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A linear-quadratic control problem of uncertain discrete-time switched systems
1. | School of Science, Nanjing Forestry University, Nanjing 210037, China |
2. | School of Science, Nanjing University of Science & Technology, Nanjing 210094, China |
This paper studies a linear-quadratic control problem for discrete-time switched systems with subsystems perturbed by uncertainty. Analytical expressions are derived for both the optimal objective function and the optimal switching strategy. A two-step pruning scheme is developed to efficiently solve such problem. The performance of this method is shown by two examples.
References:
[1] |
A. Bemporad, F. Borrelli and M. Morari,
On the optimal control law for linear discrete time hybrid systems, Lecture Notes in Computer Science, Hybrid System: Computation and Control, 2289 (2002), 222-292.
doi: 10.1007/3-540-45873-5_11. |
[2] |
S. C. Benga and R. A. Decarlo,
Optimal control of switching systems, Automatica, 41 (2005), 11-27.
doi: 10.1016/j.automatica.2004.08.003. |
[3] |
F. Borrelli, M. Baotic, A. Bemporad and M. Morari,
Dynamic programming for contrained optimal control of discrete-time linear hybrid systems, Automatica, 41 (2005), 1709-1721.
doi: 10.1016/j.automatica.2005.04.017. |
[4] |
S. Boubakera, M. Djemaic, N. Manamannid and F. M'Sahlie,
Active modes and switching instants identification for linear switched systems based on discrete particle swarm optimization, Applied Soft Computing, 14 (2014), 482-488.
doi: 10.1016/j.asoc.2013.09.009. |
[5] |
H. V. Esteban, C. Patrizio, M. Richard and B. Franco,
Discrete-time control for switched positive systems with application to mitigating viral escape, International Journal of Robust and Nonlinear Control, 21 (2011), 1093-1111.
doi: 10.1002/rnc.1628. |
[6] |
Y. Gao,
Uncertain models for single facility location problems on networks, Applied Mathematical Modelling, 36 (2012), 2592-2599.
doi: 10.1016/j.apm.2011.09.042. |
[7] |
J. Gao and L. Duan,
Linear-quadratic switching control with switching cost, Automatica, 48 (2012), 1138-1143.
doi: 10.1016/j.automatica.2012.03.006. |
[8] |
D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292. Google Scholar |
[9] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.
doi: 10.1016/S0005-1098(99)00050-3. |
[10] |
F. Li, P. Shi, L. Wu, M. V. Basin and C. C. Lim,
Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems, IEEE Transactions on Industrial Electronics, 62 (2015), 2330-2340.
doi: 10.1109/TIE.2014.2351379. |
[11] |
B. Lincoln and A. Rantzer,
Relaxing dynamic programming, IEEE Transactions on Automatic Control, 51 (2006), 1249-1260.
doi: 10.1109/TAC.2006.878720. |
[12] |
B. Liu, Why is there a need for uncertainty theory, Journal of Uncertain Systems, 6 (2012), 3-10. Google Scholar |
[13] |
B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. |
[14] |
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-540-39987-2. |
[15] |
Y. Liu and M. Ha, Expected value of function of uncertain variables, Journal of Uncertain Systems, 4 (2010), 181-186. Google Scholar |
[16] |
C. Liu, Z. Gong and E. Feng,
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.
doi: 10.3934/jimo.2009.5.835. |
[17] |
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling,
Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980.
doi: 10.1016/j.automatica.2008.10.031. |
[18] |
K. L. Teo, C. J. Goh and K. H. Wong,
A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, New York, 1991. |
[19] |
C. Tomlin, G. J. Pappas, J. Lygeros, D. N. Godbole and S. Sastry,
Hybrid control models of next generation air traffic management, Hybrid Systems IV, 1273 (1997), 378-404.
doi: 10.1007/BFb0031570. |
[20] |
L. Y. Wang, A. Beydoun, J. Sun and I. Kolmanasovsky,
Optimal hybrid control with application to automotive powertrain systems, Lecture Notes in Control and Information Science, 222 (1997), 190-200.
doi: 10.1007/BFb0036095. |
[21] |
S. Woon, V. Rehbock and R. Loxton,
Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamic Systems Theory, 10 (2010), 175-188.
|
[22] |
L. Wu, D. Ho and C. Li,
Sliding mode control of switched hybrid systems with stochastic perturbation, Systems & Control Letters, 60 (2011), 531-539.
doi: 10.1016/j.sysconle.2011.04.007. |
[23] |
X. Xu and P. Antsaklis,
Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16.
