Linear Differential-algebraic Systems Research Articles

Generic observability for port-Hamiltonian descriptor systems

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems

A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.

On the Stability of a Locally One-Dimensional Difference Scheme for a First-Order Linear Differential-Algebraic System of Index (1, 0)

On the Stability of a Locally One-Dimensional Difference Scheme for a First-Order Linear Differential-Algebraic System of Index (1, 0)

Analysis and homogenisation of a linear Cahn–Larché system with phase separation on the microscale

AbstractWe consider the process of phase separation of a binary system under the influence of mechanical stress modelled by the Cahn–Larché system, where the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level. After linearisation, we prove existence and uniqueness of a weak solution by a Galerkin approach. As discretisation in space leads to a linear differential–algebraic system of equations, we adjust known solution theory for such equations to a weak setting. This approach may be of interest more generally for coupled elliptic–parabolic systems. A‐priori estimates enable us to pass to the homogenisation limit of the linear system rigorously using the concept of two‐scale convergence. A comparison with the formally homogenised full nonlinear problem shows that both systems lead to models of distributed‐microstructure type in the limit and that homogenisation and linearisation commutes.

Computation of the nearest structured matrix triplet with common null space

We study computational methods for computing the distance to singularity, the distance to the nearest high-index problem, and the distance to instability for linear differential-algebraic systems (DAEs) with dissipative Hamiltonian structure. While for general unstructured DAEs the characterization of these distances is very difficult and partially open, it has been shown in [C. Mehl, V. Mehrmann, and M. Wojtylak, Distance problems for dissipative Hamiltonian systems and related matrix polynomials, Linear Algebra Appl., 623 (2021), pp. 335–366] that for dissipative Hamiltonian systems and related matrix pencils there exist explicit characterizations. We will use these characterizations for the development of computational methods to approximate these distances via methods that follow the flow of a differential equation converging to the smallest perturbation that destroys the property of regularity, index one, or stability.

On Controllability for Linear Differential-Algebraic Control Systems Using Equivalent Form

We investigate different concepts related to the controllability of linear constant coefficient differentialalgebraic equations. Concepts like impulse controllability, controllability at infinity, strong and complete controllability are described and defined by using an equivalent form for linear time-invariant singular systems. Some real-life problems are given to illustrate the presented theory.

Lyapunov-based stability of delayed linear differential algebraic systems

The Lyapunov stability theorem for linear systems is extended to linear delay-differential algebraic systems of index one. In particular, bounds on the decay of the solution are established in terms of solvability of a certain Lyapunov-type matrix equation.

Acceleration of implicit schemes for large linear systems of differential–algebraic equations

Implicit schemes for solving large-scale linear differential–algebraic systems with constant coefficients necessitate at each integration step the solution of a linear system, typically obtained by a Krylov subspace method such as GMRES. To accelerate the convergence, an approach is proposed that computes good initial guesses for each linear system to be solved in the implicit scheme. This approach requires, at each integration step, a small dimensional subspace where a good initial guess is found using the Petrov–Galerkin process. It is shown that the residual associated with the computed initial guess depends on the dimension of the subspace, the order of the implicit scheme, and the discretization stepsize. Several numerical illustrations are reported.

On some properties of the solutions of the linear differential-algebraic systems

It has been compared the properties of the solutions of differential-algebraic systems and the solutions of systems of differential equations. The differences between the corresponding properties of these solutions are analyzed.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Distance problems for dissipative Hamiltonian systems and related matrix polynomials

We study the characterization of several distance problems for linear differential-algebraic systems with dissipative Hamiltonian structure. Since all models are only approximations of reality and data are always inaccurate, it is an important question whether a given model is close to a ‘bad’ model that could be considered as ill-posed or singular. This is usually done by computing a distance to the nearest model with such properties. We will discuss the distance to singularity, the distance to the nearest high index problem, and the distance to instability for dissipative Hamiltonian systems. While for general unstructured differential-algebraic systems the characterization of these distances are partially open problems, we will show that for dissipative Hamiltonian systems and related matrix polynomials there exist explicit characterizations that can be implemented numerically.

Solvability of Cauchy Problem for a Differential-Algebraic System with Concentrated Delay

We study the Cauchy problem for a linear differential-algebraic system of equations with concentrated delay. Our research continues investigation of solvability of linear Noether boundary value problems for systems of functional-differential equations given in the monographs by A.D. Myshkis, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko, and A.A. Boichuk; meanwhile, we use essentially the tool of Moore-Penrose inverse matrices. For a linear differential-algebraic system with concentrated delay, we find sufficient conditions for its solvability and give a construction of generalized Green’s operator for Cauchy’s problem. We also give some examples which illustrate in detail the solvability conditions and the suggested construction.

