# 3

# Linear Systems Response, State-Space Modeling, and Monte Carlo Simulation

The central problem of linear systems analysis is: Given the input, what is the output? In the deterministic case, we usually seek an explicit expression for the response or output. In the random-input problem no such explicit expression is possible, except for the special case where the input is a so-called deterministic random process (and not always in this case). Usually, in random-input problems, we must settle for a considerably less complete description of the output than we get for corresponding deterministic problems. In the case of random processes the most convenient descriptors to work with are autocorrelation function, power spectral density function, and mean-square value. We will now examine the input–output relationships of linear systems in these terms.

## 3.1 INTRODUCTION: THE ANALYSIS PROBLEM

In any system satisfying a set of linear differential equations, the solution may be written as a superposition of an initial-condition part and another part due to the driving or forcing functions. Both the initial conditions and forcing functions may be random; and, if so, the resultant response is a random process. We direct our attention here to such situations, and it will be assumed that the reader has at least an elementary acquaintance with deterministic methods of linear system analysis (1, 2).

With reference to Fig. 3.1, the analysis problem may be simply stated: Given the initial conditions ...