Figures 1 and 2 contain two graphs showing how one can use the measured velocity and the control output to generate a first and second order model respectively.  The measured or feedback velocity in red is calculated using a  simple method. The resolution of the feedback device is 0.001 inches but the velocity is calculated over a time span of two milliseconds, so the resolution of the feedback velocity is 0.5 inches per second. One can see that the estimated velocity is much smoother because it is generated by the model, but the feedback velocity from a first-order model doesn’t follow the changes in the actual model that well. On the other hand, the estimated velocity for the second order model follows the feedback velocity much more closely than the first order model, and best represents how the actual system is moving. This indicates that the system is a second order system, behaving like a mass between two springs.

So where does the model that is used to estimate the velocity and acceleration come from?

### Auto Tuning

Auto tuning is a feature that is supported by some electronic motion control systems and is necessary when using advanced features such as the second derivative gain and the jerk feed forward because few hydraulic control system designers have any experience at tuning them manually.  The two parameters can be determined roughly by trial and error but having a method of determining the gain, damping factor and natural frequency quickly is important to reduce startup time and shorten the time it may take to retune a system as the mechanics change.

### Auto Tuning

 VCCM Equation:     Where: Fl is the load force that must be overcome, PS is supply pressure, APE is the cylinder size, v is the speed of cylinder propulsion, KVPL is the degree to which the valve is open, ρv is the symmetry of the valve, and ρc is the cylinder area ratio (cap side of piston area/rod side area). Natural Frequency:                        Where: β is the bulk modulus of oil, A is the surface area of the piston, V is the volume of the cylinder, and m is the mass of the load

Auto tuning is a way of estimating the model of the actuator and load using the control output to the actuator and the position and velocity response as in the two graphs above.  Models can be very complex but usually one can achieve 95% of the benefit with 5% of the effort if the model is kept relatively simple.  A hydraulic actuator and load can be modeled simply as a mass between two springs so the model consists of a gain, damping factor and natural frequency. It is possible to estimate the gain and natural frequency at design time using the VCCM equation (See sidebar) and the formula for natural frequency (See sidebar).  The damping factor is a little more difficult to estimate, but a typical damping factor is in the range of 0.3 to 0.4. Even a rough estimate of the damping factor can allow the person doing the motion control to enter these parameters into the motion controller’s built-in simulator so that he or she can get started before the machinery is built.  Once in the field the controls person can do an auto tune to find the actual gain, damping factor and natural frequency.

Auto tuning isn’t totally automatic. There is a procedure where the actuator must be moved in a specific way, usually via open loop control.  The relationship between the control output and the position or velocity data is estimated by trying a value for each of the three parameters (gain, damping factor and natural frequency) and then checking to see how closely the estimated position or velocity follows actual recorded position or velocity.  An evaluation can be done by summing the square of all the errors between the estimated and actual position or velocity.  The three parameters mentioned above are changed in an effort to minimize the sum of the squared errors.  This is a trial and error process but a computer can do it very quickly so it seems instantaneous. When the computer is done it has found the best values of the gain, damping factor and natural frequency.