Nonlinear H2 and H∞ Control for Nonlinear Systems with Parameter Variation Using LPV Approach

Abstract

In this dissertation, we firstly propose a modeling of nonlinear systems with parameter variation (NSPV) and its physical properties. The NSPV model has bounded nonlinear function. There are real physical systems in the form of NSPV such as the permanent magnet synchronous motor (PMSM), the permanent magnet (PM) stepper motor, and Sawyer motor. The NSPV model can be transformed to a parameter-dependent linear system by linear parameter varying (LPV) synthesis. The LPV model provides performance criterion in the sense of a linear system. First, the nonlinear H2 control with passive nonlinear observer is developed to improve transient response for the PM stepper motor. However, some states of the system are not always measurable. To estimate unmeasurable states of the system, we proposed a passive nonlinear observer. In [1], LPV modeling of the PM synchronous motor was presented in a DQ rotating frame. The LPV system using DQ transformation depends on the angular velocity of rotor as a varying parameter. Therefore, a physical assumption was required that the angular velocity is bounded. To solve these problem, we design the nonlinear H2 controller with a new torque modulation without DQ transformation. Because the proposed method includes gain scheduling based on LPV synthesis, it achieves improved transient response in the position control and provides a gain-tuning guideline for position tracking of the nonlinear system. The stability of the closed-loop system was verified by input-to-state stability theory. Second, the nonlinear H2 control using LPV synthesis is proposed to regulate yaw for the Sawyer motor. Yaw regulation is important issue for a high speed motion control. Using the nonlinear H2 control based on LPV synthesis, yaw was well-regulated and then tracking performance was improved. The weighting factors enable adjustments to the tracking performance of each states of the system. In the controller design process, we assume that the full-states are measurable. A control gain scheduling is determined using H2 control based on linear matrix inequality approach. Using Lyapunov theorem, it is proven that the closed-loop system is stable. Third, the nonlinear H∞ control with the LPV based H∞state estimator is proposed to both improve velocity tracking performance and estimate load torque. For the velocity tracking in the PMSM, the load torque should be compensated. Generally, nonlinear observers were used to estimate the disturbance such as load torque in nonlinear system. However, proof of closed-loop stability is difficult and observer gain tuning is not intuitive. By the nonlinear H∞ control with the LPV based H∞ state estimator, we can investigate analytic consideration for well-known linear approaches in the view-point of frequency domain as well as time domain. The closed-loop stability was easily verified by means of the input-to-state stability theorem. The performance of all of the proposed methods was validated through simulations and experiments. From the analysis and experimental results, we concluded that the proposed methods improve the transient response and provide intuitive gain-tuning guideline for the NSPV.