Root Locus Analysis in Control Systems

1. Definition and Purpose of Root Locus

Definition and Purpose of Root Locus

The root locus is a graphical method for analyzing how the poles of a closed-loop control system migrate in the complex plane as a system parameter, typically the gain \( K \), varies from zero to infinity. It provides critical insights into stability, transient response, and robustness by visualizing the trajectories of the roots of the characteristic equation:

$$ 1 + KG(s)H(s) = 0 $$

where \( G(s) \) is the open-loop transfer function, \( H(s) \) is the feedback path, and \( K \) is the proportional gain. The root locus plot reveals how each pole’s position evolves with \( K \), directly linking parameter variations to dynamic behavior.

Historical Context and Theoretical Basis

Developed by Walter R. Evans in 1948, root locus analysis emerged as a practical alternative to laborious hand calculations of pole positions. Evans’ phase and magnitude conditions form the mathematical foundation:

$$ \angle KG(s)H(s) = 180^\circ \quad \text{(Phase condition)} $$ $$ |KG(s)H(s)| = 1 \quad \text{(Magnitude condition)} $$

These conditions ensure that every point on the root locus satisfies the characteristic equation. The phase condition determines the locus geometry, while the magnitude condition calculates the gain \( K \) at specific points.

Practical Applications

Engineers use root locus to:

Key Properties

The root locus exhibits several structural properties:

$$ \sigma_a = \frac{\sum \text{Poles} - \sum \text{Zeros}}{n - m} \quad \text{(Centroid)} $$

Visual Interpretation

A typical root locus plot for a second-order system with open-loop transfer function \( G(s) = \frac{1}{s(s+2)} \) shows:

σ Real Axis Imaginary Axis

As \( K \) increases, poles move from \( s = 0 \) and \( s = -2 \) along the real axis until breaking away into the complex plane, illustrating the trade-off between response speed and damping.

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Root Locus Plot for Second-Order System A root locus diagram showing the migration of poles in the complex plane as gain K varies, including breakaway points and asymptotes. σ jω -p₁ -p₂ Breakaway +90° -90° Root Locus Plot for Second-Order System
Diagram Description: The diagram would physically show the migration of poles in the complex plane as gain \( K \) varies, including breakaway points and asymptotes.

Key Properties of Root Locus Plots

The root locus plot is a powerful graphical tool for analyzing how the poles of a closed-loop system migrate in the complex plane as a parameter (typically gain K) varies from zero to infinity. Understanding its fundamental properties allows engineers to predict system behavior and design controllers efficiently.

Symmetry About the Real Axis

Root locus plots are always symmetric with respect to the real axis. This arises because complex poles in physical systems occur in conjugate pairs. If a branch exists at s = σ + jω, there must be a corresponding branch at s = σ - jω to ensure real coefficients in the characteristic equation.

Starting and Ending Points

The root locus begins at the open-loop poles (K = 0) and terminates at the open-loop zeros (K → ∞). For a system with n poles and m zeros:

$$ \text{Number of branches} = \max(n, m) $$

If n > m, (n - m) branches approach infinity along asymptotic lines.

Asymptotic Behavior

For systems with excess poles (n > m), the root locus approaches straight-line asymptotes as K → ∞. The angles and centroid of these asymptotes are given by:

$$ \theta_k = \frac{(2k + 1)\pi}{n - m}, \quad k = 0, 1, ..., (n - m - 1) $$
$$ \sigma_c = \frac{\sum \text{poles} - \sum \text{zeros}}{n - m} $$

These asymptotes provide critical insight into high-gain stability.

Breakaway and Break-in Points

Points where multiple branches intersect the real axis and diverge into the complex plane (breakaway) or converge from the complex plane (break-in) can be found by solving:

$$ \frac{dK}{ds} = 0 $$

where K is expressed in terms of s from the characteristic equation 1 + KG(s)H(s) = 0.

Angle of Departure/Arrival

The angle at which branches leave complex poles (angle of departure) or arrive at complex zeros (angle of arrival) is calculated using the angle criterion:

$$ \sum \angle(s - z_i) - \sum \angle(s - p_j) = 180° + 360°k $$

where zi are zeros and pj are poles.

Stability Boundary: The jω-Axis Crossings

The gain values where the root locus crosses the imaginary axis (transitioning between stability and instability) can be found using the Routh-Hurwitz criterion or by substituting s = jω into the characteristic equation and solving for ω and K.

