State-space models of linear dynamical systemsMatrices > Applications > State-space models
DefinitionMany discrete-time dynamical systems can be modeled via linear state-space equations, of the form ![]() where In effect, a linear dynamical model postulates that the state at the next instant is a linear function of the state at past instants, and possibly other ‘‘exogeneous’’ inputs; and that the output is a linear function of the state and input vectors. A continuous-time model would take the form of a differential equation ![]() Finally, the so-called time-varying models involve time-varying matrices MotivationThe main motivation for state-space models is to be able to model high-order derivatives in dynamical equations, using only first-order derivatives, but involving vectors instead of scalar quantities. The above involves second-order derivatives of a scalar function ![]() The price we pay is that now we deal with a vector equation instead of a scalar equation: ![]() The position ![]() with A nonlinear systemIn the case of non-linear systems, we can also use state-space representations. In the case of autonomous systems (no external input) for example, these come in the form ![]() where Using the first-order approximation of the map ![]() where ![]()
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