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 any technique which (a) treats the SDC matrices A(z) and B(x ) as being constant and (b) 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 [l, 21 with particular focus on the state-dependent Riccati equation (SDRE) method [ 3 ] . By recoverable it is meant that a given nonlinear state feedback law of the form U = k ( z ) , k ( 0 ) = 0, can be obtained (or recovered) from a given control design method. Conditions relating to recoverability by extended linearization control methods are provided. An example is then 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 two control techniques which are variations of the SDRE method. The first uses a state-dependent state weighting matrix which may be negative definite or even indefinite over a subset of the state space while the second uses a nonsymmetric solution of the state-dependent Riccati equation which simultaneously satisfies a symmetry condition. Even though these latter techniques are based on the extended linearization of the system, they are not extended linearization control methods since they do not guarantee that the closed-loop SDC matrix is pointwise Hurwitz.