In the actual systems which are described with polynomic systems, there are chemical ones biological ones and so on, and we can approximate general nonlinear systems by polynomic systems on an appropriate driving domain.
This paper is concerned with the servo problem of polynomic systems. The construction of the servo system in this paper is to apply Davison-Smith method to polynomic systems, to stabilize state z which is the time derivative of state z of the augmented system, and to guarantee the boundedness of state z at the same time.
In the servo system, state z must be converged to any constant vector generally under the consideration of constant disturbances and constant desired values. Then the boundedness of the state must be accomplished. Even if the state is bounded, it would be happened that the state is time-variable, for example a periodic function of time. So limt→∞ z=0 must be accomplished because of the rejection of the above possibility. For the determination of the control's law, though the order is inverse, to set a quadratic form of z(V_D), to determine the control's law from V_D<0, and to calculate the driving domain D_S. For the next time, to set a quadratic form of z(V), and to confirm V<0 in the domain D_S. It is advantage that we can induce the condition of V<0 from one of V_D<0 in polynomic systems.
The control's law of the linear part can be gotten by the method of pole assignment or a linear regulator with the condition of controllability. On the other hand, the nonlinear part can be gotten by the minimization of coefficients of the norm-evalution with real numbers ρ_i (i=1, 3, …) for even degree polynomials of V_D. The driving domain can be calculated from system parameters. Though its domain gives the sufficient condition for the system to drive stably, the practical one is wider than the theoretical one like as to show in a simulation of a numerical example.
From these considerations this paper's method can give an useful control's law for the servo problem of polynomic systems.