1 Introduction

Port-Hamiltonian (pH) systems have been increasingly used in recent years as a unified structured framework for energy-based modeling of systems, see e.g. [1,2,3,4,5,6,7]. The pH formulation has gained interest from engineers and mathematicians due to its modeling flexibility and robustness properties [2, 8,9,10]. Specifically, pH systems are used in coupled networks of systems and multiphysics simulation and control. System coupling often imposes additional algebraic constraints on the system which naturally lead to linear time-invariant descriptor systems in the state-space form presented as

$$\begin{aligned} \begin{aligned} \tfrac{\hbox {d}}{\hbox {d}t} Ex(t)&= Ax(t) + Bu(t),\quad x(0) = x_0,\\ y(t)&= Cx(t) + Du(t), \end{aligned} \end{aligned}$$
(1)

where \(u: \mathbb {R}\rightarrow \mathbb {K}^m\), \(x: \mathbb {R}\rightarrow \mathbb {K}^n\), \(y: \mathbb {R}\rightarrow \mathbb {K}^m\) are the input, state, and output of the system, and \(E,A\in \mathbb {K}^{n\times n}, B\in \mathbb {K}^{n\times m}\), \(C\in \mathbb {K}^{m\times n}\), \(D\in \mathbb {K}^{m\times m}\), and \(\mathbb {K}=\mathbb {R}\) or \(\mathbb {K}=\mathbb {C}\). The system (1) will be concisely denoted by \(\Sigma =(E,A,B,C,D)\) and throughout it is assumed that the pair (E, A) is regular which means that \(\lambda E-A\) is invertible for some \(\lambda \in \mathbb {C}\).

In addition, we consider the following notation: For a matrix \(A\in \mathbb {K}^{n \times m}\) let \(A^{\top },A^H,A^{-H}\) denote the transpose, conjugate transpose, inverse of \(A^H\), respectively. Note that in the real case \(A^\top = A^H\). The identity matrix of dimension n is denoted by \(I_n\). For a Hermitian matrix \(A \in \mathbb {K}^{n \times n}\), we use \(A > 0\) \((A \ge 0)\) if A is positive (semi-) definite. Furthermore, we denote the set of eigenvalues of a matrix pencil \(sE-A\) by

$$\begin{aligned} \sigma (E,A):=\{\lambda \in \mathbb {C}~\mid ~\textrm{ker}(\lambda E-A)\ne \{0\}\}. \end{aligned}$$

PH systems are then defined as follows:

  1. (pH)

    The system \(\Sigma \) is port-Hamiltonian if there exists \(J,R,Q\in \mathbb {K}^{n\times n}\), \(G,P\in \mathbb {K}^{n\times m}\), and \(S,N\in \mathbb {K}^{m\times m}\) such that

    $$\begin{aligned} \begin{aligned} \begin{bmatrix} A&{}\quad B\\ C&{}\quad D \end{bmatrix}&=\begin{bmatrix}(J-R)Q&{}\quad G-P\\ (G+P)^HQ&{}\quad S+N\end{bmatrix},\quad Q^HE=E^HQ\ge 0,\\ \Gamma&:= \begin{bmatrix} J &{}\quad G \\ -G^H&{}\quad N\end{bmatrix} = - \Gamma ^H,\quad W := \begin{bmatrix} Q^HRQ &{}\quad Q^HP\\ P^HQ &{}\quad S \end{bmatrix} =W^H \ge 0. \end{aligned} \end{aligned}$$
    (2)

Here the quadratic function \(\mathcal {H}(x):=\frac{1}{2} x^HE^HQx\) is called the Hamiltonian which can often be interpreted as the energy of the system. Note that recently in [11,12,13] also a geometric pH framework was developed which is based on monotone, Dirac and Lagrangian subspaces and enlarges the class of pH systems. Furthermore, some references assume that the matrix Q in (2) is positive definite. In this case, the Q in the matrix W given by (2) is often replaced by the identity.

It is well-known that pH descriptor systems satisfy the following dissipation inequality which is referred to as passivity in the literature [14].

  1. (Pa)

    The system \(\Sigma \) is passive if there exists \(Q\in \mathbb {K}^{n\times n}\) such that \(Q^HE=E^HQ\) holds and if \(\mathcal {S}(x)=\tfrac{1}{2}x^H Q^HEx\), called storage function, satisfies for all \(T\ge 0\) the following inequality

    $$\begin{aligned} \mathcal {S}(x(T))-\mathcal {S}(x(0))\le \int _{0}^{T} {{\,\textrm{Re}\,}}y(\tau )^Hu(\tau ) d \tau ,\quad \mathcal {S}(x(T))\ge 0, \end{aligned}$$
    (3)

for all consistent initial values \(x(0)=x_0\) and all functions x, u, y whose derivatives of arbitrary order \(k\in {\mathbb {N}}\) exist and fulfill (1).

Although we only consider in this paper smooth functions in the dissipation inequality (3), it can easily be extended to inputs u which are weakly differentiable up to some order by using the density of smooth functions in the Lebesgue space \(L^1([0,T],\mathbb {K}^m)\), see e.g. [15].

The property (Pa) is hard to verify in practice since one would have to consider all possible solution trajectories. It is more convenient to solve a linear matrix inequality called KYP (discovered independently by Kalman, Yakubovich and Popov) which is equivalent to (Pa) and can be obtained by differentiation of (3). An overview is presented in [16, p. 81] for standard systems, i.e. \(E=I_n\) and in [17, 18] for descriptor systems.

  1. (KYP)

    The system \(\Sigma \) has a solution \(Q\in \mathbb {K}^{n\times n}\) to the generalized KYP inequality if

    $$\begin{aligned} \begin{bmatrix} -A^HQ-Q^HA&{}\quad C^H-Q^HB\\ C-B^HQ&{}\quad D+D^H \end{bmatrix}\ge 0,\quad E^HQ=Q^HE\ge 0. \end{aligned}$$
    (4)

In many applications, only input–output data is given and hence an important question is whether we can decide if a system is pH from this data and even more, we want to obtain a pH representation (2) of the system. The typical approach is to apply a Laplace transformation to (1) which leads to the transfer function

$$\begin{aligned} {\mathcal {T}}(s):=C(sE-A)^{-1}B+D \end{aligned}$$
(5)

that describes the input–output behavior in the frequency domain. It is well-known for standard systems that the passivity implies that its transfer function is positive real, see [19, 20].

  1. (PR)

    The system \(\Sigma \) with transfer function \({\mathcal {T}}\) given by (5) is called positive real if \({\mathcal {T}}\) has no poles for all \(s\in \mathbb {C}\), \({{\,\textrm{Re}\,}}s>0\) and satisfies \({\mathcal {T}}(s)+{\mathcal {T}}(s)^H\ge 0\) for all \(s\in \mathbb {C}\) and \({{\,\textrm{Re}\,}}s>0\).

As mentioned above, it is well-known that pH descriptor systems are passive [14], denoted by (Pa), and that passive systems are positive real (PR). Moreover, the passivity is implied by the existence of solutions to KYP inequalities, see Proposition 1. The main goal of this note is to investigate under which assumptions also the converse implications hold. The study of these implications requires the use of controllability, observability and minimality notions.

Given a transfer function, \({\mathcal {T}}\), a realization is finding the matrices (E, A, B, C, D) in a descriptor state-space form (1) such that (5) is satisfied. In addition, the realization is called minimal if the number of states n in (1) needed to represent \({\mathcal {T}}\) is minimal.

A system is called controllable (observable) if and only if

$$\begin{aligned} {{\,\textrm{rk}\,}}[\lambda I_n-A,B]=n \quad ({{\,\textrm{rk}\,}}[(\lambda I_n-A)^\top ,C^\top ]=n),\quad \text {for all}\, \lambda \in \mathbb {C}. \end{aligned}$$
(6)

In the case of standard systems, minimality is equivalent to the system being both controllable and observable.

Furthermore, one can define a weaker property stabilizability (detectability) of controllability (observability) such that

$$\begin{aligned} {{\,\textrm{rk}\,}}[\lambda I_n-A,B]=n\quad ({{\,\textrm{rk}\,}}[(\lambda I_n-A)^\top ,C^\top ]=n),\quad \text {for all}\, \lambda \in \mathbb {C},\, {{\,\textrm{Re}\,}}\lambda \ge 0. \end{aligned}$$
(7)

In [21, 22] the conditions on minimality were generalized to descriptor systems by showing that a realization (E, A, B, C, D) of a transfer function \({\mathcal {T}}\) is minimal if and only if it fulfills the following conditions

$$\begin{aligned}&{{\,\textrm{rk}\,}}[\lambda E-A,B]=n, \quad {{\,\textrm{rk}\,}}[(\lambda E-A)^\top ,C^\top ]=n,{} & {} \text {for all}\, \lambda \in \mathbb {C}, \end{aligned}$$
(8)
$$\begin{aligned}&{{\,\textrm{rk}\,}}[E,B]={{\,\textrm{rk}\,}}\begin{bmatrix} E\\ C \end{bmatrix}=n, \quad A\ker E\subseteq {{\,\textrm{ran}\,}}E.{} & {} \end{aligned}$$
(9)

The first (second) property in (8) defines the behavioral controllability (behavioral observability) of the system. If the system fulfills, in addition to behavioral controllability, \({{\,\textrm{rk}\,}}[E,B]=n\), it is called completely controllable. If the system is behaviorally observable and \({{\,\textrm{rk}\,}}[E^\top ,C^\top ]=n\) holds, it is completely observable.

For standard systems, i.e. \(E=I_n\) in (1), which are controllable and observable, it is well-known that (pH), (Pa), (KYP), and (PR) are equivalent. For uncontrollable or unobservable standard systems a detailed study of the relation between (Pa), (KYP) and (PR) has been conducted in [16, Chapter 3] but without including (pH). The connection between (Pa) and (pH) was discussed in [23]. Another recent survey for standard systems was given in [24] see p. 59 therein for a discussion on unobservable and uncontrollable systems. An overview of standard systems is presented in Fig. 1.

Fig. 1
figure 1

The relationship between (pH), (KYP), (Pa) and (PR) for a standard system (\(E= I\)). Implications without additional assumptions are marked blue and implications with additional assumptions are marked black. The counterexamples, if assumptions are not fulfilled, are colored in red (color figure online)

Full size image

For descriptor systems the relations between (pH), (Pa), (KYP) and (PR) were already studied in numerous works [17, 18, 25,26,27,28,29]. However, not all four properties have been investigated at the same time and often the minimality of the descriptor system is assumed.

As a first step, we combine the aforementioned results to obtain

$$\begin{aligned} \text {(pH)}\quad \Longrightarrow \quad \text {(KYP)}\quad \Longrightarrow \quad \text {(Pa)}\quad \Longrightarrow \quad \text {(PR)} \end{aligned}$$

which holds without observability or controllability assumptions and we provide examples showing that the converse implications do not hold.

Our aim is to provide sufficient conditions for the converse implications to hold. Hence the remaining questions which we will answer are

  1. (Q1)

    When do solutions to the KYP inequality lead to a pH formulation?

  2. (Q2)

    When does passivity lead to solutions of the KYP inequality and can we realize passive systems as pH systems?

  3. (Q3)

    Can every positive real transfer function be realized as a pH system?

The answer to question (Q1) is related to observability properties of the system. For standard systems it was shown in [23, p. 55] that only those solutions \(Q\in \mathbb {K}^{n\times n}\) to (KYP) which additionally satisfy

$$\begin{aligned} \ker Q\subseteq \ker A\cap \ker C \end{aligned}$$
(10)

lead to a pH formulation. Conversely, the Q used in (pH) automatically satisfies (10). We show that the same condition is true for descriptor systems. If the system is behaviorally observable, then \(\ker A\cap \ker C=\{0\}\) and hence the existence of a (pH) is equivalent to the existence of invertible solutions to (KYP).

