Magnitude optimum and symmetric optimum: controller settings for PT1, PT2 and I-T1 plants

What is calculated?

The magnitude optimum demands that the magnitude of the reference transfer function |G_w(jω)| stays close to 1 over the widest possible frequency range, which leads to a horizontal tangent at ω = 0 (eq. 8.24) and, for lag plants, to the optimum damping D = 1/√2 (eq. 8.25): 4.3 % overshoot, rise time ≈ 4.7·T₁, settling time ≈ 11·T₁. Basic type A is an I-T1 loop G₀ = K_PR·K_S·K_IS/(s(1 + sT₁)) – e.g. a PI controller whose Tn has cancelled the large time constant – with K_PRopt = 1/(2·K_S·K_IS·T₁). Basic type B is a PT2 plant with a P controller, K_PRopt = (T₁ + T₂)²/(2·K_S·T₁·T₂) − 1/K_S, with a remaining error 1/(1 + K_PR·K_S). The symmetric optimum applies to plants with integral action and lag controlled by PI (eq. 8.26): the phase is maximal at the crossover when ω_d is the geometric mean of the corner frequencies 1/Tn and 1/T₁. With k = Tn/T₁ follow K_PRopt = 1/(√k·K_S·K_IS·T₁), ω_d = 1/(√k·T₁) and the phase margin from k = cot²((90° − φ_R)/2) (eq. 8.36). Kessler recommends k = 4: φ_R = 37°, D = 0.5, 43.4 % overshoot, T_an ≈ 3.1·T₁, T_aus ≈ 18·T₁ (Bild 8.19); k = 9 gives the aperiodic limit with a triple pole.

Equations

Betragsoptimum: d|G_w(jω)|/dω = 0 bei ω = 0 (Gl. 8.24) → D = 1/√2 (Gl. 8.25)

Grundtyp A: K_PRopt = 1/(2·K_S·K_IS·T₁); x_m = 4,3 %, T_an ≈ 4,7·T₁, T_aus ≈ 11·T₁

Grundtyp B: K_PRopt = (T₁ + T₂)²/(2·K_S·T₁·T₂) − 1/K_S; e(∞) = 1/(1 + K_PR·K_S)

Symmetrisches Optimum: Tn = k·T₁, K_PRopt = 1/(√k·K_S·K_IS·T₁), ω_d = 1/(√k·T₁) (Gl. 8.33–8.35)

k = cot²((90° − φ_R)/2) (Gl. 8.36); D = (√k − 1)/2; k = 4: φ_R = 37°, x_m = 43,4 %

Assumptions and typical mistakes

The formulas assume the named basic types; additional time constants must be compensated (Tn = T_large, Tv = T₂) or lumped into T_E (T₁ ≥ 5·T₂). The figures hold for ideal controllers without limits and for setpoint steps; disturbance behaviour differs. Dead time is not included.

Entering K_IS as a time constant instead of a gain in 1/s. Using the large time constant as T₁ – T₁ is the small, uncompensated one. Applying the symmetric optimum to a PT1 plant without integral action.

Which method when?

Magnitude optimum for self-regulating plants or I-T1 loops where the controller adds no second integration; symmetric optimum as soon as two integrators are in series (integrating plant + PI controller) – there the magnitude optimum would be unstable. For PI on a PT1 with a dominant time constant: set Tn = T_large and use type A with K_IS = 1/Tn (row e of the table).

What you enter – and where the values come from

Method

Magnitude optimum basic type A (I-T1 plant or compensated PI loop), basic type B (PT2 plant with P controller) or symmetric optimum (I-T1 plant with PI controller). Choose by plant structure, see the method note.

Plant gain K_S

Proportional gain of the plant (dimensionless or in controlled per manipulated units), > 0.

Integral gain K_IS

Gain of the plant's integral part in 1/s (type A, symmetric optimum): ramp slope per unit of manipulated variable. For a compensated PT1 plant with PI controller set K_IS = 1/Tn. > 0.

Time constant T₁ (small time constant)

Remaining uncompensated lag of the loop (actuator, sensor, sum of small time constants T_E = T₁ + T₂ when T₁ ≥ 5·T₂). It sets speed and settling time. > 0.

Time constant T₂ (type B)

Second time constant of the PT2 plant. For T₂ ≫ T₁ the approximation K_PRopt ≈ T₂/(2·K_S·T₁) holds (row e of the table).

Factor k = Tn/T₁

Bandwidth factor of the symmetric optimum, > 1. k = 4 is Kessler's standard; larger k gives more phase margin and less overshoot but slower disturbance rejection; k = 9 is aperiodic.

What each value means

K_PRopt

Optimum controller gain for the selected rule. Transfer it to the step-response or Bode calculator to verify.

Tn

Reset time of the PI controller (symmetric optimum: k·T₁; type A: equal to the compensated large time constant).

Damping D and phase margin φ_R

Closed-loop figures: D = 0.707 for the magnitude optimum; for the symmetric optimum D = (√k − 1)/2 and φ_R from eq. 8.36 (37° at k = 4; 45° corresponds to D = 0.707 at k ≈ 5.8).

Overshoot, T_an, T_aus

Expected performance figures per Zacher/Reuter Bild 8.16/8.19: 4.3 %, 4.7·T₁ and 11·T₁ for the magnitude optimum; 43.4 %, 3.1·T₁ and 18·T₁ for the symmetric optimum with k = 4. For other k only D and φ_R are given.

Steady-state error (type B)

e(∞) = 1/(1 + K_PR·K_S) – the P controller without integral action leaves a residual error. The other methods contain an integrator and settle completely.

Worked example from the literature

Symmetric optimum with K_S·K_IS = 0.5 s⁻¹, T₁ = 2 s and k = 4: Tn = 8 s, K_PRopt = 1/(2·0.5·2) = 0.5, ω_d = 0.25 s⁻¹, φ_R = 37°, D = 0.5. The step-response calculator confirms 43 % overshoot with the same values. Magnitude optimum type A with K_S = 0.5, K_IS = 2 s⁻¹, T₁ = 3 s: K_PRopt = 1/(2·0.5·2·3) = 0.167, overshoot 4.3 %, T_an ≈ 14 s, T_aus ≈ 33 s.

What this calculator is for

Standard methods of drive engineering: current loops (magnitude optimum) and speed loops (symmetric optimum) in cascade structures, position control with integral behaviour, fast electrical and hydraulic actuator loops with known small time constants.

How to proceed

1. Identify the plant structure: I-T1 (type A or SO), PT2 (type B) or PT1 with compensating PI (Tn = large time constant, then type A with K_IS = 1/Tn).
2. Enter gains and time constants; for SO choose k.
3. Read K_PRopt and Tn and verify in the step-response calculator with the full plant.
4. If overshoot is too large (SO), add a setpoint filter or increase k.

Technical basis

Zacher/Reuter, Regelungstechnik für Ingenieure, 17th ed. 2024: sec. 8.4.1 magnitude optimum (eq. 8.24–8.25, table basic types A/B, Bild 8.16), sec. 8.4.2 symmetric optimum (eq. 8.26–8.39, Bild 8.17–8.19). Kessler, C.: Über die Vorausberechnung optimal abgestimmter Regelkreise, Regelungstechnik 1955 (cited via Zacher/Reuter).

The sources support the equations and worked examples; the reference examples are recomputed in the calculator's automated tests. The calculator does not replace a simulation with the complete nonlinear plant model.

Last updated: 2026-09-20

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