Many problems arising in control require the determination of a mathematical model of the application. This has often to be performed starting from input-output data, leading to a task known as system identification in the engineering literature. One emerging topic in this field is estimation of networks consisting of several interconnected dynamic systems. We consider the linear setting assuming that system outputs are the result of many correlated inputs, hence making system identification severely ill-conditioned. This is a scenario often encountered when modeling complex cybernetics systems composed by many sub-units with feedback and algebraic loops. We develop a strategy cast in a Bayesian regularization framework where any impulse response is seen as realization of a zero-mean Gaussian process. Any covariance is defined by the so called stable spline kernel which includes information on smooth exponential decay. We design a novel Markov chain Monte Carlo scheme able to reconstruct the impulse responses posterior by efficiently dealing with collinearity. Our scheme relies on a variation of the Gibbs sampling technique: beyond considering blocks forming a partition of the parameter space, some other (overlapping) blocks are also updated on the basis of the level of collinearity of the system inputs. Theoretical properties of the algorithm are studied obtaining its convergence rate. Numerical experiments are included using systems containing hundreds of impulse responses and highly correlated inputs.

翻译：控制领域中出现的许多问题需要确定应用的数学模型。这通常需要从输入输出数据出发进行求解，从而在工程文献中形成一项称为系统辨识的任务。该领域的一个新兴课题是估计由多个互联动态系统构成的网络。我们考虑线性设定，假设系统输出是大量相关输入的结果，这使得系统辨识严重病态。这是在建模由许多带有反馈和代数回路的子单元组成的复杂控制论系统时经常遇到的情景。我们提出了一种基于贝叶斯正则化框架的策略，其中任何脉冲响应都被视为零均值高斯过程的一次实现。任意协方差由所谓的稳定样条核定义，该核包含平滑指数衰减的信息。我们设计了一种新颖的马尔可夫链蒙特卡洛方案，能够通过有效处理共线性问题重建脉冲响应的后验分布。该方案基于吉布斯采样技术的一种变体：除了考虑构成参数空间分区的块之外，还根据系统输入的共线性程度更新一些（重叠的）块。我们研究了该算法的理论性质，得到了其收敛速率。数值实验在包含数百个脉冲响应和高度相关输入的系统中进行。