Quantized state systems method: Difference between revisions

Revision as of 09:43, 6 July 2015

Revision as of 09:54, 6 July 2015

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:<math> \dot{x}(t) = f(x(t), t), \quad x(t_0) = x_0. </math>

The first-order QSS method, known as QSS1, approximates the above system by

:<math> \dot{x}(t) = f(q(t))</math>

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where <math>\Delta Q</math> is called a ''quantum''.

This formulation therefore approximates the state by a piecewise constant function, <math>q(t)</math>, that updates its value wheneveras soon as the state deviates from this approximation by one quantum.

==High-order QSS methodmethods – QSS2 and QSS3==

The second-order QSS method, QSS2, follows the same principle as QSS1, except that it defines <math>q(t)</math> as a [[piecewise linear function|piecewise linear]] approximation of the trajectory <math>x(t)</math> that updates its trajectory as soon as the two differ from each other by one quantum.

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The pattern continues for higher-order approximations, which define the quantized state <math>q(t)</math> as successively higher-order polynomial approximations of the system's state.

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 solved for 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 can prove sufficient for many problems and the use of higher-order methods results in little benefit.

==Backward QSS method – BQSS==