Root Locus Analysis System G(s)H(s) Circle Center Derivation
Introduction to Root Locus and System Analysis
In control systems engineering, root locus is a graphical representation illustrating how the poles of a closed-loop system migrate in the complex s-plane as a system parameter, typically the gain K, varies. This powerful tool provides vital insights into the stability and performance characteristics of a feedback control system. Understanding the root locus plot enables engineers to design and fine-tune systems to meet specific performance requirements, such as stability, settling time, and overshoot. The root locus method, introduced by Walter R. Evans, remains a cornerstone in control systems analysis and design. In this context, we delve into a system characterized by the open-loop transfer function G(s)H(s) = K(s+3)/(s(s+2)) to demonstrate that a portion of its root locus forms a circle with a center at (-3, 0) and a radius of √3. We will explore the theoretical underpinnings and the step-by-step derivation to reach this conclusion, showcasing the practical application of root locus techniques in system analysis.
The root locus plot visually maps the trajectory of the closed-loop poles as the gain K varies from 0 to infinity. The closed-loop poles are the roots of the characteristic equation, which is given by 1 + G(s)H(s) = 0. By analyzing the root locus, engineers can determine the range of gain values for which the system remains stable and meets desired performance criteria. Key parameters that can be inferred from the root locus include the damping ratio (ζ), natural frequency (ωn), and settling time. The shape and location of the root locus branches reveal how the system’s transient response will change with varying gain. For example, if the root locus branches move towards the right-half plane as K increases, it indicates that the system's stability margin is decreasing. Therefore, understanding and interpreting root locus plots is crucial for effective control system design and analysis. The following sections will demonstrate the specific analysis for the given transfer function, providing a detailed explanation of how the circular root locus is derived.
Understanding the behavior of closed-loop poles is essential for designing stable and high-performing control systems. The characteristic equation, 1 + G(s)H(s) = 0, plays a central role in determining these poles. The roots of this equation are the closed-loop poles, and their location in the complex s-plane dictates the system’s stability and transient response. The root locus plot is a graphical representation of these roots as the gain K varies, offering valuable insights into how the system's behavior changes. In our case, G(s)H(s) = K(s+3)/(s(s+2)), and we will show that part of the root locus is a circle. This demonstration involves algebraic manipulation and geometric interpretation, highlighting the interplay between pole-zero configurations and the resulting root locus shape. By understanding these principles, engineers can predict and control system performance, making informed design decisions to achieve desired outcomes.
Problem Statement and Approach
The objective is to demonstrate that the part of the root locus for a system with an open-loop transfer function G(s)H(s) = K(s+3)/(s(s+2)) forms a circle. Specifically, we aim to prove that this circle has its center at (-3, 0) and a radius of √3. To achieve this, we will use the magnitude and angle conditions derived from the characteristic equation. The magnitude condition provides a relationship between the gain K and the pole-zero locations, while the angle condition dictates the permissible angles for points on the root locus. By manipulating these conditions, we will derive an equation representing the root locus, which will then be shown to be the equation of a circle. This approach combines algebraic techniques with geometric interpretations, allowing us to visualize and understand the behavior of the closed-loop poles as the gain K varies.
The methodology involves first setting up the characteristic equation for the given system, which is 1 + G(s)H(s) = 0. Substituting G(s)H(s) = K(s+3)/(s(s+2)) into this equation, we obtain 1 + K(s+3)/(s(s+2)) = 0. Next, we will manipulate this equation to isolate the gain K and apply the magnitude condition. The angle condition, which states that the sum of the angles from the open-loop poles and zeros to any point on the root locus must be an odd multiple of 180 degrees, will also be crucial. By expressing a point on the root locus as s = x + jy, where x and y are real and imaginary components respectively, we can rewrite the characteristic equation in terms of x, y, and K. The subsequent steps involve algebraic simplification and rearrangement to arrive at an equation that resembles the standard form of a circle. This equation will clearly show the center and radius of the circular portion of the root locus, thereby completing the proof.
