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Topic overview
Control systems problems require chained calculations where one error propagates through every subsequent step. The most damaging errors occur at the transfer function derivation stage and in the Routh-Hurwitz stability criterion. Getting either of these wrong makes every downstream result incorrect.
Common mistakes
Mistake 1: Using G(s) instead of G(s)H(s) in the characteristic equation
Why it happens: Students recall the closed-loop formula T(s) = G/(1+G) from unity feedback examples and use it even when H(s) is not 1.
Correct approach: For a non-unity feedback system, the characteristic equation is 1 + G(s)H(s) = 0. The closed-loop transfer function is G(s) / (1 + G(s)H(s)). Always substitute H(s) before forming the characteristic polynomial.
Example: G(s) = 10/(s+2), H(s) = 0.5. Wrong: characteristic equation = 1 + 10/(s+2) = 0, giving s + 12 = 0. Correct: 1 + G(s)H(s) = 1 + 5/(s+2) = 0, giving s + 7 = 0. The closed-loop pole is at s = -7, not s = -12.
Mistake 2: Sign error in the first column of the Routh array causes a wrong stability conclusion
Why it happens: Students divide by the pivot element but forget that dividing by a negative number flips signs in the next row. They count sign changes in the first column incorrectly.
Correct approach: Write the pivot clearly for each row. After computing each element of the new row, check the sign of the first column entry against the row above. Count the number of sign changes in the completed first column. That count equals the number of right-half-plane poles.
Example: Characteristic polynomial: s^3 + 2s^2 - s + 4. Routh array row 2 is [2, 4], row 3 first element = (2·(-1) - 1·4)/2 = (-2-4)/2 = -3. Wrong: student computes (-1-4)/2 = -2.5, missing the cross-multiplication. Correct first column: [1, 2, -3, ...]. Two sign changes mean two RHP poles.
Mistake 3: Computing the steady-state error using the wrong system type number
Why it happens: Students count the total poles of G(s), not just the poles at s = 0. A pole at s = -2 does not contribute to the system type.
Correct approach: System type is the number of poles of G(s)H(s) located exactly at s = 0. Count only integrators. Then use the correct error coefficient: Kp for type 0, Kv for type 1, Ka for type 2.
Example: G(s) = 5 / (s(s+3)(s+5)). Wrong: student counts three poles total and says type 3, then claims zero steady-state error for a ramp input. Correct: only one pole at s=0, so type 1. Kv = lim s·G(s) as s→0 = 5/15 = 1/3. Steady-state error for ramp = 1/Kv = 3.
Mistake 4: Forgetting to convert gain K to the correct form before applying the magnitude condition in root locus
Why it happens: Students pick a test point on the root locus and apply |G(s)H(s)| = 1 without checking whether K appears in the numerator or has already been absorbed into the poles and zeros.
Correct approach: Write the open-loop transfer function in Evans form: K · N(s)/D(s), where N and D are monic polynomials. K must multiply the entire numerator. Evaluate the magnitude of N(s)/D(s) at the test point and take the reciprocal to get K.
Example: G(s) = K(s+2) / (s^2+4s+8). Test point s = -1+j2. Wrong: student computes |G| at the test point and sets it to 1 including K, solving K = 1 directly from |G|=1 without isolating K. Correct: compute |s+2| / |s^2+4s+8| at s=-1+j2, then K = reciprocal of that magnitude.
Debugging tips
- Before starting any stability or error analysis, write out G(s) and H(s) separately. Then form G(s)H(s) explicitly. This one step prevents the most common unity-feedback assumption error.
- When building the Routh array, compute each entry using the 2x2 determinant formula and show the numerator and denominator separately. Cancel after confirming the arithmetic.
- To verify your steady-state error result, simulate the step or ramp response using the final value theorem on the error signal E(s) = R(s)/(1 + G(s)H(s)). The limit as s→0 of s·E(s) must equal your calculated error.
- For root locus gain calculation, substitute the selected test point numerically into every pole and zero factor. Compute the angle condition first to confirm the point is on the locus, then compute the magnitude.
- After finding the closed-loop poles from the characteristic equation, check that their real parts are negative for a stable system. If any real part is positive or zero, revisit the Routh array or characteristic polynomial derivation.
Exam warnings
- A question may ask for the range of K for stability and give G(s)H(s) with K in both numerator and denominator. Factor K out completely before forming the Routh array. Leaving K embedded in a denominator term leads to algebraic errors in the array rows.
- Steady-state error questions sometimes specify the input as a unit parabolic signal r(t) = t^2/2. The Laplace transform is 1/s^3. Use Ka (acceleration error constant) for a type 2 system. Using Kv for a parabolic input gives an infinite error answer that is incorrect.
- When a Bode plot question asks for the gain margin, it is 0 - |G(jw)| in dB at the phase crossover frequency, not the gain at the gain crossover frequency. Students frequently swap these two frequencies.
- Phase margin is measured at the gain crossover frequency where |G(jw)| = 1 (0 dB). Add 180 degrees to the phase angle at that frequency. If the phase is already more negative than -180 degrees, the phase margin is negative and the system is unstable.