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Newton’s method for coupled continuous-time algebraic Riccati equations

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Abstract

We seek the solution of the coupled continuous-time algebraic Riccati equations, arising in the optimal control of Markovian jump linear systems. Newton’s method is applied to construct the solution, under a mild and natural stabilizability assumption, leading to some coupled Lyapunov equations. Iterative methods of \(O(n^3)\) computational complexity for the coupled Lyapunov equations and the corresponding Newton’s methods for the coupled continuous-time algebraic Riccati equations are analyzed. Illustrative examples are presented.

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Notes

  1. In some literature on control theory, the feedback is defined with a negative sign in front, with the corresponding closed-loop system matrix \(D_i^{(\nu )} \equiv D_i + B_i F_i^{(\nu )}\).

References

  1. Abou-Kandil, H., Freiling, G., Junk, G.: Solution and asymptotic behaviour of coupled Riccati equations in jump linear systems. IEEE Trans. Automat. Control 39, 1631–1636 (1994)

    MathSciNet  Google Scholar 

  2. Abou-Kandil, H., Freiling, G., Junk, G.: On the solution of discrete-time Markovian jump linear quadratic control problems. Automatica 31, 765–768 (1995)

    MathSciNet  Google Scholar 

  3. Abou-kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkhäuser Verlag, Basel (2003)

    Google Scholar 

  4. Bagnara, R.: A unified proof for the convergence of Jacobi and Gauss–Seidel methods. SIAM Rev. 37, 93–97 (1995)

    MathSciNet  Google Scholar 

  5. Bini, D.A., Iannazzo, B., Meini, B.: Numerical Solution of Algebraic Riccati Equations. SIAM, Philadelphia (2012)

    Google Scholar 

  6. Borno, I.: Parallel computation of the solutions of coupled algebraic Lyapunov equations. Automatica 31, 1345–1347 (1995)

    MathSciNet  Google Scholar 

  7. Buoni, J.J., Varga, R.S.: Theorems of Stein–Rosenberg type. Numer. Math. 49, 65–75 (1979)

    MathSciNet  Google Scholar 

  8. Costa, O.L.V., Aya, J.C.C.: Temporal difference methods for the maximal solution of discrete-time coupled algebraic Riccati equations. J. Optim. Theory Appl. 109, 289–309 (2001)

    MathSciNet  Google Scholar 

  9. Costa, E.F., Do Val, J.B.R.: An algorithm for solving a perturbed algebraic Riccati equation. Eur. J. Control 10, 576–580 (2004)

    MathSciNet  Google Scholar 

  10. Costa, O.L.V., Fragoso, M.D., Marques, R.P.: Discrete-Time Markov Jump Linear Systems. Springer, London (2005)

    Google Scholar 

  11. Costa, O.L.V., Fragoso, M.D., Todorov, M.G.: Continuous-Time Markov Jump Linear Systems. Springer, Berlin (2013)

    Google Scholar 

  12. Costa, O.L.V., Marques, R.P.: Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic Riccati equations. Math. Control Signal. 12, 167–195 (1999)

    MathSciNet  Google Scholar 

  13. Damm, T.: Rational Matrix Equations in Stochastic Control, Lecture Notes in Control and Information Sciences, vol. 297. Springer, Berlin (2004)

    Google Scholar 

  14. Damm, T., Hinrichsen, D.: Newton’s method for a rational matrix equation occurring in stochastic control. Linear Algebra Appl. 332–334, 81–109 (2001)

    MathSciNet  Google Scholar 

  15. Datta, B.: Numerical Methods for Linear Control Systems. Elsevier Academic Press, Cambridge (2004)

    Google Scholar 

  16. Dehghan, M., Shirilord, A.: A new approximation algorithm for solving generalized Lyapunov matrix equations. J. Comput. Appl. Math. 404, 113898 (2022)

    MathSciNet  Google Scholar 

  17. Dehghan, M., Shirilord, A.: On the Hermitian and skew-Hermitian splitting-like iteration approach for solving complex continuous-time algebraic Riccati matrix equation. Appl. Numer. Math. 170, 109–127 (2021)

    MathSciNet  Google Scholar 

  18. Dehghan, M., Shirilord, A.: The use of homotopy analysis method for solving generalized Sylvester matrix equation with applications. Eng. Comput. 38, 2699–2716 (2022)