Derivation of the Root Locus Equation
To begin, we set up the characteristic equation for the given system. The characteristic equation is defined as 1 + G(s)H(s) = 0. Substituting the open-loop transfer function G(s)H(s) = K(s+3)/(s(s+2)), we get:
1 + K(s+3) / (s(s+2)) = 0
Multiplying through by s(s+2) to clear the denominator, we have:
s(s+2) + K(s+3) = 0
Expanding and rearranging the terms, we obtain:
s^2 + 2s + Ks + 3K = 0
s^2 + (2+K)s + 3K = 0
Now, let's express the complex variable s as s = x + jy, where x and y are the real and imaginary parts, respectively. Substituting s = x + jy into the equation, we get:
(x + jy)^2 + (2+K)(x + jy) + 3K = 0
Expanding the terms, we have:
(x^2 + 2jxy - y^2) + (2x + 2jy + Kx + Kjy) + 3K = 0
Grouping the real and imaginary parts, we get:
(x^2 - y^2 + 2x + Kx + 3K) + j(2xy + 2y + Ky) = 0
For this complex equation to hold, both the real and imaginary parts must be equal to zero. Therefore, we have two equations:
Real part: x^2 - y^2 + 2x + Kx + 3K = 0
Imaginary part: 2xy + 2y + Ky = 0
From the imaginary part equation, we can factor out y:
y(2x + 2 + K) = 0
This gives us two possibilities: y = 0 or 2x + 2 + K = 0. The case y = 0 represents the root locus on the real axis, which we will address later. For now, we focus on the second case, where:
2x + 2 + K = 0
Solving for K, we get:
K = -2x - 2
Now, we substitute this expression for K into the real part equation:
x^2 - y^2 + 2x + (-2x - 2)x + 3(-2x - 2) = 0
Expanding and simplifying, we have:
x^2 - y^2 + 2x - 2x^2 - 2x - 6x - 6 = 0
-x^2 - y^2 - 6x - 6 = 0
Multiplying by -1, we get:
x^2 + y^2 + 6x + 6 = 0
Proving the Circular Root Locus
Now that we have the equation x^2 + y^2 + 6x + 6 = 0, we can rewrite it in the standard form of a circle equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. To do this, we complete the square for the x terms:
(x^2 + 6x) + y^2 + 6 = 0
To complete the square for x^2 + 6x, we need to add and subtract (6/2)^2 = 9:
(x^2 + 6x + 9) + y^2 + 6 - 9 = 0
Now we can rewrite the equation as:
(x + 3)^2 + y^2 - 3 = 0
Rearranging the terms, we get:
(x + 3)^2 + y^2 = 3
This equation is in the standard form of a circle equation, (x - h)^2 + (y - k)^2 = r^2. Comparing our equation with the standard form, we can identify the center and radius:
Center: (h, k) = (-3, 0)
Radius: r = √3
Therefore, we have shown that a part of the root locus of the system with G(s)H(s) = K(s+3)/(s(s+2)) is indeed a circle with a center at (-3, 0) and a radius of √3. This confirms the circular nature of a segment of the root locus, which arises from the specific pole-zero configuration of the open-loop transfer function. This geometric insight helps in understanding the system's behavior as the gain K varies.
Analysis of Root Locus on the Real Axis
In the previous sections, we focused on deriving the circular portion of the root locus. However, to fully understand the root locus plot, it is essential to analyze the segments that lie on the real axis. Recall that the imaginary part equation y(2x + 2 + K) = 0 gave us two possibilities: y = 0 or 2x + 2 + K = 0. We have already explored the case where 2x + 2 + K = 0, which led to the circular root locus. Now, let's consider the case where y = 0. This condition corresponds to the root locus segments lying on the real axis of the complex s-plane.
When y = 0, the characteristic equation s^2 + (2+K)s + 3K = 0 becomes a quadratic equation in terms of s, where s is a real number. To determine the regions on the real axis that are part of the root locus, we use the angle condition. The angle condition states that for a point s on the root locus, the sum of the angles from the open-loop poles and zeros to that point must be an odd multiple of 180 degrees. For the given open-loop transfer function G(s)H(s) = K(s+3)/(s(s+2)), the open-loop poles are at s = 0 and s = -2, and the open-loop zero is at s = -3.
Consider a test point on the real axis to the left of the zero at s = -3. The angle contribution from the zero at s = -3 is 0 degrees. The angles from the poles at s = 0 and s = -2 are both 180 degrees. The sum of these angles is 0 + 180 + 180 = 360 degrees, which is an even multiple of 180 degrees. Therefore, the real axis to the left of s = -3 is not part of the root locus.
Now, consider a test point on the real axis between the pole at s = -2 and the zero at s = -3. The angle contribution from the zero at s = -3 is 0 degrees. The angle from the pole at s = 0 is 180 degrees, and the angle from the pole at s = -2 is also 180 degrees. However, since the test point is to the right of the pole at s = -2, this angle contribution becomes 0 degrees. The sum of these angles is 0 + 180 + 0 = 180 degrees, which is an odd multiple of 180 degrees. Thus, the real axis segment between s = -3 and s = -2 is part of the root locus.
Finally, consider a test point on the real axis between the poles at s = 0 and s = -2. The angle contribution from the zero at s = -3 is 180 degrees. The angle from the pole at s = 0 is 180 degrees, and the angle from the pole at s = -2 is 0 degrees. The sum of these angles is 180 + 180 + 0 = 360 degrees, which is an even multiple of 180 degrees. Therefore, the real axis segment between s = -2 and s = 0 is not part of the root locus.
To summarize, the root locus on the real axis consists of the segment between s = -3 and s = -2. This segment starts at the open-loop zero at s = -3 and terminates at the open-loop pole at s = -2. This real-axis segment, combined with the circular portion we derived earlier, provides a complete picture of the root locus for the given system. Understanding the behavior of the root locus on the real axis is crucial for determining the stability and performance characteristics of the closed-loop system.