Due to the interesting properties of pH systems, ideally, we want to obtain (pH) for any passive system given in DAE form or given time domain data measurements [30]. In order to solve that problem we need to answer the question (Q2). This problem was already studied in [25, 27] where it was shown that (Pa) only guarantees (KYP) to hold on certain subspaces and as a consequence, we can only derive (pH) on a subspace. However, if the system has index at most one, we can derive a modified KYP inequality that is solvable for passive systems and which leads for behaviorally observable systems to a pH formulation of the system on the whole space.

The question (Q3) arises when one has to reconstruct a system from input–output data. If the system is expected to be passive, then the transfer function is positive real. If the data does not allow us to conclude (PR), e.g. due to measurement errors, then one can compute the nearest positive real transfer function [31]. Therefore, the remaining task is to find (pH) of this positive real transfer function, i.e. one has to compute the system matrices (E, A, B, C, D). We show that the computation of the system matrices is always possible for a given positive real transfer function and this is based on the well-known representation, see, e.g. [20, Section 5.1]

$$\begin{aligned} {\mathcal {T}}(s)=M_1s+{\mathcal {T}}_p(s) \end{aligned}$$

for some positive semidefinte \(M_1\in \mathbb {R}^{n\times n}\) and a proper positive real rational function \({\mathcal {T}}_p\). The summand \(M_1s\) can be realized as an index two subsystem which is combined with a pH realization based on a minimal realization of \({\mathcal {T}}_p\).

Fig. 2
figure 2

Overview of the main results in this note. The implications with additional assumption are colored black and the ones without are colored blue. Moreover, if assumptions are not fulfilled, counterexamples are highlighted in red (color figure online)

Full size image

The paper is organized as follows: In Sect. 2 we summarize which relations between the basic notions (pH), (KYP), (Pa), (PR) are known for descriptor systems. In Sect. 3, it is shown that (10) can be used to define a pH realization which answers (Q1). Besides that, a possible generalization of the solutions of (KYP) is discussed. The answer to (Q1) will be used in Sect. 4 where we consider (Q2). In particular, we derive a pH formulation for passive index one systems that are (behaviorally) observable. Finally, in Sect. 5 we show how a pH formulation can be derived from a given real-valued positive real transfer function which answers (Q3). As a summary, an overview of the main results is presented in Fig. 2.

2 Literature review and combination of known results

Below, we give the first main result on the relationship between (pH), (Pa), (KYP) and (PR) for descriptor systems. Here we combine the results of [17], who studied (PR) and (KYP), and [31] who studied the relation between (PR), (pH) and (KYP) for invertible Q. In addition, we include the relation to passivity.

Proposition 1

Let (E, A, B, C, D) define a linear time-invariant descriptor system (1). Then the following holds

$$\begin{aligned} (pH) \quad \Longrightarrow \quad (KYP ) \quad \Longrightarrow \quad (PR) ~~\wedge ~~ (Pa) . \end{aligned}$$
(11)

Furthermore, every \(Q\in \mathbb {K}^{n\times n}\) fulfilling (pH) is a solution to \((KYP )\) and every solution Q to \((KYP )\) leads to a storage function in (Pa). Moreover, the following holds

$$\begin{aligned} (KYP ) ~ \text {with invertible }Q \quad&\Longrightarrow \quad (pH) , \end{aligned}$$
(12)
$$\begin{aligned} (Pa) \wedge E\text { invertible} \quad&\Longrightarrow \quad (KYP ). \end{aligned}$$
(13)

Proof

Step 1: We prove the implications (11). If (pH) holds for some Q, it solves the KYP inequality (4) since \(E^HQ=Q^HE\ge 0\) and

$$\begin{aligned}&\begin{bmatrix} -A^HQ-Q^HA&{}&{}\quad C^H-Q^HB\\ C-B^HQ&{}&{}D+D^H \end{bmatrix} \nonumber \\ {}&\quad =\begin{bmatrix} -Q^H(J-R)^HQ-Q^H(J-R)Q &{}&{}\quad Q^H(G+P)-Q^H(G-P)\\ (G+P)^HQ-(G-P)^HQ&{}&{}2S \end{bmatrix}\qquad \end{aligned}$$
(14)
$$\begin{aligned}&\quad =2\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}\begin{bmatrix} R&{}\quad P\\ P^H&{}\quad S \end{bmatrix}\begin{bmatrix} Q&{}0\\ 0&{} I_m \end{bmatrix}=2W\ge 0. \end{aligned}$$
(15)

Hence Q fulfills (KYP). Next, we show that any \(Q\in \mathbb {K}^{n\times n}\) which fulfills (KYP) defines a storage function \(\mathcal {S}(x):=\tfrac{1}{2}x^HQ^HEx\) which fulfills (Pa). The basic idea goes back to [32] for standard systems. For sufficiently smooth u, consistent initial value \(x_0\) and for all \(t\ge 0\), it holds that

$$\begin{aligned} \nonumber 2 \tfrac{\hbox {d}}{\hbox {d}t}\mathcal {S}(x(t))&=\tfrac{\hbox {d}}{\hbox {d}t}(Ex(t))^HQx(t)+x^H(t)Q^H\tfrac{\hbox {d}}{\hbox {d}t}Ex(t)\\ {}&=(Ax(t)+Bu(t))^HQx(t)+x^H(t)Q^H(Ax(t)+Bu(t))\nonumber \\ {}&=\begin{bmatrix} x(t)\\ u(t) \end{bmatrix}^H\begin{bmatrix} A^HQ+Q^HA&{}&{}\quad Q^HB-C^H\\ B^HQ-C&{}&{}\quad -D-D^H \end{bmatrix}\begin{pmatrix} x(t)\\ u(t) \end{pmatrix}\nonumber \\ {}&~~~~+\begin{bmatrix} x(t)\\ u(t) \end{bmatrix}^H\begin{bmatrix} 0&{}&{}\quad C^H\\ C&{}&{}\quad D+D^H \end{bmatrix}\begin{bmatrix} x(t)\\ u(t) \end{bmatrix}\nonumber \\&\le \begin{bmatrix} x(t)\\ u(t) \end{bmatrix}^H\begin{bmatrix} 0&{}&{}\quad C^H\\ C&{}&{}\quad D+D^H \end{bmatrix}\begin{bmatrix} x(t)\\ u(t) \end{bmatrix}=2{{\,\textrm{Re}\,}}y(t)^Hu(t). \end{aligned}$$
(16)

Integration of (16) leads to (Pa). Finally, it was shown in [17, Theorem 3.1] that (KYP) implies (PR). This completes the proof of (11).

Step 2: To prove (12), let Q be an invertible solution of (KYP). Then we can define a pH system via

$$\begin{aligned} J&:= \frac{1}{2} (A Q^{-1} - Q^{-H} A^H),&R&:=- \frac{1}{2} (A Q^{-1} + Q^{-H} A^H),\\ G&:= \frac{1}{2}(Q^{-H} C^H + B),&P&:= \frac{1}{2}(Q^{-H} C^H - B), \\ S&:= \frac{D + D^H}{2},&N&:= \frac{D - D^H}{2}. \end{aligned}$$

Hence by definition it holds that \(-\Gamma =\Gamma ^H\), \(W =W^H\) and

$$\begin{aligned} (J-R)Q = A,\quad G-P = B, \quad (G+P)^H Q = C,\quad S+N = D. \end{aligned}$$

Furthermore, \(W\ge 0\) follows from (15) and hence (pH) is satisfied. This proves (12).

Step 3: We continue with the proof of (13). Let \(Q\in \mathbb {K}^{n\times n}\) be such that \({\mathcal {S}}(x):=\tfrac{1}{2}x^HQ^HEx\) defines a storage function. To show (KYP) we verify the left inequality in (4) first. Since E is invertible the solutions of (1) are given by the solutions of the standard system \(\dot{x}(t)=E^{-1}Ax(t)+E^{-1}B u(t)\), \(x(0)=x_0\). In particular, for every choice of \((x_0,u(0))\in \mathbb {K}^{n}\times \mathbb {K}^m\) there exists a solution which fulfills (16) as a consequence of (Pa). Using now \(t=0\) in (16) this implies the left inequality in (KYP). Furthermore, by choosing \(T=0\) in (Pa) and the previous observation, that for every initial value \(x_0\in \mathbb {K}^n\) and \(u=0\) there exists a solution we conclude from the inequality on the right-hand side in (Pa) that \(Q^HE\ge 0\) holds. In summary, this proves that Q solves (KYP) and hence the implication (13). \(\square \)

However, we show with the following example that (KYP) does not necessarily imply (pH).

Example 2

Consider the real system given by \((E,A,B,C,D)=(1,-1,1,0,0)\). Then this system is asymptotically stable, i.e. all eigenvalues of matrix A have negative real parts, hence using (6) and (7), we conclude that it is detectable and controllable but not observable. Furthermore, (KYP) which is given by

$$\begin{aligned} \begin{bmatrix} 2Q&{}\quad -Q\\ {}-Q&{} \quad 0 \end{bmatrix}\ge 0 \end{aligned}$$

has only the trivial solution \(Q=0\). Indeed, for \(Q\ne 0\) the above matrix is indefinite. To construct (pH) notice that \((J-R)Q=A=-1\) which implies \(R=Q^{-1}\) and

$$\begin{aligned} B=G-P=1,\quad C=G+P=0,\quad D=S+N=0. \end{aligned}$$

This yields \(G=\frac{1}{2}=-P\), \(N=S=0\) and hence

$$\begin{aligned} \begin{bmatrix} Q^\top RQ&{}&{}\quad Q^\top P\\ P^\top Q&{}&{}\quad S \end{bmatrix}=\begin{bmatrix} R^{-1}&{}-\frac{1}{2}R^{-1}\\ {}-\frac{1}{2}R^{-1}&{}0 \end{bmatrix} \end{aligned}$$

which is indefinite. Therefore the system does not fulfill (pH).

Next, we present an example that shows that \(\mathrm {(PR)}\wedge \mathrm {(Pa)}\nRightarrow \mathrm {(KYP)}\) for descriptor systems.

Example 3

Consider the real system \((E,A,B,C,D)=(0,1,0,1,0)\). Then the system dynamics is given by \(x(t)= 0\). Hence, \(x_0=0\) is the only consistent initial value. Therefore for all \(u\in L^1([t_0,t_1],\mathbb {R})\), \(0\le t_0\le t_1\) we have

$$\begin{aligned} \int _{t_0}^{t_1} y(\tau )u(\tau )\hbox {d}\tau =\int _{t_0}^{t_1} x(\tau )u(\tau )\hbox {d}\tau =0, \end{aligned}$$

which implies passivity. On the other hand, the corresponding LMI of this system for some \(x\in \mathbb {R}\) is given by

$$\begin{aligned} \begin{bmatrix} -2x&{}&{}\quad 1\\ 1&{}&{}\quad 0 \end{bmatrix}\ge 0. \end{aligned}$$

However, this cannot be valid since the matrix is indefinite for all \(x\in \mathbb {R}\). Furthermore, the system is positive real with transfer function \({\mathcal {T}}(s)=C(sE-A)^{-1}B+D=0\). Moreover, the system is behaviorally controllable but not minimal since \({{\,\textrm{rk}\,}}[E,B]=0\ne 1\) and \(A\ker E\nsubseteq {{\,\textrm{ran}\,}}E\). The proof that the system does not fulfill (pH) is similar to the calculation in Example 2.

In 3 we use that Note that it was important in the Examples 2 and 3 that the feedthrough term D is zero. It remains an open question if a similar construction is possible with a nonsingular feedthrough term.

Below, we briefly review known results:

  1. (i)

    In Theorem 2 of [18], the authors show that (KYP) with the condition \(E^HQ\ge 0\) is equivalent to the so-called extended strict positive realness and \(D+D^H>0\) under the condition that the systems are assumed to have index one and be asymptotically stable.

  2. (ii)

    In [17], it was shown that (KYP) implies (PR) and the converse is also true for certain minimal realizations of the system.

