Adaptive backstepping control for permanent magnet linear motors against uncertainties and disturbances

In practical operation, the system parameters of a permanent magnet linear motor are affected by unknown factors, including nonlinear friction, sudden load changes, thrust fluctuations, and so on. Therefore, a controller designed for fixed parameters often cannot produce satisfactory results. To solve this problem, a novel design method for permanent magnet linear motor systems based on backstepping is proposed in this article. The proposed control scheme does not require the values or range of changes of the system parameters in advance but constructs an update law to perform online parameter estimation. A difficulty encountered in controller design and stability analysis is the estimation of the unknown coefficient of a single-variable state, which can be considered as a virtual input. In this article, the estimated virtual control coefficient is introduced into the coordinate transformation to construct a novel coordinate transformation. However, introducing the estimated coefficient into the virtual control makes the derivative of the Lyapunov function used in the stability analysis more complex. This novel nonlinear dynamic term can be precisely cancelled by changing the update law of the coefficient. Finally, simulations are performed to assess the performance of the proposed control scheme against that of a traditional proportional–integral–derivative controller, and the simulation results show that the proposed control law can effectively improve the system performance.