Research Article  Open Access
Li Wang, "An Improved Iterative Method for Solving the Discrete Algebraic Riccati Equation", Mathematical Problems in Engineering, vol. 2020, Article ID 3283157, 6 pages, 2020. https://doi.org/10.1155/2020/3283157
An Improved Iterative Method for Solving the Discrete Algebraic Riccati Equation
Abstract
The discrete algebraic Riccati equation has wide applications, especially in networked systems and optimal control systems. In this paper, according to the damped Newton method, two iterative algorithms with a stepsize parameter is proposed to solve the discrete algebraic Riccati equation, one of which is an extension of Algorithm (4.1) in Dai and Bai (2011). A numerical example demonstrates the convergence effect of the presented algorithm.
1. Introduction and Preliminaries
The discrete algebraic Riccati equation plays an important part in engineering, such as optimal control systems [1], modified filtering [2, 3], and networked systems[4–7]. Consider the following discretetime linear system:where is the state variable, is the input variable, is the input matrix, and is the system matrix and is always invertible [8]. The optimal state feedback controller of (1) iswhich minimizes the quadratic performance index of (1) and is closely related to the discrete algebraic Riccati equation (DARE):where is semipositive definite, is positive definite, and is the positive definite solution of the DARE (3). Let . According to the matrix identity,equation (3) can be transformed into
Due to the wide applications of the DARE, many works have been proposed to discuss the DARE. Various bounds and solutions about the DARE have been provided, such as upper and lower solution bounds [9–14], bounds about sum and product of eigenvalues [15, 16], determinant of the solution [17], and the existence of the solution [18–21]. However, in an optimal control system, we often need to compute the solution of the DARE to find the optimal state feedback controller which minimizes the quadratic performance index. It is very difficult to solve the DARE, especially when the dimensions of the coefficient matrices are high. So, many researchers provide a lot of iterative methods to solve this equation. Komaroff present a fixedpoint iterative algorithm that needs to compute twice matrix inversion at each step [22]. By Newton’s method, Guo derived the maximal symmetric solution of the DARE in [23]. The structurepreserving doubling algorithms are discussed in [24–27]. The Schur method is adopted to solve algebraic Riccati equations [28]. Recently, Dai and Bai propose an iterative algorithm that partially avoids computing the matrix inversions by making use of the Schulz iteration [29].
In Section 2, we propose two iterative algorithms with a stepsize parameter to solve the DARE by the damped Newton method. One of the iterative algorithms is an extension of Algorithm 4.1 in [29]. Numerical example is given in Section 3 to demonstrate the convergence effect of our algorithms.
We first introduce some symbol conventions. denotes the real number field. denotes the set of real matrices. For , let , , , and denote the transpose, inverse, spectral norm, and the minimal eigenvalue of the matrix , respectively. The inequality means X is a symmetric positive (semi) definite matrix; and the inequality means is a symmetric positive (semi) definite matrix. The identity matrix with appropriate dimensions is represented by .
Lemma 1. (see [30]). If are symmetric positive definite matrices, then
Lemma 2. (see [31]). Let and be Hermitian matrices of the same order and let . Then,
Lemma 3. (see [32]). Let be symmetric positive definite matrices. Then,
2. Improved Iterative Algorithms for Solving the DARE
To find the positive definite solution of the DARE (5), Dai and Bai, in [29], proposed an algorithm that partially avoids computing the matrix inversion as follows.
Algorithm 1. (see [29]). Take . For , computeIn this section, we propose two iterative algorithms to solve the DARE (5), which are motivated by the damped Newton method [33] and the methods in [34, 35]. Let us recall the damped Newton method to find the root of :where is a stepsize parameter. If the initial matrix is near the solution of the problem, the unit stepsize can be accepted in the local Newton method. However, it is not suitable to choose if the initial matrix is far from the solution of the problem [33].
The DARE (5) can be translated into , whereLet and . Then, to find the root of is equivalent to find the root of , we can solve the DARE (5) by constructing an iterative scheme. According to (10), we present the following iterative algorithms for the DARE (5).
Algorithm 2. Step 1: set , and . Step 2: compute
Algorithm 3. Step 1: set , and . Step 2: computeAbout Algorithms 2 and 3, we have the following results.
Theorem 1. Let be the positive definite solution of the DARE (5) and . The iterative sequences and are generated by Algorithm 2 with ; then,
Proof. We first prove and are monotone increasing by induction. Since is positive definite solution of DARE (5), thenThus, .(i)Since then by Lemma 1, we obtain From (17), we also obtain ; then, By Lemma 2 and (20), we have By (20) and Lemma 1, we obtain thereby, By (21), we obtain Thus, from the abovementioned proof, we have(ii)Assume thatFrom (26), we get ; then,Thus,By (28) and (29), we haveSo, we obtainThus, the proof of induction is completed. Moreover, as and are monotone increasing and they are bounded, then and exist. Taking limits in Algorithm 2 gives and .
