Adaptive Vibration Isolation for Flexible Structures
BACKGROUND
When flexible structures are excited by external disturbances, vibration control often focuses on isolation of quiet parts of the structure from the disturbance. Confinement of vibrations to relatively unimportant areas in the structure allows high precision pointing and/or positioning in the presence of unknown disturbances. The earliest vibration control attempts began with passive methods. Karnopp (1995) provides a comprehensive review of optimization of spring rates and damping and the addition of vibration absorbers to achieve vibration isolation. Increasingly complex and demanding applications, however, require the higher levels of isolation provided by active approaches. One of the first active vibration isolation methods uses feedback to change the eigenstructure of the system. Vibration energy is redistributed among the altered mode shapes, thus creating quiet parts in the structure. For example, Choura (1995) used position, velocity, and acceleration feedback to suppress vibrations in a sensitive area of a third order harmonic oscillator using an exact model knowledge controller. Clark and Robertshaw (1997) developed an adaptive truss as part of a steel beam flexible structure assembly to control vibrations without the use of a system model. Feedforward approaches have also been investigated. Factors limiting the performance of adaptive Least Mean Squared vibration isolation in machines were investigated by Jenkins, et al. (1993). Although LMS control schemes do not require exact model knowledge, they require a priori knowledge of the disturbance fundamental frequencies, so this technique is best suited for narrowband disturbances. Guigou et al. (1994) combined an adaptive structure and LMS algorithm to control transmitted vibrations from a machine to a receiving structure. Results showed that nonresonant vibrations were more difficult to isolate. The application of APFF (Adaptive Pseudofeedforward) control effectively reduced vibrations in a lever system despite drifting fundamental frequencies (Hillerstr\"{o}m, 1996). Recently, adaptive control has been applied to a number of flexible systems. Canbolat, et al. (1996) adaptively regulate the vibration of a flexible cable. In Canbolat, et al. (1997), adaptive boundary control stabilizes cantilevered beam vibration. Adaptive control can also be used to stabilize the displacement of an axially moving string using a two-degree of freedom actuator (Queiroz, et al., 1997). New adaptive vibration isolation control strategies are developed based on Lyapunov theory. Specifically, regulation and tracking controllers cancel unknown bounded disturbances while compensating for parametric uncertainty. Experimental results verify the theoretical predictions.RESEARCH OBJECTIVE
Vibration isolation control for flexible structures restricts the response resulting from external disturbances to areas not requiring high precision positioning and/or pointing. This paper introduces adaptive feedback isolation controllers, based on Lyapunov theory, that regulate and allow tracking of the undisturbed (controlled) coordinates in a flexible structure. Under assumptions of inertially decoupled controlled and uncontrolled coordinates, symmetric and positive definite mass and stiffness matrices for the controlled subsystem, asymptotically stable eigenvalues for the uncontrolled subsystem, and bounded disturbances, an adaptive regulator asymptotically drives the controlled coordinate rates to zero. Under similar assumptions, an adaptive tracking algorithm drives the controlled coordinates to desired time trajectories. Experimental results on a three mass system compare the response of the adaptive isolation controllers with standard PID control.EXPERIMENTAL SETUP
The experimental setup consists of a mass/spring carriage system that is controlled, via Educational Control Products, Inc. communication software and control hardware, by a DSP board in a 486DX-66 PC. The plant is comprised of three masses, connecting springs, and an actuator motor. Optical encoders measure the mass positions via rack and pinions with a resolution of approximately .01 mm. A rigid link connects the actuator, a brushless DC motor, to the middle mass. An optical encoder on the motor provides measurements for motor commutation.
Experimental Setup
ECP Executive Software
Controller servo Amplifier Box
Controller DSP Board
RESULTS
The open loop response of the system to an initial impulse input to mass 1 and repeated impulsive inputs to mass 3 was observed. The 165 N magnitude of the impulse inputs is the same for all experiments. The controlled coordinates, mass 1 and mass 2 position, closely mirror the uncontrolled mass 3 position, producing a steady, oscillatory error of approximately 10 mm.REGULATION PROBLEM
For comparison purposes, the system response using simple PID control on mass 2 was obtained. The control limits the response of mass 2 to approximately .5 mm. The response of mass 1 slowly decays to a steady-state error of less than .4 mm under the repeated impulsive inputs. As seen in the system response, the adaptive controller exhibits a 33% quicker response than PID to the initial mass 1 impulse and eliminates steady state error in mass 1. From the response of mass 2, it can be seen that once the controller learns the system parameters, the motion of mass 2 is regulated to a setpoint with a significantly lower error of .05 mm. The control effort is considerably smaller than for PID and simplifies to a simple harmonic signal proportional to the mass 3 response. The adaptation period of approximately 550 ms can be substantially reduced if the parameter estimates are initialized close to the actual values, unlike the zero initialization used. The inset plots show the parameters converging to final values of 400 N/m and 200 N/m/s for {k}{3} and {b}{3}, respectively.TRACKING PROBLEM
The system response to simple PID control on mass 2 limits the magnitude of the tracking error to approximately .2mm during the repeated impulsive inputs applied to mass 1 and mass 3. The system response with the adaptive controller shows the adherence of the controlled coordinate to the desired trajectory with a 50% lower error of approximately .1 mm under the same repeated impulsive inputs to mass 1 and mass 3. Additionally, the control effort is reduced by approximately 30%. Convergence of the system parameters from zero to actual values of 400 N/m, 400 N/m, 800 N/m, and 4 Kg for k2hat, k3hat, k23hat, and m2hat, respectively can also be seen.
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