  3. (iii)

    In [26] it was shown that a nonnegative Popov function leads to a Hermitian solution to (KYP). It is assumed that there are no eigenvalues on the imaginary axis and that the system has index one.

  4. (iv)

    The author of [33] studies a more general behavioral approach and shows the equivalence of passivity and positive real pairs associated with the behavior. Although this approach contains descriptor systems, they were not investigated explicitly.

  5. (v)

    In [31], the authors have shown that (pH) \(\Rightarrow \) (KYP) \(\Rightarrow \) (PR) and that (KYP) \(\Rightarrow \) (pH) holds for invertible solutions Q of the KYP inequality. However, it remains open whether non-invertible solutions to the KYP inequality also lead to a pH formulation or if (PR) also implies (KYP).

  6. (vi)

    In [14, Section II], it is proven that the implication (pH) \(\Rightarrow \) (Pa) also holds for descriptor systems whose coefficients depend on the time t and the state x and not necessarily quadratic Hamiltonians.

  7. (vii)

    In [34], the authors consider stable systems, i.e. for all consistent initial values \(x_0\in \mathbb {K}^n\) the solution x to \(E\dot{x}(t)=Ax(t)\), \(x(0)=x_0\) fulfills \(\sup _{t\ge 0}\Vert x(t)\Vert <\infty \). It was shown that there exists \(Q\in \mathbb {K}^{n\times n}\) based on the solution of a generalized Lyapunov equation such that \(A=(J-R)Q\) holds on a certain subspace.

  8. (viii)

    In [35], the authors generalize the relationship between (KYP) and (PR) for standard systems. Moreover, for descriptor systems, our definition of (KYP) and (PR) can be seen as the special case where Q is replaced with QE. They show equivalence in the strict inequality case between (KYP) and (PR) if \(\det (s E -A) \ne 0\) for all \({{\,\textrm{Re}\,}}s \ge 0\), E nonsingular, and Q positive definite.

In the remainder of this section, we will show that passivity implies positive realness, i.e. (Pa) implies (PR). For descriptor systems, the state-control pair \((x,u)\in \mathbb {K}^{n}\times \mathbb {K}^m\) will in general not attain all values in \(\mathbb {K}^{n}\times \mathbb {K}^m\) which is the reason why we cannot deduce (KYP) from (Pa) for descriptor systems. Also, positive realness together with certain controllability assumptions will only lead to solutions of the KYP inequality on a subspace. In the following, we will use a more compact way of writing the KYP inequality for all \((x,u)\in \mathbb {K}^n\times \mathbb {K}^m\) as

$$\begin{aligned}&{{\,\textrm{Re}\,}}\begin{bmatrix} Qx\\ u \end{bmatrix}^H\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}\nonumber \\ {}&\quad =\frac{1}{2}\begin{bmatrix} Qx\\ u \end{bmatrix}^H\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}+\frac{1}{2}\begin{bmatrix} x\\ u \end{bmatrix}^H\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}^H\begin{bmatrix} Qx\\ u \end{bmatrix}\nonumber \\ {}&\quad =\frac{1}{2}\begin{bmatrix} x\\ u \end{bmatrix}^H\begin{bmatrix} Q^HA+A^HQ&{}&{}\quad Q^HB-C^H\\ B^HQ-C&{}&{}\quad -D-D^H \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}\nonumber \\ {}&\quad \le 0 \end{aligned}$$
(17)

Further, for Hermitian \(A\in \mathbb {K}^{n\times n}\) and a subspace \({\mathcal {V}}\) of \(\mathbb {K}^n\) the relation \(A\ge _{{\mathcal {V}}}0\) means that \(v^HAv\ge 0\) for all \(v\in {\mathcal {V}}\). Then we consider the following restricted versions of (4):

(KYP|\({\mathcal {V}}\)):

There exists \(Q\in \mathbb {K}^{n\times n}\) with \(Q^HE=E^HQ\) and

$$\begin{aligned} {{\,\textrm{Re}\,}}\begin{bmatrix} Qx\\ u \end{bmatrix}^H\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}\le 0,\quad x^HE^HQx\ge 0,\quad \text {for all }\begin{bmatrix} x\\ u \end{bmatrix}\in {\mathcal {V}}.\qquad \end{aligned}$$
(18)
(KYP\(_{E}\)|\({\mathcal {V}}\)):

There exists \(Y=Y^H\in \mathbb {K}^{n\times n}\) such that (18) holds for some \(Q=YE\).

There are several authors who consider KYP inequalities restricted to suitable subspaces:

  1. (i)

    In [25], the authors use Ax instead of \(\dot{x}\) in the (KYP) inequality (4) which leads to a \(3\times 3\) block matrix where each block corresponds to one of the entries in \((\dot{x},x,u)\). To show the existence of (KYP) inequality solutions on a certain subspace for a positive real transfer function they assume minimality. Furthermore, the passivity notion is different since they allow for arbitrary nonnegative storage functions which must not vanish on \(\ker E\).

  2. (ii)

    The KYP inequality (KYP\(_{E}\)|\({\mathcal {V}}\)) is considered in [28] for the dual system restricted to the right deflating subspaces of the regular pair (E, A). The authors of [28] show that behavioral controllability and observability imply the existence of solutions to this inequality. However, the passivity notion used in [28] is only equivalent to (Pa) for behaviorally controllable and observable systems.

  3. (iii)

    In [27, 29], the authors consider KYP inequalities restricted to the so-called system space \(\mathcal {V}_{\textrm{sys}}\), which is the smallest subspace where the system trajectories evolve. It is shown in that case that behavioral controllability and (PR) imply (KYP|\(\mathcal {V}_{\textrm{sys}}\)).

  4. (iv)

    The authors in [15] consider the relation between storage functions, the feasibility of linear quadratic optimal control problems on an infinite time horizon and the existence of a solution to KYP inequalities on the system space \(\mathcal {V}_{\textrm{sys}}\). However, the definition of storage function is slightly different from the notion used in (Pa) and allows negative function values. Furthermore, in the definition of the system space \(\mathcal {V}_{\textrm{sys}}\) a slightly different solution concept is used.

Summarizing the literature review on restricted KYP inequalities given in (i)-(iv), we focus in the following on the most general approach presented in [27, 29].

The system space \(\mathcal {V}_{\textrm{sys}}\) can be characterized in terms of a limit of subspaces. It was shown in [27, Proposition 3.3] that the following sequence terminates after finitely many steps

$$\begin{aligned} {\mathcal {V}}_{k+1}:=[A,B]^{-1}([E,0]{\mathcal {V}}_k),\; k\ge 1,\quad {\mathcal {V}}_0:=\mathbb {K}^{n}\times \mathbb {K}^m, \end{aligned}$$
(19)

where the pre-image of [A, B] is used. The resulting terminal subspace is the system space \(\mathcal {V}_{\textrm{sys}}\) and it fulfills \(\mathcal {V}_{\textrm{sys}}={\mathcal {V}}_{k}\) for all \(k\ge k_0\) and some \(k_0\ge 0\).

The restricted KYP inequalities (KYP|\(\mathcal {V}_{\textrm{sys}}\)) and (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)) involve the semi-definiteness condition \(E^HQ\ge _{\mathcal {V}_{\textrm{sys}}^{x}} 0\), where \(\mathcal {V}_{\textrm{sys}}^{x}\) denotes the projection of \(\mathcal {V}_{\textrm{sys}}\subset \mathbb {K}^{n+m}\) to onto the first n components.

Compared to this, [27, 29] use the slightly different condition \(E^HQ\ge _{{\mathcal {V}}_{\textrm{diff}}} 0\), where the positive definiteness should hold on the subspace

$$\begin{aligned} {\mathcal {V}}_{\textrm{diff}}:=\{x_0\in \mathbb {R}^n ~\mid ~ {(x,u)\in L^2_{loc}(\mathbb {R},\mathbb {K}^{n+m})\hbox { solves }(1)\hbox { with }Ex(0)=Ex_0}\}. \end{aligned}$$

Here a solution of (1) means that x is differentiable almost everywhere and fulfills (1) almost everywhere.

In the appendix, we show that (KYP|\(\mathcal {V}_{\textrm{sys}}\)) and (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)), are equivalent to the restricted KYP inequalities used in [27, 29].

We have the following results from [27, Theorem 4.1, Proposition 4.4] and [29, Theorem 4.3] on the relation between (PR), (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)) and (KYP|\(\mathcal {V}_{\textrm{sys}}\)).

Proposition 4

Let \(\Sigma = (E,A,B,C,D)\) be a descriptor system of the form (1) and transfer function \({\mathcal {T}}(s)=C(sE-A)^{-1}B+D\). Then the following holds:

  1. (a)

    If there exists a solution \(Y=Y^H\) to (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)) then \({\mathcal {T}}\) is positive real.

  2. (b)

    If the system is behaviorally controllable and \({\mathcal {T}}\) is positive real, then there exists a solution \(Y=Y^H\) to (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)).

  3. (c)

    If Y satisfies (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)) then \(Q:=EY\) satisfies (KYP|\(\mathcal {V}_{\textrm{sys}}\)).

  4. (d)

    If Q satisfies (KYP|\(\mathcal {V}_{\textrm{sys}}\)) then there exists a solution Y to (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)) with \(E^HYE=E^HQ\).

In Proposition 4 (b), we only consider the condition of behavioral controllability. The question that arises is whether the dual property, behavioral observability, can be used as a condition for the existence of solutions to the KYP inequality. To answer this question, we consider the following example.

Example 5

Consider \(\dot{x}=x\), \(y=cx+u\) for some \(c\ne 0\). Then this system is observable but not controllable. Furthermore, \({\mathcal {T}}(s)=1\) for all \(s\in \mathbb {C}\) which is positive real. However, (KYP) which is given by

$$\begin{aligned} \begin{bmatrix} -2Q&{}&{} {\overline{c}}\\ c&{}&{}2 \end{bmatrix}\ge 0 \end{aligned}$$

is only solvable for sufficiently small \(Q<0\). Hence, positive realness and observability is not enough to guarantee a solution to (KYP).

Example 5 shows that behavioral observability of \(\Sigma =(E,A,B,C,D)\) together with a positive real transfer function does not imply (KYP\(_{E}\)|\(\mathcal {V}_{\textrm{sys}}\)) or (KYP|\(\mathcal {V}_{\textrm{sys}}\)). However, the behavioral observability of \(\Sigma \) is equivalent to the behavioral controllability of the dual system \(\Sigma '=(E^H,A^H,C^H,B^H,D^H)\) which leads us to the following result.

Corollary 6

Let \(\Sigma =(E,A,B,C,D)\) be a positive real and behaviorally observable descriptor system. Then the dual system \(\Sigma '=(E^H,A^H,C^H,B^H,D^H)\) is positive real and behaviorally controllable and there exists a solution to the generalized KYP inequality

$$\begin{aligned} \begin{bmatrix} -AQ-Q^HA^H&{}&{}\quad B-Q^HC^H\\ B^H-CQ&{}&{}\quad D+D^H \end{bmatrix}\ge 0,\quad EQ=Q^HE^H\ge 0. \end{aligned}$$
(20)

Proof

The behavioral controllability of the dual system holds by definition and (8). Since for all \(\lambda \in \mathbb {C}\) with \({{\,\textrm{Re}\,}}\lambda \ge 0\) it holds that

$$\begin{aligned} B^H(\lambda E^H-A^H)^{-1}C^H+D^H+C({\overline{\lambda }} E-A)^{-1}B+D={\mathcal {T}}({\overline{\lambda }})+{\mathcal {T}}({\overline{\lambda }})^H\ge 0, \end{aligned}$$

the dual system is positive real and behaviorally controllable. Hence the existence of solutions to (20) follows from Proposition 4 (b),(c). \(\square \)

From Proposition 4 and (16) we immediately obtain the following corollary.

