Reduction of triangular controlled systems to linear systems without changing the control
Single-input, triangular control systems of the form \[ \dot x_i= f_i(x_1,\dots, x_{i+1}),\quad\text{for }i= 1,2,\dots, n-1,\quad \dot x_n= f_n(x_1,\dots, x_n, u),\tag{\(*\)} \] with bounded controls \(| u|\leq 1\), are examined. The main result of this paper (Theorem 3) states and proves necessary and sufficient conditions for making the system \((*)\) equivalent by a state space diffeomorphism to the normal form \[ \dot z_i= z_{i+1},\quad i= 1,2,\dots, n-1,\quad \dot z_n= a_1 z_1+\cdots+ a_n z_n+ u.\tag{\(**\)} \] These conditions impose some constraints on the functions \(f_i\), \(i= 1,\dots, n\), and provide an explicit definition of the state space diifeomorphism in terms of differential operators \[ A_i= f_1(x_1, x_2){\partial\over\partial x_1}+\cdots+ f_i(x_1,\dots, x_{i+1}){\partial\over\partial x_i},\quad i= 1,2,\dots, n-1. \] In particular, these conditions allow one to decide when \((**)\) takes the form \(\dot z_i= z_{i+1}\), \(i= 1,2,\dots, n-1\), \(\dot z_n= u\), which is useful in optimal control.
scientific article; zbMATH DE number 1916005