Linear-quadratic optimal control for discrete-time stochastic descriptor systems

Yadong Shu; Bo Li

doi:10.3934/jimo.2021034

Article Contents

2022, Volume 18, Issue 3: 1583-1602. Doi: 10.3934/jimo.2021034

This issue Previous Article Pricing and energy efficiency decisions by manufacturer under channel coordination Next Article Effects of take-back legislation on pricing and coordination in a closed-loop supply chain

Linear-quadratic optimal control for discrete-time stochastic descriptor systems

Yadong Shu^1,2, , and
Bo Li^1,2,

1.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2.
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China

^* Corresponding author
^* Corresponding author

Received: September 30, 2020

Revised: November 30, 2020

Published: May 2022

Abstract Full Text(HTML) Figure(2) / Table(1) Related Papers Cited by

Abstract

In this paper, an optimal control model ruled by a class of linear discrete-time stochastic descriptor systems is considered under quadratic index performance. Employing dynamic programming method, a recurrence equation to simplify the optimal control problem is presented provided that the descriptor systems are both regular and impulse-free. When the objective function is quadratic, according to the recurrence equation, a discrete-time linear-quadratic optimal control problem is completely settled, that is, optimal controls and optimal values of the problem are both obtained through analytical expressions. At last, a numerical example about linear-quadratic optimal control for a discrete-time stochastic descriptor system is provided to illustrate the validness of the results derived.

Keywords:

Mathematics Subject Classification: 49N10, 93E20.

Citation:

Full Text(HTML)

Figure 1. Trajectories of optimal controls $ u^*(k) $ for problem (11)

Download: Full-size image PowerPoint slide

Figure 2. Trajectories of state vectors $ x(k) $ for problem (11)

Download: Full-size image PowerPoint slide

Table 1. The optimal results of problem (11)

