The controller algorithms are described in the following with these signals used in the formulas:

r(t) - Setpoint

y(t) - Measured Variable

u(t) - Output

x(t) - Output Tracking Variable

e(t) - Error between Setpoint and Measured Variable

T - Sample Interval

**1. ANN**

The detailed algorithm for the 1-For-3 ANN controller is lengthy and patented. We will just give a conceptual formula in the following:

where Kr is the ANN Response Knob. The ANN weighting factors are updating at every sample through some learning algorithms and need not to be tuned manually.

**2. PID Standard**

where e(t) = r(t) - y(t); This formula is the analog version in time domain. The digital version of the PID is as follows:

where e[0], e[1], and e[2] are the time sampled error signal e(t), e[2] is the current sample of e(t), and u[1] is the current sample of u(t). Kp is the Proportional Gain, Ki is the Integral Gain in second/repeat, and Kd is the Derivative Gain in repeat/second.

**3. PID Error-Squared**

and,

where SGN is the sign function.

**4. PID Error-Squared**

where LLVu is the Lower Limit Value of u, and

ULVu is the Upper Limit Value of u.

Notice that the controller output is always in range of 0 to 100 percent no matter what the LLVu and ULVu are.

**5. Ramp**

**6. Integrator**

where r(t) is the input. The formula is an analog version, and its digital version is as follows:

where, u[1] and r[1] are the current samples of u(t) and r(t), respectively.

**7. Lead-Lag**

where Gc(S) = U(S)/R(S), which is the Laplace transfer function of the lead-lag block.

8. Second-Order Filter

where Gc(S) is the Laplace transfer function of the block.

**9. Dead-time**

where Gc(S) is the Laplace transfer function of the block, K is the DC gain, Tc is the time constant, and t (Tau) is the dead time.

**10. Linear**

**11. Ratio**

**12. High Select**

**13. Low Select**

**14. Middle Select**

**15. Limit**

**16. Loader**