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.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.
They can also have many advantages compared to classical algorithms.[1] 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.
By their nature, QSS methods are therefore neatly modeled by the DEVS formalism, a discrete-event model of computation, in contrast with traditional methods, which form discrete-time models of the continuous-time system. They have therefore been implemented in [PowerDEVS], a simulation engine for such discrete-event systems.
A
B
\vec{e}(t)
\left|\vec{e}(t)\right|\leq \left|V\right| \left|\Re\left(Λ\right)-1Λ\right| \left|V-1\right| \Delta\vec{Q}+ \left|V\right| \left|\Re\left(Λ\right)-1V-1B\right| \Delta\vec{u}
where
\Delta\vec{Q}
\Delta\vec{u}
VΛV-1=A
A
\left| ⋅ \right|
It is worth noticing that this remarkable error bound comes at a price: the global error for a stable LTI system is also, in a sense, bounded below by the quantum itself, at least for the first-order QSS1 method. This is because, 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 discrete time simulation algorithms.
Let an initial value problem be specified as follows.
| x |
(t)=f(x(t),t), x(t0)=x0.
The first-order QSS method, known as QSS1, approximates the above system by
| x |
(t)=f(q(t),t), q(t0)=x0.
where
x
q
q(t)=\begin{cases}x(t)&if\left|x(t)-q(t-)\right|\geq\DeltaQ\ q(t-)&otherwise\end{cases}
where
\DeltaQ
x(t)
q(t-)
This formulation therefore approximates the state by a piecewise constant function,
q(t)
The multidimensional formulation of this system is almost the same as the single-dimensional formulation above: the
kth
qk(t)
xk(t)
\vec{x}(t)
\vec{q}(t)
\vec{x}(t)=f(\vec{q}(t),t)
The second-order QSS method, QSS2, follows the same principle as QSS1, except that it defines
q(t)
x(t)
q(t)
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,
t
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.