An approximate discrete-time numerical solution of Eq.
) is provided by
Then we can diagram the time-update as in Fig.1.7
this form, it is clear that predicts
the next state
as a function of the current state
. In the field of computer science,
computations having this form are often called finite state
(or simply state machines
), as they compute the
next state given the current state and external inputs.
Discrete-time state-space model viewed as
a state predictor, or finite state machine.
This is a simple example of numerical integration
, where in this case the ODE is given by Eq.
) (a very
general, potentially nonlinear
, vector ODE). Note that the initial
is required to start Eq.
) at time zero;
the initial state thus provides boundary conditions
for the ODE at
time zero. The time sampling interval
may be fixed for all
(as it normally is in linear, time-invariant
digital signal processing
systems), or it may vary adaptively
according to how fast the system is changing (as is often needed for
nonlinear and/or time-varying systems). Further discussion of
nonlinear ODE solvers is taken up in §7.4
, but for most of
this book, linear, time-invariant systems will be emphasized.
Note that for handling switching states
(such as op-amp
comparators and the like), the discrete-time state-space formulation
) is more conveniently applicable than the
continuous-time formulation in Eq.
Next Section: State DefinitionPrevious Section: State-Space Model of a Force-Driven Mass