Let , , and be the sampling indices for the control, attack, and sensing, respectively, be the terminal time, , , , , , , , , , , , , , ,
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(1) |
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(2) |
and
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(4) |
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(5) |
Discretization () assuming the topology is fixed during :
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(6) |
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(7) |
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(8) |
where is the impulse response of the hold device. Note that since the impulse response at time as a result of the discrete-time control at time is given by by definition, and thus (1) is just a superposition of these impulse responses. In particular, when we have a constant time interval and
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(9) |
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(10) |