Subtractive 0th Order linear feedback control (proportional controller)


x(t): Setpoint

y(t): Controlled Variable

yR(t): Measured Variable

e(t): Error

V1 and V2: Linear Gain Factors


e(t) = x(t) - yR(t)(1)

y(t) = V1e(t) = V1[x(t) - yR(t)](2)

yR(t) = V2y(t)(3)

y(t) = V1x(t) - V1V2y(t)Recursive Solution (4)

y(t) = V1x(t) / [1 + V1V2]Equifinal Solution (5)



References:
Dietrich, J. W. (1999-2003). Subtractive 0th order linear feedback control. Medizinische Kybernetik | Medical Cybernetics. http://www.medizinische-kybernetik.de/ips/fc0.html (21 Jul. 2003).
Röhler, R. (1973). Biologische Kybernetik. Stuttgart, B. G. Teubner.

Subtractive 0th Order linear feedback control with load


x(t): Setpoint

y(t): Controlled Variable

yR(t): Measured Variable

z(t): Load (Sum of all Disturbance Variables)

e(t): Error

V1 and V2: Linear Gain Factors


e(t) = x(t) - yR(t)(1)

yS(t) = V1e(t) = V1[x(t) - yR(t)](2)

y(t) = yS(t) + z(t) = V1[x(t) - yR(t)] + z(t)(3)

yR(t) = V2y(t)(4)

y(t) = V1x(t) - V1V2y(t) + z(t)Recursive Solution (5)

y(t) = [V1x(t) + z(t)] / [1 + V1V2]Equifinal Solution (6)



References:
Dietrich, J. W. (1999-2003). Subtractive 0th order linear feedback control. Medizinische Kybernetik | Medical Cybernetics. http://www.medizinische-kybernetik.de/ips/fc0l.html (21 Jul. 2003).
Röhler, R. (1973). Biologische Kybernetik. Stuttgart, B. G. Teubner.

Subtractive 1st Order linear feedback control with load


x(omega): Setpoint

y(omega): Controlled variable

yR(omega): Measured variable

z(omega): Load (Sum of all disturbance variables)

e(omega): Error

V1 and V2: Linear gain factors

omega: Angular Frequency of signal

alpha, beta: Time constants


e(omega) = x(omega) - yR(omega)(1)

yS(omega) = V1 e(omega) = V1[x(omega) - yR(omega)](2)

y(omega) = alpha / (omega + beta) * [yS(omega) + z(omega)](3)

yR(omega) = V2y(omega)(4)

y(omega) = alpha / (omega + beta) * V1x(omega) - V1V2y(omega) + z(omega)Recursive Solution (5)

y(omega) = [alpha V1x(omega) + alpha z(omega)] / [omega + beta + alpha V1V2]Equifinal Solution (6)



References:
Dietrich, J. W. (1999-2003). Subtractive 1st order linear feedback control. Medizinische Kybernetik | Medical Cybernetics. http://www.medizinische-kybernetik.de/ips/fc1l.html (27 Jun. 2004).
Röhler, R. (1973). Biologische Kybernetik. Stuttgart, B. G. Teubner.

Divisive 0th Order feedback control with load


x(t): Setpoint

y(t): Controlled variable

yR(t): Measured variable

z(t): Load (Sum of all disturbance variables)

e(t): Error

V1 and V2: Linear gain factors


e(t) = x(t) / yR(t)(1)

yS(t) = V1 e(t) = V1 x(t) / yR(t)(2)

y(t) = yS(t) + z(t) = V1 x(t) / yR(t) + z(t)(3)

yR(t) = V2y(t)(4)

y(t) = V1x(t) / [V2y(t)] + z(t)Recursive Solution (5)

y(t)1,2 = z(t)/2 +/- sqrt[V22 z(t)2 + 4 V1 V2 x(t)]/[2 V2]Equifinal Solution (6)



Reference:
Dietrich, J. W. (1999-2003). Quotient 0th Order feedback control with load. Medizinische Kybernetik | Medical Cybernetics. http://www.medizinische-kybernetik.de/ips/qfc0l.html (27 Jun. 2003).

ASIA-Element (Analog Signal Memory with Intrinsic Adjustment)


x(t): Input Signal

y(t): Output Signal

alpha: Gain Factor for Input Signal

beta: Clearance Exponent

y: Steady State Solution

Y(omega): Response to periodic input signals

yD: Deleted information



dy/dt = alpha x(t) - beta y(t)(1)

y = alpha x(t) / beta(2)

Y(omega) = alpha / (omega + beta) X(omega)(3)

t1 = 1/beta(4)

yD = y0[1 - exp(-beta t)](5)



Reference:
Dietrich, J. W. (2000). "Signal Storage in Metabolic Pathways: The ASIA Element." kybernetiknet 1 (3, 2000): 1-9.

Lateral inhibition (subtractive, feedforward, without weighting)

xr(t): Input Signal

yr(t): Output Signal
yr(t) = xr(t) - [xr-1(t) + xr+1(t)](1)

y(t)=Bx(t)
Vector Form (2)

B = (1-1...0)
-11...0
0-1...0
............
00...1
Connection Matrix (3)

Legend



Addition Element

y = x2 - x1

Filled sectors denote inversion of the function (subtraction), empty input signals are neutral
(0 for addition and subtraction).


Multiplication Element

y = x2 / x1

Filled sectors denote inversion of the function (division), empty input signals are neutral
(1 for multipliation and division).


Gain Factor

y = Gx


Special Function

y = sqrt(x)