Nonsquare Spectral Factorization for Nonlinear Control Systems

This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton–Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system.

Generalized Perron--Frobenius Theorem for Nonsquare Matrices

The celebrated Perron--Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. However, many real-life scenarios give rise to nonsquare matrices. A natural question is whether the PF Theorem (along with its applications) can be generalized to a nonsquare setting. Our paper provides a generalization of the PF Theorem to nonsquare matrices. The extension can be interpreted as representing client-server systems with additional degrees of freedom, where each client may choose between multiple servers that can cooperate in serving it (while potentially interfering with other clients). This formulation is motivated by applications to power control in wireless networks, economics and others, all of which extend known examples for the use of the original PF Theorem. We show that the option of cooperation between servers does not improve the situation, in the sense that in the optimal solution no cooperation is needed, and only one server needs to serve each client. Hence, the additional power of having several potential servers per client translates into \emph{choosing} the best single server and not into \emph{sharing} the load between the servers in some way, as one might have expected. The two main contributions of the paper are (i) a generalized PF Theorem that characterizes the optimal solution for a non-convex nonsquare problem, and (ii) an algorithm for finding the optimal solution in polynomial time

A new look at pencils of matrix valued functions

AbstractMatrix pencils depending on a parameter and their canonical forms under equivalence are discussed. The study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential-algebraic equations (DAEs). Based on the Weierstrass-Kronecker canonical form of the underlying matrix pencil, one gets existence and uniqueness results for linear constant coefficients DAEs. In order to study the solution behavior of linear DAEs with variable coefficients one has to look at new types of equivalence transformations. This then leads to new canonical forms and new invariances for pencils of matrix valued functions. We give a survey of recent results for square pencils and extend these results to nonsquare pencils. Furthermore we partially extend the results for canonical forms of Hermitian pencils and give new canonicalforms there, too. Based on these results, we obtain new existence and uniqueness theorems for differential-algebraic systems, which generalize the classical results of Weierstrass and Kronecker

Extensions of Retrospective Cost Adaptive Control: Nonsquare Plants, and Robustness Modifications.

Controllers that have adjustable parameters, and an update law for adjusting these parameters, are called ``adaptive controllers''. Adaptive controllers typically entail assumptions about the dynamic order, relative degree or transmission zeros of the system, and may not be applicable to systems that are not positive real, passive, or minimum phase. In 1980s, the fragility of adaptive controllers to violations in these assumptions has been demonstrated through various counterexamples. This motivated the development of robustness modifications for adaptive controllers, which is the main goal of this dissertation. In this dissertation, we focus on retrospective cost adaptive control (RCAC), which is a direct, digital adaptive control algorithm. RCAC is applicable to MIMO, nonminimum-phase (NMP) systems, but it assumes that the NMP zeros of the plant, if any, are known. The main contribution of this work includes theory and analysis for retrospective cost adaptive control of nonsquare systems and development of a modified, robust RCAC update law for maintaining stability and convergence in the presence of unmodeled NMP zeros.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/97926/1/dogan_1.pd