In current industrial control applications, the proportional + integral + derivative (PID) control is still used as the leading tool, but constructing controller requires precise mathematical model of plant, and tuning the parameters of controllers is not simple to implement. Motivated by the gap between theory and practice in control problems, linear active disturbance rejection control (LADRC) addresses a set of control problems in the absence of precise mathematical models. LADRC has two parameters to be tuned, namely, a closed-loop bandwidth and observer bandwidth. The performance of LADRC depends on the quick convergence of a unique state observer, known as the extended state observer, proposed by Jinqing Han ( 1994 Han, J . 1994. Nonlinear state error feedback control. Control and Decision, 10: 221–225. [Google Scholar] ). Only one parameter, observer bandwidth, significantly affects the tracking speed of extended state observer. This paper studies numerically the optimal fast tracking observer bandwidth and the absolute tracking error estimation for a class of non-linear and uncertain motion control problems by finite difference method.

Most existing methods for phylogenetic analysis involve developing an evolutionary model and then using some type of computational algorithm to perform multiple sequence alignment. There are two problems with this approach: (1) different evolutionary models can lead to different results, and (2) the computation time required for multiple alignments makes it impossible to analyse the phylogeny of a whole genome. This motivates us to create a new approach to characterize genetic sequences. To each DNA sequence, we associate a natural vector based on the distributions of nucleotides. This produces a one-to-one correspondence between the DNA sequence and its natural vector. We define the distance between two DNA sequences to be the distance between their associated natural vectors. This creates a genome space with a biological distance which makes global comparison of genomes with same topology possible. We use our proposed method to analyze the genomes of the new influenza A (H1N1) virus, human rhinoviruses (HRV) and mammalian mitochondrial. The result shows that a triple-reassortant swine virus circulating in North America and the Eurasian swine virus belong to the lineage of the influenza A (H1N1) virus. For the HRV and mammalian mitochondrial genomes, the results coincide with biologists' analyses. Our approach provides a powerful new tool for analyzing and annotating genomes and their phylogenetic relationships. Whole or partial genomes can be handled more easily and more quickly than using multiple alignment methods. Once a genome space has been constructed, it can be stored in a database. There is no need to reconstruct the genome space for subsequent applications, whereas in multiple alignment methods, realignment is needed to add new sequences. Furthermore, one can make a global comparison of all genomes simultaneously, which no other existing method can achieve.

Graphical representation of DNA sequence provides a simple way of viewing, sorting and comparing various gene structures. A new two‐dimensional graphical representation method using a two‐ quadrant Cartesian coordinates system has been derived for mathematical denotation of DNA sequence. The two‐dimensional graphic representation resolves sequences’ degeneracy and is mathematically proven to eliminate circuit formation. Given x‐projection and y‐projection of any point on the graphical representation, the number of A, G, C and T from the beginning of the sequence to that point could be found. Compared with previous methods, this graphical representation is more in‐line with the conventional recognition of linear sequences by molecular biologists, and also provides a metaphor in two dimensions for local and global DNA sequence comparison.