Root Locus: Design Applications

Root locus is more than just a plotting technique; it is a powerful visual design tool that allows control engineers to shape a system's dynamic response by strategically placing its closed-loop poles. By understanding how gain moves these poles, you can directly target performance specifications like overshoot and settling time. When simple gain adjustment falls short, the method guides the design of dynamic compensators—adding poles and zeros to bend the root locus branches through precisely desired regions of the s-plane.

Foundational Review: The Root Locus and System Performance

The root locus is a graph in the s-plane showing all possible locations of the closed-loop poles of a system as a single parameter, typically a gain $K$, varies from zero to infinity. These pole locations are the roots of the system's characteristic equation, $1+KG(s)H(s)=0$. The path each root takes as $K$ changes dictates the system's transient response.

Why does this matter for design? The position of a complex conjugate pole pair directly translates to time-domain performance metrics. A pole at $s=-\sigma +j{\omega }_d$ has a real part $\sigma$ that determines the exponential decay rate. The settling time $T_s$ (to within 2% of final value) is approximated by $T_s\approx 4/\sigma$. The imaginary part ${\omega }_d$ relates to the damped frequency of oscillation. The radial distance from the origin, ${\omega }_n=\sqrt{{\sigma }^2+{\omega }_d^2}$, is the natural frequency. The angle $\theta$ this radial line makes with the negative real axis defines the damping ratio $\zeta$, where $\cos \theta =\zeta$. A desired percent overshoot (%OS) corresponds to a specific damping ratio, which in turn corresponds to a constant $\zeta$-line (or radial line) in the s-plane.

Therefore, a design goal like "less than 16% overshoot" translates to "all dominant closed-loop poles must lie within the region where $\zeta >0.5$." The root locus shows you if and for what gain $K$ this condition is met.

Gain Selection to Meet Transient Specifications

The most direct application of root locus is selecting a controller gain $K$ to achieve a desired transient response. The process is systematic. First, you translate your performance specifications into a desired region in the s-plane. For example, a specification for settling time $T_s$ requires ${\sigma }_{min}=4/T_s$, defining a vertical boundary: all poles must have a real part less than $-{\sigma }_{min}$. A specification for percent overshoot defines a radial $\zeta$-line.

Second, you plot the root locus of the uncompensated system (e.g., $KG(s)$). Third, you identify the point on the root locus branches that intersects your desired performance region. This point represents a specific complex conjugate pole pair, $s_d$. Finally, you use the magnitude condition of the root locus to solve for the required gain $K$ that places the poles at $s_d$. The magnitude condition states that at any point $s$ on the root locus, $K|G(s)H(s)|=1$, therefore $K=1/|G(s)H(s)|$. You simply evaluate this at your desired point $s_d$.

Worked Example Concept: Suppose a system's open-loop function is $G(s)=\frac{1}{s(s+4)}$ and you need $\zeta =0.707$. The $\zeta =0.707$ line is at 45°. Drawing this line on the root locus, it intersects the locus's circular branch. At the intersection point $s_d\approx -2+j2$, you calculate $K=1/|G(s_d)|=|s_d(s_d+4)|=|(-2+j2)(2+j2)|=8$. This gain $K=8$ yields the desired damping.

Dynamic Compensation: Reshaping the Locus with Lead and Lag Networks

Often, the uncompensated system's root locus does not pass through the region required to meet all specifications simultaneously. Perhaps the branches are too far to the right, yielding slow response, or they lie in a low-damping region. You cannot fix this with gain alone; you must change the root locus plot itself by modifying the open-loop transfer function $KG(s)H(s)$. This is done by adding a compensator $G_c(s)$ in series with the plant.

• Lead Compensation: A lead compensator has the transfer function $G_c(s)=K_c\frac{s+z_c}{s+p_c}$, where $|p_c|>|z_c|$. It adds an open-loop zero and a pole, with the zero closer to the origin. Its primary purpose is to improve transient response: it introduces phase lead, which can be used to bend the root locus branches to the left, into a region of higher natural frequency (faster response) and better damping. In effect, you place the zero and pole of the compensator to attract the existing locus toward your desired pole location $s_d$. The design often involves calculating the required phase angle addition from the compensator to make $s_d$ a point on the new, compensated root locus.

