Connections to Controls Concepts: Balance Arm
Prof. R. H. Bishop
University of South Florida
The self-balancing robot assembly provides insight into two key feedback control system concepts: (i) relative stability
and robust stability of an unstable open-loop system and (ii) fundamental controller design tension in the presence of
plant changes, external disturbances, and measurement noise.
Plant
Self-balancing robot
Sensor
Accelerometer, gyroscope, and encoder sensors
Actuator
DC servo motors
Performance
Robust stability, disturbance rejection, and measurement noise attenuation
Design objectives
Tune the control system by adjusting PD gain constants & stabilize the robot at the
vertical
Reference inputs
Regulation to zero angle (vertical pose)
The self-balancing robot and two geared, DC motor-driven wheels are the plant and actuators, respectively. The sensor
suite includes a gyroscope that measures angular velocity and an accelerometer that measures non-gravitational
acceleration. Since the accelerometer and gyro are known to have systematic errors (bias, drifts, etc.) and are noisy, the
signals from the sensors are combined to produce an accurate measurement of the orientation angle of the robot—
more accurate than can be achieved with either sensor individually. The external reference commands enter the
control system via the host computer either through keyboard entry or front panel commands sent over the network
connection to the myRIO which hosts the PD controller code. The design goal is the balance the robot in an upright
pose in the presence of external disturbances and to be robustly stable to plant variations (e.g., remain stable even
when a large mass is placed on the robot). The controller is a PD controller.
Relative stability and robust stability
A closed-loop feedback system is either stable or not stable. We say that a feedback control system that displays a
bounded response to a bounded input is stable. Note that this means that the closed-loop feedback system must have
a bounded output to every bounded input. This is known as absolute stability. We might wonder about marginally
stable closed-loop systems wherein the response remains bounded, but does not decay with time. Typically, our
design specifications require the closed-loop feedback system tracking error response, represented in the frequency
domain as E(s)=Y(s)-R(s), to decay to zero—not just remain bounded—and this is the focus of this discussion.
Obviously, it is not practical to test the response to every bounded input; however, a necessary and sufficient
condition for the closedloop feedback system to be stable is that all the poles of the closed-loop transfer function lie in
the left-half s-plane—this is a primary requirement of the controller design process. Once we have expertly designed
the controller so that the closed-loop feedback system is stable, we can further quantify the degree of stability,
often referred to as relative stability. So, a closedloop feedback system can be more (or less stable) with a given
controller design than the same system with a different controller design. This all relates to the concept of a robust
feedback control system. That is, a robust feedback control system is one that maintains acceptable performance in
the presence of plant uncertainty, external disturbances, and measurement noise. From the point of view of relative
stability, a robust controller typically displays a greater relative stability than a controller with less ability to retain
closed-loop stability in the presence of plant uncertainty, external disturbances, and measurement noise. Consider
the PD controller for the self-balancing robot that allows the robot to remain stable even when a large load (such as a
book) is placed on the top thereby significantly changing the plant or when an external gentle push is applied. A well-
designed PID controller can be very robust and explains why they are utilized in industry to such a large extent. Note
that the PD controller is a PID controller with the integral gain set to zero.
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Self Balancing Robot Assembly
Summary of Contents for 41427
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