Introduction to System Science with MATLAB, 2nd Edition

7Analysis Methods for Systems with Linear Kernels

Linear system kernels represent such a significant sector of system science that we will devote this entire chapter to the description and analysis of models that have linear kernels or can be approximated by linear kernels. As we will observe, essentially all quantitative system kernels can be approximated by linear system kernels over some appropriate range of definitions.

7.1 Background and Justification

As should be apparent by now, the simplest continuous system models are those with linear system kernels. Systems with constant kernels are simpler but much less useful and may be considered a restricted class of linear system kernels. By the classification “linear,” we mean system kernels whose mathematical expressions are linear functions of the dependent variables only. For example, consider the general multiple‐input, multiple‐output system equation where we assign the independent vector variable to be c and the dependent vector variable to be e

(7.1.1)

As usual, de is the (n × 1) differential output column matrix for the dependent variable e, g(e, c) is the (n × m) system kernel matrix, and dc is the (m × 1) differential input column matrix for the independent variable c. The system kernel matrix g(e, c) is linear if each component of the matrix g(e, c) for all i and j such that i = 1,…,n and j = 1,…,m can be expressed ...