The closed-loop pole trajectories as a function of the feedback gain k are returned by the root locus (assuming negative feedback). The impact of changing feedback gains on closed-loop pole placements are studied using root loci. On the root locus plot, the poles are marked by x, while the zeros are denoted by o. The closer the poles and zeros are to the right and left sides of the complex plane respectively, the more stable the system is.
To determine the root locus, first define an initial estimate for the roots: e.g., 10 points uniformly distributed over the unit circle. This estimate can be improved upon by using more points (e.g., 100), but this increases computation time significantly. Next, compute the inverse discrete Fourier transform (IDFT) of the system transfer function H(z). Multiply the result by the initial estimate of the roots, and then sum across the frequency axis.
The root locus depicts the poles of the closed loop transfer function as a function of a gain parameter in the complex s-plane (see pole-zero plot). A graphical approach that employed a special protractor called a "spirule" to estimate angles and depict the root loci was formerly used. Modern software can compute the root locus automatically.
Pole placement is important in control theory because it determines what type of control system we need. If the zero is placed too close to the origin, then the system will be underdamped and have poor stability. On the other hand, if the zero is placed too far from the origin, then the system will be overdamped and not respond quickly enough to transient signals. The optimal position depends on the application.
Closed-loop systems are always stable but may have different performance characteristics depending on the position of their poles and zeros. An unstable system would have at least one pole within the unit circle (or else it would be a stable system with no damping), while a perfectly stable system would have all its poles outside the unit circle (or else it would be an overdamped system with infinite gain).
Open-loop systems are usually less efficient than closed-loop systems but they do not require feedback for operation. An open-loop system generates forces that try to make the object being controlled move toward a desired position, but it cannot correct its own errors.
The root-locus map clearly displays how each open-loop pole and zero contribute to the position of the closed-loop poles. The root-locus graphic also demonstrates how to modify the open-loop poles and zeros such that the response matches system performance standards. The closer the curve follows the vertical line, the better the match will be between the actual behavior of the system and its model.
In this example, the root locus plot shows that there is one unstable pole and two stable poles. This means that if we only change the gain on the feedback path, the system will oscillate between following the input signal or resisting it. To make the system more stable, we need to change both the gain on the feedback path and the open-loop gain so that they produce less oscillation.
The root locus plot also indicates that there are three zeros in this system: one stable zero near the origin and two unstable zeros outside the unit circle. This means that if we only change the gain on the feedforward path, the system will follow the input signal until it reaches one of the zeros, at which point it will stop.
To make the system more stable, we need to change the gain on both paths so that there are fewer extremes in behavior. We could try increasing the open-loop gain but that would just increase the rate of convergence to steady state.
The root locus approach should yield a graph indicating where the system's poles are for all values of K. The system is unstable when any or all of D's roots are in the unstable area. The system is barely stable if any of the roots are in the slightly stable area (oscillatory). It is perfectly stable if all of its roots are in the strongly stable area (real).
Root loci can also be used to determine how much feedback a system needs in order to have an equilibrium point. If you draw a horizontal line from each root of D back to the origin, then the number of lines crossing the imaginary axis determines the system's degree of instability (or stability). A system that requires more negative feedback than positive will always have a stable equilibrium point; one that requires equal amounts of both types of feedback will be either stable or unstable depending on the specifics of D.
Finally, root loci can be used to determine whether a system has any oscillations. If any of the roots cross over to the other side of the plane, then there are oscillations associated with this system. Oscillations are most likely if several roots cross into the unstable region at once. When this happens, the system will periodically change direction, causing it to oscillate between stable and unstable regions.
These examples should give you an idea of how root loci can be used to analyze stability and determine other properties of systems of differential equations.
A root locus diagram is a figure that depicts how the eigenvalues of a linear (or linearized) system vary in response to a single parameter (usually the loop gain). Root locus diagrams are briefly discussed in Chapter 4: Dynamic Behavior.
If the angle of the open loop transfer function at a given position is an odd multiple of 1800, the point is on the root locus. If there are an odd number of open loop poles and zeros to the left of a real axis point, that point is on the root locus branch. Otherwise, it's on the other branch.
To find the root locus poles and zeros, first determine if the open-loop gain is less than or greater than 1. If it's less than 1, the root locus is inside the unit circle. If it's greater than 1, the root locus is outside the unit circle.
Next, consider each pole and zero of the open loop separately. If the angle between any two poles or zeros is an even multiple of 180 degrees, they're on the same branch of the root locus. Otherwise, they're on different branches.
Using this information, draw a diagram to help visualize the root locus. You can use either poles or zeros as the determining factor for its branches. If the open loop gain is less than 1, all roots are inside the unit circle. If it's greater than 1, some roots are outside the unit circle while others are inside it.
Finally, use numerical methods to find the root with the largest magnitude. This will be one of the poles of the root locus.
Use these guidelines to build a root locus.
There is one primary root in the taproot system from which lateral root branches grow. The main cylindrical root, also known as the primary root, is the first root to emerge from the radicle after germination. It develops vertically downward. The secondary roots are produced from the primary root and they too develop vertically downward. The tertiary roots are produced from the secondary roots and so on. When there are no more secondary roots to produce, then the plant has reached maturity and stopped growing.
The taproot is the central root of a plant that extends down into the soil. It is usually thick at the bottom where it contacts the soil and becomes thinner toward the top. The primary purpose of the taproot is to absorb any moisture and nutrients that are below ground level. It does this by extending down into the soil and connecting with these substances through its network of vasculature (the xylem and phloem). The taproot is responsible for bringing water and minerals up from deep within the soil and distributing them throughout the plant.
Without a strong taproot, many plants would not be able to reach such great heights or spread out over such large areas. The photo above shows carrots with thin, stubby taproots. These plants would not be able to withstand drought like their taller counterparts that have developed deeper, wider taproots.