A linear-quadratic control problem of uncertain discrete-time switched systems

Hongyan Yan; Yun Sun; Yuanguo Zhu

doi:10.3934/jimo.2016016

Article Contents

2017, Volume 13, Issue 1: 267-282. Doi: 10.3934/jimo.2016016

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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

* Corresponding author
* Corresponding author

Received: December 31, 2014

Revised: November 30, 2015

Published: August 2017

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 49N10, 49L20; Secondary: 65K05.

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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: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$

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Table 1. Size of $\tilde{H}_{k}$ and $\hat{H}_{k}$ for Example 3

$k$	1	2	3	4	5	6	7	8	9	10
$\|\tilde{H}_{k}\|$	2	5	4	4	7	7	4	7	7	7
$\|\hat{H}_{k}\|$	2	2	2	3	3	2	3	3	3	3

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Table 2. The optimal results of Example 3

$k$	$y^{*}(k)$	$r_{k}$	$x(k)$	$u^{*}(k)$	$J(k,x_{k})$
0	2	-	$(3,-1)^{\tau}$	-0.7861	12.9774
1	2	0.6294	$(1.2768,-0.5093)^{\tau}$	-0.2579	2.5122
2	2	0.8116	$(0.5908,-0.1764)^{\tau}$	-0.1749	0.6456
3	2	-0.7460	$(0.1649,-0.1864)^{\tau}$	0.0761	0.2084
4	2	0.8268	$(0.1373,0.0270)^{\tau}$	-0.1143	0.1582
5	1	0.2647	$(0.0765,-0.0108)^{\tau}$	-0.0495	0.1137
6	2	-0.8049	$(0.0122,-0.1408)^{\tau}$	0.1221	0.1113
7	2	-0.4430	$(-0.0503,-0.0685)^{\tau}$	0.0918	0.0765
8	2	0.0938	$(-0.0716,0.0057)^{\tau}$	0.005	0.0443
9	2	0.9150	$(0.0846,0.0953)^{\tau}$	-0.1444	0.0678

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Table 3. Size of $\tilde{H}_{k}$ and $\hat{H}_{k}$ for Example 4

$k$	1	2	3	4	5	6	7	8	9	10
$\|\tilde{H}_{k}\|$	2	5	12	9	9	9	9	9	9	9
$\|\hat{H}_{k}\|$	2	4	3	3	3	3	3	3	3	3

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Table 4. The optimal results of Example 4

$k$	$y^{*}(k)$	$r_{k}$	$x(k)$	$u^{*}(k)$	$J(k,x_{k})$
0	5	-	$(3,-1)^{\tau}$	-0.7273	11.0263
1	1	0.6294	$(0.3356,-0.1190)^{\tau}$	-0.1808	0.6251
2	2	0.8116	$(0.4526,-0.2186)^{\tau}$	-0.0892	0.5116
3	1	-0.7460	$(0.0702,-0.2376)^{\tau}$	0.1421	0.2063
4	2	0.8268	$(0.1276,-0.0128)^{\tau}$	-0.0608	0.1428
5	2	0.2647	$(0.0805,0.0069)^{\tau}$	-0.0523	0.1121
6	2	-0.8049	$(-0.0454,-0.0908)^{\tau}$	0.1105	0.1113
7	2	-0.4430	$(-0.0700,-0.0503)^{\tau}$	0.0931	0.0690
8	1	0.0938	$(-0.0178,0.0250)^{\tau}$	-0.0062	0.0426
9	2	0.9150	$(0.0747,0.1103)^{\tau}$	-0.1522	0.0615

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