doi: 10.1109/TAC.2003.821417. |
[24] |
H. Yan and Y. Zhu,
Bang-bang control model for uncertain switched systems, Applied Mathematical Modelling, 39 (2015), 2994-3002.
doi: 10.1016/j.apm.2014.10.042. |
[25] |
H. Yan and Y. Zhu,
Bang-bang control model with optimistic value criterion for uncertain switched systems, Journal of Intelligent Manufacturing, (2015), 1-8.
doi: 10.1007/s10845-014-0996-2. |
[26] |
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. |
[27] |
W. Zhang, J. Hu and J. Lian,
Quadratic optimal control of switched linear stochastic systems, Systems & Control Letters, 59 (2010), 736-744.
doi: 10.1016/j.sysconle.2010.08.010. |
[28] |
X. Zhang and X. Chen,
A new uncertain programming model for project scheduling problem, Information: An International Interdisciplinary Journal, 15 (2012), 3901-3910.
|
[29] |
Y. Zhu,
Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems: An International Journal, 41 (2010), 535-547.
doi: 10.1080/01969722.2010.511552. |
[30] |
Y. Zhu, Functions of uncertain variables and uncertain programmin, Journal of Uncertain Systems, 6 (2012), 278-288. Google Scholar |
show all references
References:
[1] |
A. Bemporad, F. Borrelli and M. Morari,
On the optimal control law for linear discrete time hybrid systems, Lecture Notes in Computer Science, Hybrid System: Computation and Control, 2289 (2002), 222-292.
doi: 10.1007/3-540-45873-5_11. |
[2] |
S. C. Benga and R. A. Decarlo,
Optimal control of switching systems, Automatica, 41 (2005), 11-27.
doi: 10.1016/j.automatica.2004.08.003. |
[3] |
F. Borrelli, M. Baotic, A. Bemporad and M. Morari,
Dynamic programming for contrained optimal control of discrete-time linear hybrid systems, Automatica, 41 (2005), 1709-1721.
doi: 10.1016/j.automatica.2005.04.017. |
[4] |
S. Boubakera, M. Djemaic, N. Manamannid and F. M'Sahlie,
Active modes and switching instants identification for linear switched systems based on discrete particle swarm optimization, Applied Soft Computing, 14 (2014), 482-488.
doi: 10.1016/j.asoc.2013.09.009. |
[5] |
H. V. Esteban, C. Patrizio, M. Richard and B. Franco,
Discrete-time control for switched positive systems with application to mitigating viral escape, International Journal of Robust and Nonlinear Control, 21 (2011), 1093-1111.
doi: 10.1002/rnc.1628. |
[6] |
Y. Gao,
Uncertain models for single facility location problems on networks, Applied Mathematical Modelling, 36 (2012), 2592-2599.
doi: 10.1016/j.apm.2011.09.042. |
[7] |
J. Gao and L. Duan,
Linear-quadratic switching control with switching cost, Automatica, 48 (2012), 1138-1143.
doi: 10.1016/j.automatica.2012.03.006. |
[8] |
D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 263-292. Google Scholar |
[9] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for optimal discrete-valued control problems, Automatica, 35 (1999), 1401-1407.
doi: 10.1016/S0005-1098(99)00050-3. |
[10] |
F. Li, P. Shi, L. Wu, M. V. Basin and C. C. Lim,
Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems, IEEE Transactions on Industrial Electronics, 62 (2015), 2330-2340.
doi: 10.1109/TIE.2014.2351379. |
[11] |
B. Lincoln and A. Rantzer,
Relaxing dynamic programming, IEEE Transactions on Automatic Control, 51 (2006), 1249-1260.
doi: 10.1109/TAC.2006.878720. |
[12] |
B. Liu, Why is there a need for uncertainty theory, Journal of Uncertain Systems, 6 (2012), 3-10. Google Scholar |
[13] |
B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. |
[14] |
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-540-39987-2. |
[15] |
Y. Liu and M. Ha, Expected value of function of uncertain variables, Journal of Uncertain Systems, 4 (2010), 181-186. Google Scholar |
[16] |
C. Liu, Z. Gong and E. Feng,
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Journal of Industrial and Management Optimization, 5 (2009), 835-850.
doi: 10.3934/jimo.2009.5.835. |
[17] |
R. Loxton, K. L. Teo, V. Rehbock and W. K. Ling,
Optimal switching instants for a switched-capacitor DC/DC power converter, Automatica, 45 (2009), 973-980.
