Static optimal decoupling control for linear over-actuated systems regarding time-varying references

Abstract

We address static decoupling control for linear over-actuated systems and time-varying references given by exogenous systems with arbitrary eigenvalues. Based on mild assumptions, additional degrees of freedom in form of an input are provided. Then an optimal tracking problem for quadratic integral cost is formulated. Despite the time dependency of the cost and dynamics, we derive a static feedback and pre-filter satisfying necessary optimality conditions for infinite final time. These can be calculated by the solution of an algebraic Riccati equation and a Sylvester equation, respectively. In spite of its simplicity in derivation as well as implementation - offering great convenience for practical use - we prove optimal transient behavior to a unique optimal stationary trajectory of the system states. Or, more precisely, of the internal dynamics which are proven to exist. Moreover, the static control law is verified to be a close approximation of the computationally expensive finite time optimal solution if simple qualitative criteria are met. An application to a helicopter model reveals the high efficiency of our approach compared to others.