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Numerical linear algebra methods for linear differential-algebraic equations. (English) Zbl 1343.65100

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, 117-175 (2015).
In this survey, the authors study linear constant coefficient differential-algebraic equations (DAEs) and descriptor systems.
Linear constant coefficient DAEs take the form \[ E\dot{x}(t)=Ax(t)+f(t), \quad x(0)=x_0, \tag{1} \] and linear time-invariant descriptor systems have the form of constant coefficient DAEs \[ E\dot{x}(t)=Ax(t)+Bu(t), \quad x(0)=x_0, \]
\[ y(t)=Cx(t)+Du(t), \] which are combined for a uniform presentation in the form \[ E\dot{x}(t)=Ax(t)+Bu(t)+f(t), \quad x(0)=x_0, \tag{2} \]
\[ y(t)=Cx(t)+Du(t), \] where \(E,A \in \mathbb{R}^{k,n}\), \(B\in \mathbb{R}^{k,m}\), \(C\in \mathbb{R}^{p,n}\) and \(D\in \mathbb{R}^{p,m}\); \(x: [0,\infty) \to \mathbb{R}^n \) represents the state, \(u:[0,\infty) \to \mathbb{R}^m \) denotes a control input signal, \(y:[0,\infty) \to \mathbb{R}^p \) is the output signal, and \(f:[0,\infty) \to \mathbb{R}^k \) is a given inhomogeneity.
First of all, the authors briefly discuss the existence and uniqueness solutions together with the consistency of initial values.
Therefore Section 2 “Solvability theory” begins with the solvability theory for the System (1). This can done in terms of Kronecker canonical forms (KCF) of the matrix pencil \(sE-A \in \mathbb{R}[s]^{k,n}\) [S. L. Campbell, Singular systems of differential equations. San Francisco-London-Melbourne: Pitman Advanced Publishing Program (1980; Zbl 0419.34007)]. Note here that the first article devoted to DAEs is from N. N. Luzin [Avtom. Telemekh. 1940, 4–66 (1940; Zbl 0061.17301)].
In the regular case (\(k=n\), \(\det(\lambda E-A)\neq 0\)) the KCF specializes to the Weierstraß canonical form (WCF).
To ensure a smooth response for every continuous input \(u(\cdot)\) and every consistent initial value, it is necessary for the system to be regular and to have an index less than or equal to one. The presented existence and uniqueness results are useful from a theoretical point of view, but it is well known that arbitrary small perturbations can radically change the kind and number of the Kronecker blocks, and thus it is problematic to compute the KCF or WCF with a numerical algorithm in finite precision arithmetic [G. W. Stewart and J.-g. Sun, Matrix perturbation theory. Boston etc.: Academic Press, Inc. (1990; Zbl 0706.65013)]. Therefore, in the general case, when some of the conditions do not hold, a regularization of the system is required. In Section 3 “Regularization and derivative arrays” this regularization procedure is summarized for the linear constant coefficient case.
After discussion of the analysis and regularization techniques, the authors proceed to more advanced control and optimization applications for description systems.
In Section 4 “Staircase forms and properties of descriptor systems” the authors discuss the theoretical properties of descriptor systems and present the staircase forms that allow to check these properties. Main attention is payed on concepts like controllability, stabilizability and related dual notions of observability and detectability. These are introduced for the case of square systems and systems where the feed-through term \(D\) has been removed, so it is assumed that the system is already in the special form as generated by the regularization procedure of Section 3. Also, instead of defining these properties in system theoretical terms, equivalent algebraic characterizations are directly introduced.
All applications of Section 4 lead to generalized eigenvalue problems for even matrix pencils. Therefore, in Section 5, their structured considered forms as well as the appropriate numerical methods are developed.
In Section, 6 the linear quadratic optimal control problem of minimizing \[ \mathcal{J}(x(\cdot),u(\cdot))=\frac{1}{2}\int\limits_{0}^{\infty}(x(t)^TQx(t)+2x(t)^TSu(t)+u(t)^TRu(t))dt \tag{3} \] with \(Q=Q^T\in \mathbb{R}^{n,n}\), \(S\in \mathbb{R}^{n,m}\), and \(R=R^T\in \mathbb{R}^{m,m}\) subjected to the linear descriptor system of the Form (1) with initial value \(x(0)=x_0\) and the stabilization condition \(\lim_{t\to \infty}x(t)=0\).
If an output Equation (2) is also given, then the cost functional is usually given as \[ \mathcal{\widetilde{J}}(x(\cdot),u(\cdot))=\frac{1}{2}\int\limits_{0}^{\infty}(x(t)^T\widetilde{Q}x(t)+2x(t)^T\widetilde{S}u(t)+u(t)^T\widetilde{R}u(t))dt \] with \[ \widetilde{Q}:=C^TQC,\; \widetilde{S}:=C^TQD+C^TS,\; \widetilde{R}:=D^TQD+D^TS+S^D+R \] which can then easily be transformed to the form given in (3) by inserting the output equation.
In Section 7, the so-named \(\mathcal{H}_{\infty}\) optimal control problem is considered. Section 8 is devoted to the computation of the \(\mathcal{L}_{\infty}\)-norm for continuous-time descriptor systems, and finally Section 9 – to the dissipativity checking problem.
In conclusion, some open problems are stated in Section 9.
For the entire collection see [Zbl 1333.65004].

MSC:

65L80 Numerical methods for differential-algebraic equations
15A21 Canonical forms, reductions, classification
15A22 Matrix pencils
34A09 Implicit ordinary differential equations, differential-algebraic equations
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
93C05 Linear systems in control theory
93D09 Robust stability
65K10 Numerical optimization and variational techniques
93B05 Controllability
93C15 Control/observation systems governed by ordinary differential equations
93B36 \(H^\infty\)-control
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
49N10 Linear-quadratic optimal control problems
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