11.2.1. Error Response

This video introduces the error response for a controlled system and characterizes the error response in terms of its steady-state error and its transient response (overshoot and settling time).

In the previous video, we saw that the controller compares the desired behavior to the actual behavior to produce its control signals. If the control objective is motion control, then the desired behavior is given by the desired motion, theta_d of t. This is also called the reference input. The actual motion is theta of t. We define the error to be theta_e equal to theta_d minus theta. The error dynamics are the equations that describe the evolution of theta_e of the controlled system. A good controller would create error dynamics that drive any initial error to zero, or nearly zero, as quickly as possible. At least the controller should be stable, meaning that initial errors do not grow.

To measure the performance of a controller, let's focus on a robot with a single joint, since the ideas generalize easily. Let's define the unit step error response as the evolution of the error when the initial error is 1. As an example, imagine that the desired angle of your elbow joint is zero, and the actual angle matches it exactly. Then suddenly you request a constant joint angle of 1 radian. At that instant, which I'll call time zero, the error is 1 radian. If the controller is a good one, over time it should reduce the error.

Here is a plot of a typical error response. The controller succeeds in decreasing the initial error, but never eliminates it completely. As time grows large, the error becomes constant, and we define e_ss to be the steady-state error. We can also see that the error response overshoots its steady-state value before settling. Finally, we can judge how fast a controller responds by measuring the first time that the error comes close to its final error, say within 2 percent of the total steady-state reduction of error, and stays there for all time. We call this the settling time.

Visualizing the error response with my elbow, we would get a motion something like this. My arm comes to rest with a small steady-state error. A better response would have no steady-state error, no overshoot, and a faster settling time.

In summary, the error response can be characterized by its steady-state response, which refers to the final error achieved, and its transient response, which consists of the overshoot and settling time. A good controller would have zero or small steady-state error, no overshoot or oscillation, and a short settling time.

Usually we approximate the error dynamics of a controlled system by linear differential equations, so in the next video we'll take a closer look at this case.