High dimensional model representation

Abstrakt: In practical applications of the control theory, there is lack of approximation techniques applicable to intractable dynamic-programming equations describing the optimality controller. In the paper, we consider use of the technique coming from chemistry and called high dimensional model representation (HDMR). Its main advantages are finite order of expansion and rapid convergence for “well-defined” systems. The system model is “well-defined” if higher-order variable correlations are weak, permitting the model to be captured by the first few low-order terms of expansion. In fact, this is the only assumption for a meaningful application of HDMR. Provided it is satisfied, HDMR could play a role similar to neural networks. However it has clear mathematical background, which increases chance for success and offers novel opportunities for applications and theoretical research. Use of the HDMR expansion to Bellman function – a solution of the the dynamic programming – is tempting. It separates original high dimensional input–output mapping into sum of low-order (possibly non-linear) mappings acting on orthogonal subspaces. The presented example indicates the way how the HDMR can be tailored to the control design and serves for inspection whether the basic HDMR assumption is applicable.