On Observers For Nonlinear Differential-Algebraic Systems

We extend a recent approach to observer design for linear differential-algebraic systems to systems with Lipschitz nonlinearities. The observer design further extends the standard Luenberger-type observer design. We show that the design parameters for the observer can be obtained by the solution of a Riccati-type inequality. The solutions of the latter can in turn be obtained by solving a set of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs) which provides a computational procedure. A feature of the presented observer design is the possibility of reformulation as an ordinary differential equation.

Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations

We consider a linear differential-algebraic system of PDEs with special matrix coefficients. Two cases are investigated. In the first one, the system has a small index, and the matrix at unknown vector-function, while the system written in the canonical form, is arbitrary. In the second case, the system has an arbitrary index, while a matrix at the small term is triangular. In both the cases, using the methods of characteristics and successive approximations, we prove the existence of a unique classical solution of mixed problems for the considered differential-algebraic systems of PDEs.

Optimal control of differential‐algebraic equations from an ordinary differential equation perspective

SummaryWe study the optimal control problem (OCP) for regular linear differentialalgebraic systems. To this end, we introduce the input index, which allows, on the one hand, to characterize the space of consistent initial values in terms of a Kalman‐like matrix and, on the other hand, the necessary smoothness properties of the control. The latter is essential to make the problem accessible from a numerical point of view. Moreover, we derive an augmented system as the key to analyze the OCP with tools well known from optimal control of ordinary differential equations. The new concepts of the input index and the augmented system provide easily checkable sufficient conditions, which ensure that the stage costs are consistent with the differential‐algebraic system.

An extended-observer approach to robust stabilisation of linear differential-algebraic systems

ABSTRACTThis paper addresses the problem of robustly stabilising a class of linear differential-algebraic systems characterised by autonomous and asymptotically stable zero dynamics, in spite of parameter uncertainties ranging over a priori fixed bounded sets. We exploit recent results related to the structural properties and normal forms of this class of systems and propose a robust control that asymptotically recovers, in practical terms, the performance of a nominal, though non-implementable, stabilising control. The proposed control combines a partial output feedback control, aimed at letting the system behave as a regular system, and a robust control, based on an extended observer, using which the dynamic of the closed loop system is rendered arbitrarily close to the one of a properly selected stable system. The extended observer, originally conceived in the context of standard differential systems, is here shown to be the key ingredient for robustly stabilising the targeted class of differential-algebraic systems.

On a Reduction of the Order in a Differential-Algebraic System

The conditions of solvability and the structure of a generalized Green operator of the Cauchy problem for a linear differential-algebraic system are found. The sufficient conditions of reducibility of a differential-algebraic equation to a sequence of systems joining differential and algebraic equations are constructed. An original classification and a single scheme of construction of the solutions of differentialalgebraic equations are proposed.

Robust Output Regulation for a Class of Linear Differential-Algebraic Systems

This letter addresses the problem of robust output regulation of systems described by linear differential-algebraic equations (DAEs). Taking advantage of some recent relevant results concerned with the development of normal forms for general DAEs, it is shown how an internal model can be designed in such a way that the problem in question is reduced to the problem of robustly stabilizing an augmented system modeled by DAEs. Then, a suitable enhancement—within the context of systems modeled by DAEs—of the theory of extended observers is proposed, by means of which such robust stabilization problem is solved.

Regularization and index reduction for linear differential–algebraic systems

In this paper, a necessary and sufficient condition for the existence of a proportional semistate feedback is established such that the closed loop system is regular and of index at most two. The condition is characterized by a rank condition that involves only the original system coefficient matrices. Also, a new rank condition ensuring that a given linear time-invariant descriptor system is regular and of index at most some specific value is also derived. The developed theory is illustrated through physical and numerical examples.

Model reduction of constrained mechanical systems in M-M.E.S.S.

We discuss balanced truncation model reduction of constrained mechanical systems. The incorporation of the algebraic constraints leads to linear differential-algebraic control systems of index 2 or 3. Then for the square-root method in balanced truncation, the solution of projected Lyapunov equations is required. In this paper we discuss how to avoid the explicit construction of the projectors and the formulation of a projection-avoiding balanced truncation method which is suitable for efficient numerical computations. We have implemented this method in the Matlab package M-M.E.S.S. and present some numerical results for illustration.