TABLE OF CONTENTS In this last chapter, causal discrete-time state-space models of linear time-invariant (LTI) systems are introduced mainly as a means of providing methods to discretize continuous-time LTI systems. The topic of system discretization is an important one for at least two reasons. First, engineers often need to simulate the behavior of continuous-time systems, and analytical solutions can be very difficult to obtain for systems of order three or more. Second, many filter design and controller design techniques have been developed in continuous time. Discretizing the resulting system often yields a sampled-data implementation that performs just as well as the intended continuous-time design.
Of course, discrete-time LTI (DLTI) state-space models are of interest in their own right, particularly in advanced multivariable discrete-time filter design and controller design techniques, which are beyond the scope of this textbook. An important word on notation before we move on: for discrete-time state-space systems, the input signal is conventionally written as u[ n] (not to be confused with the unit step) instead of x[ n], as the latter is used for the vector of state variables. Hence, in this chapter we will use q[ n] to denote the unit step signal.
In general, an N th-order linear constant-coefficient causal difference equation with M ? N has the form
| (17.1) |  |
which can be expanded into
| (17.2) |  |
Just like the continuous-time case, we can derive a state-space model...
