Abstract
In this article, new iterative algorithms for solving the discrete Riccati and Lyapunov equations are derived in the case where the transition matrix is diagonalizable with real eigenvalues. It is shown that the proposed iterative algorithms are faster than the classical ones, even for a small number of iterations.
1 Introduction
The discrete time Riccati equation arises by implementing the discrete time Kalman filter, which is the most famous estimation algorithm associated with time invariant models of the form [1]:
for
In these models,
Kalman filter produces iteratively the state prediction with the associated prediction error covariance as well as the state estimation with the associated estimation error covariance. The equations of the discrete time Kalman filter result in the discrete time Riccati equation:
Riccati equation relates two successive values of the prediction error covariance
It is well known [1] that if the system is asymptotically stable, i.e., all eigenvalues of the transition matrix
This limiting solution satisfies the algebraic Riccati equation (or steadystate Riccati equation):
It is worth to note that inverse of the matrix
The nonsingularity of
where
The limiting solution
In the infinite measurement noise case, the discrete time Lyapunov equation is derived:
with a limiting solution
The importance of the Riccati and Lyapunov equations is undoubtable: the Riccati equation plays a very important role in various problems of stochastic filtering, statistics, ladder networks, and dynamic programming [1,2], and has many applications in the process of obtaining optimal control and determining system stability [3,4]; Lyapunov equations play a very important role in the stability theory of discrete systems [1,5].
Significant bibliography exists concerning iterative as well as noniterative solutions of the Riccati and Lyapunov equations [1,2,6,7,8,9,10,11,12,13]. The existence of the unique solution of the Riccati equation and of the Lyapunov equation requires the knowledge of the eigenvalues of the transition matrix. The classical iterative solution of the Riccati equation implements the iterative Riccati equation (3) or the iterative transformed Riccati equation (6) till the steadystate solution is reached. Similarly, the classical iterative solution of the Lyapunov equation implements the iterative Lyapunov equation (7) till the steadystate solution is reached.
The aim of this article is to study the optimal algorithm for solving the discrete Riccati and Lyapunov equations, in the case where the transition matrix is diagonalizable with real eigenvalues. The required condition appears in various application areas, see the eigenvalues of the transition matrix in the radar tracking system [14], in the eye movement prediction model [15], and in the mobile position tracking model [16,17]; the state and parameter estimation has been derived using a transition matrix, which is a diagonal itself [18]. The novelty of this article concerns the exploitation of the knowledge of the eigenvalues and eigenvectors of the transition matrix, which leads to the development of the new algorithms for iterative solving of the Riccati and Lyapunov equations, which are superior to classical algorithms, since the proposed algorithms are in general faster than the classical ones.
This article is organized as follows: In Section 2, new iterative solutions of the Riccati and Lyapunov equations are derived by the eigenvalues and eigenvectors of the transition matrix. In Section 3, the proposed algorithms are compared to the classical ones with respect to their computational burdens. Section 4 summarizes the conclusions.
2 New iterative solution algorithms for Riccati equation and Lyapunov equation
Recall that the model parameters
In the following, we consider that the
Consider the diagonalization formula of the transition matrix F given by
where
It is worth noting that equation (11) is essential for the derivation of the new iterative algorithms for solving the discrete Riccati and Lyapunov equations, since they use the eigenvalues and eigenvectors of the transition matrix instead of the transition matrix itself, reducing the computations due to the similarity of the matrices
Concerning the Riccati equation, by multiplying the lefthand side of equation (3) by
Setting in the latter equality
arises the modified Riccati equation.
It is important to note that the Riccati equation (3) with parameters
Then, the limiting solution of (15) is
and satisfies the modified algebraic Riccati equation:
It is clear that the solution of the Riccati equation (3) can be computed by the solution of the modified Riccati Equation (15) due to (16) and (12):
Concerning the transformed Riccati equation, by multiplying the lefthand side of equation (6) by
Setting in the latter equality
It is important to note that the transformed Riccati equation (6) with parameters
Then, using (16)
arises the limiting solution of (21), which satisfies the algebraic Riccati equation:
It is clear that the solution of the transformed Riccati equation (6) can be computed by the solution of the Riccati equation (22) due to (16) and (19):
Concerning the Lyapunov equation, by multiplying the lefthand side of (9) by
Setting in the latter equality
arises
It is important to note that the Lyapunov equation (9) with parameters
Then, using (16)
arises the limiting solution of (25), which satisfies the modified algebraic Lyapunov equation:
It is clear that the solution of the Lyapunov equation (9) can be computed by the solution of the modified Lyapunov equation (25) due to (16) and (24):
Table 1 summarizes the classical and the derived iterative algorithms for solving the Riccati and Lyapunov equations.
Equation  Algorithm  

