Trading the stability of finite zeros for global stabilization of nonlinear cascade systems

This paper analyzes the stabilizability properties of nonlinear cascades in which a nonminimum phase linear system is interconnected through its output to a stable nonlinear system. It is shown that the instability of the zeros of the linear system can be traded with the stability of the nonlinear system up to a limit fixed by the growth properties of the cascade interconnection term. Below this limit, global stabilization is achieved by smooth static state feedback. Beyond this limit, various examples illustrate that controllability of the cascade may be lost, making it impossible to achieve large regions of attractions.

Robust stabilization of a nonlinear cement mill model

Plugging is well known to be a major cause of instability in industrial cement mills. A simple nonlinear model able to simulate the plugging phenomenon is presented. It is shown how a nonlinear robust controller can be designed in order to fully prevent the mill from plugging.

Slow peaking and low-gain designs for global stabilization of nonlinear systems

This paper presents an analysis of the slow-peaking phenomenon, a pitfall of low-gain designs that imposes basic limitations to large regions of attraction in nonlinear control systems. The phenomenon is best understood on a chain of integrators perturbed by a vector field up(x, u) that satisfies p(x, 0) = 0. Because small controls (or low-gain designs) are sufficient to stabilize the unperturbed chain of integrators, it may seem that smaller controls, which attenuate the perturbation up(x, u) in a large compact set, can be employed to achieve larger regions of attraction. This intuition is false, however, and peaking may cause a loss of global controllability unless severe growth restrictions are imposed on p(x, u). These growth restrictions are expressed as a higher order condition with respect to a particular weighted dilation related to the peaking exponents of the nominal system. When this higher order condition is satisfied, an explicit control law is derived that achieves global asymptotic stability of x = 0. This stabilization result is extended to more general cascade nonlinear systems in which the perturbation p(x, v) v, v = (ξ, u) T, contains the state ξ and the control u of a stabilizable subsystem ξ = a(ξ, u). As an illustration, a control law is derived that achieves global stabilization of the frictionless ball-and-beam model.

Global stabilization of feedforward systems with exponentially unstable Jacobian linearization

The global stabilization of a class of feedforward systems having an exponentially unstable Jacobian linearization is achieved by a high-gain feedback saturated at a low level. The control law forces the derivatives of the state variables to small values along the closed-loop trajectories. This "slow control" design is illustrated with a benchmark example and its limitations are emphasized. © 1999 Elsevier Science B.V. All rights reserved.

Lyapunov functions for stable cascades and applications to global stabilization

This paper generalizes recent Lyapunov constructions for a cascade of two nonlinear systems, one of which is stable rather than asymptotically stable. A new cross-term construction in the Lyapunov function allows us to replace earlier growth conditions by a necessary boundedness condition. This method is instrumental in the global stabilization of feedforward systems, and new stabilization results are derived from the generalized construction.

Some remarks about periodic feedback stabilization

Periodic feedback stabilization is a very natural solution to overcome the topological obstructions which may occur when one tries to asymptotically (locally) stabilize a (locally) controllable nonlinear system around an equilibrium point. The object of this paper is to give a simple geometric interpretation of this fact, to show that one obtains a weakened form of those obstructions when periodic feedback is used, and to illustrate the success of periodic feedback stabilization on a representative system which contains a drift.