    Google Scholar 

  19. Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)

    Google Scholar 

  20. Do Val, J.B.R., Geromel, J.C., Costa, O.L.V.: Uncoupled Riccati iterations for the linear quadratic control problem of discrete-time Markov jump linear systems. IEEE Trans. Automat. Control 43, 1727–1733 (1998)

    MathSciNet  Google Scholar 

  21. Fan, H.-Y., Weng, P.C.-Y., Chu, E.K.-W.: Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control. Numer. Algorithms 71, 245–272 (2016)

    MathSciNet  Google Scholar 

  22. Feingold, D.G., Varga, R.S.: Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem. Pac. J. Math. 12, 1241–1250 (1962)

    MathSciNet  Google Scholar 

  23. Freiling, G., Hochhaus, A.: Basic properties of a class of rational matrix differential equations. Proceedings of the European Control Conference, Porto (2001)

  24. Freiling, G., Hochhaus, A.: Properties of the solutions of rational matrix difference equations. Comput. Math. Appl. 45, 1137–1154 (2003)

    MathSciNet  Google Scholar 

  25. Freiling, G., Hochhaus, A.: On a class of rational matrix differential equations arising in stochastic control. Linear Algebra Appl. 379, 43–68 (2004)

    MathSciNet  Google Scholar 

  26. Gajic, Z., Borno, I.: Lyapunov iterations for optimal control of jump linear systems at steady stata. IEEE. Trans. Automat. Control 40, 1971–1975 (1995)

    MathSciNet  Google Scholar 

  27. Gajic, Z., Losada, R.: Monotonicity of algebraic Lyapunov iterations for optimal control of jump parameter linear systems. Syst. Control Lett. 41, 175–181 (2000)

    MathSciNet  Google Scholar 

  28. Guo, C.-H.: Iterative methods for a linearly perturbed algebraic matrix Riccati equation arising in stochastic control. Numer. Funct. Anal. Optim. 34, 516–529 (2013)

    MathSciNet  Google Scholar 

  29. Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)

    Google Scholar 

  30. Ivanov, I.G.: On some iterations for optimal control of jump linear equations. Nonlinear Anal. 69, 4012–4024 (2008)

    MathSciNet  Google Scholar 

  31. Ivanov, I.G.: Stein iterations for the coupled discrete-time Riccati equations. Nonlinear Anal. 71, 6244–6253 (2009)

    MathSciNet  Google Scholar 

  32. Ji, Y., Chizeck, H.J.: Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control. IEEE Trans. Automat. Control 35, 777–788 (1990)

    MathSciNet  Google Scholar 

  33. Kleinman, D.: On an iterative technique for Riccati equation computations. IEEE Trans. Automat. Control 13, 114–115 (1968)

    Google Scholar 

  34. Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Clarendon Press, Oxford (1995)

    Google Scholar 

  35. Li, F., Shi, P., Wu, L.: Control and Filtering for Semi-Markovian Jump Systems, Studies in Systems, Decision and Control 81, Springer, Cham (2016)

  36. Li, Z.-Y., Zhou, B., Lam, J., Wang, W.: Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps. Appl. Math. Comput. 217, 8179–8195 (2011)

    MathSciNet  Google Scholar 

  37. Liu, J., Wang, L., Zhang, J.: New upper matrix bounds with power form for the solution of the continuous coupled algebraic Riccati matrix equation. Asian J. Control 19, 739–747 (2017)

    MathSciNet  Google Scholar 

  38. Liu, J.Z., Zhang, J., Luo, F.: Newton’s method for the positive solution of the coupled algebraic Riccati equation applied to automatic control. Comput. Appl. Math. 39, 113 (2020)

    MathSciNet  Google Scholar 

  39. Mariton, M.: Jump Linear Systems in Automatic Control. Marcel Dekker Inc., New York (1990)

    Google Scholar 

  40. Mariton, M., Bertrand, P.: A homotopy algorithm for solving coupled Riccati equations. Optimal Control Appl. Methods 6, 351–357 (1985)

    MathSciNet  Google Scholar 

  41. Mathworks, MATLAB User’s Guide (2010)

  42. Mehrmann, V.L.: The Autonomous Linear Quadratic Control Problem, Lecture Notes in Control and Information Sciences, vol. 163. Springer, Berlin (1991)

