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Linear-quadratic optimal control for discrete-time stochastic descriptor systems
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 |
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.
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. |
show all references
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. |


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