MethodWhat is calculated?
The characteristic equation is the denominator polynomial of the reference transfer function set to zero, i.e. 1 + G₀(s) = 0 over a common denominator (Zacher/Reuter eq. 6.8 and 6.21). A loop is stable when all its roots have negative real parts. Hurwitz showed that the roots need not be known: necessary is that all coefficients are present and share the same sign (condition a); sufficient is that all leading principal minors of the Hurwitz matrix are positive (condition b, eq. 6.16). For n = 3 this reduces to a₁a₂ − a₀a₃ > 0 (eq. 6.17), for n = 4 to a₁a₂a₃ − a₀a₃² − a₁²a₄ > 0 (eq. 6.19). If the last determinant is exactly zero, a pole pair lies on the imaginary axis: the loop oscillates permanently with ω = √(a₀/a₂) (n = 3) – the stability limit needed for the Ziegler–Nichols rules. As a cross-check the calculator also computes the roots numerically.
Equations
aₙsⁿ + aₙ₋₁sⁿ⁻¹ + … + a₁s + a₀ = 0 (Gl. 6.8)
a) alle aᵢ vorhanden und gleichen Vorzeichens
b) D₁ = a₁ > 0, D₂ = a₁a₂ − a₀a₃ > 0, …, Dₙ > 0 (Gl. 6.16)
n = 3: a₁a₂ − a₀a₃ > 0 (Gl. 6.17); n = 4: a₁a₂a₃ − a₀a₃² − a₁²a₄ > 0 (Gl. 6.19)
Stabilitätsgrenze: Dₙ₋₁ = 0, Dauerschwingung mit ω² = a₀/a₂ (n = 3)
LimitsAssumptions and typical mistakes
Only for rational transfer functions; dead time makes the characteristic equation transcendental (Zacher/Reuter sec. 6.2) – use Nyquist then. The criterion only says stable/unstable, nothing about performance: a loop just inside the limit is stable but poorly damped.
Using the coefficients of the open instead of the closed loop. Mixing time units (T₁ in s, T₂² in min²). Overlooking a single negative coefficient – then the determinant check is unnecessary, the loop is already unstable.
ExampleWorked example from the literature
Zacher/Reuter Beispiel 6.1: PT2 plant K_S = 0.5, T₁ = 30 s, T₂² = 200 s² with a PI controller K_PR = 10, Tn = 4 s. Characteristic equation: 800 s³·s³ + 120 s²·s² + 24 s·s + 5 = 0. All coefficients are positive, but a₁a₂ − a₀a₃ = 2880 − 4000 = −1120 s³ < 0 – unstable. The stability limit lies at Tn = 5.55 s where the loop oscillates with ω = 0.173 s⁻¹. Alternatively a D term with Tv = 2.33 s reaches the limit.
ContextWhat this calculator is for
Quick stability proof without root finding, deriving parameter limits (up to which K or Tn the loop stays stable), checking exercises and exam calculations, pre-computing the stability limit before commissioning (Dubbel sec. 5.3).
How to proceed
- Set up the characteristic equation: bring 1 + G_R·G_S = 0 over one denominator and sort by powers of s.
- Enter the coefficients a₀ … aₙ (or take them from the loop form).
- Check condition a and the determinants; at D = 0 read the oscillation frequency.
- For limit values of single parameters (Tn, Tv, K) solve D = 0 for the parameter as in Beispiel 6.1.
SourceTechnical basis
Zacher/Reuter, Regelungstechnik für Ingenieure, 17th ed. 2024: sec. 6.1 Hurwitz stability criterion (eq. 6.1–6.21), Beispiel 6.1, Aufgabe 6.1. Heinrich, Grundlagen Regelungstechnik, 6th ed., sec. 7.2. Dubbel, ch. X, sec. 5.3.
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
FAQFrequently asked questions
Is condition a sufficient on its own?
No. For n ≤ 2 yes (Zacher/Reuter sec. 6.1); from n = 3 all coefficients can be positive while the loop is unstable – exactly the case in Beispiel 6.1.
What is the difference to Routh?
The Routh array is a tabular form of the same statement; both give identical verdicts. Zacher/Reuter uses the Hurwitz determinants, so the calculator follows that presentation.
How do I get the coefficients from plant and controller?
Multiply 1 + G_R(s)·G_S(s) by the denominators of both elements and sort by s (Zacher/Reuter eq. 6.5). The calculator does this for PTn plants and PID controllers when you use the loop form.