Boundary Control of a Flexible Cable
with Actuator Dynamics

Background

Cables are often utilized in many different types of engineering applications due to their inherent characteristics of low weight, flexibility, strength, storability,etc. e.g., many marine applications use cables to: i) moored buoyant structures, ii) tow vehicles and arrays, and/or iii) tether remotely operated vehicles). While cables offer unique advantages in many applications, cables often transmit vibration to connecting mechanical structures resulting in undesirable or even catastrophic effects. Specifically, since the transverse stiffness of a cable depends on its tension and length, a cable structure which spans a long distance is subject to large vibrations under relatively small disturbances. This vibration can degrade the performance of the connected electrical and/or mechanical subsystems and ultimately lead to failure. For example, ice covered power lines tend to "gallop" in high winds

The cable suspension Tacoma Narrows bridge exhibited high amplitude oscillations in a steady wind that lead to its collapse. In 1979, vortex shedding induced vibration caused pipe fatigue failures on the jack-up oil rig "Offshore Mercury" resulting in over five million dollars in losses. In marine structures, vortices can excite large vibration response that can significantly degrade performance. In cable hydrophone arrays, vortices shed by the cable may induce a "strumming" effect which can severely degrade the performance of the hydrophones.

One method for reducing cable vibration is to increase the cable tension; however, this remedy induces high stresses which reduces the life of the cable. Another method for reducing cable vibration is the application of active boundary control. While the design of boundary controllers for cable systems has received little attention, many researchers have proposed boundary controllers for many other types of flexible systems. For example, the motion of flexible gantry robots can be regulated with boundary control. The vibration associated with flexible beam-like structures e.g., aircraft wings or space structures can be regulated via boundary control techniques. In addition, the vibration in a rotating flexible system can also be reduced or eliminated by using boundary control.

We developed new control strategies for the distributed cable model. Specifically, we developed an exact model knowledge controller which exponentially stabilizes the position of the cable given exact knowledge of the mechanical system parameters and measurements of the slope, slope-rate, and velocity at the cable's actuated boundary. We also designed an adaptive controller which asymptotically stabilizes the position of the cable while compensating for parametric uncertainty. The salient features of our control approach is that: i) the stability analysis utilizes relatively simple mathematical tools to illustrate the exponential and asymptotic stability results, ii) a new Lyapunov-like function is crafted to deal with the nonlinear tension function effects, and iii) this work seems to be among the first to fuse adaptive nonlinear control techniques with distributed boundary control techniques.

Experimental Setup

The proposed controller was implemented on a cable control system designed and built in-house . A braided polyester rope, pinned at one end, and connected to a horizontally-translating gantry at the other end was used for the experiments. A brushed dc motor (Baldor model 3300) was coupled to the gantry using a timing belt. Two 1000-count rotary encoders (Hohner) were used to measure the gantry position and the cable departure angle. A 486 ISA-based PC hosting a Texas Instruments TMS320C30 digital signal processing board served as the computational engine. An encoder interface card (Integrated Motions Inc. Model DS-2) allowed for quadrature extrapolation of the encoder signals. The DS2 board also supported two channels of 16-bit ADCs and DACs. The linear and angular velocities were obtained by applying a backwards difference algorithm to the position and angle signals respectively. To eliminate quantization noise, the velocity signals were filtered using a second-order digital filter. A mounting bracket attached to the gantry ensured that the cable was held tangent with respect to the rotary encoder. To test the response of the proposed controller, the cable was perturbed using a gravity-based drop hammer. The hammer was allowed to hit the cable only once and always from the same height. This seemed to yield a consistent input and therefore allowed for comparison between the uncontrolled and controlled swinging of the cable. The setup is shown below.
Block Diagram of the Setup

Actual setup of the Cable System

Experimental Results

Two experiments were conducted to test the performance of the proposed controller. First, the exact model knowledge controller was implemented. The results of these experiments appear below. The cable system exhibits excellent transient response under the proposed controller. The uncontrolled swinging of the cable to a similar input is shown for comparison purposes. The gantry position and the voltage signals are also shown. Other experiments were ran with the controller gain set to positive values; however, no improvement in performance was observed.

The adaptive version of was implemented with the update law given by and the parameter estimates initialized to 80% of their nominal values. The best performance was achieved using the following settings The results of these experiments appear below . In the first subplot of Figure , the transient performance of the adaptive controller to the disturbance is compared to the uncontrolled swinging of the cable for a similar input. Subsequent subplots show the gantry position and the voltage signals. As is clear from the figure, the estimates for the parameters mass and P(1) remain bounded during the closed-loop operation.

Also see a movie of the experiment in [40 MB .avi format] [1.6 MB .rm format] .

Please NOTE: [.rm formats] are for . to view them.

 

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