Theorem 2. Let be the positive definite solution of the DARE (5). After steps of iteration for Algorithm 2, we have ; then,
Proof. According to (15), we haveThen, by Algorithm 2, Lemma 3, and (34), we obtainbecause of .
As the proof method is similar to Theorem 1, we list the monotonicity and convergence of Algorithm 3 without proof.
Theorem 3. Let be the positive definite solution of the DARE (5) and . The iterative sequences and are generated by Algorithm 3 with and start from and ; then, is monotone increasing and converges to , and is monotone increasing and converges to .
Remark 1. For Algorithms 2 and 3, we find the steps of iteration for Algorithm 2 are less than Algorithm 3 and the convergence speed of the Algorithm 2 is faster than Algorithm 3 from the numerical examples. Therefore, in the following example, we only discuss the superiority and effectiveness of Algorithm 2.
3. Numerical Examples
In this section, we present the following numerical example to show the effectiveness of our results. We also discuss the performance of Algorithm 2 with different values. The whole process is carried out on Matlab 7.1 and the precision is .
Example 1. Consider the discrete system (1) withIn [29], Dai and Bai choose the starting matrix . After 17 steps of iteration, the required precision is derived, and the residual is .
For Algorithm 2, we choose and give the steps of iteration and the residual as Table 1 with a different parameter when the process is stopped under the required precision. When is near 1, we find that the steps of iteration are less than [29]. Especially, when , it only needs 10 steps for Algorithm 2 to converge to the iterative solution:with the residual , and Algorithm 2 has faster convergence speed than Algorithm 1 from Figure 1. Moreover, from Table 1, we see that Algorithm 2 is more efficient when . Although we only prove the convergence of Algorithm 2 when , in this paper, Algorithm 2 works well in practical computation when .

Data Availability
All data generated or analyzed during this study are included within this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported in part by the National Natural Science Foundation for Youths of China (11801164), National Natural Science Foundation of China (11971413), Key Project of National Natural Science Foundation of China (91430213), General Project of Hunan Provincial Natural Science Foundation of China (2015JJ2134), and the General Project of Hunan Provincial Education Department of China (15C1320).
References
 H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, WileyInterscience, New York, NY, USA, 1972.
 L. Wu and D. W. C. Ho, “Fuzzy filter design for Ito stochastic systems with application to sensor fault detection,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 1, pp. 233–242, 2009. View at: Google Scholar
 X. Su, P. Shi, L. Wu, and Y. D. Song, “A novel approach to filter design for TCS fuzzy discretetime systems with timevarying delay,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 6, pp. 1114–1129, 2012. View at: Google Scholar
 W. N. Anderson, T. D. Morley, and G. E. Trapp, “Ladder networks, fixpoints, and the geometric mean,” Circuits, Systems and Signal Processing, vol. 2, no. 3, pp. 259–268, 1983. View at: Publisher Site  Google Scholar
 M. Z. Q. Chen, L. Zhang, H. Su, and G. Chen, “Stabilizing solution and parameter dependence of modified algebraic Riccati equation with application to discretetime network synchronization,” IEEE Transactions on Automatic Control, vol. 61, no. 1, pp. 228–233, 2016. View at: Publisher Site  Google Scholar
 H. S. Su, H. Wu, and J. Lam, “Positive edgeconsensus for nodal networks via output feedback,” IEEE Transactions on Automatic Control, vol. 64, no. 3, pp. 1224–1249, 2019. View at: Publisher Site  Google Scholar
 M. Lan and S. Chand, “Solving linear quadratic discretetime optimal controls usingneural networks,” in Proceedings of the IEEE Conference on Decision & Control, IEEE, Honolulu, HI, USA, December 1990. View at: Publisher Site  Google Scholar
 R. A. Kennedy, “Linear system theory,” Automatica, vol. 30, no. 11, pp. 1811–1813, 1994. View at: Publisher Site  Google Scholar
 C.H. Lee, “Matrix bounds of the solutions of the continuous and discrete Riccati equations—a unified approach,” International Journal of Control, vol. 76, no. 6, pp. 635–642, 2003. View at: Publisher Site  Google Scholar
 H. H. Choi, “Upper matrix bounds for the discrete algebraic Riccati matrix equation,” IEEE Transactions on Automatic Control, vol. 46, no. 3, pp. 504–508, 2001. View at: Google Scholar
 R. Davies, P. Shi, and R. Wiltshire, “New upper solution bounds of the discrete algebraic Riccati matrix equation,” Journal of Computational and Applied Mathematics, vol. 213, no. 2, pp. 307–315, 2008. View at: Publisher Site  Google Scholar
