# 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 ) -1 1 ... 0 0 -1 ... 0 ... ... ... ... 0 0 ... 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)