On the stability of port-Hamiltonian descriptor systems ††thanks: H.G. acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 "Modeling, Simulation and Optimization of Fluid Dynamic Applications”.
Abstract
We characterize stable differential-algebraic equations (DAEs) using a generalized Lyapunov inequality. The solution of this inequality is then used to rewrite stable DAEs as dissipative Hamiltonian (dH) DAEs on the subspace where the solutions evolve. Conversely, we give sufficient conditions guaranteeing stability of dH DAEs. Further, for stabilizable descriptor systems we construct solutions of generalized algebraic Bernoulli equations which can then be used to rewrite these systems as pH descriptor systems. Furthermore, we show how to describe the stable and stabilizable systems using Dirac and Lagrange structures.
Keywords: Descriptor systems, port-Hamiltonian systems, stability, differential-algebraic equations, linear matrix inequalities, Dirac structure.
1 Introduction
In this note, we consider generalized port-Hamiltonian systems (pH systems) of the form
(1) |
where , and and
(2) |
see e.g. [MM19]. In comparison to classical port-Hamiltonian systems where the coefficient is the identity, here it might be singular.
If in (1) neither inputs nor outputs are present then the system reduces to the differential-algebraic equation (DAE)
(3) |
where the conditions (2) simplify to
(4) |
If we further assume , these equations are called dissipative Hamiltonian DAEs (dH DAEs), see [MMW21].
The first aim of this paper is to show that stable DAEs
(5) |
can be rewritten as dH DAEs (3) where the matrix is given by the solution of a generalized Lyapunov inequality. For this rewriting we have to restrict the coefficients of (5) to the smallest subspace where the solutions evolve, also called the system space, see [RRV15]. Furthermore, it is shown that such a reformulation as pH systems is also possible for stabilizable systems
by introducing a suitable output and using the solutions of generalized algebraic Bernoulli equations.
The second aim of the paper is to study the stability of DAEs (5) which are given in the geometric pH framework from [vdSM18]. This more general framework contains dH DAEs (3) which was recently shown in [GHR21]. Here it is shown how systems of the form (1) can be embedded into the more general geometric pH framework from [vdSM18].
The outline of the paper is as follows. After recalling some terminologies of matrix pencils in Section 2, we study the behavioral stability of DAEs in Section 3. We characterize the stability of DAEs using a generalized Lyapunov inequality and use their solutions to rewrite stable DAEs as dH DAEs. Further, we prove that dH DAEs which fulfill are stable. In Section 4 we rewrite behaviorally stabilizable DAEs as pH systems (1). Finally, in Section 5 we embedded dH DAEs (3) and pH systems (1) into the geometric framework from [vdSM18].
2 Preliminaries
Matrix pencils are linear matrix polynomials with coefficients in . We briefly write and this pencil is called regular if is non-zero. The spectrum of a matrix pencil is the set of all complex numbers for which is not invertible. An eigenvalue is called simple if has dimension one and semi-simple if the dimension of this subspace coincides with the multiplicity of as a root of .
Recall that every pencil can be transformed to quasi Kronecker form, see e.g. [BT12], i.e., there exists invertible and such that is block diagonal with the following four types of blocks
(6a) | ||||
(6b) | ||||
(6c) | ||||
(6d) |
The above indices are collected in multi-indices , , . For a multi-index recall . Note that for and one adds a zero column and a zero row to the pencil, respectively. The largest is called the index of the DAE. Furthermore, we write , and .
3 Behavioral stable and dH DAEs
In this section, we focus on DAEs
(7) |
with and denote the set of all such pairs by . Here we follow the behavioral approach from [PW97] and introduce the behavior of by
A definition with less smoothness assumptions can be found in [BR13].
We say that is stable if
The aim is to show that stable DAEs can be rewritten as a dH DAE (3), i.e.,
(8) |
To this end, we introduce the system space as follows
see also [RRV15] for an equivalent definition. To determine the system space first observe that the behavior and hence the system space transforms as follows
(9) |
for invertible . This combined with the quasi Kronecker form (6) is the following representation of the system space.
Lemma 3.1.
Below, the stability is characterized either in terms of the eigenvalues or a Lyapunov inequality on the system space. To this end, for a subspace and matrices we say that
If we further assume that and are symmetric, we say that
Proposition 3.2.
Let then the following statements are equivalent.
-
(a)
is stable.
-
(b)
is regular and there exists a symmetric with and such that
(11) -
(c)
The pencil is regular, and the eigenvalues on the imaginary axis are semi-simple.
Proof.
Since the above conditions are invariant under multiplication from left and right with invertible matrices, we can assume that is already given in Kronecker form (6). If is stable then cannot be present in the Kronecker form. Indeed if then for all , there exists a solution to
which contradicts stability. Since is square also is not present and therefore is regular. Furthermore, the stability of shows and that all eigenvalues on the imaginary axis are semi-simple. This proves (c). Clearly, (c) implies (a). To prove the equivalence of (b) and (c), we assume that is regular. Then Lemma 3.1 implies that
Furthermore, (11) holds if and only if
which is, by considering the upper-left block entries, equivalent to the existence of some with such that
(12) |
and hence to and semi-simple eigenvalues on the imaginary axis. If is a solution of (12) then fulfills and .
