A Note on Riccati Matrix Difference Equations
Pages 1393 - 1409
Abstract
Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications; however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a self-contained study of discrete time Riccati matrix difference equations. In particular, we provide a novel Riccati semigroup duality formula and a new Floquet-type representation for these equations. Due to the aperiodicity of the underlying flow of the solution matrix, conventional Floquet theory does not apply in this setting and thus further analysis is required. We illustrate the impact of these formulae with an explicit description of the solution of time-varying Riccati difference equations and its fundamental-type solution in terms of the fixed point of the equation and an invertible linear matrix map as well as uniform upper and lower bounds on the Riccati maps. These are the first results of this type for time-varying Riccati matrix difference equations.
References
[1]
H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, Basel, Switzerland, 2012.
[2]
M. S. Bartlett, An inverse matrix adjustment arising in discriminant analysis, Ann. Math. Stat., 22 (1951), pp. 107--111.
[3]
A. N. Bishop and P. Del Moral, On the stability of Kalman--Bucy diffusion processes, SIAM J. Control Optim., 55 (2017), pp. 4015--4047.
[4]
A. N. Bishop and P. Del Moral, An explicit Floquet-type representation of Riccati aperiodic exponential semigroups, Internat. J. Control, 94 (2021), pp. 258--266.
[5]
A. N. Bishop, P. Del Moral, and A. Niclas, A perturbation analysis of stochastic matrix Riccati diffusions, Ann. Inst. Henri Poincaré Probab. Statist., 56 (2020), pp. 884--916.
[6]
R. R. Bitmead, M. R. Gevers, I. R. Petersen, and R. J. Kaye, Monotonicity and stabilizability-properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples, Systems Control Lett., 5 (1985), pp. 309--315.
[7]
S. Bittanti and P. Colaneri, Floquet theory and stability, in Periodic Systems: Filtering and Control, Springer, London, 2009, pp. 81--108.
[8]
The Riccati Equation, S. Bittanti, A. J. Laub, and J. C. Willems, eds., Springer Science & Business Media, Springer, Berlin, Heidelberg, 2012.
[9]
R. S. Bucy and J. Rodriguez-Canabal, A negative definite equilibrium and its induced cone of global existence for the Riccati equation, SIAM J. Math. Anal., 3 (1972), pp. 644--646.
[10]
M. K. Camlibel, Popov--Belevitch--Hautus type controllability tests for linear complementarity systems, Systems Control Lett., 56 (2007), pp. 381--387.
[11]
J. J. DaCunha and J. M. Davis, A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems, J. Differential Equations, 251 (2011), pp. 2987--3027.
[12]
I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), pp. 227--263.
[13]
P. Del Moral and E. Horton, A Theoretical Analysis of One-Dimensional Discrete Generation Ensemble Kalman Particle Filters, preprint, arXiv:2107.01855 [math.PR], 2021.
[14]
P. Del Moral and A. Niclas, A Taylor expansion of the square root matrix function, J. Math. Anal. Appl., 465 (2018), pp. 259--266.
[15]
C. Y. Deng, A generalization of the Sherman--Morrison--Woodbury formula, Appl. Math. Lett., 24 (2011), pp. 1561--1564.
[16]
J. J. Deyst and C. F. Price, Conditions for asymptotic stability of the discrete minimum-variance linear estimator, IEEE Trans. Automat. Control, 13 (1968), pp. 702--705.
[17]
A. Ferrante and L. Ntogramatzidis, Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: The discrete-time case, Systems Control Lett., 54 (2005), pp. 693--703.
[18]
G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Sci. Éc. Norm. Supér., 12 (1883), pp. 47--88.
[19]
H. V. Henderson and S. R. Searle, On deriving the inverse of a sum of matrices, SIAM Rev., 23 (1981), pp. 53--60.
[20]
L. Guttmann, Enlargement methods for computing the inverse matrix, Ann. Math. Statist., 17 (1946), pp. 336--343.
[21]
W. W. Hager, Updating the inverse of a matrix, SIAM Rev., 31 (1989), pp. 221--239.
[22]
K. L. Hitz, T. E. Fortmann, and B. D. O. Anderson, A note on bounds on solutions of the Riccati equation, IEEE Trans. Automat. Control., 17 (1972), p. 178.
[23]
A. H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, Cambridge, MA, 1970.
[24]
T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation, Prentice Hall, Hoboken, NJ, 2000.
[25]
R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mex., 5 (1960), pp. 102--119.
[26]
R. E. Kalman, Further Remarks on: “A Note on Bounds on Solutions of the Riccati Equation'', IEEE Trans. Automat. Control, 17 (1972), pp. 179--180.
[27]
G. Kitagawa, An algorithm for solving the matrix equation $X= FXF'+ S$, Internat. J. Control, 25 (1977), pp. 745--753.
[28]
V. Kucera, The discrete Riccati equation of optimal control, Kybernetika, 8 (1972), pp. 430--447.
[29]
V. Kucera, A review of the matrix Riccati equation, Kybernetika, 9 (1973), pp. 42--61.
[30]
P. Lancaster and L. Rodman, Algebraic Riccati equations, Oxford University Press, Oxford, UK, 1995.
[31]
P. S. Maybeck, Stochastic Models, Estimation and Control: Volume 1. Academic Press, Cambridge, MA, 1979.
[32]
K. M. Mikkola, Riccati Equations and Optimal Control of Well-Posed Linear Systems, preprint, arXiv:1602.08618v1 [math.02], 2016.
[33]
A. Prach, Faux Riccati Equation Techniques for Feedback Control of Nonlinear and Time-Varying Systems, Ph.D. thesis, Middle East Technical University, 2015.
[34]
A. Prach, O. Tekinalp, and D. S. Bernstein, Infinite-horizon linear-quadratic control by forward propagation of the differential Riccati equation, IEEE Control Syst., 35 (2015), pp. 78--93.
[35]
G. C. Rota and W. G. Strang, A note on the joint spectral radius, Indag. Math., 63 (1960), pp. 379--381.
[36]
M. Rhudy, Y. Gu, and M. Napolitano, Relaxation of initial error and noise bounds for stability of GPS/INS attitude estimation, in AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, 2012.
[37]
J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statist., 21 (1950), pp. 124--127.
[38]
J. Sreedhar and P. Van Dooren, Discrete-time periodic systems: A Floquet approach, in Proceedings of the Conference on Information Sciences and Systems, 1993, pp. 194--195.
[39]
J. N. Tsitsiklis and V. D. Blondel, The Lyapunov exponent and joint spectral radius of pairs of matrices are hard-when not impossible-to compute and to approximate, Math. Control Signals Systems, 10 (1997), pp. 31--40.
[40]
H. K. Wimmer, Hermitian solutions of the discrete-time algebraic Riccati equation, Internat. J. Control, 63 (1996), pp. 921--936.
[41]
H. K. Wimmer, The set of positive semidefinite solutions of the algebraic Riccati equation of discrete-time optimal control, IEEE Trans. Automat. Control, 41 (1996), pp. 660--671.
[42]
H. K. Wimmer and M. Pavon, A comparison theorem for matrix Riccati difference equations, Systems Control Lett., 19 (1992), pp. 233--239.
[43]
M. A. Woodbury, Inverting modified matrices, Memorandum Report 42, Statistical Research Group, Princeton University, Princeton, NJ, 1950.
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DOI:10.1137/sjcodc.60.3
Issue’s Table of Contents© 2022, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2022
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