Research Article  Open Access
Li Wang, "Numerical Algorithms of the Discrete Coupled Algebraic Riccati Equation Arising in Optimal Control Systems", Mathematical Problems in Engineering, vol. 2020, Article ID 1841582, 8 pages, 2020. https://doi.org/10.1155/2020/1841582
Numerical Algorithms of the Discrete Coupled Algebraic Riccati Equation Arising in Optimal Control Systems
Abstract
The discrete coupled algebraic Riccati equation (DCARE) has wide applications in robust control, optimal control, and so on. In this paper, we present two iterative algorithms for solving the DCARE. The two iterative algorithms contain both the iterative solution in the last iterative step and the iterative solution in the current iterative step. And, for different initial value, the iterative sequences are increasing and bounded in one algorithm and decreasing and bounded in another. They are all monotonous and convergent. Numerical examples demonstrate the convergence effect of the presented algorithms.
1. Introduction and Preliminaries
The discrete coupled Riccati equation is usually encountered in optimal control and filter design problems in control theory [1–9], particularly in the jumplinear quadratic optimal control problem [10]. Consider the following jumplinear system:with initial state , , where is the plant state, is the control vector, and is the process output. Here, is the time index, is the form process taking values in the finite set , and is a finitestate discretetime Markov chain with transition probabilities.
Minimizing the cost criterion of system (1) reduces to solving coupled algebraic Riccatilike equations. After some transformation, the coupled algebraic Riccatilike equations turn the following discrete coupled algebraic Riccati equation (DCARE)where is a constant matrix, , is a symmetric positive definite matrix, , is the coupled term, are real nonnegative constants defined as with the properties , , and , and denotes the symmetric positive definite solution of the DCARE. Applying Woodbury matrix equalityDCARE (3) turns to
Because of the importance of Riccati equations in control theory and control engineering, a lot of research studies about Riccati equations have been devoted to this field, such as solution bounds [11–15], trace and eigenvalue bounds [16–23], and the existence and uniqueness [24–26]. Besides these results, numerical solutions of Riccati equations are very important and have been studied by many scholars [27–34] because the numerical solutions of the Riccati equations are necessary in some practical engineering, such as finding the optimal state feedback controller in the optimal control system. Especially, for the DCARE, fixed point iterative algorithms are given in [24–26]. Stein iterations are presented in [35] which are based on the properties of a Stein equation. Among these results, we find less work has been done to discuss the numerical solution of the DCARE. Considering the importance and necessity of the numerical solutions of the DCARE, we propose two algorithms to discuss the numerical solution of the DCARE.
In this paper, we first propose an iterative algorithm with a parameter for solving the DCARE and prove its monotonically convergence. Second, we give an upper solution bound of the DCARE, by which another iterative algorithm is presented, and the proof of its monotonically convergence is given. For different initial values, the iterative sequences are increasing and bounded in one algorithm and decreasing and bounded in another. Last, numerical examples are given to illustrate the convergence effect of the two algorithms.
We first introduce some symbol conventions. denotes the real number field. denotes the set of real matrices. For , let and denote the transpose and inverse of the matrix , respectively. The inequality means X is a symmetric positive or semidefinite matrix, and the inequality means is a symmetric positive (semi) definite matrix. The identity matrix with appropriate dimensions is represented by .
Lemma 1 (see [36]). If are symmetric positive definite matrices, then
Lemma 2 (see [22]). Let matrices with , , and . Then,with strict inequality if is nonsingular, and .
2. Main Results
In [25, 26], the authors have derived several solution bounds by which iterative algorithms have been proposed, but there are many restrictions in these algorithms. In this part, we first present an iterative algorithm for DCARE (5) which do not have any restrictions.
Algorithm 1. Step 1: set , , , . Step 2: computeFrom Algorithm 1, we get an increasing and bounded iterative sequence, which is convergent to the positive definite solution of DCARE (5).
Theorem 1. Let be the positive definite solution of DCARE (5) and . The iterative sequences and are generated by the iterative (8) with , and then
Proof. Since is positive definite solution of DCARE (5), thenso . Therefore,Since and (12), we getthat is, Suppose thatAccording to (16) and Lemma 2, we getFrom (15), (17), (18), we getThus, the proof of induction is completed. Because and are monotonically increasing and they are bounded, then and exist. As , Algorithm 1 givesThus, .
Theorem 2. Let be the positive definite solution of DCARE (5), then
Proof. If , then by Lemma 1, we getWhen , ; therefore, (22) changes to (21) in this case.
