A true test is to move the actuator using a sine wave motion profile.  The actual or true velocity and acceleration will smoothly go through a range of values and the goal is to estimate the velocity and acceleration accurately so the actuator can follow the target accurately. Low resolution feedback will make the quantizing effect obvious because the velocity calculated directly from the feedback will change in steps. The goal of the model based control is to estimate smooth velocities as shown below.

The graph in Figure 3 shows how an actuator with a gain of 3 inches per second, damping factor of 0.4 and a natural frequency of 10 Hz can be controlled using a PID with a second derivative gain. The feedback is truncated to 0.001 inches to simulate a start-stop linear magnetostrictive displacement transducer (LMDT) rod with a position feedback of 0.001 inches. As noted above, the resolution of the measured velocity would be 1 inch per second and the resolution of the acceleration would be 1000 inches per second squared.  The graph shows that the estimated position is so close to the actual position that they look like one line.  The estimated velocity is relatively smooth compared to what could be achieved doing simple velocity calculations where the resolution would be one inch per second. The control output is basically doing a 1 volt peak to peak sine wave to control the actuator.  There is a little noise on the control output due to the errors in estimating the velocity and acceleration but it isn’t bad compared to what would happen without estimating the velocity and acceleration with the model. 

If the model wasn’t used the control output would simply be changing from +10 to -10 volt because of the inability to estimate the velocity and acceleration.  Without the model based velocity and accelerations, the PID and second derivative gains would need to be drastically reduced in order to keep the control output from swinging 10 volts peak to peak.

Figure 4 contains a graph of the actual and estimated velocities. There are actually two lines in the plot, but the estimated velocity is almost identical to the actual velocity, so the estimated velocity line is on top of the actual velocity line.  Again, notice that there is no 1 inch per second quantizing.

As expected, the estimated acceleration(shown in Figure 5) does shows the effects of feedback quantizing, but spikes of four or five inches per second squared are a lot better than errors of 1000 inches per second squared. The estimated acceleration curve still does a good job of following the actual acceleration.

Just for comparison, compare the model based control above to control without the model (shown in Figure 6).  The same gains are used as in Figures 3-5 above.  The control output isn’t shown because it changes from -10 to +10 volts due to the affect the quantized velocity and acceleration has on the control output.  Since the control output is often saturated, the actuator is being controlled as if the valve is a simple on-off directional control valve.

Conclusion

Implementing model based control with an electrohydraulic motion controller that can run second-order algorithms allows one to control systems that would otherwise be uncontrollable because the derivative gains couldn’t be used or the gain values would need to be keep low so the response will be slow.  The alternative is to design systems so that the damping factor and natural frequency are high but this increases the system cost.