Synthesis of the Complete Root Locus Plot
Having derived the circular portion and analyzed the segments on the real axis, we can now synthesize the complete root locus plot for the system with G(s)H(s) = K(s+3)/(s(s+2)). The root locus plot provides a graphical representation of the possible locations of the closed-loop poles as the gain K varies from 0 to infinity. The complete root locus plot consists of the following components:
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Circular Segment: We have shown that a part of the root locus is a circle with its center at (-3, 0) and a radius of √3. This circle is a significant feature of the root locus, indicating a particular behavior of the closed-loop poles as K varies.
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Real Axis Segment: The root locus includes the segment on the real axis between the zero at s = -3 and the pole at s = -2. This segment represents the movement of closed-loop poles along the real axis for certain values of K.
To sketch the root locus, we start by plotting the open-loop poles and zeros on the complex s-plane. The open-loop poles are at s = 0 and s = -2, and the open-loop zero is at s = -3. The root locus starts at the open-loop poles (K = 0) and terminates at the open-loop zeros (K = ∞) or at infinity. The branches of the root locus are symmetrical about the real axis.
The segment of the root locus on the real axis lies between s = -3 and s = -2. As K increases from 0, one branch of the root locus starts at the pole s = -2 and moves towards the zero at s = -3. Another branch starts at the pole s = 0. These two branches will eventually meet and break away from the real axis to form the circular root locus. The breakaway point can be found by determining where the gain K is maximized along the real axis between s = -2 and s = 0. However, for the purpose of this discussion, we focus on sketching the general shape of the root locus.
The breakaway point is the point where the two branches coming from the poles at 0 and -2 meet and depart from the real axis. The two branches then form the circular root locus we derived, centered at (-3, 0) with a radius of √3. As K increases further, the closed-loop poles move along this circular path. This circular locus indicates that the system's damping ratio and natural frequency change in a specific manner as K varies.
The final step in synthesizing the root locus plot is to consider the behavior as K approaches infinity. As K approaches infinity, the root locus branches terminate at the open-loop zeros or at infinity. In this case, one branch terminates at the zero at s = -3, and the other branch moves along the circular path. The circular path eventually intersects the real axis, indicating that the closed-loop poles can become complex conjugates, leading to oscillatory behavior in the system's response.
By combining the circular segment and the real axis segment, we obtain a complete picture of the root locus. This graphical tool provides valuable insights into the stability and performance characteristics of the closed-loop system as the gain K is varied. The root locus plot allows engineers to select appropriate gain values to achieve desired system behavior, such as stability, settling time, and overshoot.
Conclusion: Significance of Root Locus in Control Systems
In this comprehensive analysis, we have demonstrated that a portion of the root locus for a system with the open-loop transfer function G(s)H(s) = K(s+3)/(s(s+2)) forms a circle. Specifically, this circle is centered at (-3, 0) with a radius of √3. The derivation involved setting up the characteristic equation, expressing the complex variable s as s = x + jy, and using the magnitude and angle conditions to arrive at the equation of the root locus. By completing the square, we transformed the equation into the standard form of a circle, thereby proving the circular nature of a segment of the root locus.
Furthermore, we analyzed the segments of the root locus that lie on the real axis. We found that the root locus includes the segment between the open-loop zero at s = -3 and the open-loop pole at s = -2. This analysis, combined with the derivation of the circular portion, allowed us to synthesize a complete root locus plot for the system. The complete root locus plot provides a graphical representation of the possible locations of the closed-loop poles as the gain K varies from 0 to infinity.
The root locus technique is a powerful tool in control systems engineering, offering valuable insights into the stability and performance characteristics of feedback control systems. By analyzing the root locus, engineers can determine the range of gain values for which the system remains stable and meets desired performance criteria. The root locus plot visually maps the trajectory of the closed-loop poles as the gain K varies, enabling engineers to understand how the system's transient response changes with varying gain. This understanding is crucial for designing and fine-tuning control systems to meet specific requirements.
The significance of the root locus method extends beyond theoretical analysis. It has practical applications in the design and implementation of control systems across various industries, including aerospace, robotics, and process control. By leveraging the root locus, engineers can optimize system performance, ensuring stability, minimizing overshoot, and achieving desired settling times. The ability to visualize the behavior of closed-loop poles as system parameters change is invaluable in the design process, making the root locus a cornerstone of control systems engineering.
In conclusion, the detailed derivation and analysis presented in this article underscore the importance of the root locus technique in understanding and designing control systems. The circular root locus observed in this specific system exemplifies the rich insights that can be gained through root locus analysis, highlighting its continued relevance and utility in modern control engineering practices. This method allows for a systematic approach to designing robust and effective control systems, making it an indispensable tool for engineers in the field.
