Parametric enhancement of state-dependent Riccati equation based control

In the state-dependent Riccati equation (SDRE) method for nonlinear regulation, the nonlinear system is first brought to a linear structure having state-dependent coefficient (SDC) matrices, i.e., x˙=A(x)x+B(x)u. An SDRE is then solved to obtain a nonlinear controller of the form u=-R^-1(x)B^T(x)P(x)x, where P(x) is the solution of the SDRE. It is known that there are an infinite number of ways to bring the nonlinear dynamics to the SDC form and this nonuniqueness allows the SDC matrix A to be parameterized as A(x,α(x)). If one is able to solve a certain partial differential equation, then α(x) can be determined such that all of the necessary conditions for optimality are satisfied. However, one cannot expect such a solution to be real-time implementable. In this paper, by using a certain SDC structure and integral control, we show how α(x) can be updated via feedback to enhance design performance