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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) && \text{if } \left|x(t) - q(t)\right| \geq \Delta Q \\ q(t^{-}) && \text{otherwise}\end{cases}</math>
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>.
This formulation therefore approximates the state by a piecewise constant function, <math>q(t)</math>, that updates its value as soon as the state deviates from this approximation by one quantum.
The [[Multidimensional system|multidimensional]] formulation of this system is almost the same as the single-dimensional formulation above: the <math>k^\text{th}</math> quantized state <math>q_k(t)</math> is a function of its corresponding state, <math>x_k(t)</math>, and the state vector <math>\vec{x}(t)</math> is a function of the entire quantized state vector, <math>\vec{q}(t)</math>:
:<math>\vec{x}(t) = f(\vec{q}(t), t)</math>
==High-order QSS methods – QSS2 and QSS3==
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