Recursive pointwise design for nonlinear systems

This work presents a recursive pointwise design (RPD) method for a class of nonlinear systems represented by dx(t)/dt = f(x(t)) + g(x(t))u(t). A main feature of the RPD method is to recursively design a stable controller by using pointwise information of a system. The design philosophy is that f(x(t)) and g(x(t)) can be approximated as constant vectors in very small local state spaces. Based on the design philosophy, we numerically determine constant control inputs in very small local state spaces by solving linear matrix inequalities (LMIs) derived in this paper. The designed controller switches to another constant control input when the states move to another local state space. Although the design philosophy is simple and natural, the controller does not always guarantee the stability of the original nonlinear system dx(t)/dt = f(x(t)) + g(x(t))u(t). Therefore, this paper gives ideas of compensating the approximation caused by the design philosophy. After addressing outline of the pointwise design, we provide design conditions that exactly guarantee the stability of the original system. The controller design conditions requires to give the maximum and minimum values of elements in the functions f(x(t)) and g(x(t)) in each local state space. Therefore, we also present design conditions for unknown cases of the maximum and minimum values. Furthermore, we construct an effective design procedure using the pointwise design. A feature of the design procedure is to subdivide only infeasible regions and to solve LMIs again only for the subdivided infeasible regions. The recursive procedure saves effort to design a controller. A design example demonstrates the utility of the RPD method.