doi: 10.1016/j.automatica.2008.10.031. |
[18] |
K. L. Teo, C. J. Goh and K. H. Wong,
A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, New York, 1991. |
[19] |
C. Tomlin, G. J. Pappas, J. Lygeros, D. N. Godbole and S. Sastry,
Hybrid control models of next generation air traffic management, Hybrid Systems IV, 1273 (1997), 378-404.
doi: 10.1007/BFb0031570. |
[20] |
L. Y. Wang, A. Beydoun, J. Sun and I. Kolmanasovsky,
Optimal hybrid control with application to automotive powertrain systems, Lecture Notes in Control and Information Science, 222 (1997), 190-200.
doi: 10.1007/BFb0036095. |
[21] |
S. Woon, V. Rehbock and R. Loxton,
Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamic Systems Theory, 10 (2010), 175-188.
|
[22] |
L. Wu, D. Ho and C. Li,
Sliding mode control of switched hybrid systems with stochastic perturbation, Systems & Control Letters, 60 (2011), 531-539.
doi: 10.1016/j.sysconle.2011.04.007. |
[23] |
X. Xu and P. Antsaklis,
Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49 (2004), 2-16.
doi: 10.1109/TAC.2003.821417. |
[24] |
H. Yan and Y. Zhu,
Bang-bang control model for uncertain switched systems, Applied Mathematical Modelling, 39 (2015), 2994-3002.
doi: 10.1016/j.apm.2014.10.042. |
[25] |
H. Yan and Y. Zhu,
Bang-bang control model with optimistic value criterion for uncertain switched systems, Journal of Intelligent Manufacturing, (2015), 1-8.
doi: 10.1007/s10845-014-0996-2. |
[26] |
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. |
[27] |
W. Zhang, J. Hu and J. Lian,
Quadratic optimal control of switched linear stochastic systems, Systems & Control Letters, 59 (2010), 736-744.
doi: 10.1016/j.sysconle.2010.08.010. |
[28] |
X. Zhang and X. Chen,
A new uncertain programming model for project scheduling problem, Information: An International Interdisciplinary Journal, 15 (2012), 3901-3910.
|
[29] |
Y. Zhu,
Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems: An International Journal, 41 (2010), 535-547.
doi: 10.1080/01969722.2010.511552. |
[30] |
Y. Zhu, Functions of uncertain variables and uncertain programmin, Journal of Uncertain Systems, 6 (2012), 278-288. Google Scholar |
Algorithm 1:(Two-step pruning scheme) |
1: Set $\tilde{H}_{0}=\{(Q_{f}, 0)\}$; |
2: for $k=0$ to $N-1$ do |
3: for all $(P, \gamma)\in \tilde{H}_{k}$ do |
4: $\Gamma_{k}(P, \gamma)=\emptyset$; |
5: for i=1 to m do |
6: $P^{(i)}=\rho_{i}(P)$, |
7: $\gamma^{(i)}=\gamma+\frac{1}{3}\|\sigma_{N-k}\|^{2}_{P}$, |
8: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\bigcup\{(P^{(i)}, \gamma^{(i)})\}$; |
9: end for |
10: for i=1 to m do |
11: if $(P^{(i)}, \gamma^{(i)})$ satisfies the condition in Lemma 5.3, then |
12: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\backslash\{(P^{(i)}, \gamma^{(i)})\}$; |
13: end if |
14: end for |
15: end for |
16: $\tilde{H}_{k+1}=\bigcup\limits_{(P, \gamma)\in \tilde{H}_{k}}\Gamma_{k}(P, \gamma)$; |
17: $\hat{H}_{k+1}=\tilde{H}_{k+1}$; |
18: for i=1 to $|\hat{H}_{k+1}|$ do |
19: if $(\hat{P}^{(i)}, \hat{\gamma}^{(i)})$ satisfies the condition in Lemma 5.4, then |
20: $\hat{H}_{k+1}=\hat{H}_{k+1}\backslash\{(\hat{P}^{(i)}, \hat{\gamma}^{(i)})\}$; |
21: end if |
22: end for |
23: end for |
24: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$ |