Riccati equation  Classical  input:

Proposed  input:


Transformed Riccati equation  Classical  input:

Proposed  input:


Lyapunov equation  Classical  input:

Proposed  input:

3 Comparison of classical and proposed algorithms
It is established that the proposed iterative algorithms for solving the Riccati and Lyapunov equations have been derived by the classical iterative algorithms. Thus, the classical and the derived algorithms are equivalent with respect to their behavior since they calculate theoretically the same solutions executing the same number of iterations
We are going to compare the classical and the derived algorithms with respect to their calculation burdens. Scalar operations are involved in matrix manipulation operations, which are needed for the implementation of the algorithms. Table 2 summarizes the calculation burdens of needed matrix operations. The computation of the eigenvalues is achieved by solving the characteristic equation using the NewtonRaphson method, Laguerre’s method, RegulaFalsi method or Bernoulli’s method, because each method requires different computations for solving the characteristic equation. In the following, we consider that all the algorithms apply the same method; thus the computation of the eigenvalues has the same calculation burden, which is denoted as
Matrix operation  Matrix dimensions  Calculation burden 










^{(1)}






^{(1)}



^{(2)}



^{(2)}



^{(3)}



^{(1)}
^{(2)}
^{(3)} For the general multidimensional case, where
Table 3 summarizes the calculation burdens of the classical and the proposed algorithms, for the general multidimensional case, where
Equation  Algorithm  Calculation burden 

Riccati equation  Classical 

Proposed 


Transformed Riccati equation  Classical 

Proposed 


Lyapunov equation  Classical 

Proposed 

Concerning the Riccati equation solution algorithms, from Table 3, we can write:
Note that in the case
since
The aforementioned notation leads to the conclusion that the proposed algorithm is faster than the classical algorithm for a small number of iterations, especially when the state and measurement dimensions are large.
Figure 1 depicts the faster algorithm with respect to the state dimension
Concerning the transformed Riccati equation solution algorithms, from Table 3, we obtain:
Note that in the case
since
The aforementioned notation leads to the conclusion that the proposed algorithm is faster than the classical algorithm for a small number of iterations. In fact, the proposed algorithm is always faster than the classical algorithm, when
Figure 2 depicts the faster algorithm with respect to the state dimension
Concerning the Lyapunov equation solution algorithms, from Table 3, we obtain:
Note that in the case
since
The aforementioned notation leads to the conclusion that the proposed algorithm is faster than the classical algorithm for a small number of iterations. In fact, the proposed algorithm is always faster than the classical algorithm, when
Figure 3 depicts the faster algorithm with respect to the state dimension
4 Conclusions
Riccati and Lyapunov equations are very important in many fields of science. The existence of their solutions involves the knowledge of the eigenvalues of the transition matrix. We have developed new fast iterative algorithms for solving the Riccati and Lyapunov equations. The proposed algorithms take advantage of the knowledge of the eigenvalues and eigenvectors of the transition matrix. The proposed algorithms hold in the case where the transition matrix is diagonalizable in the form
More specifically,
Concerning the Riccati equation, the proposed solution is faster than the classical one for a small number of iterations, especially when the state and measurement dimensions are large.
Concerning the transformed Riccati equation, the proposed solution is always faster than the classical one when the number of iterations is greater than 7; the proposed algorithm outperforms the classical one when the number of iterations is greater than 4, as the state dimension increases.
Concerning the Lyapunov equation, the proposed solution is always faster than the classical one when the number of iterations is greater than 5; the proposed algorithm outperforms the classical one when the number of iterations is greater than 3, as the state dimension increases.
It is evident that the proposed iterative algorithms outperform the classical ones even for a small number of iterations.
Acknowledgements
The authors thank the anonymous reviewers for their helpful comments.

Funding information: The authors state that no funding was involved.

Author contributions: The authors applied the SDC approach for the sequence of authors. NA: conceptualization and performed the simulation. MA: performed the formal analysis and prepared the manuscript.

Conflict of interest: The authors state no conflict of interest.

Data availability statement: All data generated or analyzed during this study are included in this published article.
Appendix
The calculation burdens of the classical and the proposed algorithms for solving the Riccati and Lyapunov equations, for the general multidimensional case (where
Riccati equation – classical iterative algorithm
Initialization  



