Model-based control theory

Our lab operates under the philosophy that if we have, or can develop, an accurate dynamic model of the system under consideration (which is certainly the case), then it is generally to our advantage to use it; thus, all of the vehicles developed at our lab are coordinated leveraging control strategies designed around (or at least tuned based on) such models, including gain-scheduled PID & lead-lag control, (infinite-horizon, finite-horizon, and time-periodic) LQG & H_∞, and MPC, the details of which we will not get into here. Note that accurate dynamic modeling of such systems is doable, though not necessarily easy; in particular, iceCube has a rich range of possible motions that must be handled carefully and with a singularity-free state description. Offline and online identification of model parameters is also sometimes necessary in such problems.

In our experience, PID, lead-lag, LQG & H_∞, and MPC are hearty workhorses that go a long way towards the effective control of robotic systems, which are typically fraught with complex trigonometric nonlinearities. Richard Bellman is said to have once compared one who designs linear controls for nonlinear systems with one whom, "having lost his watch in a dark alley, is searching for it under a lamp post." Erudite comments of this sort are often taken far too seriously, as all differentiable systems are linear when considered as small perturbations about a nominal position or trajectory. Indeed, "nonlinear control theories" (Lyapunov-based approaches, backstepping, etc.), though elegant when they can be applied, often represent boutique luxuries that are inapplicable to the classes of nonlinearities present in practical systems of interest. As just one example, consider the double-pendulum swing-up and stabilization problem demonstrated above: though dominantly nonlinear, our lab has solved this reference problem with a straightforward combination of MPC trajectory planning (via successive linearizations about candidate trajectories) and LQG stabilization; we are in fact unaware of anyone else who has solved this problem, and all of the so-called "more sophisticated" nonlinear control methods available appear to be inapplicable.