The flexibility of DAE formulations.

*(English)*Zbl 1343.65101
Ilchmann, Achim (ed.) et al., Surveys in differential-algebraic equations III. Cham: Springer (ISBN 978-3-319-22427-5/pbk; 978-3-319-22428-2/ebook). Differential-Algebraic Equations Forum, 1-59 (2015).

This article demonstrates the usefulness of differential-algebraic equations (DAEs) as the unique way how many complex systems can naturally be modelled. Therefore, the author concentrates his attention on the problems, where the extra flexibility of the DAE formulation permits their resolving, which would be hard to solve otherwise.

Section 2 gives some examples from control theory and particularly the observer design on the continuous time deterministic case. The design of the observes can be used in two different examples demonstrating the flexibility of the DAE formulation. In this part of the article, the author introduces the differential index notion, separately considering the observers for the DAEs of indices one and two with some results of numerical simulation. Here it is shown that the usage of the DAEs observer allows to get linear error dynamics, which is very needed in observer design. Then the estimation of disturbances is discussed.

Further, the author turns to the examination of optimal control problems. It turns out that the advantages or disadvantages of the DAE formulation are closely dependent on the type of the used numerical methods. Optimal control problems of the sufficiently general form \[ \min \mathcal{L}(x,u),\quad F(\dot{x}, x,u,t)=0,\quad g(x,u,t)=0, \] are considered together with the inequality case for dynamics: \(0\leq g(x,u,t)\); \(x(0)=x_0\).

The conclusive Section 4 is devoted to the optimal control of the delayed system with the cost functional \[ J=\phi(t_f)+\int\limits_{t_0}^{t_f}L(x,u,x(\omega(t)),u(\eta(t)),t,p)dt \] and constraints \[ \dot{x}=f(x,t,x(\omega(t)),u,u(\eta(t)),t,p),\; t_0 \leq t \leq t_f, \] \[ 0=g(x,t,x(\omega(t)),u,u(\eta(t)),t,p),\; t_0 \leq t \leq t_f, \] \[ x=\alpha (t), \; -r \leq t<0,\; x_0=q, \] \[ u=\beta(t), \; -s\leq t<0. \] It contains the following subsections: 4.1 Direct transcription algorithm; 4.2 DAEs and delay and containing a key in dealing with some types of the delays that is to be able to work with reformulations which are DAEs models.

For the entire collection see [Zbl 1333.65004].

Section 2 gives some examples from control theory and particularly the observer design on the continuous time deterministic case. The design of the observes can be used in two different examples demonstrating the flexibility of the DAE formulation. In this part of the article, the author introduces the differential index notion, separately considering the observers for the DAEs of indices one and two with some results of numerical simulation. Here it is shown that the usage of the DAEs observer allows to get linear error dynamics, which is very needed in observer design. Then the estimation of disturbances is discussed.

Further, the author turns to the examination of optimal control problems. It turns out that the advantages or disadvantages of the DAE formulation are closely dependent on the type of the used numerical methods. Optimal control problems of the sufficiently general form \[ \min \mathcal{L}(x,u),\quad F(\dot{x}, x,u,t)=0,\quad g(x,u,t)=0, \] are considered together with the inequality case for dynamics: \(0\leq g(x,u,t)\); \(x(0)=x_0\).

The conclusive Section 4 is devoted to the optimal control of the delayed system with the cost functional \[ J=\phi(t_f)+\int\limits_{t_0}^{t_f}L(x,u,x(\omega(t)),u(\eta(t)),t,p)dt \] and constraints \[ \dot{x}=f(x,t,x(\omega(t)),u,u(\eta(t)),t,p),\; t_0 \leq t \leq t_f, \] \[ 0=g(x,t,x(\omega(t)),u,u(\eta(t)),t,p),\; t_0 \leq t \leq t_f, \] \[ x=\alpha (t), \; -r \leq t<0,\; x_0=q, \] \[ u=\beta(t), \; -s\leq t<0. \] It contains the following subsections: 4.1 Direct transcription algorithm; 4.2 DAEs and delay and containing a key in dealing with some types of the delays that is to be able to work with reformulations which are DAEs models.

For the entire collection see [Zbl 1333.65004].

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

65L80 | Numerical methods for differential-algebraic equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

93B07 | Observability |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

34A40 | Differential inequalities involving functions of a single real variable |

93C15 | Control/observation systems governed by ordinary differential equations |