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Non-recursive Haar Connection Coefficients Based Approach for Linear Optimal Control

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Abstract

In the present paper, two-fold contributions are made. First, non-recursive formulations of various Haar operational matrices, such as Haar connection coefficients matrix, backward integral matrix, and product matrix are developed. These non-recursive formulations result in computationally efficient algorithms, with respect to execution time and stack-and-memory overflows in computer implementations, as compared to corresponding recursive formulations. This is demonstrated with the help of MATLAB PROFILER. Later, a unified method is proposed, based on these non-recursive connection coefficients, for solving linear optimal control problems of all types, irrespective of order and nature of the system. This means that the single method is capable of optimizing both time-invariant and time-varying linear systems of any order efficiently; it has not been reported in the literature so far. The proposed method is applied to solve finite horizon LQR problems with final state control. Computational efficiency of the proposed method is established with the help of comparison on computation-time at different resolutions by taking several illustrative examples.

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References

  1. Paraskevopoulos, P.N.: Modern Control Engineering. Dekker, New York (2002)

    Google Scholar 

  2. Hsu, N., Cheng, B.: Analysis and optimal control of time-varying linear systems via block pulse functions. Int. J. Control 33(6), 1107–1122 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C.F., Hsiao, C.H.: Design of piecewise constant gains for optimal control via Walsh functions. IEEE Trans. Autom. Control AC–20(5), 596–603 (1975)

    Article  Google Scholar 

  4. Jin, L., Watanabe, A., Kawata, S.: The linear-quadratic-Gaussian control design using an improved product formula of Walsh functions. In: Proc. of the 34th SICE Annual Conf., SICE’95, Japan, July 26–28, pp. 1375–1378 (1995)

    Chapter  Google Scholar 

  5. Razzaghi, M.: Optimal control of linear time-varying systems via Fourier series. J. Optim. Theory Appl. 65(2), 375–384 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hsiao, C.H., Wang, W.J.: Optimal control of linear time-varying systems via haar wavelets. J. Optim. Theory Appl. 103, 641–655 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36, 961–1005 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, C.F., Hsiao, C.H.: Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc., Control Theory Appl. 144(1), 87–94 (1997)

    Article  MATH  Google Scholar 

  9. Chen, C.F., Hsiao, C.H.: Wavelet approach to optimizing dynamic systems. IEE Proc., Control Theory Appl. 146(2), 213–219 (1999)

    Article  Google Scholar 

  10. Hsiao, C.H., Wang, W.J.: Short communications in state analysis and optimal control of linear time-varying systems via Haar wavelets. J. Optim. Control Appl. Math. 19, 423–433 (1998)

    Article  MathSciNet  Google Scholar 

  11. Karimi, H.R., Lohmann, B., Maralani, P.J., Moshiri, B.: A computational method for solving optimal control and parameter estimation of linear systems using Haar wavelets. Int. J. Comput. Math. 81(9), 1121–1132 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karimi, H.R., Moshiri, B., Lohmann, B., Maralani, P.J.: Haar wavelet-based approach for optimal control of second-order linear systems in time domain. J. Dyn. Control Syst. 11(2), 237–252 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Oishi, M., Moro, S., Matsumoto, T.: A modified method for circuit analysis using Haar wavelet transform with adaptive resolution—for circuits with waveform with sharp convex ranges. In: Eur. Conf. Circuit Theory Design, ECCTD 2009, pp. 299–302 (2009)

    Chapter  Google Scholar 

  14. Garg, M., Dewan, L.: A novel method of computing Haar connection coefficients for analysis of HCI systems. In: Proc. Second Int. Conf. Intelligent Human Computer Interaction, IHCI 2010. Lecture Notes in Control and Information Sciences (LNCS), pp. 360–365. Springer, Berlin (2010). ISBN 978-81-8489-540-7

    Google Scholar 

  15. Wu, J.L., Chen, C.H., Chen, C.F.: Numerical inversion of Laplace transform using Haar wavelet operational matrices. IEEE Trans. Circuits Syst. I 48(1), 120–122 (2001)

    Article  Google Scholar 

  16. Wu, J.L., Chen, C.H., Chen, C.F.: A unified derivation of operational matrices for integration in systems analysis. In: Proc. IEEE Comp. Society Int. Conf. Infor. Tech., ITCC’00, Coding and Computing, USA, March 27–29, pp. 436–442 (2000)

    Google Scholar 

  17. Chen, S.L., Lai, H.C., Ho, K.C.: Identification of linear time varying systems by Haar wavelet. Int. J. Syst. Sci. 37(9), 619–628 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Monika Garg.

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Communicated by Elijah Polak.

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Garg, M., Dewan, L. Non-recursive Haar Connection Coefficients Based Approach for Linear Optimal Control. J Optim Theory Appl 153, 320–337 (2012). https://doi.org/10.1007/s10957-011-9976-2

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  • DOI: https://doi.org/10.1007/s10957-011-9976-2

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