Control Design for Nonlinear System Described by Differential-Algebraic Equations

在这项研究中，我们研究了由微分代数方程(DAE)描述的非线性系统的控制器设计。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。其次，我们给出了两种从高指标非线性DAE系统到冗余常量系统的转换方法，其中DAE系统可能包含脉冲模式。一种是微分方法，其中代数方程随着输入的动态扩展而被微分。另一种是正则化方法的扩展，它利用状态反馈消除脉冲模式。后一种方法和原有的正则化方法不适用于具有部分观测的系统，利用前一种方法，我们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。由于系统可以转化为常微分方程组，因此可以采用多种状态反馈设计方法。然而，对于冗余常微分方程系统，可能不满足观测器设计的条件，例如存在覆盖非线性动态的多面体。因此，我们将指数为1的系统的方法推广到这种情况。此外，利用控制器所包含的输入积分器的值来降低观测器动态的维度。所提出的观测器可以估计所有冗余状态变量，其中被估计变量的代数方程所描述的约束在有限时间内满足。

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.