Corollary 7

Let (E, A, B, C, D) be a descriptor system of the form (1). Then the following holds:

$$\begin{aligned} (Pa) \quad \Longleftrightarrow \quad (KYP {|}\mathcal {V}_{\textrm{sys}}) \quad \Longleftrightarrow \quad (KYP _{E}{|}\mathcal {V}_{\textrm{sys}}) \quad \Longrightarrow \quad (PR) . \end{aligned}$$

Moreover, if the system is behaviorally controllable then (PR) implies (Pa).

It was shown in [29, Theorem 4.3] that (PR) together with the conditions

$$\begin{aligned} \begin{aligned} {{\,\textrm{rk}\,}}[\lambda E-A,B]=n,\quad \text {for all }\lambda \in \mathbb {C}, {{\,\textrm{Re}\,}}\lambda \ge 0,\\ {\mathcal {T}}(i\omega )+{\mathcal {T}}(i\omega )^H>0,\quad \text {for all }i\omega \in i\mathbb {R}\setminus \sigma (E,A)\text {,} \end{aligned} \end{aligned}$$
(21)

implies (KYP|\(\mathcal {V}_{\textrm{sys}}\)) and hence (Pa). In the case of standard systems the controllability assumptions on a positive real system to fulfill (Pa) or equivalently (KYP), can be relaxed to the sign-controllability, i.e.

$$\begin{aligned} {{\,\textrm{rk}\,}}[\lambda I_n-A,B]=n \quad \text {or}\quad {{\,\textrm{rk}\,}}[-{\overline{\lambda }} I_n-A,B]=n\quad \text {for all }\lambda \in \mathbb {C}, \end{aligned}$$

see [36] and [16, Section A.4]. In particular, the first condition in (21) implies sign-controllability. For descriptor systems [27] shows that behavioral sign-controllability which means

$$\begin{aligned} {{\,\textrm{rk}\,}}[\lambda E-A,B]=n \quad \text {or}\quad {{\,\textrm{rk}\,}}[-{\overline{\lambda }} E-A,B]=n\quad \text {for all }\lambda \in \mathbb {C}, \end{aligned}$$

together with a full rank assumption on the Popov function

$$\begin{aligned} {{\,\textrm{rk}\,}}\begin{bmatrix} (-{\overline{s}}E-A)^{-1}B\\ I_m \end{bmatrix}^H\begin{bmatrix} 0&{}&{}\quad C^H\\ C&{}&{} \quad \tfrac{1}{2}(D+D^H) \end{bmatrix}\begin{bmatrix} (sE-A)^{-1}B\\ I_m \end{bmatrix}=m \end{aligned}$$

where the rank is computed over the quotient field \(\mathbb {K}(s)\), implies that there is a Hermitian solution \(Q^HE=E^HQ\) to the generalized KYP inequality which is not necessarily nonnegative. The nonnegativity and hence (Pa) can be concluded from [29, Theorem 4.3 (c)] is the system is in addition behaviorally detectable meaning that

$$\begin{aligned} {{\,\textrm{rk}\,}}[\lambda E^\top -A^\top ,C^\top ]=n,\quad \text {for all }\lambda \in \mathbb {C}, {{\,\textrm{Re}\,}}\lambda \ge 0, \end{aligned}$$

holds.

In addition to the presented results, the following example from [37] demonstrates that invertible solutions on the system space do not lead to (pH).

Example 8

Let \(E=\begin{bmatrix} 1&{}0\\ 0&{}0 \end{bmatrix}\), \(A=\begin{bmatrix} -1&{}\quad 0\\ 0&{}\quad 1 \end{bmatrix}\), \(B=\begin{bmatrix}1\\ 0 \end{bmatrix}\), \(C=\begin{bmatrix}1&1\end{bmatrix}\), \(D=0\). Then this system is behaviorally observable, behaviorally controllable, positive real with transfer function \({\mathcal {T}}(s)=\tfrac{1}{s+1}\) and the system space is \(\mathcal {V}_{\textrm{sys}}=\{(x_1,0,u)^\top \mid x_1,u\in \mathbb {R}\}\). However, it is not minimal, since \(A\ker E\nsubseteq {{\,\textrm{ran}\,}}E\) and \({{\,\textrm{rk}\,}}[E,B]=1\). Furthermore, the (KYP) has no solution but (KYP|\(\mathcal {V}_{\textrm{sys}}\)) holds since the system is behaviorally controllable.

3 When does (KYP) imply (pH)?

In order to study if (KYP) implies (pH), we first need to consider KYP solutions for DAEs. Below we show that for DAEs with invertible E the observability of the system guarantees invertible solutions of (KYP). This result was already obtained in [38] for positive semi-definite solutions of standard systems, but the proof can easily be extended to descriptor systems.

Proposition 9

Let \(\Sigma =(E,A,B,C,D)\) be a descriptor system of the form (1) with E invertible and (E, A, C) behaviorally observable. If \(Q\in \mathbb {K}^{n\times n}\) satisfies

$$\begin{aligned} \begin{bmatrix}-A^HQ-Q^HA&{}&{}\quad C^H-Q^HB\\ C-B^HQ &{}&{}\quad D + D^H \end{bmatrix}\ge 0,\quad Q^HE=E^HQ, \end{aligned}$$

then Q is invertible.

Proof

First, one shows that \(\ker Q\) is an (A, E)-invariant subspace, i.e. \(A\ker Q\subseteq E\ker Q\). Let \(z=Av\) with \(v\in \ker Q\) then

$$\begin{aligned} 0=-v^HA^H Qv-v^H Q^HAv=v^H\underbrace{(-A^H Q- Q^HA)}_{\le 0}v. \end{aligned}$$

Hence \(v\in \ker (-A^H Q- Q^HA)\) which implies with \(v\in \ker Q\) that \(v\in \ker Q^HA\). Therefore \(Q^Hz= Q^HAv=0\), i.e. \(z\in \ker Q^H\). This shows \(A\ker Q\subseteq \ker Q^H\). Moreover, since E is invertible one has

$$\begin{aligned} \ker Q^H=EE^{-1}\ker Q^H=E\ker Q^HE=E\ker (E^HQ)=E\ker Q. \end{aligned}$$

Hence \(\ker Q\) is an (A, E)-invariant subspace. Next, we show that \(\ker Q\subseteq \ker C\). Let \(z\in \ker Q\). Then (KYP) implies for all \(w\in \mathbb {R}^m\) and \(\alpha \in \mathbb {R}\) that

$$\begin{aligned} 0&\le \begin{bmatrix} z\\ \alpha w \end{bmatrix}^H\begin{bmatrix}-A^HQ-Q^HA&{}&{} \quad C^H-Q^HB\\ C-B^HQ &{}&{} \quad D + D^H \end{bmatrix}\begin{bmatrix} z\\ \alpha w \end{bmatrix}\nonumber \\ {}&=z^HC^H\alpha w+\alpha w^HCz+\alpha ^2w^H(D+D^H)w. \end{aligned}$$
(22)

Assume that \(Cz\ne 0\). Then all \(\alpha >0\) sufficiently small would violate (22). This contradiction leads to \(z\in \ker C\) which implies \(\ker Q\subseteq \ker C\). Hence Proposition 7.2 in [39] implies \(\ker Q=\{0\}\). \(\square \)

In Proposition 9 we do not require the positive semi-definiteness of \(Q^HE\) in (KYP). Hence as a special case, we obtain the following result.

Corollary 10

Let \(\Sigma =(E,A,B,C,D)\) be a descriptor system of the form (1) with E invertible and (E, A, C) behaviorally observable. If \(Q\in \mathbb {K}^{n\times n}\) satisfies (KYP) then Q is invertible.

Example 2 shows that the detectability of the system will in general not guarantee that the solutions of (KYP) are invertible.

Next, we extend the following result for standard systems [23, p. 55] which indicates that (pH) for a passive system can be established without the observability assumption.

Proposition 11

Let \(\Sigma =(E,A,B,C,D)\) define a descriptor system of the form (1). Then (pH) holds if and only if (KYP) holds for some \(Q\in \mathbb {C}^{n\times n}\) with \(\ker Q\subseteq \ker C\cap \ker A\).

Proof

If (pH) holds, then \(A=(J-R)Q\) and \(C=(G+P)^HQ\) for some \(J,R,Q\in \mathbb {K}^{n\times n}\) and \(G,P\in \mathbb {K}^{n\times m}\). Hence \(\ker Q\subseteq \ker A\cap \ker C\) and by Proposition 1, Q satisfies (KYP). To prove sufficiency assume that (KYP) holds for some \(Q\in \mathbb {K}^{n\times n}\) with \(\ker Q\subseteq \ker C\cap \ker A\). Then \(Q^HE\ge 0\) follows and we can introduce \(\Theta \in \mathbb {K}^{(n+m)\times (n+m)}\) via

$$\begin{aligned} \begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}=\Theta \begin{bmatrix} Q&{}\qquad 0\\ 0&{}\qquad I_m \end{bmatrix}. \end{aligned}$$
(23)

Note that \(\Theta \) is well defined since \(\ker Q\subseteq \ker A\cap \ker C\) holds. To show that the system (E, A, B, C, D) fulfills (pH) we need to verify (2). To this end, we consider the trivial decomposition

$$\begin{aligned} \Theta =\frac{1}{2}(\Theta + \Theta ^H)+\frac{1}{2}(\Theta -\Theta ^H) \end{aligned}$$

and therefore the matrices \(J,R\in \mathbb {K}^{n\times n}\), \(P,G\in \mathbb {K}^{n\times m}\), \(S,N\in \mathbb {K}^{m\times m}\) in (2) are given by

$$\begin{aligned} \begin{bmatrix} J&{}\quad G\\ {}-G^H&{}\quad -N \end{bmatrix}=\frac{1}{2}(\Theta - \Theta ^H),\quad \begin{bmatrix} R&{}\quad P\\ P^H&{}\quad S \end{bmatrix}:=-\frac{1}{2}(\Theta + \Theta ^H). \end{aligned}$$

Since N is skew-Hermitian, so is \(-N\) and therefore \(\Gamma \) given by (2) is skew-Hermitian.

Furthermore, (KYP) as it is rewritten in (17) yields

$$\begin{aligned} W=-\begin{bmatrix} Q&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}^H(\Theta +\Theta ^H)\begin{bmatrix} Q&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}\ge 0 \end{aligned}$$
(24)

which finally proves (pH). \(\square \)

Remark 12

If Q is positive definite then (24) is equivalent to \(\Theta +\Theta ^H\le 0\) which also appears in some references as a definition of pH (descriptor) systems. If Q is singular, then \(\Theta +\Theta ^H\le 0\) cannot be concluded from (24) in general. Instead, we can redefine \(\Theta \) in such a way that it fulfills (23). To this end, we use the space decomposition \(\mathbb {K}^n={{\,\textrm{ran}\,}}Q\oplus \ker Q^H\) and letting

$$\begin{aligned} \Theta =\begin{bmatrix} \Theta _1&{}\quad \Theta _2\\ \Theta _3&{}\quad \Theta _4 \end{bmatrix}\in \mathbb {K}^{(n+m)\times (n+m)},\quad \Theta _1=\begin{bmatrix} P_{{{\,\textrm{ran}\,}}Q}\Theta _1\mid _{{{\,\textrm{ran}\,}}Q}&{}\quad P_{{{\,\textrm{ran}\,}}Q}\Theta _1\mid _{\ker Q^H}\\ P_{\ker Q^H}\Theta _1\mid _{{{\,\textrm{ran}\,}}Q}&{}\quad P_{\ker Q^H}\Theta _1\mid _{\ker Q^H} \end{bmatrix}, \end{aligned}$$

where \(P_{{{\,\textrm{ran}\,}}Q}\) is the orthogonal projector onto \({{\,\textrm{ran}\,}}Q\). We can redefine \(\Theta _1\) as

$$\begin{aligned} {{\hat{\Theta }}}_1:=\begin{bmatrix} P_{{{\,\textrm{ran}\,}}Q}\Theta _1\mid _{{{\,\textrm{ran}\,}}Q}&{}\quad -(P_{\ker Q^H}\Theta _1\mid _{{{\,\textrm{ran}\,}}Q})^H\\ P_{\ker Q^H}\Theta _1\mid _{{{\,\textrm{ran}\,}}Q}&{}\quad 0 \end{bmatrix} \end{aligned}$$

without changing the product on the right-hand side of (23). Hence the matrix \({{\hat{\Theta }}}\) which is obtained after replacing the block \(\Theta _1\) in \(\Theta \) with \({{\hat{\Theta }}}_1\) fulfills (23) and \({{\hat{\Theta }}}+{\hat{\Theta ^H\le }} 0\).

