2022-10-22 01:54:47 -04:00
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The continuous form of a PI loop is
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A(t) = P e(t) + I ∫e(t')dt'
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where e(t) is the error (setpoint - measured), and the integral goes
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from 0 to the current time 't'.
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In digital systems the integral must be approximated. The normal way
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of doing this is a first-order approximation of the derivative of
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A(t).
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dA(t)/dt = P de(t)/dt + Ie(t)
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A(t_n) - A(t_{n-1}) ≅ P (e(t_n) - e(t_{n-1})) + Ie(t_n)Δt
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A(t_n) ≅ A(t_{n-1}) + e(t_n)(P + IΔt) - Pe(t_{n-1})
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Using α = P + IΔt, and denoting A(t_{n-1}) as A_p,
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A ≅ A_p + αe - Pe_p.
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The formula above is what this module implements. This way, the
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controller only has to store two values between each run of the loop:
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the previous error and the previous output. This also reduces the
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amount of (redundant) computations the loop must execute each
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iteration.
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Calculating α requires knowing the precise timing of each control loop
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cycle, which in turn requires knowing the ADC and DAC timings. This
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is done outside the Verilog code. and can be calculated from
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simulating one iteration of the control loop.
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2022-10-22 01:55:56 -04:00
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# Fixed Point Integers
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2022-10-22 01:54:47 -04:00
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A regular number is stored in decimal: 123056.
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This is equal to
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6*10^0 + 5*10^1 + 0*10^2 + 3*10^3 + 2*10^4 + 1*10^5.
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A whole binary number is only ones and zeros: 1101, and is equal to
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1*2^0 + 0*2^1 + 1*2^2 + 1*2^3.
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Fixed-point integers shift the exponent of each number by a fixed
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amount. For instance, 123.056 is
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6*10^-3 + 5*10^-2 + 0*10^-1 + 3*10^0 + 2*10^1 + 1*10^2.
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Similarly, the fixed point binary integer 11.01 is
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1*2^-2 + 0*2^-1 + 1*2^0 + 1*2^1.
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To a computer, a whole binary number and a fixed point binary number
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are stored in exactly the same way: no decimal point is stored. It is
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only the interpretation of the data that changes.
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Fixed point numbers are denoted WHOLE.FRAC or [WHOLE].[FRAC], where
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WHOLE is the amount of whole number bits (including sign) and FRAC is
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the amount of fractional bits (2^-1, 2^-2, etc.).
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The rules for how many digits the output has given an input is the
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same for fixed point binary and regular decimals.
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Addition: W1.F1 + W2.F2 = [max(W1,W2)+1].[max(F1,F2)]
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Multiplication: W1.F1W2.F2 = [W1+W2].[F1+F2]
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