digital controller design and mapping analog signal to digital integer values
There are actually quite a few references out there on converting floating point algorithms to fixed point. Google "convert floating point algorithm to fixed point" for lots of examples. Some software such as MATLAB can help (not cheap though, you'd need a few toolboxes). 

The general approach is to scale the numbers at each operation (by shifts) so that you maintain the maximum number of bits (at least "enough" bits anyway) without any chance of overflowing. In some cases (such as integration) where the numbers can grow without bound under certain conditions you may find it useful to use saturating fractional fixed point math rather than the stock integer algorithms you'll find in C or whatever language you are using. 

It will not necessarily be much faster than floating point, it depends on the processor the number of bits you need and so on. For example, if you want to multiply a 24-bit number by a scaling coefficient that is close to 1.000 for calibration, you will most likely have to use long long (64 bit integers) in C because you want to have at least 24 bits in each operand and that will result in a product that can be ~48 bits (you could then discard the lower 24 (or so) bits), but if you try to multiply two 24 bit multipicands in 32 bit math you will get overflow. A 32 bit floating point multiply (with a 24 bit mantissa) may execute faster than a 64 bit integer multiply. 

You can think of the scaling as manually handling the exponent in a floating point number.