recently, Chen et al. (Systems Control Lett. 24 (1995) 19) proposed conditions for D-stability and strong D-stability in terms of structured singular values. In this paper, simpler conditions for the strong D-stability are derived.     

Characterization and efficient computation of the structured singular value

The concept of structured singular value was recently introduced by Doyle [1] as a tool for the analysis and synthesis of feedback systems with structured uncertainties. It is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements theH^{infty}approach to control system design. in this paper, it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems. Properties of these optimization problems are exhibited, leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3, and often does for block structures of larger size.     

Lower and upper bounds for the mixed capacitated arc routing problem

This paper presents a linear formulation, valid inequalities, and a lower bounding procedure for the mixed capacitated arc routing problem (MCARP). Moreover, three constructive heuristics and a memetic algorithm are described. Lower and upper bounds have been compared on two sets of randomly generated instances. Computational results show that the average gaps between lower and upper bounds are 0.51% and 0.33%, respectively.     

Improved upper bounds on synchronizing nondeterministic automata

We show that i-directable nondeterministic automata can be i-directed with a word of length O(2ⁿ) for i=1,2, where n stands for the number of states. Since for i=1,2 there exist i-directable automata having i-directing words of length Ω(2ⁿ), these upper bounds are asymptotically optimal. We also show that a 3-directable nondeterministic automaton with n states can be 3-directed with a word of length , improving the previously known upper bound O(2ⁿ). Here the best known lower bound is .     

The real structured singular value is hardly approximable

This paper investigates the problem of approximating the real structured singular value (real μ). A negative result is provided which shows that the problem of checking if μ=0 is NP-hard. This result is much more negative than the known NP-hard result for the problem of checking if μ<1. One implication of our result is that μ is hardly approximable in the following sense: there does not exist an algorithm, polynomial in the size n of the μ problem, which can produce an upper bound μ¯ for μ with the guarantee that μ⩽μ¯⩽K(n)μ for any K(n)>0 (even exponential functions of n), unless NP=P. A similar statement holds for the lower bound of μ. Our result strengthens a recent result by Toker, which demonstrates that obtaining a sublinear approximation for μ is NP-hard     

On the norms used in computing the structured singular value

Different norms are considered to replace the Euclidean norm in an algorithm given by M.H.K. Fan and A.L. Tits (ibid., vol.33, p.284-9, 1988), which is used for the computation of the structured singular value of any matrix. The algorithm is explained, and it is shown that the l₁ norm is the best possible norm in a certain sense