Research Papers

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

[+] Author and Article Information
Pranav A. Bhounsule

Department of Mechanical Engineering,
University of Texas at San Antonio,
One UTSA Circle,
San Antonio, TX 78249
e-mail: pranav.bhounsule@utsa.edu

Andy Ruina

Mechanical Engineering,
Cornell University,
Ithaca, NY 14853
e-mail: ruina@cornell.edu

Gregg Stiesberg

Department of Physics,
Cornell University,
Ithaca, NY 14853
e-mail: grs26@cornell.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 13, 2014; final manuscript received October 15, 2014; published online December 18, 2014. Assoc. Editor: M. Porfiri.

J. Dyn. Sys., Meas., Control 137(5), 051012 (May 01, 2015) (9 pages) Paper No: DS-14-1118; doi: 10.1115/1.4028851 History: Received March 13, 2014; Revised October 15, 2014; Online December 18, 2014

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.

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Grahic Jump Location
Fig. 1

Schematic example. (a) The nominal (solid red) and deviated (dashed blue) trajectory, for some dynamic variable x of interest. We measure the state x at the start of a continuous interval, namely, at section n. (b) The new deviated trajectory in target variables z after switching on our feedback controller. In this example, feedback controller nulls (zeros) the output z at the end of the interval, illustrating a dead-beat controller. (c) The feedback motor program has two control actions: a sinusoid for first half cycle and a hat function for the second half of the cycle. These shapes are arbitrary and different from each other in form only for illustrative purposes. They could overlap in time. We choose the amplitudes U1 and U2 of the two functions at the start of the interval depending on the error (x-x¯). By a proper choice of the amplitudes, U1 and U2 deviations are, in this example, fully corrected in between measurements. The choice of trigger for event n, the choice of sensor measurements x, the choice of output variables z, and the control shape functions f(t) are offline design choices.

Grahic Jump Location
Fig. 2

(a) Inverted pendulum and (b) simple pendulum. The pendulum in both cases consists of mass m at G and attached to a massless rod of length ℓ and controlled by a motor via a torque Tm at the hinge joint H. Gravity is g.

Grahic Jump Location
Fig. 3

Simulation results for balance of a simple inverted pendulum. The first column (a) uses constant control function and second column (b) uses sinusoidal control functions. The first, second, and third rows correspond to joint angles versus time, joint rate versus time, and control torque versus time, respectively. In both cases, the system is let go from an initial position of 0.4 rad and initial velocity of 0. We used the linearized governing equation to do these simulations.

Grahic Jump Location
Fig. 4

Experimental verification of event-based intermittent controller to balance a simple inverted pendulum. We measured the pendulum state—the angle and angular speed—once per second and used constant control functions active for half a second each. We were able to balance the inverted pendulum over a range of ±0.5 rad. Note that the sensing is 1 s, and the controller bandwidth is 0.5 s, and is slower than the characteristic time scale of 0.32 s of the simple inverted pendulum (see Ref. [25] for a video).

Grahic Jump Location
Fig. 6

Definition of delay for discrete control of a continuous system. One sensible definition of delay would be the time t1 between when the system is sensed and when the start of the related control action. This does not respect an invariance described in the text. A sensible definition of delay is the time t+t1 between the start of an actuation interval and the most recent sensing that had an effect, through previous actuation, on the starting state.

Grahic Jump Location
Fig. 5

Simulated result for pumping the swing of a simple pendulum. The pendulum starts from rest from the hanging position. (a) Pendulum angle versus time, (b) pendulum velocity versus time, and (c) torque versus time.



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