Control Design for Nonlinear System Described by Differential-Algebraic Equations

在本研究中，我们研究由微分代数方程（DAE）所描述的非线性系统的控制器设计。在非线性情况下，由于非线性代数方程组一般不能解析求解，因此很难将系统无冗余地转化为常微分方程组（ODE）.首先，针对指标为1的非线性DAE系统，提出了一种精确I/O线性化控制器和观测器的设计方法，并给出了观测器的全局稳定性条件.观测器中的状态估计量在有限时间内收敛到由代数方程组确定的不变流形上。其次，给出了两种将高指标非线性DAE系统转化为冗余常微分方程系统的方法，其中DAE系统可能包含脉冲模态。一种是微分方法，其中代数方程微分与动态扩展的输入。另一种是正则化方法的推广，利用状态反馈消除脉冲模态。后一种方法和原有的正则化方法不适用于部分观测系统，本文利用前一种方法，给出了非线性高指标DAE系统的I/O线性化控制器和观测器的设计。由于该系统可以转化为常微分系统，因此许多状态反馈设计方法可以应用于该系统。然而，对于冗余常微分方程系统，观测器设计的条件，例如存在多面体覆盖非线性动力学，可能不被满足。因此，我们将指数为1的系统的方法扩展到这种情况。此外，观测器动态的维数减少，通过使用的输入积分器的值包括由控制器。所提出的观测器可以估计所有冗余的状态变量，其中所估计的变量的代数方程所描述的限制是在有限的时间内满足。

In this research, we study controller design of nonlinear systems described by differential-algebraic equations (DAE). In nonlinear cases, it is hard to convert the system to ordinary differential equations (ODE) without redundancy, because nonlinear algebraic equations cannot be solved analytically in general.First, for nonlinear DAE systems with index one, a design method of exact I/O-linearizing control and observer is proposed, where the global-stability condition of the observer is also obtained. The estimated state variables in the observer converge to an invariant manifold determined by the algebraic equations in finite time.Secondly, we show two methods transforming from a nonlinear DAE system with high index to a redundant ODE system, where the DAE system may include an impulsive mode. One is a differentiation method in which algebraic equations are differentiated with dynamic extension of input. The other is an extension of regularizing method that cancels impulsive modes using state feedback. The latter method and the original regularizing method are not applicable for systems with partial observation.By using the former method, we show an I/O-linearizing controller and an observer design for nonlinear high-index DAE systems. Since the system can be transformed to ODE system, many state feedback design methods can be applied to the system. However, for the redundant ODE system, conditions of the observer design, for example existence of polytope covering nonlinear dynamics, may not be satisfied. Therefore, we extend the method for systems with index one to this case. Moreover, the dimension of the observer dynamics is reduced by using the values of input-integrators included by the controller. The proposed observer can estimate all redundant state variables, where the restriction described by the algebraic equations for the estimated variables is satisfied in finite time.