    Google Scholar 

  43. Qian, Y.-Y., Pang, W.-J.: An implicit sequential algorithm for solving coupled Lyapunov equations of continuous-time Markovian jump systems. Automatica 60, 245–250 (2015)

    MathSciNet  Google Scholar 

  44. Rajasingam, P., Xu, J.: On the convergence of the accelerated Riccati iteration method. Int. J. Control 93, 1800–1809 (2020)

    MathSciNet  Google Scholar 

  45. Shirilord, A., Dehghan, M.: Combined real and imaginary parts method for solving generalized Lyapunov matrix equation. Appl. Numer. Math. 181, 94–109 (2022)

    MathSciNet  Google Scholar 

  46. Shirilord, A., Dehghan, M.: Closed-form solution of non-symmetric algebraic Riccati matrix equation. Appl. Math. Lett. 131, 108040 (2022)

    MathSciNet  Google Scholar 

  47. Shi, P., Li, F.: A survey on Markovian jump systems: modeling and design. Int. J. Control Automat. Syst. 13, 1–16 (2015)

    Google Scholar 

  48. Sworder, D.D.: Feedback control of a class of linear systems with jump parameters. IEEE Trans. Automat. Control 14, 9–14 (1969)

    MathSciNet  Google Scholar 

  49. Tian, Z., Fan, C.M., Deng, Y., Wen, P.H.: New explicit iteration algorithms for solving coupled continuous Markovian jump Lyapunov matrix equations. J. Frankl. Inst. 355, 8346–8372 (2018)

    MathSciNet  Google Scholar 

  50. Wang, Q., Lam, J., Wei, Y., Chen, T.: Iterative solutions of coupled discrete Markovian jump Lyapunov equations. Comput. Math. Appl. 55, 843–850 (2008)

    MathSciNet  Google Scholar 

  51. Wonham, W.M.: Random differential equations in control theory. In: Bharucha-Reid, A.T. (Ed.) Probabilistic Methods in Applied Mathematics, 2, 131–212, Academic Press, New York (1970)

  52. Wu, A.G., Duan, G.R.: New iterative algorithms for solving coupled Markovian jump Lyapunov equations. IEEE Trans. Automat. Control 60, 289–294 (2015)

    MathSciNet  Google Scholar 

  53. Wu, A.-G., Wang, X., Sreeram, V.: Iterative algorithms for solving continuous stochastic Lyapunov equations. IET Control Theory Appl. 11, 73–80 (2017)

    MathSciNet  Google Scholar 

  54. Wu, A.-G., Sun, H.-J., Zhang, Y.: a novel iterative algorithms for solving coupled Riccati equations. Appl. Math. Comput. 364, 124645 (2020)

    MathSciNet  Google Scholar 

  55. Zhang, L., Fan, H.-Y., Chu, E.K.-W., Wei, Y.: Homotopy for rational Riccati equations arising in stochastic control. SIAM J. Sci. Comput. 37, B103–B125 (2015)

    MathSciNet  Google Scholar 

  56. Zhang, J., Liu, J., Luo, F.: A class of fixed point iteration for the coupled algebraic Riccati equation. J. Appl. Math. Comput. 68, 4119–4133 (2022)

    MathSciNet  Google Scholar 

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Acknowledgements

Part of the work occurred when the first author visited Monash University and the second author visited the School of Mathematical Sciences at Fudan University and the Department of Mathematics at the National Taiwan Normal University. The first author is supported by the National Natural Science Foundation of China (No. 12001146) and the Natural Science Foundation of Zhejiang Province (No. LQ21A010006). The second author is supported by Fudan University and the Ministry of Science and Technology of China under grant G2023132005L.

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  1. Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou, 310000, People’s Republic of China

    Ting-Ting Feng

  2. School of Mathematics, Monash University, Melbourne, VIC, 3800, Australia

    Eric King-Wah Chu

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Feng, TT., Chu, E.KW. Newton’s method for coupled continuous-time algebraic Riccati equations. J. Appl. Math. Comput. 70, 1023–1042 (2024). https://doi.org/10.1007/s12190-024-01990-z

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