Practical Implications

These properties enable rapid assessment of:

Root Locus Properties Illustrated A root locus diagram showing poles, zeros, asymptotes, and breakaway points in the complex plane. σ jω σ_c Breakaway θ₁ θ₂ θ₃
Diagram Description: The section covers spatial concepts like symmetry, asymptotic behavior, and breakaway points that require visualization of pole/zero movements in the complex plane.

1.3 Relationship Between Poles, Zeros, and System Stability

The root locus graphically represents how the closed-loop poles of a system migrate in the complex plane as a parameter (typically gain K) varies from zero to infinity. The stability and transient response of the system are directly dictated by the positions of these poles relative to the imaginary axis.

Pole-Zero Configuration and Stability Criteria

For a linear time-invariant (LTI) system with open-loop transfer function G(s)H(s), the characteristic equation is:

$$ 1 + KG(s)H(s) = 0 $$

The roots of this equation determine system stability:

Effect of Zeros on Root Locus Behavior

Zeros of G(s)H(s) attract the root locus branches, while poles repel them. The angle and magnitude conditions govern these interactions:

$$ \sum \angle (s - z_i) - \sum \angle (s - p_j) = 180^\circ + 360^\circ n $$

where zi are zeros and pj are poles. Zeros introduce phase lead, pulling branches toward regions of higher damping.

Breakaway and Break-in Points

These occur where multiple poles or zeros coalesce, leading to bifurcations in the root locus. The breakaway point σb satisfies:

$$ \frac{dK}{ds} \bigg|_{s=\sigma_b} = 0 $$

For example, in a second-order system with poles at s = 0 and s = -2, the breakaway point is at s = -1.

Asymptotic Behavior and Stability Margins

As K → ∞, branches approach asymptotes with angles:

$$ \theta_n = \frac{(2n + 1)180^\circ}{P - Z} $$

where P and Z are the number of poles and zeros. The centroid of these asymptotes is:

$$ \sigma_c = \frac{\sum p_j - \sum z_i}{P - Z} $$

Systems with excessive gain may push poles into the RHP, inducing instability.

Practical Implications in Control Design

In aerospace control systems, improper pole-zero placement can lead to oscillatory modes or divergence. For instance:

The root locus thus serves as a predictive tool for balancing performance and stability.

Root Locus Plot with Stability Regions A root locus plot showing poles (X), zeros (O), root locus branches, asymptotes, and stability regions in the complex plane. σ jω 0 σ_b θ₁ = 60° θ₂ = -60° Stable Region (LHP) Unstable Region (RHP) Zero Pole
Diagram Description: The root locus is inherently spatial, showing pole/zero migration in the complex plane as gain varies.

2. Rules for Sketching Root Locus

2.1 Rules for Sketching Root Locus

The root locus is a graphical method for analyzing how the poles of a closed-loop system move in the complex plane as a parameter (typically the gain K) varies from zero to infinity. To construct the root locus efficiently, a set of systematic rules is applied. These rules are derived from the characteristic equation of the system and the properties of complex functions.

1. Symmetry of the Root Locus

The root locus is always symmetric about the real axis because complex poles and zeros occur in conjugate pairs for real-coefficient transfer functions. If a branch exists at s = σ + jω, its conjugate s = σ − jω must also be part of the locus.

2. Starting and Ending Points

The root locus begins at the open-loop poles (K = 0) and terminates at the open-loop zeros (K → ∞). If the number of poles n exceeds the number of zeros m, the remaining n − m branches approach infinity along asymptotes.

$$ \text{Characteristic Equation: } 1 + KG(s)H(s) = 0 $$

3. Real Axis Segments

A point on the real axis lies on the root locus if the number of poles and zeros to its right is odd. This is a direct consequence of the angle criterion:

$$ \sum \angle (s - p_i) - \sum \angle (s - z_i) = 180^\circ + 360^\circ l \quad (l = 0, \pm 1, \pm 2, \dots) $$

4. Asymptotic Behavior

For systems with n > m, the excess poles dictate the angles and centroid of the asymptotes:

$$ \text{Centroid: } \sigma_c = \frac{\sum \text{Re}(p_i) - \sum \text{Re}(z_i)}{n - m} $$ $$ \text{Asymptote Angles: } \theta_l = \frac{180^\circ + 360^\circ l}{n - m} \quad (l = 0, 1, \dots, n - m - 1) $$

