Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls
Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls
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DOI：
10.1109/9.863597
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发表时间：
2000-06
期刊：
影响因子：
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通讯作者：
M. A. Rami;X. Zhou
中科院分区：
文献类型：
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作者：
M. A. Rami;X. Zhou
研究了一个无限时间范围内的最优随机线性二次控制问题，其中动力学扩散项既依赖于状态变量，也依赖于控制变量。与确定性情况相反，我们允许代价函数中的控制和状态加权矩阵是不确定的。这导致了一个不确定的LQ问题，由于所涉及的不确定性的深层性质，它可能仍然是很好的。这个问题产生了随机代数里卡蒂方程（SARE），然而，由于LQ问题的不确定性，它与经典的代数里卡蒂方程有着根本的不同。为了分析SARE，我们引入线性矩阵不等式（lmi），其可行性被证明等同于SARE的可解性。此外，我们通过与lmi相关的半确定规划开发了一种计算方法。最后，通过数值实验对该方法进行了验证。
This paper deals with an optimal stochastic linear-quadratic (LQ) control problem in infinite time horizon, where the diffusion term in dynamics depends on both the state and the control variables. In contrast to the deterministic case, we allow the control and state weighting matrices in the cost functional to be indefinite. This leads to an indefinite LQ problem, which may still be well posed due to the deep nature of uncertainty involved. The problem gives rise to a stochastic algebraic Riccati equation (SARE), which is, however, fundamentally different from the classical algebraic Riccati equation as a result of the indefinite nature of the LQ problem. To analyze the SARE, we introduce linear matrix inequalities (LMIs) whose feasibility is shown to be equivalent to the solvability of the SARE. Moreover, we develop a computational approach to the SARE via a semi-definite programming associated with the LMIs. Finally, numerical experiments are reported to illustrate the proposed approach.