Remark 3.3.
Oftentimes asymptotically stable DAEs are of interest which are defined by
Analogously to Proposition 3.2, such DAEs can be characterized by a strict inequality in (11) or by . Related characterizations with Lyapunov equations were previously given e.g. in [Sty02] and recently in [AAM21, Theorem 4].
The proposition below provides a dH formulation (8) of stable DAEs on the system space. Here we use the pseudo-inverse of a matrix .
Proposition 3.4.
Proof.
Remark 3.5.
If is stable with index at most one then we can redefine in Proposition 3.4 in such a way that the relations in (b) hold on . Here we assume for simplicity that and are already given in quasi Kronecker form. Then and . Let satisfy . Then we define which is invertible and with and we have and . If has index greater than one then this extension of is still possible but leads to .
The following example from [MMW18] shows that not every dH DAE given by (8) is stable. Consider
which is port-Hamiltonian and has a Jordan block of size 2 at zero and is therefore unstable. The above example is an ordinary differential equation and is not a Lyapunov function.
It will be shown below, that an additional assumption which guarantees the stability of dH DAEs is
(13) |
An interpretation of this condition is given later in the geometric formulation of DAEs in Section 5 where it ensures that the only trajectories of a system with vanishing effort are the purely algebraic solutions.
Proposition 3.6.
Proof.
If is regular then . This together with implies and hence is invertible. This proves (a).
Next, observe that
(14) |
If then and this implies . Hence and therefore . The reverse inclusion is trivial.
To characterize the stability of (8), we use Proposition 3.2 (c). If (8) is stable then is regular. Hence, using the invertibility of and (14) we find
(15) |
Conversely, assume that (15) holds. The pencil is semi-dissipative Hamiltonian and according to [MMW21, Thm. 2] in the quasi-Kronecker form (6) one has and the eigenvalue conditions in Proposition 3.2 (c) are satisfied for the regular part (6a). Since
the multi-index cannot be present in the quasi-Kronecker form of . Hence is not present implying that is regular and therefore stable.
Remark 3.7.
Similar considerations can be done for systems which are stable backwards in time, i.e., is stable. A characterization similar to Proposition 3.2 is straightforward and as a consequence, every DAE which has semi-simple eigenvalues on the imaginary axis can be rewritten on the system space in the form (8) but with instead of .
4 Stabilizable systems as pH systems
In this section, we consider descriptor systems
with , for and we write . The behavior and system space of are given by
In the following it is shown that systems which can be stabilized in behavioral sense can be rewritten as pH systems (1) on the system space after introducing a suitable output.
We say that the system is behaviorally stabilizable if
(16) |
Furthermore, the class of behavioral stabilizable systems for which is regular and only has semi-simple eigenvalues on the imaginary axis are denoted by .
If then applying a Jordan decomposition on the first block entry in the quasi Kronecker form (6) we see that there exists some invertible , with such that
(17) |
Furthermore, is controllable.
Lemma 4.1.
Proof.
Clearly, (18) has a unique positive definite solution if and only if
has a unique positive definite solution. Since is controllable and such a solution exists by [TSH01, Thm. 3.28]. It remains to show that is a stabilizing feedback, i.e., that (16) holds. Note that is also controllable and that (18) is also equivalent to
Now invoking [TSH01, Thm. 3.28] again shows . Moreover, with
there exist such that for all which are smaller than the index of
Therefore, we have for some such that for all which are smaller then the index of
(19) |
Since is stable, the variation of constants formula implies that the solution of are bounded. Next, we only consider the first block-entry (if any) of , , since the others are treated analogously. With the solution of fulfills
and inspecting the last row leads to
which tends to zero as . Similarly, the penultimate row leads to
which is again exponentially bounded by (19). Repeating the last step with the remaining rows starting with the th row shows that as which completes the proof that stabilizes the solution of .
Below, we obtain another characterization of stabilizability using solutions to certain matrix equalities on the system space which allows us to reformulate the system in a port-Hamiltonian way. Similar equations for stabilization of DAEs have been studied under the name generalized algebraic Bernoulli equation in [BBQO07].
Lemma 4.2.
Let . Then there exists some such that the following holds.
(20) | |||
(21) |
with and .
Proof.
All conditions are invariant under transformations of the form for all invertible . Hence we can assume without restriction that is already given in the block diagonal form on the right-hand side of (17). Introduce , and then we set , where is a solution of (18) and where is a solution of the Lyapunov inequality . Clearly, and they satisfy (20) and (21), respectively. Furthermore, and hence map into itself with .
Based on this result, we show how to interpret a stabilizable system as a port-Hamiltonian system (1) which can be viewed as an analogue to Proposition 3.4.
