On stability of time-varying linear differential-algebraic equations

We develop a stability theory for time-varying linear differential algebraic equations (DAEs). Standard stability concepts for ODEs are formulated for DAEs and characterized. Lyapunov’s direct method is derived as well as the converse of the stability theorems. Stronger results are achieved for DAEs which are transferable into standard canonical form; in this case the existence of the generalized transition matrix is exploited

Algebraic theory of time-varying linear systems: a survey

The development of the algebraic theory of time-varying linear systems is described. The class of systems considered consists of differential-algebraic equation in kernel presentation. This class encompasses time-varying state space, descriptor systems as well as Rosenbrock systems, and time-invariant systems in the behavioural approach.One difference between time-varying and time-invariant systems is that, since the coefficients of the differential equations are time-varying function, the differential operator does not commute with the coefficients. However, the main difficulty is that solutions may exhibit a finite escape time. Hence there is a conflict between the class of time-varying coefficients and the class of admissible solution spaces. All contributions to time-varying systems have to cope with this.As an efficient tool in linear, time-invariant system theory, Kalman introduced in the 1960s elementary module theory over principal ideal rings. This tool proved efficient also for time-varying systems. Although from then on, the field of time-varying linear systems has never been a ``hot topic" in systems theory, there has been an ongoing evolution which led to a rather substantial theory. Not surprisingly, the theory is mainly restricted to linear systems and most results are on such properties as controllability, and not on stability. Recent results use successfully tools from module theory and homological algebra

Robust stability of differential-algebraic equations

This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated

Stabilization of structure-preserving power networks with market dynamics

This paper studies the problem of maximizing the social welfare while stabilizing both the physical power network as well as the market dynamics. For the physical power grid a third-order structure-preserving model is considered involving both frequency and voltage dynamics. By applying the primal-dual gradient method to the social welfare problem, a distributed dynamic pricing algorithm in port-Hamiltonian form is obtained. After interconnection with the physical system a closed-loop port-Hamiltonian system of differential-algebraic equations is obtained, whose properties are exploited to prove local asymptotic stability of the optimal points.Comment: IFAC World Congress 2017, accepted, 6 page

Reduced order modeling of convection-dominated flows, dimensionality reduction and stabilization

We present methodologies for reduced order modeling of convection dominated flows. Accordingly, three main problems are addressed. Firstly, an optimal manifold is realized to enhance reducibility of convection dominated flows. We design a low-rank auto-encoder to specifically reduce the dimensionality of solution arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Also, considering that the latent variables are often hard to interpret, many of these methods are dismissed in the reduced order modeling of dynamical systems governed by partial differential equations (PDEs). This deficiency is of importance to the extent that linear methods, such as principle component analysis (PCA) and Koopman operators, are still prevalent. Accordingly, we propose an interpretable nonlinear dimensionality reduction algorithm. An unsupervised learning problem is constructed that learns a diffeomorphic spatio-temporal grid which registers the output sequence of the PDEs on a non-uniform time-varying grid. The Kolmogorov n-width of the mapped data on the learned grid is minimized. Secondly, the reduced order models are constructed on the realized manifolds. We project the high fidelity models on the learned manifold, leading to a time-varying system of equations. Moreover, as a data-driven model free architecture, recurrent neural networks on the learned manifold are trained, showing versatility of the proposed framework. Finally, a stabilization method is developed to maintain stability and accuracy of the projection based ROMs on the learned manifold a posteriori. We extend the eigenvalue reassignment method of stabilization of linear time-invariant ROMs, to the more general case of linear time-varying systems. Through a post-processing step, the ROMs are controlled using a constrained nonlinear lease-square minimization problem. The controller and the input signals are defined at the algebraic level, using left and right singular vectors of the reduced system matrices. The proposed stabilization method is general and applicable to a large variety of linear time-varying ROMs