Internal stabilization and external L p stabilization of linear systems subject to constraints

During the last decade stabilization and other control design problems for linear systems subject to constraints, especially those with actuator saturation, have received much attention. Whenever the magnitude of the control input is bounded, internal stabilization in either global or semiglobal sense is possible if and only if the open-loop system is asymptotically null controllable with bounded control (i.e. if and only if the open-loop system is stabilizable and all its poles are in the closed left-half plane). During the 1990’s, control design problems for linear systems with actuator saturation were mostly studied in the framework of semiglobal and global stabilization and hence attentions were focused only on asymptotically null controllable systems. Besides internal stabilization, another control design problem of interest is external stabilization or the requirement of L p stability, i.e. the requirement of the controlled output being in L p whenever the external signals are in L p and the initial conditions are zero. However, in many cases internal stabilization by itself does not automatically guarantee external stabilization. One has to design a new controller for simultaneous internal and external stabilization. A number of simultaneous internal and external stabilization problems in either global or semiglobal or regional sense have been formulated and studied (see [2, 5] and the references cited there). A standard topology in all these works pertains to the case where the external signal w is input additive. For such a configuration, the study of different types of simultaneous internal and external stabilization problems with or without finite gain (finite gain implies that the induced norm of the mapping from the external input to the controlled output is finite) is complete [2, 5] when state feedback controllers are used, and leads to the following result: