A computationally efficient Lyapunov-based scheduling procedure for control of nonlinear systems with stability guarantees

Abstract

We propose an alternative to gain scheduling for stabilization of nonlinear systems. For a useful class of nonlinear systems, the characterization of a region of stability based on a control Lyapunov function is computationally tractable, in the sense that computation times vary polynomially with the state dimension for a fixed number of scheduling variables. Using this fact, we develop a procedure to expand the region of stability by constructing control Lyapunov functions to various trim points of the system. A Lyapunov-based control synthesis algorithm is used to construct a control law that guarantees closed-loop stability for initial conditions in the expanded region of state space. This control asymptotically recovers the optimal stability margin in the sense of a Lyapunov derivative, which in turn can be seen as a performance measure. Robustness to bounded disturbances and stabilization under bounded control are easily incorporated into this framework. In the worst case, the computational complexity of the analysis problem that develops in the new method is increased by an exponential in the disturbance dimension. Similarly, we can handle control constraints with an increase in computational complexity of no more than an exponential in the control dimension. We demonstrate the new control design procedure on an example.