Construction of reduced order controllers for nonlinear systems with periodic coefficients

Abstract

This study provides a methodology for reduced order controller design for nonlinear dynamic systems with time-periodic coefficients. The proposed methodology is quite general in a sense that it can be easily modified for nonlinear systems with constant coefficients. The equations of motion are represented by quasi-linear differential equations in state space, containing a time-periodic linear part and nonlinear monomials of states with periodic coefficients. The Lyapunov—Floquet (L-F) transformation is used to transform the time-varying linear part of the system into a time-invariant form. Eigenvalue decomposition of the time-invariant linear part can then be used to identify the dominant/nondominant dynamics of the system. The nondominant states of the system are expressed as a nonlinear, time-periodic, manifold relationship in terms of the dominant states. As a result, the original large system can be expressed as a lower order system represented only by the dominant states. A reducibility condition is derived to provide conditions under which a nonlinear order reduction is possible. Then a proper coordinate transformation and state feedback can be found under which the reduced order system is transformed into a linear, time-periodic, closed-loop system. This permits the design of a time-varying feedback controller in linear space to guarantee the stability of the system. The case of systems with constant coefficients is treated as a subset of its periodic counterpart. In this case one does not have to use the L-F transformation; moreover, the manifold and all other transformations are purely spatial. The proposed methodology is illustrated by designing a two dimensional reduced order controller for a 4-degrees of freedom, inverted pendulum subjected to a periodic force for which the equations of motion are time-periodic. An example involving a 2-dof spring-mass-damper system is also presented which yields the equations of motion with constant coefficients.