16-08-2012, 02:17 PM
Time Response of First- and Second-order Dynamical Systems
Preamble
How physical systems, e.g., aircraft, respond to a given input, such as a gust wind,
under given initial conditions, like cruising altitude and cruising speed, is known
as the time response of the system. Here, we have two major items that come into
the picture when determining the time response, namely, the mathematical model
of the system and the history of the input. The mathematical model, as studied in
Chap. 1, is given as an ODE in the generalized coordinate of the system. Moreover,
this equation is usually nonlinear, and hence, rather cumbersome to handle with the
purpose of predicting how the system will respond under given initial conditions
and a given input. However, if we first find the equilibrium states of the system, e.g.,
the altitude, the aircraft angle of attack, and cruising speed in our example above,
and then linearize the model about this equilibrium state, then we can readily obtain
the information sought, as described here.
We will thus start by assuming that the system model at hand has been linearized
about an equilibrium state. That is, we will be concerned mainly with the time
response of linear, time-invariant (LTI) dynamical systems, also termed linear timeinvariant
systems (LTIS), when subjected to arbitrary initial conditions and inputs.
This class of systems is also known as stationary systems and systems with constant
coefficients, and so, we will use these terms interchangeably.We will discuss in this
chapter only first- and second-order systems, i.e., systems that are described by firstand
second-orderODEs.