Port-Hamiltonian systems and monotonicity

Port-based modeling of physical systems leads to their description as port-Hamiltonian systems. Such models have turned out to be powerful for purposes of analysis, simulation and control, see e.g. [13, 1, 2]. On the other hand, during the last decades the concept of monotonicity has attracted much attention from multiple points of view. In relation with the current paper the following two aspects of monotonicity are most relevant. First, monotonicity has been a key notion in the study of nonlinear electrical circuits and general nonlinear network dynamics; see e.g. the recent paper [3] for a historical context and references. From a systems and control point of view this view on monotonicity is strongly related to notions of incremental passivity [4] and contraction [5]. Second, monotonicity has evolved as a key concept in convex optimization (see e.g. [6] and the references therein), as well as in nonlinear analysis (see e.g. [7, 8]).

The present paper takes a closer look at the connections between port-Hamiltonian systems and monotonicity, and explores overarching notions. Already in our paper [9], inspired by [10], we defined a new class of dynamical systems, coined as incrementally port-Hamiltonian systems. This was done by replacing the composition of the Dirac structure and the energy-dissipating relation in the standard definition of port-Hamiltonian systems by a general (maximal) monotone relation. Furthermore, in [9] it was shown how monotone relations share the same compositionality property as Dirac structures, and sufficient conditions for the composition of two maximally monotone relations to be again maximally monotone were given. Moreover, the connections between incrementally port-Hamiltonian systems and the notions of incremental and differential passivity were briefly discussed. In the current paper this line of research is continued by developing a full-fledged theory of composition of maximally monotone relations. Furthermore, we take an in-depth look at (maximal) cyclically monotone relations in the context of port-Hamiltonian systems modeling. In particular, we show how, under mild technical conditions, the composition of two maximal cyclically monotone relations is again a maximal cyclically monotone relation. Maximal cyclically monotone relations are of special interest because they correspond to extended convex functions, in the sense that any maximal cyclically monotone relation is given as the subdifferential of a convex function, called the generating function of the relation. We show how the composition of two maximal cyclically monotone relations can be directly computed via their generating functions. Also, we define an appealing class of incrementally port-Hamiltonian systems which are defined by convex functions of the state and input. Obviously, this connects the theory of incrementally port-Hamiltonian systems to convex optimization. In fact, simple examples are continuous-time gradient algorithms for convex functions, and primal-dual gradient algorithms in case of minimization under affine constraints. Furthermore, we show how the equilibrium of interconnections of incrementally port-Hamiltonian systems defined by maximal cyclically monotone relations can be computed by convex optimization; thereby extending the innovative work of [11]. Finally, another connection with convex analysis appears if we assume the Hamiltonian function of the incrementally port-Hamiltonian system to be convex. This leads to shifted passivity [1] of steady states, and in particular to (maximal) equilibrium independent passivity [12, 11].

The organization of the paper is as follows. In Section 2, we quickly review the concepts of Dirac structures and standard port-Hamiltonian systems. This is followed by the definition of incrementally port-Hamiltonian systems in Section 3, and the discussion of a number of examples. In Section 4, we prove that under mild technical conditions the composition of two (maximal) (cyclically) monotone relations is (maximal) cyclically monotone, and thus the power-conserving interconnection of incrementally port-Hamiltonian systems is again an incrementally port-Hamiltonian system. In Section 5 we investigate the various relationships and differences between incrementally port-Hamiltonian and standard port-Hamiltonian systems. As a direct application of the established compositionality theory we study in Section 6 the structure of the set of steady states of incrementally port-Hamiltonian systems, and the computation of the equilibrium of interconnected maximal cyclically monotone port-Hamiltonian systems via convex optimization. Furthermore, assuming convexity of the Hamiltonian, the relations with (maximal) equilibrium independent passivity are investigated. Finally in Section  7 the relations of incrementally port-Hamiltonian systems with incremental passivity and differential passivity are discussed, continuing upon the explorations in [9]. The conclusions are in Section 8.

In order to motivate the definition of incrementally port-Hamiltonian systems we first review the definition of ‘ordinary’ port-Hamiltonian systems; cf. [2, 13, 1] for more details and ramifications.