 J. Liu, L. Wang, and J. Zhang, “The solution bounds and fixed point iterative algorithm for the discrete coupled algebraic Riccati equation applied to automatic control,” IMA Journal of Mathematical Control and Information, pp. 1–22, 2016. View at: Google Scholar
 J. Liu, L. Wang, and J. Zhang, “New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation,” International Journal of Control, vol. 90, no. 11, pp. 2326–2337, 2017. View at: Publisher Site  Google Scholar
 N. Komaroff, “Upper bounds for the solution of the discrete Riccati equation,” IEEE Transactions on Automatic Control, vol. 37, no. 9, pp. 1370–1373, 1992. View at: Publisher Site  Google Scholar
 J. Zhang, J. Liu, and Y. Zha, “The improved eigenvalue bounds for the solution of the discrete algebraic Riccati equation,” IMA Journal of Mathematical Control and Information, pp. 1–20, 2016. View at: Google Scholar
 N. Komaroff and B. Shahian, “Lower summation bounds for the discrete Riccati and Lyapunov equations,” IEEE Transactions on Automatic Control, vol. 37, no. 7, pp. 1078–1080, 1992. View at: Publisher Site  Google Scholar
 M. T. Tran and M. E. Sawan, “On the discrete Riccati matrix equation,” SIAM Journal on Algebraic Discrete Methods, vol. 6, no. 1, pp. 107108, 1985. View at: Publisher Site  Google Scholar
 J. Liu and J. Zhang, “The existence uniqueness and the fixed iterative algorithm of the solution for the discrete coupled algebraic Riccati equation,” International Journal of Control, vol. 84, no. 8, pp. 1430–1441, 2011. View at: Publisher Site  Google Scholar
 R. Huang, J. Z. Liu, and L. Zhu, “Accurate solutions of diagonally dominant tridiagonal linear systems,” BIT Numerical Mathematics, vol. 54, no. 3, pp. 711–727, 2014. View at: Google Scholar
 Q. H. Liu, X. X. Li, and J. Yan, “On the large time behaviour of solutions for a class of timedependent HamiltonJacobi equations,” Science China Mathematics, vol. 59, no. 5, pp. 875–8890, 2016. View at: Publisher Site  Google Scholar
 Z.H. He, “Some new results on a system of Sylvestertype quaternion matrix equations,” Linear and Multilinear Algebra, pp. 1–23, 2019. View at: Publisher Site  Google Scholar
 N. Komaroff, “Iterative matrix bounds and computational solutions to the discrete algebraic Riccati equation,” IEEE Transactions on Automatic Control, vol. 39, no. 8, pp. 1676–1678, 1994. View at: Publisher Site  Google Scholar
 C.H. Guo, “Newton’s method for discrete algebraic Riccati equations when the closedloop matrix has eigenvalues on the unit circle,” SIAM Journal on Matrix Analysis and Applications, vol. 20, no. 2, pp. 279–294, 1998. View at: Publisher Site  Google Scholar
 W.W. Lin and S.F. Xu, “Convergence analysis of structurepreserving doubling algorithms for riccatitype matrix equations,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 1, pp. 26–39, 2006. View at: Publisher Site  Google Scholar
 E. K.W. Chu, H.Y. Fan, W.W. Lin, and C.S. Wang, “Structurepreserving algorithms for periodic discretetime algebraic Riccati equations,” International Journal of Control, vol. 77, no. 8, pp. 767–788, 2004. View at: Publisher Site  Google Scholar
 L.Z. Lu, W.W. Lin, and C. E. M. Pearce, “An efficient algorithm for the discretetime algebraic Riccati equation,” IEEE Transactions on Automatic Control, vol. 44, no. 6, pp. 1216–1220, 1999. View at: Publisher Site  Google Scholar
 T.M. Hwang, E. K.W. Chu, and W.W. Lin, “A generalized structurepreserving doubling algorithm for generalized discretetime algebraic Riccati equations,” International Journal of Control, vol. 78, no. 14, pp. 1063–1075, 2005. View at: Publisher Site  Google Scholar
 A. Laub, “A Schur method for solving algebraic Riccati equations,” IEEE Transactions on Automatic Control, vol. 24, no. 6, pp. 913–921, 1979. View at: Publisher Site  Google Scholar
 H. Dai and Z. Z. Bai, “On eigenvalue bounds and iteration methods for discrete algebraic Riccati equations,” Journal of Computational Mathematics, vol. 29, no. 3, pp. 341–366, 2011. View at: Google Scholar
 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 2012.
 X. Zhan, “Computing the extremal positive definite solutions of a matrix equation,” SIAM Journal on Scientific Computing, vol. 17, no. 5, pp. 1167–1174, 1996. View at: Publisher Site  Google Scholar
 A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, NY, USA, 1979.
 W. Sun and Y. Yuan, Optimization Theory and Methods, Springer Science and Business Media, LLC, New York, NY, USA, 2006.
 M. Monsalve and M. Raydan, “A new inversionfree method for a rational matrix equation,” Linear Algebra and Its Applications, vol. 433, no. 1, pp. 64–71, 2010. View at: Publisher Site  Google Scholar
 G. Schulz, “Iterative Berechung der reziproken Matrix,” ZAMM—Zeitschrift für Angewandte Mathematik und Mechanik, vol. 13, no. 1, pp. 57–59, 1933. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2020 Li Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.