Connections between control allocation and linear quadratic control for weakly redundant systems

In this paper, connections between the control allocation and linear quadratic (LQ) control frameworks for optimally distributing control inputs in weakly input redundant systems are explored. It is also shown that, for a representative class of exogenous disturbance and reference signals, the LQ control technique is identical to the so-called optimal control subspace-based (OCS) control allocation technique. However, for this equivalence to hold, the OCS control allocation technique requires evaluation of a (generally) non-causal relationship between control inputs, while the LQ control technique requires perfect knowledge of the system, disturbance and reference states. In practice, neither of these conditions is achievable; therefore, approximations are needed. In this regard, the OCS control allocation technique is superior because it is explicit about the relationship that must be accurately approximated to attain optimality; this enables good approximations of the optimal relationship to be realized. Conversely, the LQ control technique implicitly approximates the optimal relationship via estimation of states, disturbances and/or reference signals. In a comparison with a classical example based on the Kalman filter, the OCS control allocation is shown to be superior in preserving the optimal alignment of redundant control inputs, thus introducing enhanced performance with reduced cost.