We can choose (21) as starting value and get the following algorithm.
Algorithm 2. Step 1: set , , , . Step 2: compute From Algorithm 2, we get a decreasing and bounded iterative sequences, which is convergent to the positive definite solution of DCARE (5).
Theorem 3. Let be the positive definite solution of DCARE (5) and . The iterative sequences and are generated by iterative (23) with , and then
Proof. According to (21) and (23), we haveSince , with (25) and (26) we getthat is, Suppose that According to (30) and Lemma 2, we getFrom (29), (31), (32), we getThus, the proof of induction is completed. Because and are monotonically decreasing and they are bounded, then and exist. In a similar way as the proof of (20), as , Algorithm 2 gives .
Remark 1. In Algorithm 2, if is singular, we can choose a suitable so that is nonsingular, as in Theorem 2.
Remark 2. In Algorithm 1, the sequence in (8) with the initial value converges monotonically to a positive definite solution of DCARE (5), and so does the sequence in (23) with the initial value . But the two positive definite solutions may be different. Whether the positive definite solution of DCARE (5) is unique or not, a problem needs to be discussed further.
Remark 3. When , DCARE (5) changes to the discrete algebraic Riccati equation. And iterative sequences (8) and (23), respectively, in Algorithm 1 and Algorithm 2, become the iterative (17) and iterative (28) in [22], which means that the algorithms of the DCARE in this paper are generalizations of the discrete algebraic Riccati equation. Moreover, when , the iterative (8) and (23) are extensions on the discrete coupled algebraic Riccati equation of the work of [22].
Remark 4. In this paper, we only prove Algorithms 1 and 2 are convergent under the condition , but we can run the two algorithms with in practical computation. And, we have faster convergence speed if appropriate parameters are selected. We will illustrate it in the following examples.
3. Numerical Examples
In this section, the following numerical examples are presented to show the effectiveness of our results.
Example 1 (see [26]). Consider DCARE (5) withSince there are two equations in the DCARE, the superiority of the in Algorithm 1 is not obvious. So, we choose here. After 9 steps of iteration of (8), we obtain the solution of DCARE (5).and the residual is .
However, it needs 47 steps of iteration for the algorithm in [26] to get the iteration solution of DCARE (5).
Example 2. Consider DCARE (5) withBecause the restrictions of the algorithms in [25, 26] are not met for this case, the algorithms in [25, 26] cannot work.
For Algorithm 1, the steps of iteration and the residual are presented in Table 1 with different parameter . Although we only prove the convergence of Algorithm 1 with , from Table 1, we find the convergence rapid is the fastest when is 1.8. After 31 steps of iteration of (8) with , we obtain the solution of DCARE (5).and the residual is .

Example 3 (see [26]). Consider the DCARE (5) with and are the same as Example 2.
For Algorithm 2, since is singular, by choosing , then Algorithm 2 can work now. After 4 steps of iteration of (23) with , we obtain the solution of DCARE (5):and the residual is .
However, it needs 18 steps of iteration for the algorithm in [26] to get the iteration solution of DCARE (5).
Data Availability
All data generated or analyzed during this study are included in 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), the Key Project of National Natural Science Foundation of China (91430213), the 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. AbouKandil, G. Freiling, and G. Jank, “On the solution of discretetime markovian jump linear quadratic control problems,” Automatica, vol. 31, no. 5, pp. 765–768, 1995. View at: Publisher Site  Google Scholar
 C. Xu, Y. Zhao, B. Qin, and H. Zhang, “Adaptive synchronization of coupled harmonic oscillators under switching topology,” Journal of the Franklin Institute, vol. 356, no. 2, pp. 1067–1087, 2019. View at: Publisher Site  Google Scholar
 H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley, Hoboken, NJ, USA, 1972.
