Nonlinear optimal and multi-loop flatness-based control for the 5-DOF rotor and AMB system

Abstract

The 5-DOF rotor and active magnetic bearing (AMB) system exhibits strong nonlinearities and multi-variable dynamics and its stabilization and control is a non-trivial task. In this article the control problem for the 5-DOF rotor and active magnetic bearing system is solved with (i) a nonlinear optimal control method (ii) a flatness-based control approach which is implemented in successive loops. To apply method (i) that is nonlinear optimal control, the dynamic model of the 5-DOF rotor and active magnetic bearing system undergoes approximate linearization at each sampling instance with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization point is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. To implement control method (ii), that is flatness-based control in successive loops, the state–space model of the 5-DOF rotor and active magnetic bearing system is separated into two subsystems, which are connected between them in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input–output linearized flat systems. The state variables of the second subsystem become virtual control inputs for the first subsystem. In turn, exogenous control inputs are applied to the second subsystem. The whole control method is implemented in two successive loops and its global stability properties are also proven through Lyapunov stability analysis. The proposed method achieves stabilization of the dynamics of the 5-DOF rotor and active magnetic bearing system without the need of diffeomorphisms and complicated state–space model transformations.