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MPC Implementation for Vibration Control

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Model Predictive Vibration Control

Abstract

This chapter is devoted to the implementation of model predictive control (MPC) algorithms in active vibration control (AVC) applications. Even though the main area of interest is AVC, the software implementation tasks presented here are valid for any other engineering application of MPC, thus the material may be recommended to anyone interested in practical issues with MPC software deployment. Three different MPC strategies are discussed, each having its own advantages and disadvantages: the well known infinite horizon cost dual-mode quadratic programming based MPC (QPMPC), optimal and suboptimal explicit pre-computed multi-parametric programming based MPC (MPMPC) and the efficient but suboptimal Newton–Raphson MPC (NRMPC). The offline portion of the algorithms is implemented in the Matlab m-file scripting language, while the real-time controllers are realized in Simulink and subsequently transferred to the xPC Target rapid control software prototyping platform. The practical approach utilized here is focused at simplicity. An off-the shelf quadratic solver called qpOASES is used for the online implementation of QPMPC, while the MPMPC algorithm is implemented using the MPT Toolbox. As the implementation of NRMPC to physical systems is unique to this book, the most attention is devoted to the code developed for the use of Newton–Raphson MPC in the vibration damping of lightly damped structures.

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Notes

  1. 1.

    Download at http://www.qpoases.org.

  2. 2.

    Van den Broeck et al. used a computer with a Mobile Pentium 2 GHz processor with 2 GB of RAM.

  3. 3.

    See more on the topic of Sect. 9.7.3.

  4. 4.

    The illustration in Fig. 10.4 shows the variable naming conventions familiar from earlier chapters. The code implementation in Appendix B uses the following notation for variables: \({\mathbf{H}={\rm Hqp}}{,}\) \({\mathbf{G}}={\rm F}{,}\) \({\mathbf{B}}_x={\rm BxQP}{,}\) \({\mathbf{b}}_0={\rm b0QP}\) and \({\mathbf{A}}_c={\rm AcQP}{.}\)

  5. 5.

    The particular solver module featured in Fig. 10.4 is qpOAES_QProblem, which is recommended for problem formulations assuming nominal systems and fixed constraints. In case a sequential quadratic programming problem is required by the MPC formulation (e.g. in adaptive systems), the resulting optimization task can be solved more efficiently by using the sequential qpOAES_SQProblem module of qpOASES [16]. The timing analysis featured in Sect. 12.5 utilizes the sequential qpOASES solver, but with constant parameter and constraint matrices.

  6. 6.

    Software package and extended documentation is available at: http://control.ee.ethz.ch/~mpt/.

  7. 7.

    Also known as R2007b.

  8. 8.

    Also known as R2006a.

  9. 9.

    At the time of writing the book manuscript and according to the literature research performed by the authors.

  10. 10.

    See 12.7.2.6 for details.

  11. 11.

    Remember that these are in general vectors, the boldface type is reserved for the vector of predicted inputs \({\mathbf{u}}_k\) to simplify notation.

  12. 12.

    There is an additional tolerance value. \({\mathbf{L}}\) is supposed to be positive definite, although sometimes may contain negative elements too.

  13. 13.

    Matlab is in fact based on BLAS and LAPACK.

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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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Takács, G., Rohal’-Ilkiv, B. (2012). MPC Implementation for Vibration Control. In: Model Predictive Vibration Control. Springer, London. https://doi.org/10.1007/978-1-4471-2333-0_10

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