I stumbled across an interesting formula while reading a paper:

Formula with a subscript and superscript outside of some grouping symbols

I'm put off by the $Q(i)$ and $S(i)$ functions as subscripts. I've never seen that notation before, and I'm wondering what it means.

Additionally, does it change the meaning of the 2 that's above each subscript, which i would normally take to mean squaring?

Edit: The Paper is "Sampling Based Model Predictive Control with Application to Autonomous Vehicle Guidance", by Dunlap, Collins, and Caldwell. The formula is on page two.

Unless this link expires, you can find the paper here

  • 3
    $\begingroup$ Citing the paper would help a lot. In general, a subscript under the norm notation refers to a specific type of norm. $\endgroup$ – avs Sep 2 '16 at 0:30

It means weighted norm using the subscripted notation:

$$\sum_{i=1}^N ||\boldsymbol{r}(k+i)-\boldsymbol{y}(k+i)||^2_{\boldsymbol{Q}(i)}+\sum_{i=0}^{M-1} ||\boldsymbol{u}(k+i)||^2_{\boldsymbol{S}(i)}$$

is equivalent to

$$\sum_{i=1}^N (\boldsymbol{r}(k+i)-\boldsymbol{y}(k+i))^T\boldsymbol{Q}(i)(\boldsymbol{r}(k+i)-\boldsymbol{y}(k+i))+\sum_{i=0}^{M-1} (\boldsymbol{u}(k+i))^T\boldsymbol{S}(i)\boldsymbol{u}(k+i)$$

where $\square ^T$ is the matrix transpose symbol.


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