Remark 13

If \(\Sigma =(E,A,B,C,D)\) is behaviorally observable then the matrix Q in (pH) fulfills \(\ker Q\subseteq \ker A\cap \ker C=\{0\}\) by [39, Proposition 7.2, Theorem 7.3] and hence Q must be invertible.

Remark 14

Another way to use solutions Q to the KYP inequalities to obtain a representation that is quite similar to a pH formulation and does not require further assumptions is given by a left-multiplication of the state equations with \(Q^H\)

$$\begin{aligned} {{\hat{\Theta }}}=\begin{bmatrix} J-R&{}&{}\quad G-P\\ {}-(G+P)^H&{}&{}\quad -S-N \end{bmatrix}:=\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I \end{bmatrix}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}. \end{aligned}$$

However, if Q is not invertible then this multiplication might enlarge the solution set of the descriptor system. The treatment of pH systems with singular Q is described in detail in [10, Section 6.3], see also [4].

4 When does (Pa) imply (pH)?

In Sect. 3, we discussed when (KYP) implies (pH). Although all solutions Q to (KYP) fulfill \(\ker Q\subseteq \ker C\) (proof of Proposition 9), one cannot guarantee that \(\ker Q\subseteq \ker A\cap \ker C\) holds, as Example 2 shows. Hence, a pH formulation of a given passive system is not always possible without further assumptions.

If E is invertible and (Pa) holds then Proposition 1 yields (KYP) and therefore the results of Sect. 3 can be used to characterize when (pH) holds. Hence we will focus in this section on the case where E is not invertible.

As a first result, we show that for passive systems one might restrict to systems that are controllable and observable with index at most two. First, we recall the notion of index of descriptor systems (1). Since (E, A) is assumed to be regular, there exist invertible \(S,T\in \mathbb {C}^{n\times n}\) and \(r\in {\mathbb {N}}\), see, e.g. [21], such that

$$\begin{aligned} (SET,SAT)=\left( \begin{bmatrix}I_{r}&{}\quad 0\\ 0&{}\quad N\end{bmatrix},\begin{bmatrix}J&{}\quad 0\\ 0&{}\quad I_{n-r}\end{bmatrix}\right) \end{aligned}$$
(25)

where J and N are in Jordan canonical form and N is nilpotent. The block-diagonal form (25) is typically referred to as Kronecker–Weierstraß form. Based on this form, the index of the system (1) is defined as the smallest natural number \(\nu \) such that \(N^{\nu -1}\ne 0\) and \(N^{\nu }=0\).

If a given system in state-space form is not controllable or observable then one might consider the KYP inequality restricted to observable or controllable subspaces. This was proposed for standard systems in [40, 41], see also [16, Theorem 3.39].

For descriptor systems we recall a Kalman-like decomposition from [39, Theorem 8.1], see also [42] and [21, p. 51].

Proposition 15

For \(E,A\in \mathbb {K}^{l\times n}\), \(B\in \mathbb {K}^{l\times m}\) and \(C\in \mathbb {K}^{m\times n}\) there exists invertible \(S\in \mathbb {K}^{l\times l}\), \(T\in \mathbb {K}^{n\times n}\) such that

$$\begin{aligned}{} & {} [SET,SAT,SB,CT]\nonumber \\{} & {} \quad =\left[ \begin{bmatrix} E_{11}&{}\quad E_{12}&{}\quad E_{13}&{}\quad E_{14}\\ 0&{}\quad E_{22}&{}\quad 0&{}\quad E_{24}\\ 0&{}\quad 0&{}\quad E_{33}&{}\quad E_{34}\\ 0&{}\quad 0&{}\quad 0&{}\quad E_{44} \end{bmatrix},\begin{bmatrix} A_{11}&{}\quad A_{12}&{}\quad A_{13}&{}\quad A_{14}\\ 0&{}\quad A_{22}&{}\quad 0&{}\quad A_{24}\\ 0&{}\quad 0&{}\quad A_{33}&{}\quad A_{34}\\ 0&{}\quad 0&{}\quad 0&{}\quad A_{44} \end{bmatrix},\begin{bmatrix} B_1\\ B_2\\ 0\\ 0 \end{bmatrix},[0~C_2~0~C_4]\right] . \nonumber \\ \end{aligned}$$
(26)

Furthermore, the subsystems

$$\begin{aligned} \left[ \begin{bmatrix} E_{11}&{}E_{21}\\ 0&{}E_{22} \end{bmatrix},\begin{bmatrix} A_{11}&{}A_{21}\\ 0&{}A_{22} \end{bmatrix},\begin{bmatrix} B_{1}\\ B_{2} \end{bmatrix}\right] ,\quad \left[ \begin{bmatrix} E_{22}&{}\quad E_{24}\\ 0&{}\quad E_{44} \end{bmatrix},\begin{bmatrix} A_{22}&{}\quad A_{24}\\ 0&{}\quad A_{44} \end{bmatrix},\begin{bmatrix} C_{2}&C_{4} \end{bmatrix}\right] \end{aligned}$$

are completely controllable and observable, respectively.

Numerically, it is beneficial to use only unitary or orthogonal transformations S and T to obtain the Kalman-like form given in Proposition 15 as discussed in [43], see also [4, Section 7].

In the following we use the Kalman-like decomposition from Proposition 15 to obtain a completely controllable and observable realization of passive descriptor systems which has index at most two. Note that in [44] it was shown that the index of matrix pairs associated to port-Hamiltonian systems is at most two.

Corollary 16

Let \(\Sigma =(E,A,B,C,D)\) define a passive descriptor system of the form (1) with Kalman-like decomposition (26). Then the transfer function \({\mathcal {T}}(s)=C(sE-A)^{-1}B+D\) is positive real, \((E_{22},A_{22},B_2,C_2,D)\) is completely controllable and observable realization of \({\mathcal {T}}\) and the index of \((E_{22},A_{22})\) is at most two.

Proof

The matrices S and T used in (26) are invertible and therefore the following holds

$$\begin{aligned} {\mathcal {T}}(s)=C(sE-A)^{-1}B+D&=CT(sSET-SAT)^{-1}SB+D\\&=C_2(sE_{22}-A_{22})^{-1}B_2+D. \end{aligned}$$

Hence, \((E_{22},A_{22},B_2,C_2,D)\) is a realization of the transfer function \({\mathcal {T}}\). The passivity of the system together with Corollary 7 implies that \({\mathcal {T}}\) is positive real. Hence \(C_2(sE_{22}-A_{22})^{-1}B_2+D\) is positive real. Furthermore, Proposition 15 and the upper block-triangular structure implies that \((E_{22},A_{22},B_2,C_2,D)\) is completely controllable and observable. As a consequence, if \(n_2\) denotes the number of rows and columns of \(E_{22}\), then

$$\begin{aligned} {{\,\textrm{rk}\,}}[E_{22},B_2]={{\,\textrm{rk}\,}}[E_{22}^H,C_2^H]=n_2. \end{aligned}$$
(27)

It remains to show that the index of \((E_{22},A_{22})\) is at most two. By modifying the matrices S and T in (26) we can assume without restriction, that \((E_{22},A_{22})\) is already given in block diagonal form (25). We can further assume that \(E_{22}\) is a Jordan block at 0 of size \(n_2\) and that \(A_{22}\) is the identity. Using the nilpotency of \(E_{22}\) it follows that \((sE_{22}-A_{22})^{-1}=-\sum _{i=0}^{n_2-1}(sE_{22})^i\) holds. The rank condition (27) implies \(C_2E_{22}^{n_2-1}B_2\ne 0\). It is shown in [20, Section 5.1], see also Lemma 19, that the positive real function can be written as \({\mathcal {T}}(s)=C_2(sE_{22}-A_{22})^{-1}B_2+D=sM_1+{\mathcal {T}}_p(s)\) where \(\lim \limits _{s\rightarrow \infty }{\mathcal {T}}_p(s)\) exists. Therefore \(n_2\le 2\) which implies that the index of \((E_{22},A_{22})\) is at most two. \(\square \)

4.1 From (KYP|\(\mathcal {V}_{\textrm{sys}}\)) to (KYP)

It was shown in Corollary 7 that passive systems have a solution \(Q\in \mathbb {K}^{n\times n}\) to (KYP|\(\mathcal {V}_{\textrm{sys}}\)). In this section, we will rewrite this restricted KYP inequality and derive an equivalent KYP inequality that holds on the whole space. The idea is to choose a basis of \(\mathcal {V}_{\textrm{sys}}\) and redefine the system matrices in such a way that the system space of the new system will be the whole space.

Let \(M_{\mathcal {V}_{\textrm{sys}}}\in \mathbb {K}^{n\times \dim \mathcal {V}_{\textrm{sys}}}\) and \(M_{\hat{\mathcal {V}}_{\textrm{sys}}}\in \mathbb {K}^{n\times \dim \mathcal {V}_{\textrm{sys}}}\), \(\hat{\mathcal {V}}_{\textrm{sys}}=\left[ {\begin{matrix} A&{}B\\ {}-C&{}-D \end{matrix}}\right] \mathcal {V}_{\textrm{sys}}\) be matrices whose columns are a basis of \(\mathcal {V}_{\textrm{sys}}\) and a span of \(\hat{\mathcal {V}}_{\textrm{sys}}\), respectively. Using the Moore-Penrose pseudo-inverse \(M^\dagger \) of \(M\in \mathbb {K}^{k\times l}\), see, e.g. [45, Section 5.5.2], we introduce the system

$$\begin{aligned} \begin{bmatrix} A_{sys}&{}\quad B_{sys}\\ -C_{sys}&{}\quad -D_{sys} \end{bmatrix}:=M_{\hat{\mathcal {V}}_{\textrm{sys}}}^\dagger \begin{bmatrix} A&{}B\\ {}-C&{}\quad -D \end{bmatrix}M_{\mathcal {V}_{\textrm{sys}}}. \end{aligned}$$

If we consider the KYP inequality together with the projector formulas based on the pseudo-inverse

$$\begin{aligned} P_{\hat{\mathcal {V}}_{\textrm{sys}}}=M_{\hat{\mathcal {V}}_{\textrm{sys}}}M_{\hat{\mathcal {V}}_{\textrm{sys}}}^\dagger , \end{aligned}$$

then for all \((x,u) \in \mathcal {V}_{\textrm{sys}}\) with \(w \in \mathbb {K}^{\dim \mathcal {V}_{\textrm{sys}}}\) such that \((x,u)= M_{\mathcal {V}_{\textrm{sys}}} w\) we obtain

$$\begin{aligned} 0&\ge {{\,\textrm{Re}\,}}\begin{bmatrix} x\\ u \end{bmatrix}^H\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}\\&={{\,\textrm{Re}\,}}(M_{\mathcal {V}_{\textrm{sys}}} w)^H\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}M_{\hat{\mathcal {V}}_{\textrm{sys}}}M_{\hat{\mathcal {V}}_{\textrm{sys}}}^\dagger \begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}M_{\mathcal {V}_{\textrm{sys}}} w, \end{aligned}$$

and using \([A,B]\mathcal {V}_{\textrm{sys}}\subseteq [E,0]\mathcal {V}_{\textrm{sys}}\) yields

$$\begin{aligned} x^HQ^HEx&=\begin{bmatrix} x\\ u \end{bmatrix}^H\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}I_m \end{bmatrix}\begin{bmatrix}E&{}\quad 0\\ 0&{}\quad 0\end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}\\ {}&=(M_{\mathcal {V}_{\textrm{sys}}} w)^H\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}\begin{bmatrix}E&{}\quad 0\\ 0&{}\quad 0\end{bmatrix}M_{\mathcal {V}_{\textrm{sys}}} w\\&=(M_{\mathcal {V}_{\textrm{sys}}} w)^H\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}M_{\hat{\mathcal {V}}_{\textrm{sys}}}M_{\hat{\mathcal {V}}_{\textrm{sys}}}^\dagger \begin{bmatrix}E&{}\quad 0\\ 0&{}\quad 0\end{bmatrix}M_{\mathcal {V}_{\textrm{sys}}} w. \end{aligned}$$

