基于线性单级倒立摆的LQR和NMPC控制方法对比研究

Comparison Study of LQR and NMPC Control Methods Based on Linear Single-Stage

摘要: 针对目前欠驱动系统较成熟的控制方法对比缺乏的问题，研究线性单级倒立摆系统在全状态反馈控制下，采用线性二次型调节器(LQR)与非线性模型预测控制(NMPC)两种控制策略的效果。通过对比这两种控制方法的动态性能，评估它们在线性单级倒立摆系统中的适用性。首先，通过拉格朗日方程推导出系统的非线性数学模型，并进一步通过线性化理论假设得到系统的线性化数学模型。随后对其进行了稳定性、可控性和可观性分析。基于这些分析，设计了LQR和NMPC控制器。最后，在Matlab/Simulink中进行联合仿真实验。通过设置相同的初始条件和加权矩阵Q与R，对仿真结果的小车位移、摆动角度、小车速度、摆动角速度四个指标进行了对比分析。结果表明，LQR控制使小车在2.5 s内回到初始位置，比NMPC控制快37.5%，并且位移最大超调量减少了38.7%，从−0.3522 m降至−0.2160 m。同时，LQR控制还缩短了摆杆达到竖直状态的时间至2.5 s，比NMPC快28.6%，且摆杆角度的最大超调量从NMPC的−0.2381 rad减少至−0.1050 rad，降低了56%。结论指出，对于线性化后的单级倒立摆系统，LQR比NMPC更具优势，提供了更高效的控制效果，对以后欠驱动系统的控制具有一定的参考价值。

Abstract: To address the current lack of comparison between mature control methods for underactuated systems, this study investigates the performance of Linear Quadratic Regulator (LQR) and Nonlinear Model Predictive Control (NMPC) for a linear single-stage inverted pendulum system under full state feedback control. By comparing the dynamic performance of these two control strategies, the study evaluates their applicability to the linear single-stage inverted pendulum system. First, the nonlinear mathematical model of the system is derived using the Lagrangian method, and subsequently, the system’s linearized mathematical model is obtained through linearization theory. Stability, controllability, and observability analyses are then conducted. Based on these analyses, LQR and NMPC controllers are designed. Finally, joint simulation experiments are conducted using Matlab/Simulink. With the same initial conditions and weighting matrices Q and R, the simulation results for the cart displacement, pendulum angle, cart velocity, and angular velocity are compared. The results indicate that LQR control returns the cart to its initial position within 2.5 seconds, which is 37.5% faster than NMPC control. Additionally, the maximum displacement overshoot is reduced by 38.7%, from −0.3522 meters to −0.2160 meters. LQR control also shortens the time required for the pendulum to reach the vertical state to 2.5 seconds, which is 28.6% faster than NMPC, and reduces the maximum pendulum angle overshoot from −0.2381 radians to −0.1050 radians, a decrease of 56%. The conclusion highlights that, for the linearized single-stage inverted pendulum system, LQR provides more efficient control compared to NMPC, offering valuable insights for the control of underactuated systems in the future.