C. R. MacCluer

The Many Proofs and Applications of Perron's Theorem

This paper chronicles the wide dispersal of Perrons 1907 result on positive matrices into many fields of science. The many proofs given during the last 93 years are categorized and critiqued (including Perrons original two proofs), and a more natural proof is presented. This simple-to-understand result of Perron provides a unequaled vehicle for taking students on a tour of many applied areas with some depth.

A natural modal expansion for the flexible robot arm problem via a self-adjoint formulation

The equations of motion of a flexible robot arm consist of a coupled partial differential equation describing the arms transverse vibrations and an ordinary differential equation describing the hubs rigid motion. Many researchers obtained a solution using a modal expansion based on the arms equation alone, which has erroneous eigenfunctions and eigenvalues. A novel method is presented for obtaining an equivalent but self-adjoint form for the problem. This self-adjoint form leads to a natural modal expansion, where the equations decouple. This method is used to show that the effect of the hub-arm model coupling depends exclusively on the hub-inertia-to-arm-mass ratio. The need for a self-adjoint form arises in many control applications. This is because, typically, the control design is based on approximate models, and in order to guarantee robust performance, a prior estimate of the approximation error is required. When a self-adjoint form is available, obtaining approximate modes and the associated error bounds becomes an easy task. >

A Nyquist stability criterion for distributed parameter systems

A Nyquist graphical stability criterion is developed for distributed parameter, possible unstable, single-loop systems. Practical conditions are presented for existence, uniqueness, causality, and asymptotic or exponential stability of the closed-loop impulse response. The hypotheses are given for the transfer function only and do not require any knowledge of its time-domain impulse response. >

Choosing an Inner Product That Separates Variables

Under the assumptions most frequently encountered in linear distributed parameter control problems, namely when the implied operator is closed with compact resolvent, there will often exist an inner product that separates variables.

Choosing An Inner Product That Separates Variables

Under the assumptions most frequently encountered in linear distributed parameter control problems, namely when the implied operator is closed with compact resolvent, there will often exist an inner product that separates variables.

Using Operational Calculus to Obtain Transient Response Estimates in Linear Uncertain Feedback Systems

A new methodology is presented for deriving transient response estimates for a large class of uncertain linear feedback SISO systems, including unstable distributed parameter systems. It is shown that, using operational calculus, estimates exist and can be computed even in cases where a certain error norm is greater than unity. Several simple examples illustrate the application of the method.