In a more simple way, this can be rewritten with

$$\begin{aligned} {{\hat{Q}}}^H:=M_{\mathcal {V}_{\textrm{sys}}}^H\begin{bmatrix} Q^H&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}M_{\hat{\mathcal {V}}_{\textrm{sys}}},\quad {{\hat{E}}}:=M_{\hat{\mathcal {V}}_{\textrm{sys}}}^\dagger \begin{bmatrix}E&{}\quad 0\\ 0&{}\quad 0\end{bmatrix}M_{\mathcal {V}_{\textrm{sys}}} \end{aligned}$$

as

$$\begin{aligned} {{\,\textrm{Re}\,}}w^H {{\hat{Q}}}^H\begin{bmatrix} A_{sys}&{}\quad B_{sys}\\ {}-C_{sys}&{}\quad -D_{sys}\end{bmatrix} w\le 0,\quad w\in \mathbb {K}^{\dim \mathcal {V}_{\textrm{sys}}},\quad {{\hat{Q}}}^H{{\hat{E}}}\ge 0. \end{aligned}$$
(28)

This is in fact (KYP) of a standard system which is obtained by using \({{\,\textrm{ran}\,}}E\) as state space and treating the remaining variables in \(\mathbb {K}^{\dim \mathcal {V}_{\textrm{sys}}}\) as input variables. However, the matrix \({{\hat{Q}}}\) which replaces \(\left[ {\begin{matrix} Q&{}0\\ 0&{}I_m \end{matrix}}\right] \) in (KYP) of the standard system must in general not admit this diagonal structure. This seems reasonable because the controls and states are not decoupled for general descriptor systems. For pH descriptor systems the interplay between state and control variables can be seen from a staircase form which was derived in [46] and can be achieved using only unitary transformations.

Analogously to Proposition 11, a pH formulation of the system \(\Sigma _{sys}\) is given if

$$\begin{aligned} \ker {{\hat{Q}}}\subseteq \ker \begin{bmatrix} A_{sys}&{}\quad B_{sys}\\ -C_{sys}&{}\quad -D_{sys} \end{bmatrix} \end{aligned}$$

holds. With this assumption we can define a matrix \({{\hat{\Theta }}}\in \mathbb {K}^{\dim \mathcal {V}_{\textrm{sys}}\times \dim \mathcal {V}_{\textrm{sys}}}\) as in Proposition 11 by setting

$$\begin{aligned} \begin{bmatrix} A_{sys}&{}\quad B_{sys}\\ -C_{sys}&{}\quad -D_{sys} \end{bmatrix}={{\hat{\Theta }}}{{\hat{Q}}}. \end{aligned}$$

Motivated by the modified KYP inequality (28), we enlarge the class of solutions of (KYP) and show how in this case a pH formulation can be obtained. In the following we will restrict to standard systems \(\Sigma =(I_n,A,B,C,D)\) and study solutions \({{\hat{Q}}}\in \mathbb {K}^{(n+m)\times (n+m)}\)

$$\begin{aligned} {{\,\textrm{Re}\,}}\begin{bmatrix}x\\ u \end{bmatrix}^H{{\hat{Q}}}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D\end{bmatrix}\begin{bmatrix}x\\ u\end{bmatrix}\le 0,\quad {{\hat{Q}}}={{\hat{Q}}}^H\ge 0 \quad \text {for all } (x,u)\in \mathbb {K}^n \times \mathbb {K}^m. \end{aligned}$$
(29)

Although every ordinary pH system fulfills (29), the converse is not necessarily true as the following example shows.

Example 17

We consider

$$\begin{aligned} \dot{x}=x+3u,\quad y=3x+5u. \end{aligned}$$

This system does not fulfill (pH) since it is unstable. However the matrix \(\left[ {\begin{matrix} A&{}\quad B\\ {}-C&{}\quad -D \end{matrix}}\right] =\left[ {\begin{matrix} 1&{}\quad 3\\ {}-3&{}\quad -5 \end{matrix}}\right] \) used in (29) has the double eigenvalue \(\lambda _{1,2}=-2\). Hence the Lyapunov inequality (29) has a positive definite solution \({{\hat{Q}}}\in \mathbb {R}^{2\times 2}\).

In the following, we show different ways of obtaining (pH) if we additionally assume that \({{\hat{Q}}}\) in (29) is positive definite. The first way is to define

$$\begin{aligned} \begin{bmatrix} J-R&{}&{}\quad B-P\\ {}-B^H-P^H&{}&{}\quad -N-S \end{bmatrix}:=\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}{{\hat{Q}}}^{-1}. \end{aligned}$$

Since \({{\hat{Q}}}\) is not block diagonal, the new state and input variables might be given as a combination of the old state and input variables. Another way to obtain a pH formulation where either the new state variable or the new input variable can be chosen as a rescaling of the old variables is based on the following block Cholesky factorization, see, e.g. [45, Section 4.2.9],

$$\begin{aligned} {{\hat{Q}}}={\mathcal {C}}^H{\mathcal {C}}=\begin{bmatrix} C_1&{}\quad C_2\\ 0&{}\quad C_3 \end{bmatrix}^H\begin{bmatrix} C_1&{}\quad C_2\\ 0&{}\quad C_3 \end{bmatrix}. \end{aligned}$$
(30)

This can be used to rewrite (29) as follows

$$\begin{aligned} {{\,\textrm{Re}\,}}\left( {\mathcal {C}}\begin{bmatrix}x\\ u \end{bmatrix}\right) ^H\left( {\mathcal {C}}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D\end{bmatrix}{\mathcal {C}}^{-1}\right) {\mathcal {C}}\begin{bmatrix}x\\ u\end{bmatrix}={{\,\textrm{Re}\,}}\begin{bmatrix}x\\ u \end{bmatrix}^H{\mathcal {C}}^H{\mathcal {C}}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D\end{bmatrix}\begin{bmatrix}x\\ u\end{bmatrix}\le 0. \end{aligned}$$
(31)

Hence we could rewrite the system using the new variables

$$\begin{aligned} \begin{bmatrix} {{\hat{x}}}\\ {{\hat{u}}} \end{bmatrix}={\mathcal {C}}\begin{bmatrix}x\\ u \end{bmatrix}=\begin{bmatrix} C_1x+C_2u\\ C_3u \end{bmatrix}\quad \text {and}\quad \begin{bmatrix} {{\hat{A}}}&{}\quad {{\hat{B}}}\\ {}-{{\hat{C}}}&{}\quad -{{\hat{D}}}\end{bmatrix}:={\mathcal {C}}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D\end{bmatrix}{\mathcal {C}}^{-1}. \end{aligned}$$

We obtain a pH system with Hamiltonian \({{\hat{Q}}}=I_n\), state \({{\hat{x}}}\) and input \({{\hat{u}}}\). If we would choose \({\mathcal {C}}\) to be lower triangular in the Cholesky factorization (30) then the new state \({{\hat{x}}}\) would be a rescaling of the old state x whereas \({{\hat{u}}}\) would be a linear combination of the state and input variables x and u.

Using the particular structure of \({\mathcal {C}}\), we further obtain

$$\begin{aligned}&\begin{bmatrix} {{\hat{A}}}&{}\quad {{\hat{B}}}\\ {}-{{\hat{C}}}&{}-{{\hat{D}}}\end{bmatrix}={\mathcal {C}}\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D\end{bmatrix}{\mathcal {C}}^{-1}\\ {}&\quad =\begin{bmatrix} C_1A-C_2C&{}&{}\quad C_1B-C_2D\\ {}-C_3C&{}&{}\quad -C_3D\end{bmatrix}{\mathcal {C}}^{-1}\\&\quad =\begin{bmatrix} C_1A-C_2C&{}&{}\quad C_1B-C_2D\\ {}-C_3C&{}&{}\quad -C_3D\end{bmatrix}\begin{bmatrix} C_1^{-1}&{}&{}\quad -C_1^{-1}C_2C_3^{-1}\\ 0&{}&{}\quad C_3^{-1}\end{bmatrix}\\&\quad =\begin{bmatrix} C_1AC_1^{-1}-C_2CC_1^{-1}&{}&{}\quad C_1BC_3^{-1}-C_2DC_3^{-1}-C_1AC_1^{-1}C_2C_3^{-1}+C_2CC_1^{-1}C_2C_3^{-1}\\ {}-C_3CC_1^{-1}&{}&{}\quad -C_3DC_3^{-1}+C_3CC_1^{-1}C_2C_3^{-1}\end{bmatrix} \end{aligned}$$

Furthermore, the matrix \({{\hat{A}}}\) satisfies \({{\hat{A}}}+{{\hat{A}}}^H\le 0\) and hence it is stable, i.e. it has only eigenvalues with a nonpositive real part and semi-simple eigenvalues on the imaginary axis (if there are any).

4.2 Passive systems with index at most one

As we have seen in Sect. 4, passive systems have always a realization that has index of at most two and that the index two blocks can be realized separately as pH systems. Hence the question remains whether passive systems with index at most one can be realized as (pH).

In the following, we construct a pH realization for descriptor systems \(\Sigma =(E,A,B,C,D)\) which are behaviorally observable and have index at most one.

If \(\Sigma \) has index at most one then, similar to the Kronecker–Weierstraß form (25), there exist invertible \(T_l,T_r\in \mathbb {C}^{n\times n}\) such that

$$\begin{aligned} T_lET_r=\begin{bmatrix} E_1&{}0\\ 0&{}0 \end{bmatrix},\quad T_lAT_r=\begin{bmatrix} A_1&{}\quad 0\\ 0&{}\quad A_2 \end{bmatrix},\quad T_lB=\begin{bmatrix} B_1\\ B_2 \end{bmatrix},\quad CT_r=[C_1,C_2]\nonumber \\ \end{aligned}$$
(32)

with \(n = n_1 + n_2\), \(E_1\in \mathbb {K}^{n_1\times n_1}\) and \(A_2\in \mathbb {K}^{n_2\times n_2}\) invertible. Based on this transformation, we consider the following KYP inequality

$$\begin{aligned} \begin{aligned} \begin{bmatrix} -A_1^HQ_1-Q_1^HA_1&{}\quad C_1^H-Q_1^HB_1 \\ C_1-B_1^HQ_1&{}\quad (D-C_2A_2^{-1}B_2)+(D-C_2A_2^{-1}B_2)^H \end{bmatrix}&\ge 0,\quad x_1\in \mathbb {K}^{n_1}\\ \quad x_1^H E_1^HQ_1x_1&\ge 0. \end{aligned} \end{aligned}$$
(33)

Proposition 18

Let \(\Sigma =(E,A,B,C,D)\) be a descriptor system with index at most one which satisfies (32). Then the modified KYP inequality (33) has a solution \(Q_1\) if and only if \(\Sigma \) is passive. Moreover, if \(\Sigma \) is behaviorally observable then every solution \(Q_1\) of (33) is invertible.

