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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 & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
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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: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$
 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: $J(0, x_{0})=\min\limits_{(P, \gamma)\in \hat{H}_{N}}(\|x_{0}\|^{2}_{P}+\gamma).$
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
 $k$ 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
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
 $k$ $y^{*}(k)$ $r_{k}$ $x(k)$ $u^{*}(k)$ $J(k,x_{k})$ 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
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
 $k$ 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
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
 $k$ $y^{*}(k)$ $r_{k}$ $x(k)$ $u^{*}(k)$ $J(k,x_{k})$ 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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