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The '''quantized state systems''' ('''QSS''') '''methods''' are a family of numerical integration solvers based on the idea of state quantization, [[dual (mathematics)|dual]] to the traditional idea of time discretization.▼
Unlike traditional [[numerical methods for ordinary differential equations|numerical solution methods]], which approach the problem by [[Discretization|discretizing]] time and solving for the next (real-valued) state at each successive time step, QSS methods keep time as a continuous entity and instead [[Quantization (signal processing)|quantize]] the system's state, instead solving for the ''time'' at which the state deviates from its quantized value by a ''quantum''.▼
They can also have many advantages compared to classical algorithms.<ref>{{cite journal |author1=Migoni, Gustavo |author2=Ernesto Kofman |author3=François Cellier |title=Quantization-based new integration methods for stiff ordinary differential equations|year=2011 |journal = Simulation |volume=88 |issue=4 |pages=387–407 |doi=10.1177/0037549711403645 |url=http://sim.sagepub.com/content/88/4/387 |url-access=subscription }}</ref>
▲The '''quantized state systems (QSS) methods''' are a family of numerical integration solvers based on the idea of state quantization, [[dual]] to the traditional idea of time discretization.
They inherently allow for modeling discontinuities in the system due to their discrete-event nature and asynchronous nature. They also allow for explicit root-finding and detection of zero-crossing using ''explicit'' algorithms, avoiding the need for iteration---a fact which is especially important in the case of stiff systems, where traditional time-stepping methods require a heavy computational penalty due to the requirement to implicitly solve for the next system state. Finally, QSS methods satisfy remarkable global stability and error bounds, described below, which are not satisfied by classical solution techniques.
▲Unlike traditional [[numerical methods for ordinary differential equations|numerical solution methods]], which approach the problem by [[Discretization|discretizing]] time and solving for the next (real-valued) state at each successive time step, QSS methods keep time as a continuous entity and instead [[quantize]] the system's state, instead solving for the ''time'' at which the state deviates from its quantized value by a ''quantum''.
By their nature, QSS methods are therefore neatly modeled by the [[DEVS]] formalism, a [[discrete event simulation|discrete-event]] [[model of computation]], in contrast with traditional methods, which form [[Discrete time and continuous time#
==Theoretical properties==▼
In 2001, Ernesto Kofman proved<ref>{{cite journal | last=Kofman | first=Ernesto |title=A second-order approximation for DEVS simulation of continuous systems|year=2002 |journal = Simulation |volume=78 | issue=2 |pages=76–89 |
:<math>
\left| \vec{e}(t) \right| \leq
\left| V \right|\ \left| \Re\left(\Lambda\right)^{-1} V^{-1} B \right|\ \Delta\vec{u}</math>
where <math>\Delta\vec{Q}</math> is the vector of state quanta, <math>
It is worth noticing that this
==First-order QSS method – QSS1==
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where <math>x</math> and <math>q</math> are related by a ''[[Hysteresis|hysteretic]] quantization function''
:<math>q(t) = \begin{cases}x(t)
where <math>\Delta Q</math> is called a ''quantum''. Notice that this quantization function is '''hysteretic''' because it has ''memory'': not only is its output a function of the current state <math>x(t)</math>, but it also depends on its old value, <math>q(t^{-})</math>.
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It is important to note that, while in principle a QSS method of arbitrary order can be used to model a continuous-time system, it is seldom desirable to use methods of order higher than four, as the [[Abel–Ruffini theorem]] implies that the time of the next quantization, <math>t</math>, cannot (in general) be [[Explicit and implicit methods|explicitly solved]] for [[algebraic solution|algebraically]] when the polynomial approximation is of degree greater than four, and hence must be approximated iteratively using a [[root-finding algorithm]]. In practice, QSS2 or QSS3 proves sufficient for many problems and the use of higher-order methods results in little, if any, additional benefit.
▲==Theoretical properties==
▲In 2001, Ernesto Kofman proved<ref>{{cite journal | last=Kofman | first=Ernesto |title=A second-order approximation for DEVS simulation of continuous systems|year=2002 |journal = Simulation |volume=78 |pages=76–89 |url=http://sim.sagepub.com/content/78/2/76.short }}</ref> a remarkable property of the quantized-state system simulation method: namely, that when the technique is used to solve a [[Exponential stability|stable]] [[LTI system theory|linear time-invariant (LTI) system]], the global error is bounded by a constant that is proportional to the quantum, but (crucially) independent of the duration of the simulation. More specifically, for a stable multidimensional LTI system with the [[state-transition matrix]] <math>A</math>, it was shown in [[Quantized state systems method|[CK06]]] that the absolute error vector <math>\vec{e}(t)</math> is bounded above by
▲:<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 finding a reliable technique for dynamically lowering the quantum in a manner analogous to [[adaptive stepsize]] methods in traditional simulation algorithms.
==Software implementation==
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QSS methods constitute the main numerical solver for [[PowerDEVS]][[PowerDEVS#References|[BK011]]] software.
They have also been implemented in as a stand
== References ==
{{Reflist}}
* [CK06] {{cite book|
* [BK11] {{cite news|
==External links==
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