Quantized state systems method: Difference between revisions

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Revision as of 05:05, 18 August 2015

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where <math>\Delta\vec{Q}</math> is the vector of state quanta, <math>\Delta\vec{u}</math> is the vector with quanta adopted in the input signals, <math>V \Lambda V^{-1} = A</math> is the [[Eigendecomposition#Eigendecomposition of a matrix|eigendecomposition]] or [[Jordan canonical form]] of <math>A</math>, and <math>\left|\,\cdot\,\right|</math> denotes the element-wise [[absolute value]] operator (not to be confused with the [[determinant]] or [[Norm (mathematics)|norm]]).

It is worth noticing that this spectacular error bound comes at a price: the global error for a stable LTI system is also, in a sense, bounded ''below'' by a the quantum itself, at least for the first-order QSS1 method. This is due to the fact thatbecause, unless the approximation happens to coincide ''exactly'' with the correct value (an event which will [[almost surely]] not happen), it will simply continue oscillating around the equilibrium, as the state is always (by definition) guaranteed to change by exactly one quantum outside of the equilibrium. Avoiding this condition would require finding a reliable technique for dynamically lowering the quantum in a manner analogous to [[adaptive stepsize]] methods in traditional simulation algorithms.

==First-order QSS method – QSS1==