General hybrid orthogonal functions and their applications in systems and control
This monograph introduces the General Hybrid Orthogonal Functions (GHOF) defined on the domain \(\Omega=(0,T)\subset \mathbb{R}_1\) as follows: \[ G=\{\theta_{i,j}(t)\mid j\in I_m,\quad i\in I_{r_j}\},\quad \forall t\in \Omega\subset\mathbb{R}_1 \] where \[ \theta_{i,j}(t)=\begin{cases} p_{i,j}((t-t_{j-1})/T_j),\quad & t\in \Omega_j=(t_{j-1},t_j)\\ 0\quad & \text{otherwise}\end{cases} \] with \(t_j=\sum^j_{l=1} T_l\), such that \(T_j=t_j-t_{j-1}\), \(\bigcup^m_{j=1}\Omega_j=\Omega\), \(I_n=\{1,2,\dots,n\}\) and the set \(G_j=\{\theta_{i,j}(t)\mid i\in I_{r_j}\}=\{p_{ij}((t-t_{j-1})/T_j)\mid i\in I_{r_j}\}\), \(\forall j\in I_m\), can be any of the complete orthogonal continuous basis functions (e.g., a system of Legendre polynomials for a given \(j\), and sine-cosine functions for another \(j\)). The GHOF thus form a suitable hybrid system of basis functions inherently possessing mixed features of continuity and jumps. Completeness, orthogonality and a formal method for function expansion are also shown. The GHOF have been used as a flexible framework of computational tools in the following problems: 1) Analysis of SCR-controlled DC drive systems; 2) Parameter estimation and combined parameter and state estimation of continuous-time systems; 3) Continuous-time model-based self-tuning control. An important feature of the book is its coverage of recursive algorithms for real-time implementation and a completely continuous-time based self-tuning control scheme using block pulse functions which are seen to belong to the GHOF family. Other possible applications include analysis of time-delay systems, solution of functional differential equations, optimal control of lumped linear systems, distributed parameter systems.
continuous-time model-based self-tuning control
scientific article; zbMATH DE number 872793