7.1: Introduction
- Page ID
- 24269
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A central question in dealing with a causal discrete-time (DT) system with input \(u\), output \(y\), is the following:
Given the input at some time \(n\), i.e. given \(u[n]\), how much information do we need about past inputs, i.e. about \(u[k]\) for \(k < n\), in order to determine the present output, namely \(y[n]\) ?
The same question can be asked for continuous-time (CT) systems. This question addresses the issue of memory in the system. Why is this a central question? Some reasons:
- The answer gives us an idea of the complexity, or number of degrees of freedom, associated with the dynamic behavior of the system. The more information we need about past inputs in order to determine the present output, the richer the variety of possible output behaviors.
- In a control application, the answer to the above question suggests the required degree of complexity of the controller, because the controller has to remember enough about the past to determine the effects of present control actions on the response of the system.
- For a computer algorithm that acts causally on a data stream, the answer to the above question suggests how much memory will be needed to run the algorithm.
We now describe the general structure of state-space models, for which the preceding question has an immediate and transparent answer.