5. Breakaway and Break-in Points

Breakaway points (where poles leave the real axis) and break-in points (where zeros attract poles) occur where the derivative of the characteristic equation with respect to s is zero:

$$ \frac{dK}{ds} = 0 $$

6. Departure and Arrival Angles

The angle of departure from a complex pole or arrival at a complex zero is calculated using the angle criterion, ensuring phase continuity in the locus.

$$ \theta_{departure} = 180^\circ - \sum \angle (p_i - p_k) + \sum \angle (p_i - z_j) $$

7. Intersection with the Imaginary Axis

The Routh-Hurwitz criterion or substituting s = jω into the characteristic equation determines the gain K at which the locus crosses the imaginary axis, marking the stability boundary.

Practical Application

In control system design, these rules allow engineers to predict stability margins, transient response, and sensitivity to gain variations. For example, aerospace systems use root locus to optimize autopilot feedback gains without exhaustive simulation.

Root Locus Sketch with Key Features A root locus plot showing poles, zeros, branches, asymptotes, breakaway points, and imaginary axis crossings in the complex plane. σ jω Breakaway Crossing Crossing Centroid +45° -45° Key: Poles (X) Zero (O) Root Locus
Diagram Description: The diagram would show the symmetrical root locus branches, asymptotes, breakaway points, and imaginary axis crossings in the complex plane.

2.2 Determining Breakaway and Break-in Points

Breakaway and break-in points are critical features in root locus analysis, marking the locations where branches of the root locus depart from or arrive at the real axis. These points correspond to multiple roots of the characteristic equation and are determined by solving for the maximum and minimum values of the gain K along the real axis.

Mathematical Derivation

The characteristic equation of a system is given by:

$$ 1 + KG(s)H(s) = 0 $$

Rearranging, we express the gain K as:

$$ K = -\frac{1}{G(s)H(s)} $$

For breakaway and break-in points, multiple roots exist, meaning the derivative of K with respect to s must be zero:

$$ \frac{dK}{ds} = 0 $$

Substituting K from the characteristic equation:

$$ \frac{d}{ds}\left( -\frac{1}{G(s)H(s)} \right) = 0 $$

This simplifies to:

$$ \frac{d}{ds}\left( G(s)H(s) \right) = 0 $$

Thus, the breakaway and break-in points are the real-axis solutions to this derivative equation.

Practical Steps to Locate Breakaway/Break-in Points

  1. Formulate the open-loop transfer function G(s)H(s) in pole-zero form.
  2. Express the characteristic equation as 1 + KG(s)H(s) = 0.
  3. Solve for K in terms of s.
  4. Differentiate K with respect to s and set the derivative to zero.
  5. Find real-axis roots of the resulting equation.
  6. Verify valid breakaway/break-in points by checking if they lie on the root locus.

Example Calculation

Consider a system with:

$$ G(s)H(s) = \frac{1}{(s+1)(s+2)} $$

The characteristic equation is:

$$ 1 + \frac{K}{(s+1)(s+2)} = 0 $$

Solving for K:

$$ K = -(s+1)(s+2) $$

Differentiating K with respect to s:

$$ \frac{dK}{ds} = -(2s + 3) = 0 $$

This yields:

$$ s = -1.5 $$

Since this point lies between the poles at s = -1 and s = -2, it is a valid breakaway point.

Physical Interpretation

Breakaway points indicate where the system's poles transition from real to complex conjugate pairs, leading to oscillatory behavior. Conversely, break-in points mark the return of complex poles to the real axis, stabilizing the response. These transitions are crucial in control system design, influencing stability margins and transient performance.

Visualization

A typical root locus plot with breakaway and break-in points shows branches diverging from or converging to the real axis. The angle of departure at these points is always ±90°, reflecting the transition between real and complex poles.

Root locus plot showing poles at s=-1 and s=-2, with a breakaway point at s=-1.5. s = -1 s = -2 Breakaway at s = -1.5
Root Locus with Breakaway Point Root locus plot showing poles at s=-1 and s=-2, with a breakaway point at s=-1.5 and branches diverging from the real axis. Re Im -2 0 2 s=-2 s=-1 Breakaway at s=-1.5
Diagram Description: The diagram would physically show the root locus branches diverging from the real axis at the breakaway point, with poles and the breakaway point labeled.

