By inverse-filtering the control signal applied to a plant by the estimated OTF of the control function, we can compensate for arbitrarily poor stiffness, up to limits imposed by the control-output bounds and the Q factor of series resonances in the system which impede our ability to impose rapid changes on it. The estimated OTF can be updated moment by moment from incoming sensor data, which permits compensation for mild nonlinearities in the plant; for example, the resonant frequencies of a robot arm may change as it is being extended. High-Q-factor series resonances impose notch filters on the spectrum of the OTF, making it poorly conditioned, thus requiring large components in the inverse filter. These large components can easily cause the inverse-filtered signal to have a very poor signal-to-noise ratio or to exceed the limits of the control actuators, requiring for example very large forces, velocities, or displacements. This problem can be ameliorated somewhat by using nonlinear optimization algorithms, rather than simply solving a linear system, in the control loop. However, in many cases, it may be better to change the design of the plant to damp the high-Q resonances. For example, in a mechanical system, these high-Q resonances can be damped and broadened by adding dashpots or other dissipative elements, thus trading off efficiency for precision control. Counterintuitively, the common approach of increasing rigidity can worsen the controllability of the system when using such an adaptive control algorithm, as it increases the Q factor of the system’s vibrational modes, even as it moves them to higher frequencies. As more data is gathered about the system, it becomes possible to empirically estimate the variation of the control-feedback OTF over the plant’s parameter space, thus enabling compensation for OTF nonlinearities in parts of the parameter space we expect to visit in the near future. If the control system can learn an OTF that is locally nonlinear or stateful, then nonlinear optimization algorithms in the control loop could potentially compensate even for such phenomena as gear backlash. Finally, using a self-validating analysis system such as reduced affine arithmetic, the control system can optimize not only to reduce the expected deviation between the plant’s state and the commanded result, but even for the uncertainty in the plant’s state.
All of the above is, to me at least, still somewhat speculative. I have a strong intuition that all of it is true, but a lot of work is needed to verify it in practice. It’s about four times too long for a paper abstract!
How to investigate? Well, one thing to try is to simulate a simple physical system, like maybe a linear one-dimensional mass-spring-dashpot system connected to another mass-spring-dashpot system plus a little bit of measurement noise, and try to command it to make some movements with some different kinds of control systems:
That should provide some kind of evidence that this is a good idea, before I get into more complicated plants.
The obvious way to do the simulation at this point is with SVG and JS in Chrome, using direct-mode solutions with maybe a second-degree approximation for the integrals, which will also make it easy to demonstrate to other people. Maybe I can make it game-like, which should make it easy to draw a desired motion with the mouse or a touchscreen in order to see how different control systems respond.
Maybe I can try some experiments first with Jupyter notebooks with Numpy to see what inverse-filtering a noisy signal looks like. Maybe I need to review how to compute the OTF of a linear mass-spring- dashpot system, because I’m pretty sure that’s like a closed-form kind of thing.