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Sparse feedback in linear control systems

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Abstract

We consider a classical problem of linear static state feedback design in the linear system ẋ = Ax + Bu subject to a nonstandard constraint that the control vector u = Kx has as many zero components as possible.

A simple approach to approximate solutions of such kind of nonconvex problems is proposed, which is based on convexification. The problem reduces to the minimization of special matrix norms subject to the constraints in the form of linear matrix inequalities (LMIs).

The approach can be generalized to numerous problems of robust and optimal control that admit a “sparse” reformulation. To the best of our knowledge, both the solution and the problem formulation are new.

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References

  1. Barabanov, A.E. and Granichin, O.N., Optimal Controller for Linear Plant with Bounded Noise, Autom. Remote Control, 1984, vol. 45, no. 5, part 1, pp. 578–584.

    MATH  MathSciNet  Google Scholar 

  2. Tibshirani, R., Regression Shrinkage and Selection via the Lasso, J. Royal Statist. Soc., 1996, vol. 58, no. 1, pp. 267–288.

    MATH  MathSciNet  Google Scholar 

  3. Donoho, D.L., Compressed Sensing, IEEE Trans. Inf. Theory, 2006, vol. 52, pp. 1289–1306.

    Article  MATH  MathSciNet  Google Scholar 

  4. Kim, S.-J., Koh, K., Boyd, S., et al., ℓ 1-Trend Filtering, SIAM Rev., 2009, vol. 51, no. 2, pp. 339–360.

    Article  MATH  MathSciNet  Google Scholar 

  5. Lin, F., Fardad, M., and Jovanović, M., Sparse Feedback Synthesis via the Alternating Direction Method of Multipliers, Proc. Am. Control Conf., Washington, 2012, pp. 4765–4770.

    Google Scholar 

  6. Lin, F., Fardad, M., and Jovanović, M., Augmented Lagrangian Approach to Design of Structured Optimal State Feedback Gains, IEEE Trans. Autom. Control, 2011, vol. 56, no. 12, pp. 2923–2929.

    Article  Google Scholar 

  7. Fradkov, A.L., Cybernetical Physics: From Control of Chaos to Quantum Control, New York: Springer, 2007.

    Google Scholar 

  8. Graham, S. and Kumar, P.R., The Convergence of Control, Communication, and Computation, in Personal Wireless Communications, vol. 2775 of Lecture Notes in Computer Science, Conti, M., Giordano, S., Gregori, E., and Olariu, S., Eds., Heidelberg: Springer-Verlag, 2003, pp. 458–475.

    Chapter  Google Scholar 

  9. Matveev, A.S. and Savkin, A.V., Estimation and Control over Communication Networks, Boston: Birkhäuzer, 2008.

    Google Scholar 

  10. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., An LMI Approach to Structured Sparse Feedback Design in Linear Control Systems, Proc. 12 Eur. Control Conf., Zürich, Jul. 2013, pp. 833–838.

    Google Scholar 

  11. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Design of Sparse Feedback in Linear Control Systems, XII All-Russia Conf. Control Problems, Moscow, Jun. 2014, pp. 218–229.

    Google Scholar 

  12. Tropp, J.A., Algorithms for Simultaneous Sparse Approximation. Part II: Convex Relaxation, Signal Proc. (special issue “Sparse Approximations in Signal and Image Processing”), 2006, vol. 86, pp. 589–602.

    MATH  Google Scholar 

  13. Quattoni, A., Carreras, X., Collins M., et al., An Efficient Projection for ℓ 1,∞ Regularization, Proc. 26 Int. Conf. Machine Learning, Montreal, Jun. 2009, pp. 857–864.

    Google Scholar 

  14. Boyd, S., El Ghaoui, L., Feron, E., et al., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.

    Book  MATH  Google Scholar 

  15. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems Subject to Exogenous Disturbances: The Linear Matrix Inequalities Technique), Moscow: LENAND, 2014.

    Google Scholar 

  16. Grant, M. and Boyd, S., CVX: Matlab Software for Disciplined Convex Programming (web page and software), URL http://stanford.edu/boyd/cvx.

  17. Syrmos, V.L., Abdallah, C.T., Dorato, P., et al., Static Output Feedback: A Survey, Automatica, 1997, vol. 33, pp. 125–137.

    Article  MATH  MathSciNet  Google Scholar 

  18. Leibfritz, F. and Lipinski, W., Description of the Benchmark Examples in COMPleib 1.0, Technical report. Univ. of Trier, 2003, URL www.complib.de.

    Google Scholar 

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    B. T. Polyak, M. V. Khlebnikov & P. S. Shcherbakov

Authors
  1. B. T. Polyak
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  2. M. V. Khlebnikov
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  3. P. S. Shcherbakov
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Correspondence to B. T. Polyak.

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Original Russian Text © B.T. Polyak, M.V. Khlebnikov, P.S. Shcherbakov, 2014, published in Avtomatika i Telemekhanika, 2014, No. 12, pp. 13–27.

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Polyak, B.T., Khlebnikov, M.V. & Shcherbakov, P.S. Sparse feedback in linear control systems. Autom Remote Control 75, 2099–2111 (2014). https://doi.org/10.1134/S0005117914120029

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