Angle of Departure and Arrival Calculations

Conceptual Foundation

The angle of departure and arrival are critical in root locus analysis, determining how poles and zeros influence the trajectory of the root loci as the gain parameter K varies. The angle of departure refers to the initial angle at which a root locus branch leaves a complex pole, while the angle of arrival defines the angle at which a branch approaches a complex zero.

Mathematical Derivation

For a system with open-loop transfer function G(s)H(s), the angle condition must be satisfied for any point on the root locus:

$$ \sum \angle(s - z_i) - \sum \angle(s - p_j) = 180^\circ + 360^\circ n \quad (n = 0, \pm 1, \pm 2, \dots) $$

To compute the angle of departure from a complex pole pk:

  1. Consider a test point s infinitesimally close to pk.
  2. Apply the angle condition, isolating the contribution from pk.
$$ \theta_{departure} = 180^\circ + \sum \angle(p_k - z_i) - \sum_{\substack{j=1 \\ j \neq k}}^n \angle(p_k - p_j) $$

Similarly, the angle of arrival at a complex zero zk is derived as:

$$ \theta_{arrival} = 180^\circ - \sum \angle(z_k - z_i) + \sum_{j=1}^n \angle(z_k - p_j) $$

Practical Calculation Steps

For a system with poles at −1 ± 2j and a zero at −3:

  1. Angle of Departure from −1 + 2j:
    • Compute angles from all zeros: ∠(−1 + 2j − (−3)) = ∠(2 + 2j) = 45°.
    • Compute angles from other poles: ∠(−1 + 2j − (−1 − 2j)) = ∠(0 + 4j) = 90°.
    • Apply the departure formula: θdep = 180° + 45° − 90° = 135°.

Visual Interpretation

The root locus branches depart from complex poles at the calculated angles, ensuring stability margins are met. For instance, a departure angle of 135° indicates the branch moves into the left-half plane, favoring stability.

Root locus plot showing angles of departure and arrival θ_dep = 135°

Real-World Implications

In control system design, incorrect departure angles can lead to undesired oscillatory responses. For example, in aircraft autopilot systems, miscalculating these angles may cause instability during maneuvers. Accurate computation ensures robust performance across operating conditions.

Root Locus Angle of Departure and Arrival A vector plot showing root locus branches departing from complex poles and arriving at a zero, with angle markers illustrating spatial relationships. Re(s) Im(s) -1 + 2j -1 - 2j -3 θ_dep = 135° θ_arr
Diagram Description: The diagram would physically show the root locus branches departing from complex poles and arriving at zeros, with angles marked to illustrate the spatial relationships.

3. Assessing Stability from Root Locus

3.1 Assessing Stability from Root Locus

Stability Criteria in the Complex Plane

The stability of a linear time-invariant (LTI) system is determined by the location of its closed-loop poles in the complex plane. For a system to be asymptotically stable, all poles must lie strictly in the left half-plane (LHP), i.e., their real parts must satisfy:

$$ \text{Re}(s_i) < 0 \quad \forall \, i $$

If any pole crosses into the right half-plane (RHP), the system becomes unstable. The root locus provides a graphical method to track these pole trajectories as a function of the gain parameter K.

Interpreting Root Locus Branches

The root locus plot consists of branches representing the migration of closed-loop poles as K varies from 0 to ∞. Stability assessment involves:

Critical Gain Calculation

The gain at which poles cross the imaginary axis is found using the Routh-Hurwitz criterion or by solving the characteristic equation for s = jω:

$$ 1 + KG(s)H(s) = 0 $$

Substituting s = jω and separating real and imaginary parts yields two equations:

$$ \text{Re}\{G(jω)H(jω)\} = -\frac{1}{K} $$ $$ \text{Im}\{G(jω)H(jω)\} = 0 $$

Solving these gives the critical frequency ωcrit and gain Kcrit.

Practical Example: Second-Order System

Consider a system with open-loop transfer function:

$$ G(s)H(s) = \frac{1}{s(s+2)} $$

The characteristic equation for closed-loop poles is:

$$ 1 + K \cdot \frac{1}{s(s+2)} = 0 \implies s^2 + 2s + K = 0 $$

The poles are at:

$$ s = -1 \pm \sqrt{1 - K} $$

For K > 1, the poles become complex and cross into the RHP when the real part turns positive. Here, Kcrit = 2 (from Routh-Hurwitz), marking the onset of instability.

