Why we use filter for derivative control to make it proper?
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This is basically a problem of realizability. Both the controlled plant and controller have to be realizable ( relative degree greater than or equal to zero- not more zeros than poles- , and preferably greater than zero. Assume that only the output and non the state is available for control- this is a "worst- case situation" for the problem at hand , and assume also that:
y (outpu)=G(transfer function)*u ( control)
The transfer function has relative degree
p=n(number of poles)-m( number of zeros)
derivatives of y -, tah tis -D(superj)j (y) can be got from j=0 , the output itself uptil order D(super "p")y- but not gretater since then s**(p+1)G(s) is not realizable. So there is a maximum of order of output derivatives which can be used by the controller while keeping realizability.
If there are a number of output deribvatives availbale in the loop ( i.e. , not only the output is availbale) then those ones can be used by the controller but ,if only the output is available, then output derivatives have to be generated by deriving the output but there is bound ( the relative degrre) to do that to keep realizability of the "new transfer functions" sG(s) up to (s**p)G(s). The accuracy for good filtering of low-frequency noise is to take derivatives uptil order p-1 or p-2 ( this is another point to be taken into account if noise is expected).
y (outpu)=G(transfer function)*u ( control)
The transfer function has relative degree
p=n(number of poles)-m( number of zeros)
derivatives of y -, tah tis -D(superj)j (y) can be got from j=0 , the output itself uptil order D(super "p")y- but not gretater since then s**(p+1)G(s) is not realizable. So there is a maximum of order of output derivatives which can be used by the controller while keeping realizability.
If there are a number of output deribvatives availbale in the loop ( i.e. , not only the output is availbale) then those ones can be used by the controller but ,if only the output is available, then output derivatives have to be generated by deriving the output but there is bound ( the relative degrre) to do that to keep realizability of the "new transfer functions" sG(s) up to (s**p)G(s). The accuracy for good filtering of low-frequency noise is to take derivatives uptil order p-1 or p-2 ( this is another point to be taken into account if noise is expected).
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