Proof

We apply Corollary 7 where we showed that passivity is equivalent to (KYP|\(\mathcal {V}_{\textrm{sys}}\)). The system space of the block diagonal system (32) is given by

$$\begin{aligned} \{(x_1,-A_2^{-1}B_2u,u) \mid x_1\in \mathbb {K}^{n_1}, u\in \mathbb {K}^m\} \end{aligned}$$

and therefore

$$\begin{aligned} \mathcal {V}_{\textrm{sys}}={{\,\textrm{diag}\,}}(T_r,I_m)\{(x_1,-A_2^{-1}B_2u,u) \mid x_1\in \mathbb {K}^{n_1}, u\in \mathbb {K}^m\}. \end{aligned}$$
(34)

Hence the KYP inequalities restricted to the system space are given by

$$\begin{aligned} {{\,\textrm{Re}\,}}\begin{bmatrix} x\\ u \end{bmatrix}^H\begin{bmatrix} Q&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}^H\begin{bmatrix} A&{}\quad B\\ {}-C&{}\quad -D \end{bmatrix}\begin{bmatrix} x\\ u \end{bmatrix}\le 0,\quad x^HE^HQx\ge 0,\quad \text {for all }\begin{bmatrix} x\\ u \end{bmatrix}\in \mathcal {V}_{\textrm{sys}}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} {{\,\textrm{Re}\,}}\begin{bmatrix} x_1\\ {}-A_2^{-1}B_2u\\ u \end{bmatrix}^H \begin{bmatrix} T_l^{-H}QT_r&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}^H \begin{bmatrix} A_1&{}0\quad &{}\quad B_1\\ 0&{}\quad A_2&{}\quad B_2\\ {}-C_1&{}\quad -C_2&{}\quad -D \end{bmatrix}\begin{bmatrix} x_1\\ {}-A_2^{-1}B_2u\\ u \end{bmatrix}\le 0,\\ \quad \begin{bmatrix} x_1\\ {}-A_2^{-1}B_2u \end{bmatrix}^H\begin{bmatrix} E_1^H&{}\quad 0\\ 0&{}\quad 0 \end{bmatrix}T_l^{-H}QT_r\begin{bmatrix} x_1\\ {}-A_2^{-1}B_2u \end{bmatrix}\ge 0\quad x_1\in \mathbb {K}^{n_1},~u\in \mathbb {K}^m. \end{aligned}$$

Since \(E^HQ=Q^HE\), the following matrix is Hermitian

$$\begin{aligned} \begin{bmatrix} E_1^H&{}\quad 0\\ 0&{}\quad 0 \end{bmatrix}T_l^{-H}QT_r=\begin{bmatrix} E_1^H&{}\quad 0\\ 0&{}\quad 0 \end{bmatrix}\begin{bmatrix} Q_1&{}\quad Q_2\\ Q_3&{}\quad Q_4 \end{bmatrix}=\begin{bmatrix} E_1^HQ_1&{}\quad E_1^H Q_2\\ 0&{}\quad 0 \end{bmatrix}=\begin{bmatrix} E_1^H Q_1&{}\quad 0\\ 0&{}\quad 0 \end{bmatrix}. \end{aligned}$$

The invertibility of \(E_1\) implies \( Q_2=0\), i.e. \(T_l^{-H}QT_r\) is block lower-triangular. Hence the KYP inequalities on the system space are equivalent to

$$\begin{aligned} {{\,\textrm{Re}\,}}\begin{bmatrix} x_1\\ u \end{bmatrix}^H\begin{bmatrix} Q_1&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}^H \begin{bmatrix} A_1&{}\quad B_1\\ {}-C_1&{}\quad C_2A_2^{-1}B_2-D \end{bmatrix}\begin{bmatrix} x_1\\ u \end{bmatrix}&\le 0,\quad x_1\in \mathbb {R}^{n_1},~u\in \mathbb {K}^m\\ x_1^H E_1^HQ_1x_1&\ge 0. \end{aligned}$$

If the system \(\Sigma \) is behaviorally observable, then \((E_1,A_1,C_1)\) is observable and hence, using Proposition 9, we find that \(Q_1\) is invertible. \(\square \)

Proposition 18 can be used to obtain a pH representation of \((E_1,A_1,B_1,C_1,D-C_2A_2^{-1}B_2)\) as follows. If there exists a solution \(Q_1\) to (33) with \(\ker Q_1\subseteq \ker A_1\cap \ker C_1\) then a pH representation can be obtained by considering \(\Theta \) given by

$$\begin{aligned} \begin{bmatrix} A_1&{}&{}\quad B_1\\ {}-C_1&{}&{}\quad C_2A_2^{-1}B_2-D \end{bmatrix}=\Theta \begin{bmatrix} Q_1&{}\quad 0\\ 0&{}\quad I_m \end{bmatrix}, \end{aligned}$$

and if \(Q_1\) is invertible then this simplifies to

$$\begin{aligned} \Theta =\begin{bmatrix} A_1Q_1^{-1}&{}&{}\quad B_1\\ {}-C_1Q_1^{-1}&{}&{}\quad C_2A_2^{-1}B_2-D \end{bmatrix}. \end{aligned}$$

5 When does (PR) imply (pH)?

In this section, we study whether a pH representation can be obtained from a positive real transfer function or not. This is also of particular interest when one wants to obtain a pH representation from frequency measurements of the transfer function. It was shown in [47] that if the interpolation points are chosen to be spectral zeros then one ends up with a pH realization. However, these spectral zeroes cannot be known in advance if only transfer function measurements are available. One solution proposed in [47] is to construct an intermediate realization from which the spectral zeroes can be computed. The question that arises then is what are the conditions on this intermediate system to end up in pH representation even for an index two pH descriptor system.

Given a transfer function \({\mathcal {T}}\) of a descriptor system (1) then \({\mathcal {T}}\) has a pole of finite order at \(\infty \) and if \(\nu \) is the index of the pair (E, A) then this order is at most \(\nu -1\). Hence, using the Laurent expansion of the entries of \({\mathcal {T}}\) there exists a sequence of matrices \((M_i)_{i=k}^{-\infty }\) with \(M_i\in \mathbb {C}^{n\times n}\) such that

$$\begin{aligned} {\mathcal {T}}(s)=\sum _{i=k-1}^{-\infty }M_is^i. \end{aligned}$$
(35)

If \({\mathcal {T}}(s)\) is assumed to be a real rational function, then \(M_i\in \mathbb {R}^{n\times n}\). In the following lemma, we show that for positive real rational functions, this representation can be simplified, see also [20, Section 5.1].

Lemma 19

Let \(\Sigma =(E,A,B,C,D)\) be a descriptor system with positive real transfer function \({\mathcal {T}}\) with Laurent expansion (35) then

$$\begin{aligned} {\mathcal {T}}(s)=M_{1}s+{\mathcal {T}}_p(s), \end{aligned}$$
(36)

holds for some rational function \({\mathcal {T}}_p(s)\) which fulfills \(M_0=\lim \nolimits _{s\rightarrow \infty }{\mathcal {T}}_p(s)\). Furthermore, it holds that \(M_0+M_0^H\ge 0\) and \(M_{1}=M_1^H\ge 0\). Moreover, if the system matrices E, A, B, C, D are real, then \(M_0\) and \(M_1\) are real and \({\mathcal {T}}_p(s)\) is positive real.

Proof

Since \({\mathcal {T}}\) is a rational function, it has no poles for all \(s=i\omega \) with \(|\omega |\) sufficiently large. Furthermore, the positive realness and analyticity imply that \({\mathcal {T}}(i\omega )+{\mathcal {T}}(i\omega )^H\ge 0\) holds for these values.

Using the positive realness of \({\mathcal {T}}\) and (35) for some \(k\ge 1\), we conclude for \(s=e^{i\varphi }r\) for all \(r>0\) and \(\varphi >0\) satisfying \(\tfrac{\varphi }{k}\le \tfrac{\pi }{2}\) that the following holds

$$\begin{aligned} {\mathcal {T}}(s)+{\mathcal {T}}(s)^H=M_{k}s^{k}+M_{k}^H{\overline{s}}^{k}+ \sum _{i=k-1}^{-\infty } M_is^i+M_i^H{\overline{s}}^i\ge 0. \end{aligned}$$
(37)

We consider first the case \(k\ge 2\). Considering (37) for \(r>0\) sufficiently large and \(\varphi =\tfrac{\pi }{k}\) implies

$$\begin{aligned} M_{k}+M_{k}^H\le 0. \end{aligned}$$

Furthermore, choosing \(\varphi =0\) and \(r>0\) sufficiently large in (37) yields \(M_{k}+M_{k}^H\ge 0\). Therefore \(M_{k}=-M_{k}^H\) holds. If we consider (37) with \(\varphi =\tfrac{\pi }{2k}\) and \(r>0\) sufficiently large leads to \(2iM_{k}\le 0\) and choosing \(\varphi =-\tfrac{\pi }{2k}\) yields \(-2iM_{k}\le 0\). Hence we conclude \(M_k=0\) and by repeating this argument (if necessary) we obtain \(M_k=\ldots =M_2=0\).

Therefore (35) holds with \(k=1\). Then it remains to prove that \(M_1=M_1^H\ge 0\) is satisfied. We decompose \(M_1=M_1^++M_1^-\) where \(M_1^{\pm }=\frac{1}{2}(M_1\pm M_1^H)\). Then \(M_1^+\) is Hermitian and \(M_1^-\) is skew-Hermitian and we have \(sM_1+(sM_1)^H=s(M_1^++M_1^-)+{\overline{s}}(M_1^++M_1^-)^H=2({{\,\textrm{Re}\,}}s)M_1^++2({\textrm{Im}}\,s)iM_1^-\). Hence, if we consider \(s=i\omega \) and let \(\omega \rightarrow \infty \) then this contradicts \({\mathcal {T}}(s)+{\mathcal {T}}(s)^H\ge 0\). As a consequence, \(M_1^-=0\). Hence \(M_1=M_1^H\). If \(M_1\) would have a negative eigenvalue with eigenvector \(x\in \mathbb {C}^n\) we obtain a contradiction by considering \(x^H({\mathcal {T}}(s)+{\mathcal {T}}(s)^H)x\) for \(s\rightarrow \infty \). This shows that \(M_1=M_1^\top \ge 0\). Taking the limit \(\omega \rightarrow \infty \) in the positive realness condition

$$\begin{aligned} {\mathcal {T}}(i\omega )+{\mathcal {T}}(i\omega )^H\ge 0 \end{aligned}$$

we further deduce \(M_0+M_0^\top \ge 0\).

If the system matrices are real then clearly \(M_0\) and \(M_1\) are real. Hence, it remains to conclude that \({\mathcal {T}}_p(s)\) is positive real. Since \({\mathcal {T}}(s)\) is real and positive real and \({\mathcal {T}}_p(i\omega )+{\mathcal {T}}_p(-i\omega )^\top ={\mathcal {T}}(i\omega )+{\mathcal {T}}(-i\omega )^\top \ge 0\) holds for all \(\omega \) such that \(i\omega \) is not a pole of \({\mathcal {T}}\), it follows from [20, Theorem 2.7.2] that \({\mathcal {T}}_p(s)\) is positive real. \(\square \)

Note that rational functions \({\mathcal {T}}\) for which \(\lim _{s\rightarrow \infty }{\mathcal {T}}(s)\) exists, are called proper. Hence we will refer to \({\mathcal {T}}_p\) in (36) as the proper part of a positive real transfer function.

The following example shows that positive realness of arbitrary real rational functions with polynomial growth cannot be concluded from considering the behavior on the imaginary axis alone.

Example 20

The function \({\mathcal {T}}(s)=s^3\) is analytic and hence it has no poles on the imaginary axis. Furthermore, it fulfills \({\mathcal {T}}(i\omega )+{\mathcal {T}}(-i\omega )^\top =0\). However, using the same arguments as in the proof of Lemma 19 we find that it is not positive real. Hence Theorem 2.7.2 in [20] cannot be extended to real rational functions that have a polynomial growth of order larger than two.

Furthermore, a nonnegative real part of the transfer function on the imaginary axis does not guarantee positive realness.

