Abstract: The necessity to deal with partial differential equations (PDEs) and the dissipation condition are the main adversities in the application of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC). Recently, an algebraic solution of IDA-PBC has been explored for a class of affine polynomial systems by using sum of squares (SOS) and semidefinite programming (SDP). In this work, we extend the previous method by incorporating actuator saturation (AS) and two minimization objectives in the SDP. Our results are validated on two polynomial systems.