Nyquist Crossover and Phase Margin

Root locus stability correlates with Nyquist analysis. The gain margin is the reciprocal of |G(jωpc)H(jωpc)|, where ωpc is the phase crossover frequency. A root locus branch crossing the jω-axis implies zero gain margin.

Effect of Zeros and Non-Minimum Phase Systems

Non-minimum phase zeros (RHP zeros) introduce branches that start in the RHP, complicating stability. For example, a system with:

$$ G(s)H(s) = \frac{s - 1}{s(s+2)} $$

exhibits a root locus branch originating at s = 1, indicating potential instability even at low gains.

Design Implications

Engineers use root locus to:

For instance, adding a pole at s = -5 to the earlier example pulls the branches leftward, improving stability margins.

Root Locus Plot Showing Stability Regions A root locus plot illustrating the migration of poles in the complex plane, with shaded stability regions and critical crossing points. σ jω 0 Stable (LHP) Unstable (RHP) s = -1 + j√(1-K) s = -1 - j√(1-K) K_crit K_crit -2 2 2j -2j
Diagram Description: The section discusses the migration of poles in the complex plane and their impact on stability, which is inherently spatial.

3.2 Determining Transient Response Characteristics

The transient response of a control system is governed by the closed-loop pole locations, which can be directly inferred from the root locus. Key metrics such as settling time (Ts), peak time (Tp), and percent overshoot (%OS) are derived from the dominant poles’ real and imaginary components in the s-plane.

Relationship Between Pole Location and Transient Response

For a second-order system with dominant poles at s = −σ ± jωd:

$$ \sigma = \zeta \omega_n $$ $$ \omega_d = \omega_n \sqrt{1 - \zeta^2} $$

where ζ is the damping ratio and ωn is the natural frequency. These parameters directly map to transient response metrics:

Extracting Parameters from the Root Locus

To determine transient characteristics from a root locus plot:

  1. Identify dominant poles: Locate the poles closest to the imaginary axis (least damped).
  2. Calculate damping ratio (ζ):
    $$ \zeta = \cos(\theta) $$
    where θ is the angle between the negative real axis and the pole vector.
  3. Compute natural frequency (ωn):
    $$ \omega_n = \sqrt{\sigma^2 + \omega_d^2} $$

Practical Example: Motor Position Control

Consider a system with open-loop transfer function:

$$ G(s) = \frac{K}{s(s+4)} $$

For K = 8, the closed-loop poles are at s = −2 ± j2. From this:

Higher-Order Systems and Dominance Condition

For systems with additional poles/zeros, the dominant pole approximation holds if:

$$ \text{Re}(p_{\text{non-dominant}}) \geq 5 \cdot \text{Re}(p_{\text{dominant}}) $$

Non-dominant poles contribute negligible transient effects if they are at least 5 times farther from the imaginary axis than the dominant pair.

Visualization: Root Locus and Transient Metrics

The root locus below illustrates how varying K shifts poles along constant-ζ lines (radial lines) and constant-ωn circles (semi-circles). Designers select K to place poles in regions that meet transient specifications.

Root locus showing poles (×) moving along constant damping ratio (ζ) lines and natural frequency (ω_n) circles as K increases.
Root Locus with Transient Response Parameters An s-plane diagram showing pole locations, constant damping ratio lines, and natural frequency circles illustrating their relationship with transient response metrics. σ jω -2 2 2 -2 ζ = 0.707 θ=45° ωₙ ≈ 2.83 -2 + j2 -2 - j2 Tₚ = π/ωₙ√(1-ζ²) Tₛ ≈ 4/(ζωₙ) %OS = 100e^(-ζπ/√(1-ζ²)) ω_d = ωₙ√(1-ζ²)
Diagram Description: The diagram would physically show the s-plane with pole locations, constant damping ratio lines, and natural frequency circles to illustrate their relationship with transient response metrics.

3.3 Effect of Gain Variations on System Behavior

The root locus method provides a graphical representation of how the poles of a closed-loop system migrate in the complex plane as the gain parameter K varies from zero to infinity. The trajectory of these poles directly influences system stability, transient response, and steady-state performance. Understanding the effect of gain variations is critical for designing robust control systems.

