Quantized state systems method

The quantized state systems (QSS) methods [CK06] are a family of numerical integration solvers based on the idea of state quantization, which are dual to the idea of time discretization. Unlike traditional numerical solution methods, which approach the problem by 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. QSS methods are therefore neatly modeled by DEVS, a discrete-event model of computation, in contrast with traditional methods, which form discrete-time models of the continuous-time system.

First-order QSS method – QSS1

Let an initial value problem be specified as follows.

${\dot {x}}(t)=f(x(t),t),\quad x(t_{0})=x_{0}.$

QSS1 approximates the above system by

${\dot {x}}(t)=f(q(t))$

where $x$ and $q$ are related by a hysteretic quantization function

$q(t)={\begin{cases}x(t)&&{\text{if }}\left|x(t)-q(t)\right|\geq \Delta Q\\q(t^{-})&&{\text{otherwise}}\end{cases}}$

where $\Delta Q$ is called a quantum.

This formulation therefore approximates the state by a piecewise constant function, $q(t)$, that updates its value whenever the state deviates from this approximation by one quantum.

Software implementation

The QSS Methods can be implemented as a discrete event system and simulated in any DEVS simulator.

QSS methods constitute the main numerical solver for PowerDEVS[BK011] software. They have also been implemented in as a stand alone version

References

• [CK06] Francois E. Cellier and Ernesto Kofman (2006). Continuous System Simulation (first ed.). Springer. ISBN 978-0-387-26102-7.

• [BK11] Bergero, Federico and Kofman, Ernesto (2011). "PowerDEVS: a tool for hybrid system modeling and real-time simulation" (first ed.). Society for Computer Simulation International,San Diego.{{cite news}}: CS1 maint: multiple names: authors list (link)

External links

• Stand-alone implementation of QSS Methods
• PowerDEVS at SourceForge