Mutual Direct State Controllability Analysis of Multivariable Underactuated LTI Systems

The concept of mutual direct linear state controllability (MDC) is introduced for underactuated mechanical and aerospace systems. The MDC analysis identifies the sets of state variables of a control system that are simultaneously and independently controllable by direct action of the available control authority. The analysis utilizes the controls coefficient vectors that correspond to virtual constraint state dynamics (VCD) that are imposed on the linear system dynamics. It is shown that the largest (non unique) set of mutually-directly controllable state variables of a control system has the same number of elements as the number of independent control variables of the system. The MDC assessment leads to a systematic multivariable control system design that results from successive generalized inversion of the individual VCDs of the mutually directly controllable states of the system by using the Generalized Dynamic Inversion (GDI) control law. The produced control system design is in the nested GDI control loops configuration.