Topics in input-output systems theory: Feedback systems with tapered input spaces, state and generalized adjoint systems.

Three topics in input-output systems theory are discussed in this thesis. These are: Feedback Systems With Tapered Input and Output Spaces, State and Generalized Adjoint Systems. An example of a weighted input space is a weighted $L\\sb{p}$ space. The weighted $L\\sb{p}$ norm of a time function $u$ at time $t$ is $\\vert u\\vert\\sb{t}$ = $\\left(\\int\\sbsp{-\\infty}{t}\\vert u(\au)\\vert\\sp{p}w(t-\au)d\au\\right)\\sp{1/p}$. The weighting function $w:$ $\\IR\\sp+$ $\o$ $\\IR\\sp+$ is nonincreasing so that the recent past is emphasized and the remote past is de-emphasized in computing the norm. If $\\int\\sbsp{0}{\\infty}w(\au)d\au<\\infty$ a tapered $L\\sb{p}$ norm is defined. Fitted families of seminorms, which are an abstract generalization of tapered $L\\sb{p}$ spaces are defined. Tapered spaces may be desirable or necessary to use in some situations. The causality and stability of a nonlinear feedback system with tapered or weighted input and output spaces is considered. The Nyquist and circle conditions are recovered for systems with tapered input and output spaces. State is defined for a causal possibly nonlinear and time-varying input-output system. Given the state, referred to as the natural state, the future output can be determined from it and the future input. Also, the natural state and future input determine the future state. The natural state is consistent with state in linear systems. A metric is defined on the natural state space and some continuity properties are considered. State space properties are considered. Stability is required for the natural state; for unstable systems, an approximate notion called a pseudo-state is defined. Differentiational properties analogous to the continuity properties are considered. Differential equation and integral operator examples are presented. A generalized adjoint system is defined from a causal, possibly nonlinear and time-varying input-output system using a procedure analogous to the way the adjoint of a bounded linear transformation is defined. The input space to a generalized adjoint system is a space of causal input-output systems which, by considering their system trajectories, are time functions. The generalized adjoint is linear, memoryless and bounded. A generalized adjoint map between a Banach space of input-output systems and the space of their generalized adjoints is defined. Under certain conditions this map is linear, bounded and invertible.PhDElectrical engineeringUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/161973/1/8821650.pd