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Analysis and design of singular linear systems under actuator saturation and ${ℒ}_{2}/{ℒ}_{\infty }$disturbances. (English) Zbl 1149.93030
Summary: This paper carries out an analysis of the ${ℒ}_{2}$ gain and ${ℒ}_{\infty }$ performance for singular linear systems under actuator saturation. The notion of Bounded State Stability (BSS) with respect to the influence of ${ℒ}_{2}$ or ${ℒ}_{\infty }$ disturbances is introduced and conditions under which a system is bounded state stable are established in terms of linear matrix inequalities (LMIs). The disturbance tolerance capability of the system is then measured as the bound on the ${ℒ}_{2}$ or ${ℒ}_{\infty }$ norm of the disturbances under which the system remains bounded state stable and the disturbance rejection capability is measured by the restricted ${ℒ}_{2}$ gain from the disturbance to the system output or ${ℒ}_{\infty }$ norm of the system output. Based on the BSS conditions, assessment of the disturbance tolerance and rejection capabilities of the system under a given state feedback law is formulated and solved as LMI constrained optimization problems. By viewing the feedback gain as an additional variable, these optimization problems can be readily adapted for control design. Our analysis and design reduce the existing results for regular linear systems to the degenerate case where the singular linear system reduces to a regular system, and for singular systems in the absence of actuator saturation or when the disturbance is weak enough to not cause saturation.
##### MSC:
 93D99 Stability of control systems 93C05 Linear control systems 93B51 Design techniques in systems theory