 H. Su, H. Wu, X. Chen et al., “Positive edge consensus of complex networks,” IEEE Transactions on Systems Man & Cybernetics Systems, vol. 48, no. 12, pp. 1–9, 2017. View at: Publisher Site  Google Scholar
 C. Xu, H. Xu, H. Su, and C. Liu, “Disturbanceobserver based consensus of linear multiagent systems with exogenous disturbance under intermittent communication,” Neurocomputing, vol. 404, pp. 26–33, 2020. View at: Publisher Site  Google Scholar
 H. Su, J. Zhang, and X. Chen, “A stochastic sampling mechanism for timevarying formation of multiagent systems with multiple leaders and communication delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 30, no. 12, pp. 3699–3707, 2019. View at: Publisher Site  Google Scholar
 Q. Liu, X. Li, and Y. yan, “Large time behavior of solutions for a class of timedependent hamiltonjacobi equations,” Science China Mathematics, vol. 59, no. 5, pp. 875–890, 2016. View at: Publisher Site  Google Scholar
 Q. Liu, X. Li, and D. Qian, “An abstract theorem on the existence of periodic motions of nonautonomous lagrange systems,” Journal of Differential Equations, vol. 261, no. 10, pp. 5784–5802, 2016. View at: Publisher Site  Google Scholar
 M. A. Rami and L. E. Ghaoui, “LMI optimization for nonstandard riccati equations arising in stochastic control,” IEEE Transactions on Automatic Control, vol. 41, no. 11, pp. 1666–1671, 1996. View at: Publisher Site  Google Scholar
 H. AbouKandil, G. Freiling, and G. Jank, “Solution and asymptotic behavior of coupled riccati equations in jump linear systems,” IEEE Transactions on Automatic Control, vol. 39, no. 8, pp. 1631–1636, 1994. View at: Publisher Site  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, 2008a. View at: Publisher Site  Google Scholar
 R. Davies, P. Shi, and R. Wiltshire, “Upper solution bounds of the continuous and discrete coupled algebraic riccati equations,” Automatica, vol. 44, no. 4, pp. 1088–1096, 2008. 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
 Q. Liu, C. Wang, and Z. Wang, “On littlewood’s boundedness problem for relativistic oscillators with anharmonic potentials,” Journal of Differential Equations, vol. 257, no. 12, pp. 4542–4571, 2014. 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: Publisher Site  Google Scholar
 J. Liu, J. Zhang, and Y. Liu, “Trace inequalities for matrix products and trace bounds for the solution of the algebraic riccati equations,” Journal of Inequalities & Applications, vol. 2009, no. 1, pp. 171–174, 2009. View at: Publisher Site  Google Scholar
 R. Huang and D. Chu, “Relative perturbation analysis for eigenvalues and singular values of totally nonpositive matrices,” Siam Journal on Matrix Analysis & Applications, vol. 36, no. 2, pp. 476–495, 2015. View at: Publisher Site  Google Scholar
 J. Liu and J. Zhang, “New upper and lower eigenvalue bounds for the solution of the continuous algebraic riccati equation,” Asian Journal of Control, vol. 16, no. 1, pp. 284–291, 2014. View at: Publisher Site  Google Scholar
 R. Huang, “A Periodic qdType Reduction for Computing Eigenvalues of Structured Matrix Products to High Relative Accuracy,” Journal of Scientific Computing, vol. 75, no. 3, pp. 1229–1261, 2018. 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, vol. 34, no. 3, pp. 851–870, 2016. 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: 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
 J. Liu and L. Wang, “New solution bounds of the continuous algebraic riccati equation and their applications in redundant control input systems,” Science China Information Sciences, vol. 62, no. 10, pp. 69–85, 2019. 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
 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, vol. 34, no. 4, pp. 1135–1156, 2016. View at: Publisher Site  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
 A. Wu, Y. Zhang, and T. Jiang, “Iterative algorithms for discrete periodic riccati matrix equations,” International Journal of Systems Science, vol. 1, 2019. View at: Google Scholar
 R. Huang, “A qdtype algorithm for computing the generalized singular values of BF matrices with sign regularity to high relative accuracy,” Mathematics of Computation, vol. 89, pp. 229–252, 2020. View at: Publisher Site  Google Scholar
 J. Guan, “Modified alternately linearized implicit iteration method for mmatrix algebraic riccati equations,” Applied Mathematics and Computation, vol. 347, pp. 442–448, 2019. View at: Publisher Site  Google Scholar
 R. Huang and D. Chu, “Computing singular value decompositions of parameterized matrices with total nonpositivity to high relative accuracy,” Journal of Scientific Computing, vol. 71, no. 2, pp. 682–711, 2017. View at: Publisher Site  Google Scholar
 R. Huang, J. 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: Publisher Site  Google Scholar
 R. Huang, “Factoring symmetric totally nonpositive matrices and inverses with a diagonal pivoting method,” BIT Numerical Mathematics, vol. 53, pp. 433–458, 2013. 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
 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
 I. G. Ivanov, “A method to solve the discretetime coupled algebraic riccati equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 34–41, 2008. View at: Publisher Site  Google Scholar
 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 2012.
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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.