On the recoverability of nonlinear state feedback laws by extended linearization control techniques

Extended linearization is the process of factoring a nonlinear system into a linear-like structure x˙=A(x)x+ B(x)u which contains state-dependent coefficient (SDC) matrices. An extended linearization control technique is a technique which: 1) treats the SDC matrices A(x) and B(x) as being constant, and 2) uses a linear control synthesis method on the linear-like structure to produce a closed-loop SDC matrix which is pointwise Hurwitz. This paper investigates the recoverability of nonlinear state feedback laws using extended linearization control techniques with particular focus on the state-dependent Riccati equation method. An example is presented where it is attempted to recover an optimal feedback law. It is shown that there exists no extended linearization control technique that is capable of recovering the given law. It is then shown how the feedback law can be recovered by using: a state-dependent state weighting matrix and a nonsymmetric solution of the state-dependent Riccati equation which simultaneously satisfies a symmetry condition