Discrete-Decision Continuous-Actuation Control: Balance of an Inverted Pendulum and Pumping a Pendulum Swing

In some practical control problems of essentially continuous systems, the goal is not to tightly track a trajectory in state space, but only some aspects of the state at various points along the trajectory, and possibly only loosely. Here, we show examples in which classical discrete-control approaches can provide simple, low input-, and low output- bandwidth control of such systems. The sensing occurs at discrete state- or time-based events. Based on the state at the event, we set a small set of control parameters. These parameters prescribe features, e.g., amplitudes of open-loop commands that, assuming perfect modeling, force the system to, or toward, goal points in the trajectory. Using this discrete decision continuous actuation (DDCA) control approach, we demonstrate stabilization of two examples: (1) linear “dead-beat” control of a time delayed linearized inverted pendulum and (2) pumping of a hanging pendulum. Advantages of this approach include: It is computationally cheap compared to real-time control or online optimization; it can handle long time delays; it can fully correct disturbances in finite time (dead-beat control); it can be simple, using few control gains and set points and limited sensing; and it provides low bandwidth for both sensing and actuator commands. We have found the approach is useful for controlling robotic walking.