This chapter has presented a general introduction to feedback control design by means of root locus analysis and has applied this technique to the digital control of the velocity of a DC motor. The principal ideas include the following:
A block diagram describes the structure of the feedback control system, showing the dynamic relationships between the controller, the DC motor plant to be controlled, and the feedback path that defines the error signal.
The controller design problem is defined by requirements for both the transient and the steady state performance of the closed-loop system. Specific measures of performance, such as settling time and overshoot, are derived for systems with dominant poles of second order. Stability and robustness with respect to system parameter variations are also discussed.
Root locus analysis relates the transfer function of the open-loop system to the locations of the closed-loop poles as the system gain varies from 0 to \(\infty\). As such, the root locus diagram is the basis of a general control design method. An example in this chapter illustrates the use of root locus analysis to select the gain of a proportional controller to meet a percent overshoot specification.
The proportional controller design is extended to proportional-integral (PI) control through a consideration of steady-state error. Root locus analysis is used to design a PI controller meeting a specific steady-state error requirement.
In this book, the approach to digital control design uses digital emulation, in which continuous system design methods are used to develop the controller transfer function, which is then discretized using Tustin's method.
The sample rate is a key consideration. Although the Nyquist-Shannon sampling theorem sets a lower bound on sample rate based on the system bandwidth, it is frequently far too low to adequately represent a control system subject to rapid transient response. Using the rise time corresponding to the system dominant poles at \(\omega_n\) as a guide, a rule of thumb is to select the sample frequency in the range \(\omega_s\in[20\omega_n,60\omega_n]\).
In digital control, the time between sensing the output value and updating the control actuation is called the time delay. Generally, delays can occur anywhere in the control loop and can result from the finite time necessary for computation, sensing, control actuation, and command execution. Time delays decrease system stability (loss of phase margin) and should be minimized.
Both voltage and current amplifiers may be used to control DC motors. However, current amplifier designs offer improved transient response.
The lab exercise will organize elements from previous chapters into a multithread program in which one thread exchanges set point and control parameters with the user, while the other seamlessly controls the motor/load velocity.