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Linear-quadratic optimal control for discrete-time stochastic descriptor systems

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  • In this paper, an optimal control model ruled by a class of linear discrete-time stochastic descriptor systems is considered under quadratic index performance. Employing dynamic programming method, a recurrence equation to simplify the optimal control problem is presented provided that the descriptor systems are both regular and impulse-free. When the objective function is quadratic, according to the recurrence equation, a discrete-time linear-quadratic optimal control problem is completely settled, that is, optimal controls and optimal values of the problem are both obtained through analytical expressions. At last, a numerical example about linear-quadratic optimal control for a discrete-time stochastic descriptor system is provided to illustrate the validness of the results derived.

    Mathematics Subject Classification: 49N10, 93E20.

    Citation:

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  • Figure 1.  Trajectories of optimal controls $ u^*(k) $ for problem (11)

    Figure 2.  Trajectories of state vectors $ x(k) $ for problem (11)

    Table 1.  The optimal results of problem (11)

    Stage $ (u^{*}(k))^T $ $ \mu_k $ $ x_1^T(k) $ $ x^T(k) $ $ J\big(k, x_1(k), x_2(7)\big) $
    0 $ (3.567, -2.323, -5.406) $ $ 2.517 $ $ (1, 2) $ $ (1.000, -5.434, 2.000, -4.600) $ $ -112.722 $
    1 $ (0.338, -2.541, -0.797) $ $ 1.743 $ $ (-4.756, 0.145) $ $ (-4.756, -2.699, 0.145, -4.756) $ $ -55.708 $
    2 $ (1.796, -2.599, -0.303) $ $ 2.098 $ $ (-1.416, -4.148) $ $ (-1.416, -5.019, -4.148, -1.218) $ $ 20.114 $
    3 $ (0.167, -1.651, 0.469) $ $ 0.202 $ $ (-1.429, -1.614) $ $ (-1.429, -1.769, -1.614, 0.490) $ $ 41.636 $
    4 $ (0.635, -1.194, 0.582) $ $ 1.165 $ $ (-0.931, -2.843) $ $ (-0.931, -2.743, -2.843, -0.320) $ $ 39.171 $
    5 $ (0.862, -0.338, -0.251) $ $ 0.872 $ $ (-0.362, -0.430) $ $ (-0.362, -1.988, -0.430, -1.147) $ $ 32.059 $
    6 $ (0.390, -0.274, -0.403) $ $ 1.394 $ $ (-0.505, 0.182) $ $ (-0.505, -2.598, 0.182, -2.027) $ $ 46.328 $
    7 $ u^{*}(7)\in R^3 $ $ (-1.217, 1.619) $ $ (-1.217, 0.598, 1.619, -0.217) $ $ 7.363 $
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