Stage	$ (u^{*}(k))^T $	$ \mu_k $	$ x_1^T(k) $	$ x^T(k) $	$ J\big(k, x_1(k), x_2(7)\big) $
0	$ (3.567, -2.323, -5.406) $	$ 2.517 $	$ (1, 2) $	$ (1.000, -5.434, 2.000, -4.600) $	$ -112.722 $
1	$ (0.338, -2.541, -0.797) $	$ 1.743 $	$ (-4.756, 0.145) $	$ (-4.756, -2.699, 0.145, -4.756) $	$ -55.708 $
2	$ (1.796, -2.599, -0.303) $	$ 2.098 $	$ (-1.416, -4.148) $	$ (-1.416, -5.019, -4.148, -1.218) $	$ 20.114 $
3	$ (0.167, -1.651, 0.469) $	$ 0.202 $	$ (-1.429, -1.614) $	$ (-1.429, -1.769, -1.614, 0.490) $	$ 41.636 $
4	$ (0.635, -1.194, 0.582) $	$ 1.165 $	$ (-0.931, -2.843) $	$ (-0.931, -2.743, -2.843, -0.320) $	$ 39.171 $
5	$ (0.862, -0.338, -0.251) $	$ 0.872 $	$ (-0.362, -0.430) $	$ (-0.362, -1.988, -0.430, -1.147) $	$ 32.059 $
6	$ (0.390, -0.274, -0.403) $	$ 1.394 $	$ (-0.505, 0.182) $	$ (-0.505, -2.598, 0.182, -2.027) $	$ 46.328 $
7	$ u^{*}(7)\in R^3 $		$ (-1.217, 1.619) $	$ (-1.217, 0.598, 1.619, -0.217) $	$ 7.363 $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	K. Avrachenkov, O. Habachi, A. Piunovskiy and Y. Zhang, Infinite horizon optimal impulsive control with applications to Internet congestion control, International Journal of Control, 88 (2015), 703-716. doi: 10.1080/00207179.2014.971436.
[2]	A. V. Balakrishnan, On stochastic bang-bang control, Applied Mathematics and Optimization, 6 (1980), 91-96. doi: 10.1007/BF01442885.
[3]	M. Basin and and J. Rodriguez-Gonzalez, Optimal control for linear systems with time delay in control input, IEEE Transactions on Automatic Control, 51 (2006), 91-97. doi: 10.1109/TAC.2005.861718.
[4]	A. Cairns, Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time, Astin Bulletin, 30 (2000), 19-55. doi: 10.2143/AST.30.1.504625.
[5]	S. L. Campbell, Singular Systems of Differential Equations II, Pitman, Boston, Mass.-London, 1982.
[6]	M. Chadli, H. R. Karimi and P. Shi, On stability and stabilization of singular uncertain Takagi-Sugeno fuzzy systems, Journal of the Franklin Institute, 351 (2014), 1453-1463. doi: 10.1016/j.jfranklin.2013.11.008.
[7]	L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074-1080. doi: 10.1016/j.automatica.2010.03.005.
[8]	D. Cobb, Controllability, observability, and duality in singular systems, IEEE Transactions on Automatic Control, 29 (1984), 1076-1082. doi: 10.1109/TAC.1984.1103451.
[9]	L. Dai, Singular Control Systems, Springer-Verlag, Berlin, Germany, 1989. doi: 10.1007/BFb0002475.
[10]	J.-e Feng, P. Cui and Z. Hou, Singular linear quadratic optimal control for singular stochastic discrete-time systems, Optimal Control Applications and Methods, 34 (2013), 505-516. doi: 10.1002/oca.2033.
[11]	J. Y. Ishihara and M. H. Terra, On the Lyapunov theorem for singular systems, IEEE Transactions on Automatic Control, 47 (2002), 1926-1930. doi: 10.1109/TAC.2002.804463.
[12]	R. E. Kalman, Contribution to the theory of optimal control, Boletin Sociedad Matem$\acute{a}$tica Mexicana, 5 (1960), 102–119.
[13]	Y. Kang and Y. Zhu, Bang-bang optimal control for multi-stage uncertain systems, International Journal on Information, 15 (2012), 3229-3237.
[14]	D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2012.
[15]	N. Kumaresan and P. Balasubramaniam, Optimal control for stochastic nonlinear singular system using neural networks, Computers and Mathematics with Applications, 56 (2008), 2145-2154. doi: 10.1016/j.camwa.2008.03.041.
[16]	P. Kunkel and V. Mehrmann, The linear quadratic optimal control problem for linear descriptor systems with variable coefficients, Mathematics of Control, Signals, and Systems, 10 (1997), 247-264. doi: 10.1007/BF01211506.
[17]	B. Li and and Y. Zhu, The piecewise parametric optimal control of uncertain linear quadratic models, International Journal of Systems Science, 50 (2019), 961-969. doi: 10.1080/00207721.2019.1586003.
[18]	F. Liao, Z. Ren, M. Tomizuka and J. Wu, Preview control for impulse-free continuous-time descriptor systems, International Journal of Control, 88 (2015), 1142-1149. doi: 10.1080/00207179.2014.996769.
[19]	J. Lian, C. Li and D. Liu, Input-to-state stability for discrete-time nonlinear switched singular systems, IET Control Theory Appl., 11 (2017), 2893–2899. doi: 10.1049/iet-cta.2017.0028.
[20]	W. Liu, X. Liang, Y. Ma and W. Liu, Aircraft trajectory optimization for collision avoidance using stochastic optimal control, Asian Journal of Control, 21 (2019), 2308-2320. doi: 10.1002/asjc.1855.
[21]	D. G. Luenberger, Dynamic equations in descriptor form, IEEE Transactions on Automatic Control, 22 (1977), 312-321. doi: 10.1109/tac.1977.1101502.
[22]	D. G. Luenberger and A. Arbel, Singular dynamic Leontief systems, Econometrica, 45 (1977), 991-995. doi: 10.2307/1912686.
[23]	G. P. Mantas and N. J. Krikelis, LQ optimal control for discrete descriptor systems with a complete set of boundary information, International Journal of Control, 47 (1988), 1467-1477. doi: 10.1080/00207178808906108.
[24]	D. S. Naidu, Singular perturbations and time scales in control theory and applications: An overview., Dynamics of Continuous Discrete & Impulsive Systems Ser. B Appl. Algorithms, 9 (2002), 233-278.
[25]	H. Peng, Fei. Li, J. Liu and Z. Ju, A symplectic instantaneous optimal control for robot trajectory tracking with differential-algebraic equation models, IEEE Transactions on Industrial Electronics, 67 (2020), 3819-3829. doi: 10.1109/TIE.2019.2916390.
[26]	Y. Shu and Y. Zhu, Optimistic value based optimal control for uncertain linear singular systems and application to a dynamic input-output model, ISA Transactions, 71 (2017), 235-251. doi: 10.1016/j.isatra.2017.08.007.
[27]	Y. Shu and Y. Zhu, Stability analysis of uncertain singular systems, Soft Computing, 22 (2018), 5671-5681. doi: 10.1007/s00500-017-2599-2.
[28]	K. Wang, L. Xie and Y. Liu, Characteristics of self-excited oscillation in a curved diffuser., Journal of Propulsion Technology, 39 (2018), 1955-1964.
[29]	W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM Journal on Control, 6 (1968), 681-697. doi: 10.1137/0306044.
[30]	H. Yan, Y. Sun and Y. Zhu, A linear-quadratic control problem of uncertain discrete-time switched systems, Journal of Industrial and Management Optimization, 13 (2017), 267-282. doi: 10.3934/jimo.2016016.
[31]	W. Zhang, J. Hu and A. Abate, On the value function of the discrete-time switched LQR problem, IEEE Transactions on Automatic Control, 54 (2009), 2669-2674. doi: 10.1109/TAC.2009.2031574.
[32]	W. Zhang, Y. Lin and L. Xue, Linear quadratic Pareto optimal control problem of stochastic singular systems, Journal of the Franklin Institute, 354 (2017), 1220-1238. doi: 10.1016/j.jfranklin.2016.11.021.