Underlying the definition of a port-Hamiltonian system is the geometric notion of a Dirac structure, which relates the power variables of the constituting elements of the system in a power-conserving manner. Since incrementally port-Hamiltonian systems will be defined on linear state spaces we restrict as well attention to port-Hamiltonian systems on linear state spaces, and correspondingly to constant Dirac structures on linear spaces¹¹1For the extension to port-Hamiltonian systems on manifolds, and the corresponding notions of Dirac on manifolds we refer to e.g. [14, 1, 15]..

Power variables (such as voltages and currents, and forces and velocities), appear in conjugated pairs, whose products have dimension of power. They take values in dual linear spaces, with product meaning duality product. In particular, let ${\mathcal{F}}$ be a finite-dimensional linear space and $\mathcal{E}:=\mathcal{F}^{*}$ be its dual space. We call $\mathcal{F}$ the space of flow variables, and $\mathcal{E}$ the space of effort variables. The duality product for the pair $({\mathcal{E}},{\mathcal{F}})$, denoted by $\langle\cdot\mid\cdot\rangle$, is given as

for $e\in{\mathcal{E}}$ and $f\in{\mathcal{F}}$, and is the power associated to the pair $(f,e)$. Furthermore on $\mathcal{F}\times\mathcal{E}$ an indefinite bilinear form is defined as

where $(f_{i},e_{i})\in\mathcal{F}\times\mathcal{E}$ with $i\in\{1,2\}$. For any subspace $\mathcal{S}\subset\mathcal{F}\times\mathcal{E}$, we denote its orthogonal companion with respect to this indefinite bilinear form by $\mathcal{S}^{\perp\!\!\!\perp}$.

Throughout the paper, we will work with various spaces of flow/effort variables. By convention, if $\mathcal{F}_{\bullet}$ denotes a certain space of flow variables then $\mathcal{E}_{\bullet}:=\mathcal{F}_{\bullet}^{*}$ will denote the corresponding space of effort variables.

From now on in this paper a Dirac structure will simply refer to a constant Dirac structure on a linear space.

The definition of a port-Hamiltonian system on a linear space contains the following ingredients (see e.g. [16, 1, 14, 2]). First a Dirac structure $\mathcal{D}$ defined on the space of all flow variables, that is,

Here $(f_{x},e_{x})\in\mathcal{F}_{x}\times\mathcal{E}_{x}$ are the flow and effort variables linking to the energy-storing elements, $(f_{R},e_{R})\in\mathcal{F}_{R}\times\mathcal{E}_{R}$ are the flow and effort variables linking to energy-dissipating elements, and finally $(f_{P},e_{P})\in\mathcal{F}_{P}\times\mathcal{E}_{P}$ are the flow and effort port variables. The port-Hamiltonian system is defined by specifying, next to its Dirac structure $\mathcal{D}$, the constitutive relations of the energy-dissipating elements, and of the energy-storing elements. An energy-dissipating relation is any subset $\mathcal{R}\subset\mathcal{F}_{R}\times\mathcal{E}_{R}$ with the property

The constitutive relations of the energy-storing elements are specified by a Hamiltonian $H:\mathcal{X}\to\mathbb{R}$, where $\mathcal{X}=\mathcal{F}_{x}$. Thus the total energy while at state $x$ is given as $H(x)$. This defines the following constitutive relations between the state variables $x$ and the flow and effort vectors $(f_{x},e_{x})$ of the energy-storing elements²²2Throughout this paper the vector $\frac{\partial H}{\partial x}(x)$ denotes the column vector of partial derivatives; the corresponding row vector denoted as $\frac{\partial H}{\partial x^{T}}(x)$.

Equation (4) immediately implies the energy balance $\frac{d}{dt}H=\frac{\partial H}{\partial x^{T}}(x)\dot{x}=-\langle e_{x}\mid f_{x}\rangle$. Furthermore, the composition

satisfies by the power-conserving property of the Dirac structure and (3)

for all $(f_{x},f_{P},e_{x},e_{P})\in\mathcal{D}\rightleftarrows\mathcal{R}$. Taken together this implies that

showing cyclo-passivity of any port-Hamiltonian system, and passivity if $H:\mathcal{X}\to\mathbb{R}_{+}$ [1] .

