The notion of reproducing kernel Hilbert space (RKHS) has emerged in system identification during the past decade. In the resulting framework, the impulse response estimation problem is formulated as a regularized optimization defined on an infinite-dimensional RKHS consisting of stable impulse responses. The consequent estimation problem is well-defined under the central assumption that the convolution operators restricted to the RKHS are continuous linear functionals. Moreover, according to this assumption, the representer theorem hold, and therefore, the impulse response can be estimated by solving a finite-dimensional program. Thus, the continuity feature plays a significant role in kernel-based system identification. We show that this central assumption is guaranteed to be satisfied in considerably general situations, namely when the input signal is bounded, the kernel is an integrable function, and in the case of continuous-time dynamics, continuous. Furthermore, the strong convexity of the optimization problem and the continuity property of the convolution operators imply that the kernel-based system identification admits a unique solution. Consequently, it follows that kernel-based system identification is a well-defined approach.

再生核希尔伯特空间 (RKHS) 的概念在过去十年中出现在系统识别中。在由此产生的框架中，脉冲响应估计问题被表述为定义在由稳定脉冲响应组成的无限维 RKHS 上的正则化优化。随之而来的估计问题在以下中心假设下得到明确定义，即限于 RKHS 的卷积算子是连续线性泛函。此外，根据该假设，表示定理成立，因此可以通过求解有限维程序来估计脉冲响应。因此，连续性特征在基于内核的系统识别中起着重要作用。我们表明，在相当普遍的情况下，可以保证满足这一中心假设，即当输入信号有界时，内核是一个可积函数，在连续时间动态的情况下，内核是连续的。此外，优化问题的强凸性和卷积算子的连续性意味着基于内核的系统识别承认唯一的解决方案。因此，基于内核的系统识别是一种定义明确的方法。