Michael K. H. Fan

Frequency domain robust stability criteria for linear delay systems

Some results in robust stability of differential delay systems and H/sub /spl infin// control are combined to obtain frequency domain based criteria for the robust stability of linear differential delay systems. These tests are amenable to graphical verification.<<ETX>>

Structured singular values and stability analysis of uncertain polynomials, part 2: a missing link

Abstract In Part 1 of this paper, a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one. In Part 2 of this series, we demonstrate that the framework developed in Part 1 is particularly suitable for a class of stability problems and its unifies several frequency-based stability conditions obtained via alternative approaches. Generally, these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion. Several stability conditions are derived readily from solution of the structured singular value presented in Part 1, which either extend or coincide with previously known results. In particular, for family of interval and diamond polynomials, we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of Kharitonov theorem lead to one another, thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series. A related robust stabilization problem is also investigated in this framework.

Structured singular values and stability analysis of uncertain polynomials, part 1: the generalized m

Abstract One of the primary goals in this two-part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials. Another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis. To achieve these goal, we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework. Part 1 of this series is devoted entirely to the generalized structured singular values, specifically for the case when a certain matrix representing the ‘nominal system’ is of rank one. An analytical expression for the generalized notion is derived in this case which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable. In particular, when the general framework is specialized to that of the standard structured singular value, the expression is solved explicitly. Also for several additional important cases, explicit solutions are obtained. The framework as well as results will then be used in Part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties. We demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches. For several problems of interest, we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of Kharitonov theorem can be derive from one another, and hence establish a link between two drastically different type of results.

On D-stability and structured singular values

Abstract It is shown in this paper that there are close connections between the notion of D-stability of a real square matrix and several quantities related to the structured singular value. As a main result, we show that a real square matrix is D-stable if and only if the real structured singular value of some complex matrix is less than one. This condition implies that checking D-stability may in general be an NP-hard problem. Since the exact verification of D-stability is difficult, we provide several additional conditions that are either necessary or sufficient, and these conditions are also connected to the real or complex structured singular values. These results are further extended to the notion of strong D-stability.

On minimizing the largest eigenvalue of a symmetric matrix

The problem of minimizing the largest eigenvalue over an affine family of symmetric matrices is considered. This problem has a variety of applications, such as the stability analysis of dynamic systems or the computation of structured singular values. Given in >or=0, an optimality condition is given which ensures that the largest eigenvalue is within in error bound of the solution. A novel line search rule is proposed and shown to have good descent property. When the multiplicity of the largest eigenvalue at solution is known, a novel algorithm for the optimization problem under consideration is derived. Numerical experiments show that the algorithm has good convergence behavior.<<ETX>>

On robustness analysis with non-diagonally structured uncertainty

The authors study the structured singular value for a class of robustness analysis problems in which the uncertainty is allowed to be non-diagonally structured, and the norm of each block is individually considered. The results extend previous work conducted for problems where each uncertainty block is assumed to be a scalar. Computations of this generalization notion via both the similarity and nonsimilarity scaling methods are investigated. It is shown in particular that under certain circumstances the structured singular value defined here coincides with a vector-induced norm.<<ETX>>

Eigenvalue multiplicity estimate in semidefinite programming

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable, but convex. It covers several standard problems (such as linear and quadratic programming) and has many applications in engineering. Typically, the optimal eigenvalue multiplicity associated with a linear matrix inequality is larger than one. Algorithms based on prior knowledge of the optimal eigenvalue multiplicity for solving the underlying problem have been shown to be efficient. In this paper, we propose a scheme to estimate the optimal eigenvalue multiplicity from points close to the solution. With some mild assumptions, it is shown that there exists an open neighborhood around the minimizer so that our scheme applied to any point in the neighborhood will always give the correct optimal eigenvalue multiplicity. We then show how to incorporate this result into a generalization of an existing local method for solving the semidefinite programming problem. Finally, a numerical example is included to illustrate the results.

A quadratically convergent local algorithm on minimizing sums of the largest eigenvalues of a symmetric matrix

In this paper, we consider the problem on minimizing sums of the largest eigenvalues of a symmetric matrix which depends on the decision variable affinely. An important application of this problem is the graph partitioning problem, which arises in layout of circuit boards, computer logic partitioning, and paging of computer programs. Given ∈≥0, we first derive an optimality condition which ensures that the objective function is within ∈ error bound of the solution. This condition may be used as a practical stopping criterion for any algorithm solving the underlying problem. We also show that, in a neighborhood of the minimizer, the optimization problem can be equivalently formulated as a smooth constrained problem. An existing algorithm on minimizing the largest eigenvalue of a symmetric matrix is shown to be applicable here. This algoritm enjoys the property that if started close enough to the minimizer, then it will converge quadratically. To implement a practical algorithm, one needs to incorporate some technique to generate a good starting point. Since the problem is convex, this can be done by using an algorithm for general convex optimization problems (e.g., Kelleys cutting plane method or ellipsoid methods), or an algorithm specific for the optimization problem under consideration (e.g., the algorithm developed by Cullum, Donath, and Wolfe). Such union ensures that the overall algorithm has global convergence with quadratic rate. Finally, the results presented in this paper are readily extended on minimizing sums of the largest eigenvalues of a Hermitian matrix.

Exterior minimum-penalty path-following methods in semidefinite programming

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable but convex. It covers several standard problems, such as linear and quadratic programming, and has many applications in engineering. In this paper, we introduce the notion of minimal-penalty path, which is defined as the collection of minimizers for a family of convex optimization problems, and propose two methods for solving the problem by following the minimal-penalty path from the exterior of the feasible set. Our first method, which is also a constraint-aggregation method, achieves the solution by solving a sequence of linear programs, but exhibits a zigzagging behavior around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centering and ascending steps. The global convergence of the methods is proved and their performance is illustrated by means of an example.

Convergence analysis of an interior point method in convex programming, regular constraint case

We study the convergence properties of a previously proposed algorithm in the context of solving a class of smooth nonlinear convex optimization problems. With some mild assumptions, it is shown that the algorithm has global convergence with guaranteed accuracy upon termination. Further, an upper bound for the local rate of convergence is derived. It rigorously justifies the efficiency of the algorithm in spite of the fact that the bound is in general conservative.