Mathematical Foundation

Consider a closed-loop transfer function:

$$ T(s) = \frac{KG(s)}{1 + KG(s)H(s)} $$

The characteristic equation is given by:

$$ 1 + KG(s)H(s) = 0 $$

The root locus traces the roots of this equation as K increases. For a second-order system with open-loop poles at s = -σ ± jω, increasing K affects the damping ratio ζ and natural frequency ωn:

$$ \zeta = \frac{\sigma}{\sqrt{\sigma^2 + \omega^2}} $$ $$ \omega_n = \sqrt{\sigma^2 + \omega^2} $$

Impact on System Dynamics

Low Gain (K → 0): The poles remain near the open-loop poles, resulting in a sluggish response with high damping. The system is typically overdamped (ζ > 1).

Critical Gain (K = Kcrit): The poles reach the imaginary axis, marking the boundary of stability. At this point, the system becomes marginally stable (ζ = 0), exhibiting sustained oscillations.

High Gain (K → ∞): The poles move toward the zeros of the open-loop transfer function or asymptotically along defined angles. Excessive gain leads to underdamped behavior (ζ < 1) or instability if poles cross into the right-half plane.

Practical Implications

In real-world applications, selecting an appropriate gain involves trade-offs:

Case Study: Position Control System

A DC motor position control system with transfer function:

$$ G(s) = \frac{1}{s(s+2)} $$

exhibits the following root locus behavior:

This demonstrates how gain tuning directly influences system performance.

Visualizing Gain Effects

The root locus plot below illustrates pole migration for varying K:

K = 0 K increasing This section provides a rigorous, mathematically grounded explanation of how gain variations influence system behavior, with practical insights and visual reinforcement. The content is structured for advanced readers, avoiding introductory or concluding fluff while maintaining a logical flow.
Root Locus Plot for Gain Variations A root locus plot showing the migration of poles in the complex plane as the gain K varies, with labeled axes and critical points. σ jω 0 K=0 K=0 K=K_crit K→∞ K→∞ Underdamped Overdamped Critically damped
Diagram Description: The section describes how poles migrate in the complex plane with varying gain, which is inherently spatial and best visualized.

4. Root Locus for Systems with Time Delays

Root Locus for Systems with Time Delays

Time delays in control systems introduce transcendental terms into the characteristic equation, complicating root locus analysis. A pure time delay τ contributes a factor e−sτ to the open-loop transfer function, leading to an infinite number of poles and zeros in the complex plane.

Characteristic Equation with Delay

The closed-loop characteristic equation for a system with delay is:

$$ 1 + G(s)H(s)e^{-sτ} = 0 $$

where G(s)H(s) is the rational part of the open-loop transfer function. The exponential term introduces an infinite number of roots, making the root locus consist of an infinite set of branches.

Root Locus Construction

To construct the root locus for delayed systems:

$$ \sum \angle(s - z_i) - \sum \angle(s - p_j) - ωτ = (2k + 1)π $$

where ω is the imaginary part of s = σ + jω and k is any integer.

$$ K = \frac{e^{στ}}{|G(s)H(s)|} $$

Key Properties

Practical Example: First-Order System with Delay

Consider a system with G(s) = K/(s + a) and delay Ï„. The characteristic equation becomes:

$$ 1 + \frac{Ke^{-sτ}}{s + a} = 0 $$

Applying the phase condition at s = jω:

$$ -\tan^{-1}\left(\frac{ω}{a}\right) - ωτ = (2k + 1)π $$

This transcendental equation must be solved numerically to determine stability boundaries.

Approximation Methods

For small delays (τ ≈ 0), a Padé approximation can linearize the exponential term:

$$ e^{-sτ} ≈ \frac{1 - sτ/2}{1 + sτ/2} $$

This converts the system into a rational transfer function, allowing conventional root locus techniques. However, this approximation becomes inaccurate for larger delays or higher frequencies.

Numerical Computation

Modern tools (MATLAB, Python) use numerical methods to trace the root locus for delayed systems:

Root locus plot for a delayed system showing infinite branches Re Im −a
Root Locus for Delayed System Root locus plot showing infinite branches diverging from a real pole and approaching vertical asymptotes in the complex plane. Re Im 0 -a Asymptotes jω₁ -jω₁
Diagram Description: The diagram would show the infinite branches of the root locus for a delayed system, illustrating their asymptotic behavior and periodic distribution along the imaginary axis.

