Quantized state systems method: Difference between revisions

Revision as of 11:22, 6 July 2015

Revision as of 11:23, 6 July 2015

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:<math>\left| \vec{e}(t) \right| \leq \left| V \right|\ \left| \Re\left(\Lambda\right)^{-1} \Lambda \right|\ \left| V^{-1} \right|\ \Delta \vec{Q}</math>

where <math>\Delta\vec{Q}</math> is the vector of state quanta, <math>A = V \Lambda V^{-1}</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 that, 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 derivative is always (by definition) guaranteed to change by exactly one quantum outside of the equilibrium. Avoiding this condition would require dynamically lowering the quantum in a manner analogous to [[adaptive stepsize methods]] methods in traditional simulation algorithms.

==Software implementation==