In a basic system, the system gains, typically a position gain, velocity estimate (or simply derivative) gain, and integral gain, are limited by the gain/phase present in the physical system. A large inertia mismatch causes significant resonance peaking in the system. The velocity response of the system has a lag of 90 degrees as it is the integral of the acceleration. The position response of the system lags the velocity by 90 degrees as it is the integral of the velocity. This means if the motor and control system response were perfect, the position feedback would still be 180 degrees out of phase, meaning position-only gain would always be ready to oscillate (given enough gain to overcome friction). We add the velocity feedback to improve the phase margin of the system by anticipating when we will need to apply the brakes to avoid (or minimize) overshot. At just above the resonant frequency (about 510Hz for this model) of the motor/shaft/load without damper, we have a significant rise in gain to 64 db or about 1631, with a -90-degree phase, which means the gain must be set quite low to avoid having the system oscillate. With the damper added, however, the peak gain near the mechanical resonance has been reduced to 30 db (or about 33), while the phase at this maximum gain near resonance is 14 degrees positive rather than -90 for the motor velocity response. The system with a damper does not drop to a -45-degree angle until almost 754 Hz, a substantial improvement. Note that the non-damper system would need to have the gain significantly reduced (with the resulting bandwidth reduced) to avoid oscillating at the resonant frequency. The damper (either physical or synthetic) allows a much higher gain which also extends the bandwidth to allow for faster responses and much tighter control of a high inertia system. The phase boost also significantly helps even nominally low inertia systems and makes the tuning of the system much easier, often allowing the same tuning constants for an open shaft to five times motor inertia or larger with little change in the resulting motion when the load is varied.

Let’s look at a couple of other example systems, starting with a geared or chain-fed system with backlash. These systems change their transfer function as the system is moving. That is, the load is only reflected to the motor when the teeth of the gear (or sprocket and chain) are in contact. When the motion reverses, the driving gear (sprocket) can rapidly accelerate while the teeth are disengaged (i.e., the load is decoupled). The teeth then slap, and according to the materials used, may significantly rebound. If the gain is high enough, or the load is positioned such that little torque is needed to keep it in position, a limit cycle oscillation may continue with the teeth bouncing off the adjacent teeth in both the clockwise and counterclockwise directions. This oscillation can quickly damage the gears while making much-undesired noise! When a physical damper is present, the inertia of the damper slows the acceleration of the decoupled motor. Upon contact between the teeth of the drive and load gears, the damper inertia continues at a higher rate than the motor for a short period, causing the teeth to not bounce off, or to have significantly less bounce, which allows the system to settle in without the limit-cycle oscillation. The damper effect allows the system gain to be significantly improved for better performance. The synthetic damper performs very similarly but without the added size and cost of a physical damper. In some animatronic applications, for example, synthetic inertia can be made significantly larger than the physical motor inertia to help smooth out the motions in mechanisms having some degree of backlash. The damper then acts as a flywheel but with viscous damping. The flywheel action eliminates most of the high-speed vibrations which would otherwise make the motions look artificial. In pumping applications, such as those involving a syringe-type pump, stiction may be a very significant problem. Stiction describes the fact that static friction is normally higher than dynamic friction, sometimes by a considerable degree. Stiction effects in a pump are noticed when the motion slows to a point where the seal on the moving piston begins to form mechanical bonds to the cylinder walls. This higher static friction coefficient may completely stop the piston until the control system can build up enough force to overcome the higher static friction coefficient, and then the piston lurches forward due to the lower dynamic friction coefficient. The resulting fluid flow is anything but smooth. The corrective action for this is to have sufficient gain and bandwidth in the system to rapidly adjust the forces so that the cylinder is not allowed to slow down. Rather, it can maintain the desired motion even in the presence of rapid variation in the frictional forces. Looking again at the physical damper, one might imagine a very stiff coupling grease and a large inertia that acts as a flywheel to prevent friction from stalling the motion. But this is only part of the solution, as the improved phase margin of the system with a (synthetic or physical) damper allows the gain to be significantly increased, allowing for both wider bandwidth and more powerful control system response to the friction variations, resulting in very smooth liquid dispensing even in the presence of stiction. Note that the margins described here are just the plant torque to velocity forward transfer function wth a (synthetic) damper. Additonal installments will cover additional closed loop control techniques which are not available in a PID system which significantly benefit the performance of the improved control system.