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January  2017, 13(1): 267-282. doi: 10.3934/jimo.2016016

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

* Corresponding author

Received  January 2015 Revised  December 2015 Published  March 2016

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.

Citation: Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial and Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
References:
[1]

A. BemporadF. 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. BorrelliM. BaoticA. 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. BoubakeraM. DjemaicN. 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. EstebanC. PatrizioM. 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. 

[9]

H. W. J. LeeK. L. TeoV. 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. LiP. ShiL. WuM. 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. 

[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. 

[16]

C. LiuZ. 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. LoxtonK. L. TeoV. 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. TomlinG. J. PappasJ. LygerosD. 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. WangA. BeydounJ. 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. WoonV. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamic Systems Theory, 10 (2010), 175-188. 

[22]

L. WuD. 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. ZhangJ. 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. ZhangJ. 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. 

show all references

References:
[1]

A. BemporadF. 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. BorrelliM. BaoticA. 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. BoubakeraM. DjemaicN. 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. EstebanC. PatrizioM. 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. 

[9]

H. W. J. LeeK. L. TeoV. 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. LiP. ShiL. WuM. 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. 

[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. 

[16]

C. LiuZ. 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. LoxtonK. L. TeoV. 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. TomlinG. J. PappasJ. LygerosD. 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. WangA. BeydounJ. 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. WoonV. Rehbock and R. Loxton, Global optimization method for continuous-time sensor scheduling, Nonlinear Dynamic Systems Theory, 10 (2010), 175-188. 

[22]

L. WuD. 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. ZhangJ. 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. ZhangJ. 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. 

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: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$
Table 1.  Size of $\tilde{H}_{k}$ and $\hat{H}_{k}$ for Example 3
$k$12345678910
$|\tilde{H}_{k}|$2544774777
$|\hat{H}_{k}|$2223323333
$k$12345678910
$|\tilde{H}_{k}|$2544774777
$|\hat{H}_{k}|$2223323333
Table 2.  The optimal results of Example 3
$k$$y^{*}(k)$$r_{k}$ $x(k)$$u^{*}(k)$ $J(k,x_{k})$
02- $(3,-1)^{\tau}$-0.786112.9774
120.6294 $(1.2768,-0.5093)^{\tau}$-0.25792.5122
220.8116 $(0.5908,-0.1764)^{\tau}$-0.17490.6456
32-0.7460 $(0.1649,-0.1864)^{\tau}$0.07610.2084
420.8268 $(0.1373,0.0270)^{\tau}$-0.11430.1582
510.2647 $(0.0765,-0.0108)^{\tau}$-0.04950.1137
62-0.8049 $(0.0122,-0.1408)^{\tau}$0.12210.1113
72-0.4430 $(-0.0503,-0.0685)^{\tau}$0.09180.0765
820.0938 $(-0.0716,0.0057)^{\tau}$0.0050.0443
920.9150 $(0.0846,0.0953)^{\tau}$-0.14440.0678
$k$$y^{*}(k)$$r_{k}$ $x(k)$$u^{*}(k)$ $J(k,x_{k})$
02- $(3,-1)^{\tau}$-0.786112.9774
120.6294 $(1.2768,-0.5093)^{\tau}$-0.25792.5122
220.8116 $(0.5908,-0.1764)^{\tau}$-0.17490.6456
32-0.7460 $(0.1649,-0.1864)^{\tau}$0.07610.2084
420.8268 $(0.1373,0.0270)^{\tau}$-0.11430.1582
510.2647 $(0.0765,-0.0108)^{\tau}$-0.04950.1137
62-0.8049 $(0.0122,-0.1408)^{\tau}$0.12210.1113
72-0.4430 $(-0.0503,-0.0685)^{\tau}$0.09180.0765
820.0938 $(-0.0716,0.0057)^{\tau}$0.0050.0443
920.9150 $(0.0846,0.0953)^{\tau}$-0.14440.0678
Table 3.  Size of $\tilde{H}_{k}$ and $\hat{H}_{k}$ for Example 4
$k$12345678910
$|\tilde{H}_{k}|$25129999999
$|\hat{H}_{k}|$2433333333
$k$12345678910
$|\tilde{H}_{k}|$25129999999
$|\hat{H}_{k}|$2433333333
Table 4.  The optimal results of Example 4
$k$$y^{*}(k)$$r_{k}$ $x(k)$$u^{*}(k)$ $J(k,x_{k})$
05- $(3,-1)^{\tau}$-0.727311.0263
110.6294 $(0.3356,-0.1190)^{\tau}$-0.18080.6251
220.8116 $(0.4526,-0.2186)^{\tau}$-0.08920.5116
31-0.7460 $(0.0702,-0.2376)^{\tau}$0.14210.2063
420.8268 $(0.1276,-0.0128)^{\tau}$-0.06080.1428
520.2647 $(0.0805,0.0069)^{\tau}$-0.05230.1121
62-0.8049$(-0.0454,-0.0908)^{\tau}$0.11050.1113
72-0.4430 $(-0.0700,-0.0503)^{\tau}$0.09310.0690
810.0938 $(-0.0178,0.0250)^{\tau}$-0.00620.0426
920.9150 $(0.0747,0.1103)^{\tau}$-0.15220.0615
$k$$y^{*}(k)$$r_{k}$ $x(k)$$u^{*}(k)$ $J(k,x_{k})$
05- $(3,-1)^{\tau}$-0.727311.0263
110.6294 $(0.3356,-0.1190)^{\tau}$-0.18080.6251
220.8116 $(0.4526,-0.2186)^{\tau}$-0.08920.5116
31-0.7460 $(0.0702,-0.2376)^{\tau}$0.14210.2063
420.8268 $(0.1276,-0.0128)^{\tau}$-0.06080.1428
520.2647 $(0.0805,0.0069)^{\tau}$-0.05230.1121
62-0.8049$(-0.0454,-0.0908)^{\tau}$0.11050.1113
72-0.4430 $(-0.0700,-0.0503)^{\tau}$0.09310.0690
810.0938 $(-0.0178,0.0250)^{\tau}$-0.00620.0426
920.9150 $(0.0747,0.1103)^{\tau}$-0.15220.0615
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