Effect of addition of Poles & Zeros on Root Locus - Control Systems - Electrical

Addition of Poles

The presence and location of open-loop poles and zeros determine the shape and position of the closed-loop roots shown by the root locus. An added open-loop pole typically increases the order of the characteristic equation and tends to pull one or more branches of the root locus toward the added pole. In practical terms, this often causes the root locus to shift toward the right half of the s-plane (toward less negative real parts), thereby reducing relative stability and slowing the transient response (increasing settling time).

• Qualitative effect: Adding a pole usually moves root-locus branches toward the right (less stable), increases the chance of closed-loop poles moving closer to the imaginary axis or into the right half-plane, and can increase settling time and overshoot.
• Order and branches: The number of root-locus branches equals the number of open-loop poles. Adding a pole increases the number of branches and changes asymptotes, breakaway points and angle characteristics.
• Asymptote centroid: The asymptote centroid (σa) for n poles and m zeros is σa = (Σ poles - Σ zeros)/(n - m). Adding a pole changes Σ poles and n, and so alters the centroid location.
• Angle condition: Each open-loop pole contributes -arg(s - p_i) to the net angle. An extra pole changes the angle condition arg[G(s)H(s)] = (2k + 1)π and therefore changes the locus geometry and root locations.

Mathematical illustration: effect on centroid and asymptotes

Consider an open-loop transfer function with poles and zeros. The centroid of asymptotes is given by:

σa = (Σ p_i - Σ z_j) / (n - m)

If a pole at p_new is added then the new centroid becomes

σa_new = (Σ p_i + p_new - Σ z_j) / (n + 1 - m)

When p_new is in the left half-plane but closer to the origin than the previous centroid, σa_new will be less negative (move right). This demonstrates why adding a pole often shifts the locus toward the right half-plane.

Worked example - adding a pole

Example: Start with G(s) = K / ((s + 1)(s + 3)).

Poles are at s = -1 and s = -3.

Number of poles n = 2, number of zeros m = 0.

Sum of poles Σp = (-1) + (-3) = -4.

Centroid σa = (Σp - Σz) / (n - m) = (-4 - 0) / (2 - 0) = -2.

Now add a pole at s = -0.5.

New sum of poles Σp_new = -1 + (-3) + (-0.5) = -4.5.

New number of poles n_new = 3, zeros unchanged m = 0.

New centroid σa_new = (-4.5 - 0) / (3 - 0) = -1.5.

Comparison: centroid moved from -2 to -1.5 (shifted right towards origin), indicating a tendency for root-locus branches to shift right and for the dominant closed-loop poles to become less negative.

Effect on time-domain response

• Settling time (approx.): For dominant poles with real part σd, settling time Ts ≈ 4/|σd|. If σd becomes less negative, |σd| decreases and Ts increases.
• Damping and overshoot: Adding poles can reduce damping ratio of dominant modes, increasing overshoot and oscillation tendency.
• Steady-state error: Adding poles may increase system type or change error constants depending on pole at origin; a pole at origin increases system type and can reduce steady-state error for certain inputs, but usually at the cost of slower dynamics.

Addition of Zeros

Adding a zero to the open-loop transfer function tends to pull root-locus branches toward the zero location. In many cases adding a suitably placed zero pulls branches leftwards in the s-plane (towards more negative real parts), increasing relative stability and speeding the transient response (decreasing settling time). Zeros contribute positively in the angle condition and can improve damping of dominant poles when placed appropriately (this is the principle behind lead compensation).

• Qualitative effect: A zero can attract branches and increase damping of the dominant modes, often shifting closed-loop poles left, reducing settling time and reducing overshoot.
• Centroid change: Adding a zero changes Σ z_j and (n - m), so the centroid σa shifts; if the zero is located nearer the origin than the centroid, σa typically becomes more negative (shifts left).
• Angle contribution: A zero at z_k contributes +arg(s - z_k) to arg[G(s)H(s)]. This change can move root-locus branches and breakaway/break-in behaviour.
• Compensator design: Placing a zero on the left half-plane is a standard means of improving transient response (lead network). Placing a pole near the origin (lag) improves steady-state error but slows response.

Worked example - adding a zero

Example: Use the same initial G(s) = K / ((s + 1)(s + 3)).

Poles: s = -1, -3. Sum Σp = -4. n = 2, m = 0. Centroid σa = -4/2 = -2.

Add a zero at s = -0.5 to give G_new(s) = K (s + 0.5) / ((s + 1)(s + 3)).

Sum of zeros Σz = -0.5.

New centroid σa_new = (Σp - Σz) / (n - m) = (-4 - (-0.5)) / (2 - 1) = (-3.5) / 1 = -3.5.

Comparison: centroid moved from -2 to -3.5 (shifted left away from origin), indicating the root-locus branches tend to move left and the dominant closed-loop poles become more negative, yielding faster settling.

Other important effects of added zeros

• Transient improvement: A zero placed near a dominant pole can increase damping and reduce overshoot; a properly chosen zero is the basis of lead compensation.
• Introduced high-frequency behaviour: Zeros can introduce phase lead or lag at different frequencies and affect gain and phase margins.
• Unwanted effects: Zeros located in the right half-plane (non-minimum phase zeros) can pull the locus into unfavourable regions and create inverse response or performance limits.

Root locus rules - how added poles or zeros enter the standard construction

• Number of branches: Equals the number of open-loop poles; zeros determine where branches terminate (finite zeros) or head to infinity (asymptotes) when n > m.
• Asymptote angles: For n - m asymptotes, the angles are θk = (2k + 1)π / (n - m), k = 0, 1, ..., n - m - 1.
• Centroid location: σa = (Σ poles - Σ zeros) / (n - m). Adding poles or zeros changes σa.
• Angle and magnitude conditions: The root locus satisfies arg[G(s)H(s)] = (2k + 1)π and |G(s)H(s)| = 1. Each added pole contributes -π to the total asymptotic phase at large s and each zero contributes +π, changing available branches.
• Breakaway/break-in points: Locations on the real axis where branches leave or join the real axis are found from dK/ds = 0; adding poles or zeros changes this polynomial and therefore possible break points.

Design implications and practical guidance

• Use zeros for speed and damping: To speed up response and increase damping, introduce a left half-plane zero (lead compensator). Place the zero near the dominant closed-loop pole to have strong effect.
• Use poles for steady-state improvement: To improve steady-state error, a lag compensator (pole near origin with zero closer to origin) may be used. Expect some slowing of transient response.
• Avoid adding RHP zeros: Right half-plane zeros are detrimental to transient performance and limit achievable speed and damping.
• Always check margins: After adding poles or zeros, recompute gain and phase margins and closed-loop poles to verify stability and performance requirements.

Summary

Addition of poles generally increases system order and tends to shift root-locus branches toward the right half-plane, reducing relative stability and slowing response. Addition of zeros generally attracts branches and can pull the root locus to the left, improving damping and speeding response when zeros are placed appropriately. The exact effect depends on the location of the added pole or zero; use centroid, asymptote formulas and angle/magnitude conditions to predict and design the desired behaviour.