| Scope and method of study. There are numerous physical nonlinear systems whose mathematical descriptions present difficulties when designing a globally stabilizing controller. For such systems, we often admit locally stabilizing controllers designed by applying linear system theory to suitable linearized models. Theoretically, we expect the linear controller to stabilize the nonlinear system in a region where the linear approximation is valid. However, it is traditional that we do not examine the extent of this region when designing a linear controller. After such design, simulations are employed to numerically estimate the resultant attractive region. A major drawback of such a linear controller is that the attractive region of the nonlinear system can be unsatisfactorily small. It is known that a linear controller can be designed either by relocating eigenvalues, by optimizing a performance index, or by shaping frequency-domain plots of the linearized model. However, their relationships to the attractive region are not obvious, and thus it is not clear how several design parameters in these techniques should be selected when requiring that the attractive region be large. Under mild assumptions, the Lyapunov Attractive Region Control (LARC) can be constructed from a special quadratic Lyapunov function such that the system of interest is at least locally uniformly asymptotically stable with a reasonably large attractive region. The formulation is primarily based on geometry, relative orientation of surfaces, and matrix theory. Optimization can be employed to generate a stabilizing Lyapunov Attractive Region (LAR) controller.; Findings and conclusion. Systematic procedures for generating a LARC with and without uncertainty specifications are proposed. Using these, a LARC can be generated in a timely fashion and yields satisfactory results. For uncertain systems with uncertainty specifications, our procedures typically call for an optimization routine. In this case, we provide a tool for determining reasonably good initial values for the optimization. Starting from these initial values, it appears that a simple optimization routine is sufficient to produce fast convergence to a stabilizing LAR controller, which cannot be reached from inappropriate initial conditions. When applying LARC to common examples in the literature for local stabilization, the attractive regions resulting from LARC are larger than those resulting from pole placement and LQR. For global stabilization, LARC produces the least conservative allowable uncertainty bounds when compared to those in the common examples. |