Example 21

Consider \({\mathcal {T}}(s)=-s^{-1}\), which satisfies \({\mathcal {T}}(s)+{\mathcal {T}}(s)^H=\tfrac{-2 {{\,\textrm{Re}\,}}s}{|s|^2}\le 0\) for all \(s\in \mathbb {C}\) with \({{\,\textrm{Re}\,}}s\ge 0\) and therefore does not fulfill (PR). Furthermore, a state-space realization is given by \((E,A,B,C,D)=(1,0,B,-B^{-1},0)\) for all scalar \(B\ne 0\). Hence the only solution to (KYP)

$$\begin{aligned} \begin{bmatrix} -A^HQ-Q^HA&{}&{}\quad C^H-Q^HB\\ C-B^HQ&{}&{}\quad D+D^H \end{bmatrix}=\begin{bmatrix} 0&{}\quad -B^{-H}-Q^HB\\ {}-B^{-1}-B^HQ&{}\quad 0 \end{bmatrix}\ge 0 \end{aligned}$$

is given by \(Q=-\frac{1}{B^2}\). Observe that \({\mathcal {T}}\) satisfies \({\mathcal {T}}(i\omega )+{\mathcal {T}}(i\omega )^H\ge 0\) for all \(\omega \ne 0\), but \({\mathcal {T}}\) has a negative residue it the simple pole at \(\omega _0=0\), which is the reason why it is not positive real. Another consequence of this example is the existence of Hermitian solutions to (KYP) does not imply that the transfer function of the system is positive real.

Next, we study the relation of (PR) to (KYP) by recalling the following result presented in [17].

Proposition 22

If (E, A, B, C, D) is a real-valued minimal realization of a positive real transfer function \({\mathcal {T}}(s)\) with \(D+D^\top \ge M_0+M_0^\top \) then (KYP) holds.

First, observe that one can choose a minimal realization with \(D=M_0\). Hence for every positive real transfer function, there exists a realization that has a solution to (KYP). However, as we have seen in Sect. 4 this solution has to fulfill additional requirements if we want to define a pH realization. Furthermore, note that for minimal realizations of positive real transfer functions with index two it was shown in [17, Theorem 4.1] that (KYP\(_{E}\)|\(\mathbb {K}^n\times \mathbb {K}^m\)) is never fulfilled. In particular, we cannot obtain (pH) from the solution of this KYP inequality as the following example shows.

Example 23

Consider \(E=\begin{bmatrix} 1&{}0\\ 0&{}0 \end{bmatrix}\), \(A=\begin{bmatrix} 0&{}\quad -1\\ 1&{}\quad 0 \end{bmatrix}\), \(B=\begin{bmatrix} 0\\ 1 \end{bmatrix}=C^\top \), \(D=0\). Then this is a minimal realization of the positive real transfer function \({\mathcal {T}}(s)=s\). Here the (KYP) has the solution \(Q=I_2\). However, the inequalities in (KYP\(_{E}\)|\(\mathbb {K}^n\times \mathbb {K}^m\)) have no solution.

As a second example consider the descriptor system given by \((E,A,B,C,D)=(0,-1,1,-1,-1)\). Then \({\mathcal {T}}(s)=0\) which is positive real. However, (KYP) has no solution since D is negative. If we would replace D with \(M_1=0\) then (KYP) becomes solvable.

As an alternative, we can define a minimal pH realization directly from the transfer function (36). To this end, we consider a minimal realization of the proper part \({\mathcal {T}}_p(s)\) which is given by

$$\begin{aligned} {\mathcal {T}}_p(s)=C_p(sE_p-A_p)^{-1}B_p+M_0 \end{aligned}$$

where \(E_p\) is invertible which follows from the fact that minimal realizations of descriptor systems have no index one blocks in the Kronecker–Weierstraß form (25), see, e.g. [17, Theorem 6.3]. Then the minimality conditions (8) and (9) trivially hold which means that \((E_p,A_p,B_p)\) is behaviorally controllable and that \((E_p,A_p,C_p)\) is behaviorally observable. Hence, we know from Proposition 9 that there exists invertible \(Q_p\) such that \(Q_p^HE_p\ge 0\) and

$$\begin{aligned} \begin{bmatrix} J-R &{}&{}\quad G-P\\ (G-P)^H&{}&{}\quad D \end{bmatrix}:=\begin{bmatrix} A_pQ_p^{-1}&{}&{}\quad B_p\\ C_pQ_p^{-1}&{}&{}\quad D_p \end{bmatrix}. \end{aligned}$$

Furthermore, a minimal pH realization of \(sM_{1}\) is given by

$$\begin{aligned} \begin{aligned} E_{\infty }&=\begin{bmatrix}M_{1}&{}\quad 0\\ 0&{}\quad 0\end{bmatrix},~~ A_{\infty }=\begin{bmatrix}0&{}\quad -I_m\\ I_m&{}\quad 0\end{bmatrix}\in \mathbb {R}^{2m\times 2m},~~ C_{\infty }=[0,I_m]=B_{\infty }^\top \in \mathbb {R}^{m\times 2m},\\~~ D_{\infty }&=0\in \mathbb {R}^{m\times m}. \end{aligned} \end{aligned}$$
(38)

Indeed,

$$\begin{aligned} C_{\infty }(sE_{\infty }-A_{\infty })^{-1}B_{\infty }=C_{\infty }\begin{bmatrix}0&{}&{}I_m\\ -I_m&{}&{}sM_{1}\end{bmatrix}B_{\infty }=sM_{1} \end{aligned}$$

and the minimality conditions (8) and (9) can be verified easily. Furthermore, the system (38) is pH with \(Q_{\infty }=I_{2m}\).

In the following lemma, we show that we can combine the two pH systems from the proper and the non-proper part to obtain a minimal pH system realization of a positive real transfer function.

Lemma 24

Let \(\Sigma _i=(E_i,A_i,B_i,C_i,D_i)\), \(i=1,2\), be descriptor systems which fulfill (pH) for some \(Q=Q_1\) and \(Q=Q_2\) respectively. Then the system \(\Sigma _+\) given by

$$\begin{aligned} E_+&:=\begin{bmatrix} E_1&{}0\\ 0&{}E_2 \end{bmatrix},\quad A_+:=\begin{bmatrix} A_1&{}\quad 0\\ 0&{}\quad A_2 \end{bmatrix},\quad B_+:=\begin{bmatrix} B_1\\ B_2 \end{bmatrix},\quad C_+:=\begin{bmatrix} C_1&\quad C_2 \end{bmatrix},\\ D_+&:=D_1+D_2 \end{aligned}$$

has the transfer function \({\mathcal {T}}_+={\mathcal {T}}_1+{\mathcal {T}}_2\) and fulfills (pH) with \(Q={{\,\textrm{diag}\,}}(Q_1,Q_2)\).

Proof

If systems \(\Sigma _i\), \(i=1,2\), fulfill (pH) then there exists \(J_i,R_i,Q_i\in \mathbb {K}^{n_i\times n_i}\), \(G_i,P_i\in \mathbb {K}^{n_i\times m_i}\), and \(S_i,N_i\in \mathbb {K}^{m_i\times m_i}\) such that

$$\begin{aligned} \begin{bmatrix} A_i&{}\quad B_i\\ C_i&{}\quad D_i \end{bmatrix}&=\begin{bmatrix}(J_i-R_i)Q_i&{}\quad G_i-P_i\\ (G_i+P_i)^HQ_i&{}\quad S_i+N_i\end{bmatrix},\quad Q_i^HE_i=E_i^HQ_i\ge 0,\\ \Gamma _i&:= \begin{bmatrix} J_i &{}\quad G_i \\ -G_i^H&{}\quad N_i\end{bmatrix} = - \Gamma _i^H,\quad W_i := \begin{bmatrix} R_i &{}\quad P_i\\ P_i^H &{}\quad S_i \end{bmatrix} =W_i^H \ge 0. \end{aligned}$$

By setting \(J_+ = {{\,\textrm{diag}\,}}(J_1,J_2)\), \(R_+ ={{\,\textrm{diag}\,}}(R_1,R_2)\), \(Q_+={{\,\textrm{diag}\,}}(Q_1,Q_2)\), \(G_+={{\,\textrm{diag}\,}}(G_1,G_2)\), \(P_+={{\,\textrm{diag}\,}}(P_1,P_2)\), \(S_+={{\,\textrm{diag}\,}}(S_1,S_2)\) and \(N_+={{\,\textrm{diag}\,}}(N_1,N_2)\) it holds that

$$\begin{aligned} \begin{bmatrix} A_+&{}\quad B_+\\ C_+&{}\quad D_+ \end{bmatrix}&=\begin{bmatrix}(J_+-R_+)Q_+&{}\quad G_+-P_+\\ (G_++P_+)^HQ_+&{}\quad S_++N_+\end{bmatrix},\quad Q_+^HE_+=E_+^HQ_+\ge 0,\\ \Gamma _+&:= \begin{bmatrix} J_+ &{}\quad G_+ \\ -G_+^H&{}\quad N_+\end{bmatrix} = - \Gamma _+^H,\quad W_+ := \begin{bmatrix} R_+ &{}\quad P_+\\ P_+^H &{}\quad S_+ \end{bmatrix} =W_+^H \ge 0. \end{aligned}$$

Hence \(\Sigma _+\) fulfills (pH). \(\square \)

In summary, this means that for behaviorally controllable and observable descriptor systems with positive real transfer functions, we can obtain a pH realization via

$$\begin{aligned} E_{pH}&:=\begin{bmatrix} E_p&{}0&{}0\\ 0&{}M_{1}&{}0\\ 0&{}0&{}0 \end{bmatrix},&A_{pH}&:=\begin{bmatrix} A_p&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad -I_m\\ 0&{}\quad I_m&{}\quad 0 \end{bmatrix},&B_{pH}&:=\begin{bmatrix} B_p\\ 0\\ I_m\end{bmatrix},\\ C_{pH}&:=\begin{bmatrix} C_p&\quad 0&\quad I_m\end{bmatrix},&D_{pH}&:=M_0, \quad Q:=\begin{bmatrix} Q_p&{}\quad 0\\ 0&{}\quad I_{2m} \end{bmatrix}. \end{aligned}$$

This is then already a pH descriptor system in the so-called staircase form which was studied recently in [46].

Fig. 3
figure 3

Overview of the relationship between (pH), (KYP), (Pa) and (PR) for descriptor systems. The implications with additional assumptions are colored black and the one without are colored in blue. (a) \(\ker Q \subseteq \ker C\cap \ker A\) (color figure online)

Full size image

The connections of (pH), (KYP), (Pa) and (PR) for descriptor systems are summarized in Fig. 3.

6 Conclusion

In this paper, we studied conditions for the equivalence between passive, positive real and port-Hamiltonian descriptor systems, as well as their relation to the solvability of generalized KYP inequalities. The conditions on the equivalence already available in the literature were either validated or relaxed and counterexamples were also presented in the cases where the equivalence does not hold. In addition, we considered special cases: index one descriptor systems and KYP inequalities on a subspace which are shown to be equivalent to the finiteness of the available storage. Finally, we focused on conditions to obtain a port-Hamiltonian system from either a passive system, a positive real transfer function or the solution of the KYP inequality.

As future work, the analysis conducted in this note can be extended in several directions. Namely, one can study more general passivity properties such as cyclo passivity [48] which allows for possibly negative storage functions. In addition, we only focused on continuous-time systems. A similar study could be conducted for discrete-time systems. This could also help in finding a definition analogous to (pH) for this class of systems. Finally, pH systems are dissipative in the sense of [49] with respect to a specific supply rate \(w(x,u)={{\,\textrm{Re}\,}}u^H(Cx+Du)\). The analysis conducted in this note could also be extended to systems that are dissipative with respect to other quadratic supply rates such as the scattering supply rate which is given by \(w(x,u)=\Vert u\Vert ^2-\Vert Cx+Du\Vert ^2\) and closely related to the transfer functions being bounded real.