The system can also be checked for general stability when controller parameters are varied using root-locus plots.
In a engineered system we may typically have one or more design parameters, adjustments, or user settings. It is important to determine if any of these will make the system unstable. This is generally undesirable and possibly unsafe. For example, think of a washing machine that vibrates so much that it `walks' across a floor, or a high speed aircraft that fails due to resonant vibrations. Root-locus plots are used to plot the system roots over the range of a variable to determine if the system will become unstable, or oscillate.
Recall the general solution to a homogeneous differential equation. Complex roots will result in a sinusoidal oscillation. If the roots are real the result will be e-to-the-t terms. If the real roots are negative then the terms will tend to decay to zero and be stable, while positive roots will result in terms that grow exponentially and become unstable. Consider the roots of a second-order homogeneous differential equation, as shown in Figure 12.1 Negative real roots make a system stable to Figure 12.7 Complex roots with positive real parts have growing oscillations and are unstable. These roots are shown on the complex planes on the left, and a time response is shown to the right. Notice that in these figures (negative real) roots on the left hand side of the complex plane cause the response to decrease while roots on the right hand side cause it to increase. The rule is that any roots on the right hand side of the plane make a system unstable. Also note that the complex roots cause some amount of oscillation.
Figure 12.1 Negative real roots make a system stable
Figure 12.2 Complex roots make a system oscillate
Figure 12.3 Negative real and complex roots cause decaying oscillation
Figure 12.4 More negative real and complex roots cause a faster decaying oscillation
Figure 12.5 Overlapped roots are possible
Figure 12.6 Positive real roots cause exponential growth and are unstable
Figure 12.7 Complex roots with positive real parts have growing oscillations and are unstable
Next, recall that the denominator of a transfer function is the homogeneous equation. By analyzing the function in the denominator of a transfer function the general system response can be found. An example of root-locus analysis for a mass-spring-damper system is given in Figure 12.8 A mass-spring-damper system equation. In this example the transfer function is found and the roots of the equation are written with the quadratic equation. At this point there are three unspecified values that can be manipulated to change the roots. The mass and damper values are fixed, and the spring value will be varied. The range of values for the spring coefficient should be determined by practical and design limitations. For example, the spring coefficient should not be zero or negative.
Figure 12.8 A mass-spring-damper system equation
The roots of the equation can then be plotted to provide a root locus diagram. These will show how the values of the roots change as the design parameter is varied. If any of these roots pass into the right hand plane we will know that the system is unstable. In addition complex roots will indicate oscillation.
Figure 12.9 Drill problem: Plot the calculated roots on the axes above
A feedback controller with a variable control function gain is shown in Figure 12.10 Root-locus analysis in controller design. The variable gain 'K' necessitates the evaluation of controller stability over the range of operating values. This analysis begins by developing a transfer function for the overall system. The root of the denominator is then calculated and plotted for a range of 'K' values. In this case all of the roots are on the left side of the plane, so the system is stable and doesn't oscillate. Keep in mind that gain values near zero put the control system close to the right hand plane. In real terms this will mean that the controller becomes unresponsive, and the system can go where it pleases. It would be advisable to keep the system gain greater than zero to avoid this region.
Figure 12.10 Root-locus analysis in controller design
Figure 12.11 Root-locus plotting in Scilab
Figure 12.12 Drill problem: Complete the root-locus analysis
Figure 12.13 Drill problem: Draw a root locus plot
· Root-locus plots show the roots of a transfer function denominator to determine stability
1. Draw the root locus diagram for the system below. specify all points and values.
2. The block diagram below is for a motor position control system. The system has a proportional controller with a variable gain K.
a) Simplify the block diagram to a single transfer function.
b) Draw the Root-Locus diagram for the system (as K varies). Use either the approximate or exact techniques.
c) Select a K value that will result in an overall damping factor of 1. State if the Root-Locus diagram shows that the system is stable for the chosen K.
3. Given the system transfer function below.
a) Draw the root locus diagram and state what values of K are acceptable.
b) Select a gain value for K that has either a damping factor of 0.707 or a natural frequency of 3 rad/sec.
c) Given a gain of K=10 find the steady-state response to an input step of 1 rad.
d) Given a gain of K=0.01 find the response of the system to an input step of 0.1rad.
4. A feedback control system is shown below. The system incorporates a PID controller. The closed loop transfer function is given.
a) Verify the close loop controller function given.
b) Draw a root locus plot for the controller if Kp=1 and Ki=1. Identify any values of Kd that would leave the system unstable.
c) Draw a Bode plot for the feedback system if Kd=Kp=Ki=1.
d) Select controller values that will result in a natural frequency of 2 rad/sec and damping factor of 0.5. Verify that the controller will be stable.
e) For the parameters found in the last step can the initial values be found?
f) If the values of Kd=1 and Ki=Kp=0, find the response to a unit ramp input as a function of time.
5. Draw a root locus plot for the control system below and determine acceptable values of K, including critical points.
6. The feedback loop below is for controlling a DC motor with a PID controller.
a) Find the transfer function for the system.
b) Draw a root locus diagram for the variable parameter `P'.
c) Find the response of the system in to a unit step input using explicit integration.
1. The systems below have a variable spring coefficient. For each of the systems below,
a) Write the differential equation and convert it to a transfer function.
b) If the input force is a step function of magnitude 1N, calculate the time response for `y' by solving a differential equation for a Ks value of 10N/m.
c) Draw the poles for the transfer function on a real-complex plane.
d) Draw a Bode plot for Ks = 1N/m.
2. Draw a root locus diagram for the feedback system below given the variable parameter `P'.
3. For the transfer functions below, draw the root locus plots assuming there is unity feedback, i.e., H(D) = 1. Draw an approximate time response for each for a step input.
4. Draw a root-locus plot for the following feedback control systems.