In this paper, we examine the fundamental performance limitations in the control of stochastic dynamical systems; more specifically, we derive generic $\mathcal{L}_p$ bounds that hold for any causal (stabilizing) controllers and any stochastic disturbances, by an information-theoretic analysis. We first consider the scenario where the plant (i.e., the dynamical system to be controlled) is linear time-invariant, and it is seen in general that the lower bounds are characterized by the unstable poles (or nonminimum-phase zeros) of the plant as well as the conditional entropy of the disturbance. We then analyze the setting where the plant is assumed to be (strictly) causal, for which case the lower bounds are determined by the conditional entropy of the disturbance. We also discuss the special cases of $p = 2$ and $p = \infty$, which correspond to minimum-variance control and controlling the maximum deviations, respectively. In addition, we investigate the power-spectral characterization of the lower bounds as well as its relation to the Kolmogorov-Szeg\"o formula.
翻译:在本文中,我们通过信息理论分析,审视了对气动系统控制的基本性能限制;更具体地说,我们通过信息理论分析,得出用于控制任何因果(稳定)控制器和任何气动扰动的通用值$mathcal{L ⁇ p$界限。我们首先考虑工厂(即要控制的动态系统)是线性时间变化的假设情况,一般地看,下限的特点是工厂的不稳定杆(或非最小级零)以及扰动的有条件酶。我们然后分析假设该工厂是(严格)因果的(严格)控制器和任何气动扰动的场景。我们还讨论与最低性能控制和控制最大偏差分别对应的美元=2美元和美元=\infty美元的特殊情况。此外,我们调查了下限的能量光谱特征及其与Kolmrovze公式的关系。