Routh-Hurwitz Stability Criteria

Routh-Hurwitz Stability Criteria:

• Stability in a system implies that for small changes in input or in initial conditions or in system parameters, do not result in large changes in system output.
• A stable system if disturbed from its equilibrium position returns to its state eventually after disturbance is removed.
• For an example when pendulum is disturbed from its original position by a small amount it exhibits damped oscillations about mean position and finally settles down to original mean position.

Point s to remember:

• For a causal LTI system to be stable, all the poles of H(S) must be strictly in L.H.P.
• In control system, we have to deal with Causal + LTI

• If all the coefficients of 1 ^st order polynomial have same sign then all the roots are:

Example ; T(s) = S+2 , S+2 =0, S= -2

• I f all the coefficients of 2 ^nd order polynomial are having same sign then all the roots will be in L.H.P.

Example ; T(s) = S ² +3S+2 , For T(S) = 0 ,

S = -1 , S= -2

• If all the coefficients of 3 ^rd order polynomial are of same sign then : All the real roots are in L.H.P.

Example ; D(s) =S ³ + 4S ² +6S+2 , For T(S) = 0 , For D(s) = 0

s= -2 , -1±j

Method s to calculate location of r oots for h igher order polynomials:

• The criterion is based on ordering the coefficients of the characteristic equation into an array called Routh Array.
• Example ; D(s) =a ₀ S ⁶ +a ₁ S ⁵ + a ₂ S ⁴ + a ₃ S ³ + a ₄ S ² +a ₅ S+a ₆ ,

S ⁶ a ₀ a ₂ a _{4                      a6}

S ⁵ a ₁ a ₃ a ₅ 0

S ⁴

S ³

S ²

S ¹ 0

S ⁰ a ₆

Important observations:

1. If any Row of RH-table is multiplied or divided by positive constant: Then result does not change.
2. If all the elements of first column is having same sign: Then no roots will be in R.H.P.
3. If all the coefficients of 1 ^st column are having same sign and there is no ROZ (Row of Zeros) →odd row zero →Rows corresponding to odd powers of s.
4. Then all the roots will be in L.H.P.
5. The no. of sign changes in first column indicates the no. of roots in R.H.P.