Control theory in the plane
25 years after publishing his Dynamical Systems in the Plane (Academic Press 1966) the author is concerned with control in the plane. He came to the control field from outside, and he took as the new control paradigma not to consider the feedback systems but open-loop control. Nevertheless, the main results in the field were done by R. E. Kalman for feedback LQ and LQG control. The same holds for Lyapunov stabilization and of exact linearization of nonlinear control systems. The open-loop approach, based mainly on calculus of variations, constructs the open-loop control and then is trying to implement it by a feedback and then to cheque -- not to synthetize the stability, paying little concern to the control robustness. Part 1 -- Control systems in Euclidean spaces is rather standard being concerned with examples of control systems, ordinary differential equations and control systems theory. The degree of exactness is over standard and this is coupled with carefull treatment of the examples from mechanics. Part 2 -- Control systems in the plane is the main part of the book. The planar control systems are \(\dot x= uf(x)+ (1- u)g(x)\) where the state \(x\) is planar, and open-loop control \(u\) is scalar. The phase points are classified as noncritical (without contact, weak center, lenticular) and critical (elementary, higher order). A whole chapter is devoted to the theory of Poincaré index. The last chapter treats optimal control theory in the plane.
scientific article; zbMATH DE number 108215