• Lag Compensation: A lag compensator also has the form $G_c(s)=K_c\frac{s+z_c}{s+p_c}$, but with $|z_c|>|p_c|$. Its pole and zero are placed very close together and very near the origin in the s-plane. Its primary purpose is not to reshape the locus significantly but to improve steady-state error characteristics (like reducing position or velocity error) without altering the already-satisfactory transient response. The lag compensator provides a high, low-frequency gain boost to improve error constants, while its pole-zero pair near the origin creates only a minor, negligible change to the root locus near the dominant poles.

In practice, you first use a lead compensator to achieve the desired transient response (placing dominant poles at $s_d$), then, if necessary, fine-tune steady-state performance with a lag compensator.

A Unified Design Example Workflow

Let's outline a complete design scenario. Assume a plant $G_p(s)=\frac{1}{s(s+2)}$ with unity feedback. Specifications require a settling time $T_s\le 2$ seconds and a damping ratio $\zeta \approx 0.5$.

1. Translate Specs: $T_s\le 2$ sec $⟹\sigma \ge 4/2=2$. $\zeta =0.5$ defines a 60° line. The desired pole region is where the $\sigma =2$ vertical line and the $\zeta =0.5$ radial line intersect.
2. Analyze Uncompensated Locus: The root locus for $K{G}_p(s)$ has poles at $s=0$ and $s=-2$ and a breakaway point at $s=-1$. The branches are entirely on the real axis for $K>0$, never entering the desired region. Gain adjustment alone fails.
3. Design Lead Compensator: We need to pull the locus left and into the desired region. Choose a compensator zero, say at $s=-3$, to attract the locus. Place the compensator pole further left, e.g., at $s=-10$, so the ratio $p_c/z_c$ is significant. The new open-loop function is $G_c(s)G_p(s)=K\frac{(s+3)}{s(s+2)(s+10)}$.
4. Verify and Find Gain: Plot the new root locus. The added zero pulls the circular branch from the complex poles into the left half-plane. Find the point $s_d$ where this branch crosses the $\zeta =0.5$ line with $\sigma \approx 2$. Use the magnitude condition at $s_d$ to solve for the total gain $K$.
5. Check Steady-State Performance: Evaluate the system type and error constant (e.g., $K_v$) with the designed lead compensator. If steady-state error is too large, consider adding a lag network near the origin to boost the error constant without moving $s_d$.

Common Pitfalls

1. Ignoring Actuator Saturation and Real-World Gain Limits: Solving for a gain $K=10,000$ might place poles perfectly on paper, but such a high gain will likely saturate the real system's actuator, leading to nonlinear behavior, instability, or poor performance. Always consider practical gain ranges.
2. Placing Compensator Pole/Zero Too Far or Too Close: In lead design, placing the pole excessively far to the left ($p_c>>z_c$) requires very high gain and amplifies high-frequency noise. Placing them too close together provides negligible phase lead. A typical ratio $p_c/z_c$ is between 5 and 20. In lag design, placing the pole and zero too far from the origin will significantly alter the root locus and ruin your transient design.
3. Overlooking the Role of the Third Pole: Root locus design typically focuses on placing the dominant complex pole pair. However, adding a compensator increases the system order. You must verify that the remaining closed-loop poles (the "third" or "fourth" poles) are fast and well-damped, or sufficiently far into the left half-plane so they don't significantly impact the response. A slow, real closed-loop pole near the origin can dominate the response despite your carefully placed complex pair.
4. Confusing Lead and Lag Compensator Structures: A compensator with the form $(s+z)/(s+p)$ is a lead compensator only if $|p|>|z|$ (pole is further from the origin). If $|z|>|p|$, it is a lag network. Mixing up this fundamental rule will lead to a design that achieves the opposite of your intent.

Summary

• The root locus provides a direct map between controller gain $K$ and closed-loop pole locations, which dictate transient performance metrics like settling time and overshoot.
• Gain selection involves translating performance specs into a region in the s-plane, finding where the locus intersects that region, and applying the magnitude condition to calculate the required $K$.
• When the uncompensated root locus cannot satisfy specifications, a lead compensator $(s+z_c)/(s+p_c)$ (with $p_c>z_c$) is used to reshape the locus, bending branches to the left for improved speed and damping.
• A lag compensator $(s+z_c)/(s+p_c)$ (with $z_c>p_c$) places a pole-zero pair near the origin to increase error constants and improve steady-state accuracy without altering the satisfactory transient response.
• Successful design requires checking for practical gain limits, verifying non-dominant pole locations, and precisely applying the pole/zero relationships that define lead versus lag action.