14. The double integrator with linear control laws subject to saturation

As discussed in previous chapters, for linear systems subject to actuator saturation, simultaneous external and internal stabilization is possible only for systems which are asymptotically null controllable with bounded control (ANCBC). Even then, in general, nonlinear feedback control laws are required. Of particular interest is the use of linear feedback control laws. Clearly, as shown in Chaps. 12 and 13, for open-loop neutrally stable systems, the global asymptotic stability and external L _p  (ℓ _p ) stability for p ∈ [1, ∞) with arbitrary initial conditions can be achieved by linear feedback control laws. We continue in this chapter the theme of pursuing systems other than open-loop neutrally stable ones for which such a simultaneous stabilization is feasible. We focus here on a canonical class of ANCBC systems, namely, a double integrator which is ubiquitously used. In fact, the double-integrator system is commonly seen in control applications including low-friction, free rigid-body motion, such as single-axis spacecraft rotation and rotary crane motion (see [117], and the references therein). All these issues are the motivating factors why we choose in this chapter to reexamine the notions of external and internal stabilization via linear feedback control laws for a double integrator with linear feedback control laws subject to saturation.