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

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  • 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.

    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

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    Table 2.  The optimal results of Example 3

    $k$$y^{*}(k)$$r_{k}$ $x(k)$$u^{*}(k)$ $J(k,x_{k})$
    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
     | Show Table
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    Table 3.  Size of $\tilde{H}_{k}$ and $\hat{H}_{k}$ for Example 4

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    Table 4.  The optimal results of Example 4

    $k$$y^{*}(k)$$r_{k}$ $x(k)$$u^{*}(k)$ $J(k,x_{k})$
    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
    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
     | Show Table
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