Root Locus in Digital Control Systems

Discrete-Time Root Locus Fundamentals

The root locus method in digital control systems extends the continuous-time analysis to discrete-time systems by examining the closed-loop pole locations in the z-plane. The characteristic equation for a digital control system with open-loop transfer function G(z) and compensator K(z) is given by:

$$ 1 + K(z)G(z) = 0 $$

Unlike the s-plane, where poles move along continuous trajectories, the z-plane root locus must account for the sampling period T and the mapping z = esT. Stability is now determined by whether poles lie inside the unit circle (|z| < 1).

Constructing the Root Locus in the z-Plane

The root locus rules for continuous systems largely apply, but with key adaptations:

Effect of Sampling on Root Locus

Sampling introduces aliasing and frequency folding, which distort the root locus. The primary and complementary root loci must be analyzed due to the periodic nature of z = esT. Higher sampling rates reduce distortion but increase computational load.

$$ z = e^{sT} = e^{(σ + jω)T} $$

For a stable system, σ < 0 ensures |z| < 1. The Nyquist criterion must also be considered to avoid aliasing effects.

Practical Design Considerations

Digital root locus is used in:

Case Study: Discrete PD Controller

Consider a system with open-loop transfer function:

$$ G(z) = \frac{z + 0.5}{(z - 0.2)(z - 0.8)} $$

Adding a PD compensator K(z) = K(z - 0.6), the root locus shows:

This demonstrates how digital compensators shape the root locus while respecting z-plane constraints.

s-plane to z-plane mapping with root locus A diagram illustrating the mapping between the s-plane and z-plane, showing pole trajectories and stability regions. σ jω Stable s₁ s₂ s₃ s-plane |z|=1 Stable z₁ = e^(s₁T) z₂ = e^(s₂T) z₃ = e^(s₃T) z-plane z = e^(sT)
Diagram Description: The section discusses the mapping between s-plane and z-plane, which is inherently spatial and requires visualization of pole movements and the unit circle.

4.3 Compensator Design Using Root Locus

The root locus method provides a systematic approach to designing compensators that reshape the closed-loop pole trajectories to meet desired transient and steady-state performance criteria. Compensators are typically classified as lead, lag, or lead-lag networks, each serving distinct purposes in control system tuning.

Lead Compensator Design

A lead compensator introduces a pole-zero pair to improve transient response by increasing system damping and bandwidth. Its transfer function is given by:

$$ G_c(s) = K_c \frac{s + z}{s + p}, \quad p > z $$

The compensator's phase contribution is maximized at the geometric mean of the pole and zero:

$$ \omega_m = \sqrt{z \cdot p} $$

Design steps:

Root locus showing original poles (×), compensator zero (○), and shifted closed-loop poles (●).

Lag Compensator Design

Lag compensators improve steady-state accuracy without significantly altering transient response. Their transfer function is:

$$ G_c(s) = K_c \frac{s + z}{s + p}, \quad z > p $$

Key considerations:

Lead-Lag Compensation

Combining both strategies addresses multiple performance requirements:

$$ G_c(s) = K_c \frac{(s + z_{lead})(s + z_{lag})}{(s + p_{lead})(s + p_{lag})} $$

Practical implementation often involves:

Real-World Constraints

Physical compensator realization must account for:

$$ \text{Sensitivity} \quad S_p^G = \frac{p}{G} \frac{\partial G}{\partial p} $$

Case studies in aerospace control systems demonstrate compensator designs where root locus methods reduced settling time by 40% while maintaining < 5% overshoot.

Root Locus with Compensator Poles/Zeros Root locus plot showing original poles (×), compensator zero (○), and shifted closed-loop poles (●) with root locus branches, damping ratio line, and natural frequency circle. σ jω Original Pole Compensator Zero Shifted Pole ζ ωₙ
Diagram Description: The diagram would show the root locus plot with original poles, compensator zero, and shifted closed-loop poles to visualize the pole movement and compensator effect.

5. Recommended Textbooks on Control Systems

5.1 Recommended Textbooks on Control Systems

5.2 Research Papers on Root Locus Techniques

5.3 Online Resources and Tutorials