Relaxed Stabilization Conditions via Sum of Squares Approach for the Nonlinear Polynomial Model

In this paper، stabilization conditions and controller design for a class of nonlinear systems are proposed. The proposed method is based on the nonlinear feedback، quadratic Lyapunov function and heuristic slack matrices definition. These slack matrices in null products are derived using the properties of the system dynamics. Based on the Lyapunov stability theorem and Sum of Squares (SOS) decomposition techniques، the conditions are derived in terms of SOS. This approach has two main advantages. First، using the polynomial model، the proposed method uses the polynomial state space matrices in the model description. Therefore، it does not need any existing modeling methods such as the Takagi Sugeno (T-S) fuzzy model which can be a source of conservativeness in the control design conditions، because the membership function information cannot be used completely in the derivation of the controller design conditions. Second، using slack matrices، one can find the matrices that leads to applicable controller design which this means it provides extra degrees of freedom. To show the effectiveness of the proposed method، a PMSM is considered in the numerical simulation.