By Astrem, Wittenmark
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Extra resources for Computer Controlled Systems
8) will now be calculated. 8) over one sampling period gives x(kh + h) =eAhx(kh ) + l Jr. 9) kh Because the signal u (t) is piecewise constant over the samplinginterval, the delayedsignal u(t - r) is also piecewise constant. The delayed signal will, however, change between the sampling instants (see Fig. 2). 9), it is then convenient to split the integration interval into two parts 50 u(t) • , .. 2 The relationship among u(t), the delayed signal u(t - r), and the sampling instants. 39 Sampling a Continuous-Time State-Space System Sec.
Any(k - n) where ai are the coefficients of the characteristic polynomial of «t>. The most common case in computer-control systems is that bo = 0, that is, there is no direct term in the discrete-time model. Usually j'(k) is measured first, and then u(k} is determined. Then y(k) cannot he influenced by u(k) even if the continuous-time system has a direct term. 2 when h ::: 1. 5(q-l + q-2) . 5 s. :.. 5 shows that different state-space representations can be used. Of course, this does not change the input-output model.
The use of difference equations then becomes a natural tool. It should be emphasized that computer-oriented mathematical models onlygivethe behavior at the sampling points-the physical process is still a continuous-time system. Looking at the problem this way, however, will greatly simplify the treatment. We will give formulas that allow a computation of inters ample behavior, but a fun treatment of process-oriented models, which takes continuous-time behavior into account, is given Chapter 7. 1).
Computer Controlled Systems by Astrem, Wittenmark