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Numerical Integration of General State-Space Models

An approximate discrete-time numerical solution of Eq.$ \,$(1.6) is provided by

$\displaystyle \underline{x}(t_n+T_n) \eqsp \underline{x}(t_n) + T_n\,f[\underline{x}(t_n),\underline{u}(t_n)], \quad n=0,1,2,\ldots\,. \protect$ (2.7)

Let

$\displaystyle g_{t_n}[\underline{x}(t_n),\underline{u}(t_n)] \isdefs \underline{x}(t_n) + T_n\,f_{t_n}[\underline{x}(t_n),\underline{u}(t_n)]. $

Then we can diagram the time-update as in Fig.1.7. In this form, it is clear that $ g_{t_n}$ predicts the next state $ \underline{x}(t_n+T_n)$ as a function of the current state $ \underline{x}(t_n)$ and current input $ \underline{u}(t_n)$. In the field of computer science, computations having this form are often called finite state machines (or simply state machines), as they compute the next state given the current state and external inputs.

Figure 1.7: Discrete-time state-space model viewed as a state predictor, or finite state machine.
\includegraphics{eps/statemachineg}

This is a simple example of numerical integration for solving an ODE, where in this case the ODE is given by Eq.$ \,$(1.6) (a very general, potentially nonlinear, vector ODE). Note that the initial state $ \underline{x}(t_0)$ is required to start Eq.$ \,$(1.7) at time zero; the initial state thus provides boundary conditions for the ODE at time zero. The time sampling interval $ T_n$ may be fixed for all time as $ T_n=T$ (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 of Eq.$ \,$(1.7) is more conveniently applicable than the continuous-time formulation in Eq.$ \,$(1.6).

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State-Space Model of a Force-Driven Mass