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The mystery of root locus analysis: How to graphically explore the pole movement of a system?
In the field of control theory and stability analysis, root locus analysis is a graphical method that aims to explore the root of a system as a function of a certain system parameter (usually the gain in a feedback system) changes. change. This technique is derived from the classical control theory developed by Walter R. Evans and can effectively determine the stability of the system.
The root locus plot shows the variation of the poles of the closed-loop transfer function on the complex s-plane.
The root locus can not only be used to determine the stability of the system, but also help design the damping ratio (ζ) and natural frequency (ωn) of the feedback system. By drawing straight lines of fixed damping ratio, radiating from the origin, and arcs of fixed natural frequency radiating from the origin, a point can be selected to determine the required system gain K. In this way, the designer can approach the required stability and dynamic performance, which is discussed in detail in various control textbooks.
The definition of the root locus is the graphical representation of the closed-loop poles of the system on the complex s-plane under varying specific parameter values.
Overall, the root locus analyzer enables control engineers to graphically identify and predict the behavior of a system. The root locus method is particularly effective when the designed feedback system has obvious dominant pole pairs. In real applications, many systems may not fully meet this assumption. Therefore, it is important to perform simulation verification after completing the design to ensure that actual requirements are met.
Principles of Root Locus Analysis
The operating principle of root locus analysis is based on the angular and amplitude conditions of the instrument. If there is a feedback system with input signal X(s) and output signal Y(s), then the forward path transfer function can be expressed as G (s), and the feedback path transfer function is H(s). The closed-loop transfer function is then T(s) = Y(s) / X(s) = G(s) / (1 + G(s)H(s)).
This means that the closed-loop poles with respect to the roots of the characteristic equation are
1 + G(s)H(s) = 0.
Of course, when there is no pure delay in the system, the product of G(s)H(s) can be expressed in the form of a rational polynomial. Through this analysis, combined with vector techniques to calculate the angles of the poles and zeros, we can gain insight into the behavior and dynamics of the system.
Drawing of root loci
When plotting the root locus, you first need to mark the poles and zeros of the open loop and mark the portion of the real axis to the left of all the poles and zeros. Further analysis shows that when the number of poles P is subtracted from the number of zeros Z, we get an asymptote of quantity P-Z. This asymptote will intersect the real axis at the center of gravity, and the outward angle can be calculated by the following formula:
φ_l = 180° + (l - 1) * 360° / (P - Z),α = Re(ΣP - ΣZ) / (P - Z)
In addition, the phase of the test point needs to be confirmed to find the departure angle and entry point. These processes fully demonstrate the power and application potential of the root locus method, and lead us to explore the stability of the system more deeply.
Conclusion
The plotting and analysis of root loci allow control system engineers to extract key information from complex calculations. This is not only a theoretical discussion, but also an essential skill in practice. In the face of future technological challenges, can root locus analysis help us uncover deeper mysteries of system dynamics?
