We consider the problem of estimating a continuous-time Gauss-Markov source process observed through a vector Gaussian channel with an adjustable channel gain matrix. For a given (generally time-varying) channel gain matrix, we provide formulas to compute (i) the mean-square estimation error attainable by the classical Kalman-Bucy filter, and (ii) the mutual information between the source process and its Kalman-Bucy estimate. We then formulate a novel "optimal channel gain control problem" where the objective is to control the channel gain matrix strategically to minimize the weighted sum of these two performance metrics. To develop insights into the optimal solution, we first consider the problem of controlling a time-varying channel gain over a finite time interval. A necessary optimality condition is derived based on Pontryagin's minimum principle. For a scalar system, we show that the optimal channel gain is a piece-wise constant signal with at most two switches. We also consider the problem of designing the optimal time-invariant gain to minimize the average cost over an infinite time horizon. A novel semidefinite programming (SDP) heuristic is proposed and the exactness of the solution is discussed.
翻译:我们考虑了通过矢量高斯-马尔科夫频道用可调整的频道增益矩阵观测到的连续时间高斯-马尔科夫源源进程的问题。对于给定的(一般时间变化的)频道增益矩阵,我们提供公式来计算(一)古典Kalman-Buscy过滤器所能达到的平方估计错误,和(二)源进程与Kalman-Bucy估计之间的相互信息。然后我们设计了一个新型的“最佳频道增益控制问题”,目的是从战略上控制频道增益矩阵,以最大限度地减少这两个性能计量的加权总和。为了对最佳解决方案进行深入了解,我们首先考虑在一定时间间隔内控制一个时间变化频道增益的问题。一个必要的最佳性条件根据Pontryagin的最低限度原则推导出。对于电路系统来说,我们显示最佳的频道增益是一个在最多两个开关上都具有零碎的恒定信号。我们还考虑了设计最佳时差增益问题,以便在无限的时间范围内最大限度地减少平均成本。我们先考虑如何设计一个新的半定式的解决方案。