Proposition 4.3.
Let . Then there exist such that with the choices of
(22) | ||||
the system
(23) | ||||
is port-Hamiltonian on in the sense that
(24) | ||||
Proof.
Remark 4.4.
In the context of Proposition 4.3, one can show that the solutions of correspond to the solutions of (23) when imposing . This restriction corresponds to an interconnection (cf. [CvdSBn07]) with respect to the Dirac structure which are key objects in the geometric formulation of pH systems (cf. Section 5). Furthermore, if has index one then in (17) and coincides with . If is invertible then , and .
Remark 4.5.
In Proposition 4.3 we defined a suitable output to obtain a pH system. More generally, in [GS18, Thm. 3.6] it was shown that descriptor systems with output can be rewritten as a pH system of the form (1) if there exists an invertible solution of a linear matrix inequality (typically referred to as Kalman-Yakubovich-Popov inequality). Moreover, one has in (1). Such invertible solutions can be obtained by restricting the system (23) to the space . In this case in (22) can be replaced by the invertible .
5 Stability of geometric dH DAEs
In this section we study the stability of DAEs which are given by the geometrical formulation used in [vdSM18].
In comparison to (1), the equations here are given implicitly. To this end, let and be subspaces of with range representations and for some . Then is Lagrangian and is maximally dissipative if
(25) | |||||
(26) |
Furthermore, the matrices can be chosen in such a way that and holds. These and can be used to implicitly define a DAE by demanding that the system trajectories satisfy
(27) |
These systems were called generalized pH DAE systems in [vdSM18] and and are called the state and effort, respectively. Therein, was assumed to be a so-called Dirac structure, i.e., (26) holds with equality. In [GHR21], in (27) is allowed to be dissipative and we include the port and the resistive variables already in the state for simplicity.
The DAE is then explicitly given by the range representation of the following subspace
(28) |
for some . Moreover, the functions which fulfill and (27) with coincide, see [GHR21, Section 4].
To ensure stability of given by (28), we need the additional assumption that the Lagrangian subspace is nonnegative, i.e., and, using a suitable CS-decomposition, see [MMW18, Prop. 3.1], we can choose and in such a way that .
As a main result of this section we provide sufficient conditions for the stability of DAEs given by (28). Here we use the orthogonal projector onto a subspace . The result is an immediate consequence of [GHR21, Prop. 5.1, Prop. 6.3].
Proposition 5.1.
Let be maximally dissipative and nonnegative Lagrangian with and . Further, let be the pencil given by (28). Then with the following holds.
-
(a)
is regular if and only if
-
(b)
is stable if additionally holds.
DH DAEs can be written in the form (28) using and . Here, and and hence the second condition in Proposition 5.1 (a) trivially holds.
Corollary 5.2.
Let be a dH DAE (3) then it is regular if and only if . Furthermore, is stable if additionally .
Proof.
Since stability and regularity are invariant under pencil equivalence we can use [MMW18, Prop. 3.1] and assume without restriction that , . If is regular then and hence [MMW18, Prop. 3.1] implies that is Lagrangian and without restriction , , holds. By Proposition 5.1 with the regularity is equivalent to
(29) | |||
The assumptions on and imply
and therefore (29) is equivalent to
Furthermore, (14) implies
which proves . Conversely, if this intersection is trivial then and hence by [MMW18, Prop. 3.1] the subspace fulfills and is therefore Lagrangian. Hence (29) holds and is regular by Proposition 5.1. Conversely, if holds then and hence is Lagrangian. Hence Proposition 5.1 implies that is regular. To prove stability we apply Proposition 5.1 (b) and observe that is equivalent to (13), i.e., .
The above characterization of regularity was also obtained recently in [FMP+21]. However, Proposition 5.1 characterizes the regularity and the stability for a larger class of DAEs. If is invertible then the assumptions in Proposition 5.1 and Proposition 3.6 (b) coincide.
6 Conclusion
We studied the stability of dH DAEs (3) and DAEs given by the implicit pH formulation (28). In both cases we obtained that regularity together with the assumption guarantees the stability of theses DAEs. Furthermore, we showed in Proposition 3.4 that stable DAEs can always be reformulated as dH DAEs if we restrict the coefficients of the underlying equations to the system space. This restriction is not necessary if we consider index one DAEs, see Remark 3.5, and the matrix in (3) can be obtained from the solution of the generalized Lyapunov inequality (11). Similarly, we showed in Proposition 4.3 that stabiliazble systems can be reformulated as pH systems (1) using the solutions of generalized algebraic Bernoulli equations (20) and (21). Finally, it was shown how dH DAEs (3) and pH systems (1) can be embedded into the geometric pH framework from [vdSM18].
Acknowledgements
The authors are indebted to Timo Reis for his valuable remarks and suggestions on an earlier draft of this manuscript. Furthermore, the authors would like to thank Volker Mehrmann for providing valuable references.
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