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

• * Corresponding author
• 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: • • 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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