Algorithm 1:(Two-step pruning scheme) |
1: Set $\tilde{H}_{0}=\{(Q_{f}, 0)\}$; |
2: for $k=0$ to $N-1$ do |
3: for all $(P, \gamma)\in \tilde{H}_{k}$ do |
4: $\Gamma_{k}(P, \gamma)=\emptyset$; |
5: for i=1 to m do |
6: $P^{(i)}=\rho_{i}(P)$, |
7: $\gamma^{(i)}=\gamma+\frac{1}{3}\|\sigma_{N-k}\|^{2}_{P}$, |
8: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\bigcup\{(P^{(i)}, \gamma^{(i)})\}$; |
9: end for |
10: for i=1 to m do |
11: if $(P^{(i)}, \gamma^{(i)})$ satisfies the condition in Lemma 5.3, then |
12: $\Gamma_{k}(P, \gamma)=\Gamma_{k}(P, \gamma)\backslash\{(P^{(i)}, \gamma^{(i)})\}$; |
13: end if |
14: end for |
15: end for |
16: $\tilde{H}_{k+1}=\bigcup\limits_{(P, \gamma)\in \tilde{H}_{k}}\Gamma_{k}(P, \gamma)$; |
17: $\hat{H}_{k+1}=\tilde{H}_{k+1}$; |
18: for i=1 to $|\hat{H}_{k+1}|$ do |
19: if $(\hat{P}^{(i)}, \hat{\gamma}^{(i)})$ satisfies the condition in Lemma 5.4, then |
20: $\hat{H}_{k+1}=\hat{H}_{k+1}\backslash\{(\hat{P}^{(i)}, \hat{\gamma}^{(i)})\}$; |
21: end if |
22: end for |
23: end for |
24: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$ |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | 5 | 4 | 4 | 7 | 7 | 4 | 7 | 7 | 7 | |
2 | 2 | 2 | 3 | 3 | 2 | 3 | 3 | 3 | 3 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | 5 | 4 | 4 | 7 | 7 | 4 | 7 | 7 | 7 | |
2 | 2 | 2 | 3 | 3 | 2 | 3 | 3 | 3 | 3 |
| | ||||
0 | 2 | - | | -0.7861 | 12.9774 |
1 | 2 | 0.6294 | | -0.2579 | 2.5122 |
2 | 2 | 0.8116 | | -0.1749 | 0.6456 |
3 | 2 | -0.7460 | | 0.0761 | 0.2084 |
4 | 2 | 0.8268 | | -0.1143 | 0.1582 |
5 | 1 | 0.2647 | | -0.0495 | 0.1137 |
6 | 2 | -0.8049 | | 0.1221 | 0.1113 |
7 | 2 | -0.4430 | | 0.0918 | 0.0765 |
8 | 2 | 0.0938 | | 0.005 | 0.0443 |
9 | 2 | 0.9150 | | -0.1444 | 0.0678 |
| | ||||
0 | 2 | - | | -0.7861 | 12.9774 |
1 | 2 | 0.6294 | | -0.2579 | 2.5122 |
2 | 2 | 0.8116 | | -0.1749 | 0.6456 |
3 | 2 | -0.7460 | | 0.0761 | 0.2084 |
4 | 2 | 0.8268 | | -0.1143 | 0.1582 |
5 | 1 | 0.2647 | | -0.0495 | 0.1137 |
6 | 2 | -0.8049 | | 0.1221 | 0.1113 |
7 | 2 | -0.4430 | | 0.0918 | 0.0765 |
8 | 2 | 0.0938 | | 0.005 | 0.0443 |
9 | 2 | 0.9150 | | -0.1444 | 0.0678 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | 5 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
2 | 4 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | 5 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
2 | 4 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| | ||||
0 | 5 | - | | -0.7273 | 11.0263 |
1 | 1 | 0.6294 | | -0.1808 | 0.6251 |
2 | 2 | 0.8116 | | -0.0892 | 0.5116 |
3 | 1 | -0.7460 | | 0.1421 | 0.2063 |
4 | 2 | 0.8268 | | -0.0608 | 0.1428 |
5 | 2 | 0.2647 | | -0.0523 | 0.1121 |
6 | 2 | -0.8049 | 0.1105 | 0.1113 | |
7 | 2 | -0.4430 | | 0.0931 | 0.0690 |
8 | 1 | 0.0938 | | -0.0062 | 0.0426 |
9 | 2 | 0.9150 | | -0.1522 | 0.0615 |
| | ||||
0 | 5 | - | | -0.7273 | 11.0263 |
1 | 1 | 0.6294 | | -0.1808 | 0.6251 |
2 | 2 | 0.8116 | | -0.0892 | 0.5116 |
3 | 1 | -0.7460 | | 0.1421 | 0.2063 |
4 | 2 | 0.8268 | | -0.0608 | 0.1428 |
5 | 2 | 0.2647 | | -0.0523 | 0.1121 |
6 | 2 | -0.8049 | 0.1105 | 0.1113 | |
7 | 2 | -0.4430 | | 0.0931 | 0.0690 |
8 | 1 | 0.0938 | | -0.0062 | 0.0426 |
9 | 2 | 0.9150 | | -0.1522 | 0.0615 |
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