Nonlinear Control Methods for a Class of Singularly Perturbed Interconnected Nonlinear Systems

Abstract

In this dissertation, we firstly propose a modeling of singularly perturbed interconnected nonlinear systems (SPINSs) and its physical properties. The SPINS model is a two-time scale nonlinear system composed of slow and fast subsystem. The SPINS model has interconnections between the slow and fast subsystems. The slow subsystem is driven by the state of the fast subsystem and the fast subsystem is affected by the slow subsystem. There are real physical systems in the form of SPINS such as electro-hydraulic actuators (EHAs), the permanent magnet synchronous motor (PMSM), the permanent magnet (PM) stepper motor, and direct current (DC) motors. The SPINS model enables analyzing the effects of the interconnections and unified controller design.

Based on the SPINS model, we address two different control problems: the tracking problem for a slow subsystem with direct feedback and that for a fast subsystem with indirect feedback, respectively. In order to deal with the tracking problem for a slow subsystem with direct feedback, we propose an enhanced nonlinear damping controller and a nonlinear output feedback controller with a state dependant reset integrator (SDRI).

First, the enhanced nonlinear damping control method is designed to guarantee the globally exponential convergence of the objective state, for example in terms of position or velocity. In the controller design, we intentionally hold the interconnections to give the enhanced nonlinear damping effects and weighting factors to adjust the tracking performance of each state of the slow subsystem. We analyze how the interconnections can generate the nonlinear damping effects. In addition, a nonlinear reduced order observer is proposed to estimate unmeasurable states of the slow subsystems and disturbances. Using a composite Lyapunov function, it is proven that the closed-loop system is globally exponentially stable. However, using the observer may degrade the tracking performance in a transient period of a slow subsystem. Moreover, the enhanced nonlinear damping control method requires the measurement of a fast subsystem.

Second, we develop the nonlinear output feedback control with SDRI to archive the output feedback control and to improve both transient and steady-state tracking performance of slow subsystems. Using passive properties and the nonlinear damping effects, we implement the output feedback tracking control. The state of the integrator is reset at the threshold of the output tracking error by the proposed method. Thus, the closed-loop system should be considered as a switched system. For the closed-loop stability analysis, we prove that there exists the threshold, which becomes an invariant set. The output tracking error is bounded under the invariant set in a finite time. Thus, the integrator is also no more reset after the finite time. Through the invariant set analysis, it is proven that the closed-loop system is globally exponentially stable using the common Lyapunov function. In addition, the optimization problem to minimize the threshold is proposed to improve transient performance of slow subsystems. Based on the stability analysis and the optimization of the threshold, the tuning guideline of control parameters, which ensures the globally exponentially stability, is also provided through some mathematical analysis.

In the two control methods for SPINSs proposed above, we address the tracking control problem of slow subsystems through direct measuring of the objective states, for example position tracking control with position feedback. To reduce manufacturing costs and to ensure safety, however, a sensorless control method for SPINS has been required. Thus, there have already been many studies on sensorless controller designs for slow subsystems with indirect feedback, such as position tracking control with current feedback. However, the sensorless control for the fast subsystem with indirect feedback still remains as a challenging control problem, for example torque tracking control using only position feedback in electric motors.

To resolve the tracking problem for fast subsystems with indirect feedback, therefore, we propose a nonlinear augmented observer to estimate both full-state and unknown disturbances using only the output of a slow subsystem. Then, we design an observer based tracking controller to ensure globally exponential convergence of tracking errors of fast subsystems. Using the composite Lyapunov function and changing variables, it is also proven that the closed-loop system is globally exponentially stable.

The performance of the proposed methods was validated through simulations and experiments. From these analyses and experimental results, it can be concluded that we propose unified tracking methods for the SPINS model. Moreover, the proposed methods ensure robustness against disturbances since all of the proposed methods ensure globally uniformly ultimately bounded tracking error in the presence of disturbances.