On a large-gain theorem

The aim of this paper is to give a general quantitative requirement which the loop gain must satisfy in order to stabilize a given unstable (possibly nonlinear and time-varying) plant, namely that the gain must exceed one.     

Stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second-order SISO LTI system of the form , y=Cx with feedback of the form u(x)=v(x)Cx is considered, where v(x) is real-valued and has domain which is all of . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state-space (i.e., v(x){k₁,k₂} for all x and for some k₁,k₂), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI-A)^-1B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.     

Robustness of periodic trajectories

A robustness problem for periodic trajectories is considered. A nonautonomous system with a periodic solution is given. The problem is to decide whether a stable periodic solution remains in a neighborhood of the nominal periodic solution when the dynamics of the system is perturbed. The case with a structured dynamic perturbation is considered. This makes the problem a nontrivial generalization of a classical problem in the theory of dynamical systems. A solution to the robustness problem will be obtained by using a variational system obtained by linearizing the system dynamics along a trajectory, which is uncertain but within the prespecified neighborhood of the nominal trajectory. This gives rise to robustness conditions that can be solved using integral quadratic constraints for linear time periodic systems.     

Optimal detection of symmetric mixed quantum states

We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.     

System analysis via integral quadratic constraints

This paper introduces a unified approach to robustness analysis with respect to nonlinearities, time variations, and uncertain parameters. From an original idea by Yakubovich (1967), the approach has been developed under a combination of influences from the Western and Russian traditions of control theory. It is shown how a complex system can be described, using integral quadratic constraints (IQC) for its elementary components. A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality. A systematic computational approach is described, and relations to other methods of stability analysis are discussed. Last, but not least, the paper contains a summarizing list of IQCs for important types of system components     

On ?∞, robust stability and performance with ?2 and ?∞ perturbations

In this paper we will consider systems with linear time-invariant perturbations. We will analyze robust performance in the ?₂ and ?_∞ settings. The ?₂ setting gives rise to the familiar case of structured singular values, and a stability criterion is given by the “small μ” theorem. We show that although the necessary and sufficient criterion of robust stability for the ?_∞ case (?_∞ stability with structured ?_∞-gain bounded perturbations) is the same “small μ” criterion, a system with ?₂-gain bounded perturbations is never ?_∞ stable.     

A cutting plane algorithm for robustness analysis of periodicallytime-varying systems

An algorithm for robustness analysis of periodic systems is derived. The system under consideration consists of a linear periodically time-varying plant in feedback interconnection with a structured uncertainty. Conditions for robust stability and robust performance can be formulated in terms of periodic integral quadratic constraints (IQCs). In this way, the robustness analysis becomes a problem of optimizing the parameters of the IQC. A cutting plane algorithm is suggested for solving this infinite-dimensional optimization problem     

Specialized fast algorithms for IQC feasibility and optimization problems

The conventional way to treat integral quadratic constraint (IQC) problems is to transform them into semi-definite programs (SDPs). SDPs can then be solved using interior point methods which have been proven efficient. This approach, however, is not always the most efficient since it introduces additional decision variables to the SDP, and the additional decision variables sometimes largely increase the complexity of the problem. In this paper, we demonstrate how to solve IQC problems by other alternatives. More specifically, we consider two cutting plane algorithms. We will show that in certain cases these cutting plane algorithms can solve IQC problems much faster than the conventional approach. Numerical examples, as well as some explanations from the point of view of computational complexity, are provided to support our point.     

Necessary and sufficient conditions of stability: a multiloopgeneralization of the circle criterion

A linear time-invariant system with a vector output and a vector input is described. This system is closed by an uncertain (nonlinear, time-varying) feedback. The only information on this feedback is given by several integral quadratic inequalities, i.e. the uncertainties under consideration generalize the so-called conic nonlinearities. Necessary and sufficient frequency-domain conditions of stability are obtained. An advanced version of the S-procedure losslessness theorem and some other tools of the absolute stability theory are used