Iteration (

































Riccati equation – proposed iterative algorithm
Initialization  


















Iteration (
































Finalization  







Transformed Riccati equation – classical iterative algorithm
Initialization  












Iteration (





















Transformed Riccati equation – proposed iterative algorithm
Initialization  






























Iteration (




















Finalization  







Lyapunov equation – classical iterative algorithm
Initialization  



iteration (












Lyapunov equation – proposed iterative algorithm
Initialization  















Iteration (











Finalization  







References
[1] B. D. O. Anderson and J. B. Moore, Optimal Filtering, Dover Publications, New York, 2005.Search in Google Scholar
[2] N. Assimakis and M. Adam, Fast doubling algorithm for the solution of the riccati equation using cyclic reduction method, 2020 International Conference on Mathematics and Computers in Science and Engineering (MACISE), 2020, pp. 1–5, https://doi.org/10.1109/MACISE49704.2020.00007.10.1109/MACISE49704.2020.00007Search in Google Scholar
[3] D. S. Bernstein, Matrix Mathematics: Theory Facts and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, NJ, 2005.Search in Google Scholar
[4] C. Kojima, K. Takaba, O. Kaneko, and P. Rapisarda, A characterization of solutions of the discretetime algebraic riccati equation based on quadratic difference forms, Linear Algebra Appl. 416 (2006), 1060–1082.10.1016/j.laa.2005.11.027Search in Google Scholar
[5] A. Nakhmani, Modern Control: StateSpace Analysis and Design Methods, McGraw Hill, New York, 2020.Search in Google Scholar
[6] N. D. Assimakis, D. G. Lainiotis, S. K. Katsikas, and F. L. Sanida, A survey of recursive algorithms for the solution of the discrete time Riccati equation, Nonlinear Anal. (1997), no. 4, 2409–2420.10.1016/S0362546X(97)00062XSearch in Google Scholar
[7] D. G. Lainiotis, N. D. Assimakis, and S. K. Katsikas, A new computationally effective algorithm for solving the discrete Riccati equation, J. Math. Anal. Appl. 186 (1994), no. 3, 868–895.10.1006/jmaa.1994.1338Search in Google Scholar
[8] D. G. Lainiotis, N. D. Assimakis, and S. K. Katsikas, Fast and numerically robust recursive algorithms for solving the discrete time Riccati equation: The case of nonsingular plant noise covariance matrix, Neural, Parallel Sci. Comput. 3 (1995), 565–584.Search in Google Scholar
[9] N. Assimakis, S. Roulis, and D. Lainiotis, Recursive solutions of the discrete time riccati equation, Neural, Parallel Sci. Comput. 11 (2003), 343–350.Search in Google Scholar
[10] V. Dragan, The linear quadratic optimization problem for a class of singularly stochastic systems, Int. J. Innov. Comput. Inf. Control 1 (2005), 53–63.Search in Google Scholar
[11] N. Komaroff, Iterative matrix bounds and computational solutions to the discrete algebraic Riccati equation, IEEE Trans. Autom. Control 39 (1994), 1676–1678.10.1109/9.310049Search in Google Scholar
[12] J. Zhang and J. Liu, New upper and lower bounds, the iteration algorithm for the solution of the discrete algebraic Riccati equation, Adv. Differ. Equ. 313 (2015), 1–17, https://doi.org/10.1186/s1366201506496.10.1186/s1366201506496Search in Google Scholar
[13] N. Assimakis and M. Adam, Closed form solutions of Lyapunov equations using the vech and veck operators, WSEAS Trans. Math. 20 (2021), 276–282, https://doi.org/10.37394/23206.2021.20.28.10.37394/23206.2021.20.28Search in Google Scholar
[14] M. S. Grewal and A. P. Andrews, Kalman Filtering: Theory and Practice Using MATLAB, John Wiley and Sons, Inc, New York, 2001.10.1002/0471266388Search in Google Scholar
[15] T. Grindinger, Eye movement analysis & prediction with the Kalman filter, (MS thesis), Clemson University, 2006.Search in Google Scholar
[16] J. P. Dubois, J. S. Daba, M. Nader, and C. El Ferkh, GSM position tracking using a Kalman filter, Int. J. Electron. Commun. Eng. 6 (2012), no. 8, 867–876, https://publications.waset.org/vol/68.Search in Google Scholar
[17] N. Assimakis and M. Adam, Mobile position tracking in three dimensions using Kalman and Lainiotis filters, Open Math. J. 8 (2015), 1–6, http://doi.org/10.2174/1874117701508010001.10.2174/1874117701508010001Search in Google Scholar
[18] R. Zanetti and C. D’Souza, Recursive implementations of the SchmidtKalman ‘consider’ filter, J. of Astronaut. Sci. 60 (2013), no. 3–4, 672–685, http://doi.org/10.1007/s4029501500687.10.1007/s4029501500687Search in Google Scholar
[19] R. W. Farebrother, Linear Least Squares Computations, STATISTICS: Textbooks and Monographs, Marcel Dekker, New York, 1988.Search in Google Scholar
[20] N. Assimakis and M. Adam, Discrete time Kalman and Lainiotis filters comparison, Internat. J. Math. Anal. 1 (2007), no. 13, 635–659.Search in Google Scholar
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