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Experimental Model Predictive Vibration Control

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Model Predictive Vibration Control

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Abstract

This chapter presents the results of experiments comparing different computationally efficient model predictive control (MPC) methods applied to a laboratory device, demonstrating the active vibration control (AVC) of lightly damped mechanical structures. Because of the combination of long prediction horizons, short sampling times and large actuator-disturbance asymmetry, the implementation of the predictive control strategy on lightly damped vibrating structures is highly demanding. The vibration damping effect and online timing properties of model predictive control algorithms such as infinite horizon cost dual-mode quadratic programming based MPC (QPMPC), pre-computed explicit multi-parametric programming based MPC (MPMPC), minimum-time MPMPC and the very efficient but suboptimal Newton–Raphson based MPC (NRMPC); all with guaranteed stability and constraint feasibility are analyzed in different disturbance and loading scenarios. All MPC methods along with the baseline linear quadratic (LQ) controller decrease vibration settling to equilibrium by an order of magnitude time. The damping effect of all investigated MPC strategies is comparable with a slight decrease in performance for the suboptimal minimum-time MPMPC and NRMPC controllers. Due to the excessive online computational needs of QPMPC, it is a very unlikely candidate for lightly damped vibrating systems given currently available hardware. The online timing analysis presented here demonstrates that MPMPC provides significantly shorter online execution times, however its suboptimal minimum-time version does not bring a convincing improvement. NRMPC can provide online execution times on par with linear quadratic controllers; however, its suboptimality becomes excessive with increasing prediction model orders.

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Notes

  1. 1.

    See Sect. 5.1.1 for more details.

  2. 2.

    Materializing as disproportionately high control moves, severely exceeding constraints.

  3. 3.

    See Sect. 5.5.4.2 for exact target computer specifications and Sect. 12.5 for timing properties.

  4. 4.

    See Sect. 5.5.5 for the description of the shaker and shaker amplifier.

  5. 5.

    See Sect. 5.1.4 for the designation and placement of transducers.

  6. 6.

    Note that \(\xi\) (xi) is used as a relative factor comparing the peak amplitudes in the periodogram and is not to be confused with the damping factor commonly denoted by \(\zeta\) (zeta).

  7. 7.

    Approximately 0.2 mm, also depending on the frequency.

  8. 8.

    At 40 Hz excitation, the difference between mean uncontrolled vibration and NRMPC controlled is approximately 0.01 mm. This is below the noise level normally encountered at the given laboratory environment.

  9. 9.

    Note that the timing data for QPMPC featured in this section is given for the qpOASES sequential solver module. The sequential qpOASES_SQProblem module is computationally more efficient than the constant qpOASES_QProblem, given that the MPC formulation uses variable parameters [23]. The timing experiments featured here assume fixed prediction \({\mathbf{H}}\), \({\mathbf{G}}\) and fixed constraint matrices \({\mathbf{A}}_c\), \({\mathbf{b}}_0\), \({\mathbf{B}}_x\). The task execution times shown in the following tables and figures include only the execution time of the sequential QP solver and do not enclose any additional operation, thus QPMPC is directly comparable to the other efficient MPC methods. By using the non-sequential solver (qpOASES_QProblem), in theory the QPMPC method could perform slightly better than it is suggested by the timing data featured here. Nevertheless, this does not influence any of the conclusions implied by the analysis featured in the following pages.

  10. 10.

    Compared to the experiment in Sect. 12.2, minimal execution time given by the LQ controller is increased by shaker controls and additional data logging.

  11. 11.

    This is five times longer sampling than the one utilized in this work. Moreover, Ferrau et al. utilized an 5 step prediction model instead of the 75 step assumed in this work. Note that the model in [24] has been more complex than the one used here.

  12. 12.

    See Sect. 11.1 for prediction horizon lengths and Sect. 11.2 for a simulation results and discussion involving MPMPC properties on the experimental system.

  13. 13.

    See Sects. 10.3.1, B.3.2.1 and B.2.3.2 for answers.

  14. 14.

    See Sect. 11.3 for the numerical issues causing invariance condition violation and Sect. 11.4 for optimality issues.

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  1. Faculty of Mechanical Engineering, Institute of Automation, Measurement and Applied Informatics, Slovak University of Technology in Bratislava, Námestie slobody 17, 812 31, Bratislava 1, Slovakia

    Gergely Takács & Boris Rohal’-Ilkiv

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  1. Gergely Takács
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Takács, G., Rohal’-Ilkiv, B. (2012). Experimental Model Predictive Vibration Control. In: Model Predictive Vibration Control. Springer, London. https://doi.org/10.1007/978-1-4471-2333-0_12

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