Tools › Building Systems › Controls and Commissioning › Control Loop Tuning › Example
FOPDT process K = 2, τ = 10 min, θ = 2 min: ultimate gain, six tuning rules side by side and the SIMC PI closed loop with its margins
| Tool | Control Loop Tuning |
| Source | Ziegler, J. G., and Nichols, N. B. (1942). "Optimum Settings for Automatic Controllers." Transactions of the ASME, Vol. 64, No. 8, pp. 759–768; Skogestad, S. (2003). "Simple Analytic Rules for Model Reduction and PID Controller Tuning." Journal of Process Control, Vol. 13, No. 4, pp. 291–309 (eqs. 23–25, Table 2). |
| Level | Professional |
Problem
A heating-coil discharge-air loop behaves as a first-order-plus-dead-time process with gain K = 2 (°F per % valve), time constant τ = 10 min and dead time θ = 2 min. Find the ultimate gain and period, the P, PI and PID settings of the Ziegler–Nichols reaction-curve and ultimate-gain rules, Cohen–Coon, IMC, SIMC and Tyreus–Luyben, simulate the PI loop under a unit setpoint step and a 0.5 load step, read its margins, and check the valve authority (5 psi open in a 20 psi circuit), the proportional-thermostat offset for a 4 °F throttling range and the sample time.
Inputs to Enter
- Model: K 2, τ 10 min, θ 2 min, no second lag; controller PI, ideal form, rule SIMC with τ_c = θ (blank), derivative filter N 10
- Simulation: setpoint step 1 at t = 0, load step 0.5 at t = 72 min, end 144 min (automatic), no output limits
- Valve Δp 5 psi open, circuit 20 psi, equal-percentage R = 50; thermostat throttling range 4 °F, span 100 %, load change 50 %
Answers
- Phase crossover ω_u = 0.8443 rad/min: K_u = 4.251, P_u = 7.442 min; Ziegler–Nichols classification L = P_u/4 = 1.860 min, R_1 = 8/(P_u K_u) = 0.253 (model: θ = 2, K/τ = 0.2)
- PI settings (K_c, T_i min): ZN reaction curve 2.250 / 6.667; ZN ultimate 1.913 / 6.201; Cohen–Coon 2.292 / 4.708; IMC 2.500 / 10.00; SIMC 1.250 / 10.00; Tyreus–Luyben 1.329 / 16.37
- SIMC PI closed loop: overshoot 4.1 %, rise 3.8 min, settling (2 %) 12.1 min, IAE 4.34, TV 2.58; load step: peak deviation 0.293, IAE 4.00; GM 3.14, PM 61.4°, M_s 1.59 (Skogestad's Table 2 values for τ ≤ 8θ)
- ZN ultimate PI: overshoot 44 %, settling 21 min, IAE 5.42, M_s 2.43; Cohen–Coon PI: overshoot 74 %, M_s 3.76 (the quarter-decay rules are the most aggressive)
- Valve authority N = 0.25: installed flow 0.275 at half stroke against the inherent 0.141, installed gain twice the inherent near closed; thermostat gain 25 % per °F, offset 2 °F for a 50 % load change; sample time ≤ 0.5 min (θ/4 governs)
Notes
The tool's default inputs are this case; press Submit. The process is illustrative (no printed worked example has it); test_control_loop_tuning.php checks the rules at thirteen models, the ultimate gain against scipy brentq, the margins and the simulation against an independent RK4 integration in References/Control-Loop-Tuning/checks/, and reproduces Ziegler and Nichols' own 1942 case (R 1.7 in/min, L 0.2 min, F 1.7 psi → 5 psi/in; S_u 10 psi/in, P_u 0.8 min → 4.5 psi/in and 1.5 repeats/min) and Skogestad's printed IAE 2.17 / 2.0 and TV 4.11 / 1.08 for his first-order test process.