The basic idea in the definition of an incrementally port-Hamiltonian system, as first introduced in [9], is to replace the composition $\mathcal{D}\rightleftarrows\mathcal{R}$ of a Dirac structure $\mathcal{D}$ and an energy-dissipating relation $\mathcal{R}$ by a monotone relation ${\mathcal{M}}$. To do so, we begin with a quick review of monotone relations.

Since $\langle e_{0}\mid f_{0}-f_{1}\rangle+\langle e_{1}\mid f_{1}-f_{0}\rangle=\langle e_{0}-e_{1}\mid f_{0}-f_{1}\rangle$ for all $e_{0},f_{0},e_{1},f_{1}$, every cyclically monotone relations is automatically monotone.

A simple example of a monotone relation $\mathcal{M}\subset\mathbb{R}\times\mathbb{R}$ is the graph of a monotone (i.e., non-decreasing), possibly discontinuous, function. For example, the graph of the discontinuous function $\theta:\mathbb{R}\to\mathbb{R}$ given by

is a monotone relation. This example already motivates the strengthened definition of a maximal monotone relation.

The graph of the discontinuous function $\theta$ in (9) is monotone, but not maximal monotone. In fact, its graph can be enlarged so as to obtain the following maximal monotone relation

Note that the function $\theta$ in (9) can be regarded as the description of a relay, while its closure given by the maximal monotone relation $\mathcal{M}$ defined in (10) defines for example an ideal Coulomb friction characteristic.

A few well-known facts are noteworthy. For continuous functions, monotonicity of the graph implies maximal monotonicity (see e.g. [17]). Also, every maximal monotone relation on $\mathbb{R}\times\mathbb{R}$ is maximal cyclically monotone. (Hence the above Coulomb friction characteristic in (10) is maximal cyclically monotone.) In higher dimensions, however, not every maximal monotone relation enjoys the cyclical monotonicity property. Indeed, for example the relation given by

is maximal monotone but not cyclically monotone. More generally, later on (Proposition 5.4) we will see that Dirac structures are maximal monotone, but not cyclically monotone if they are the graph of a non-zero map.

The importance of maximal cyclically monotone relations ${\mathcal{M}}$ lies in the fact that they correspond to extended real-valued convex functions. This will be briefly review next, for more details we refer to [17]. Let $\phi:{\mathcal{F}}\rightarrow(-\infty,+\infty]$ be a proper convex function. Its effective domain is defined by

its subdifferential of $\phi$ at $f$ by

and its conjugate $\phi^{\star}:{\mathcal{E}}\rightarrow(-\infty,+\infty]$ by

If, in addition, $\phi$ is lower semicontinuous, then $\phi=(\phi^{\star})^{\star}$ and

It turns out (see [17, Thm. 12.25]) that a relation $\mathcal{M}\subset{\mathcal{F}}\times{\mathcal{E}}$ is maximal cyclically monotone if and only if there exists a proper lower semicontinuous convex function $\phi$ such that

In this case, we say that $\mathcal{M}$ is generated by $\phi$, or that $\phi$ is a generating function of $\mathcal{M}$. Note that $\phi$ is determined by $\mathcal{M}$ uniquely up to an additive constant.

As an example, consider the relation given by (10). One easily verifies that $\mathcal{M}$ is generated by the convex function $\phi(x)$ given by $x\mapsto|x|$. Furthermore,

The definition of an incrementally port-Hamiltonian system as introduced in [9] is now extended as follows.

It follows that the dynamics of any incrementally port-Hamiltonian system satisfies the incremental dissipation inequality

for all quadruples $(x_{i},\dot{x}_{i},f_{P}^{i},e_{P}^{i}),i=1,2,$ satisfying $\big{(}-\dot{x}_{i},f_{P}^{i},,\frac{\partial H}{\partial x}\big{(}x_{i}\big{)},e_{P}^{i}\big{)}\in\mathcal{M}$ with $i\in\{1,2\}$. The consequences of this dynamical inequality, and especially the relation with incremental and differential passivity, will be discussed in Section 7.

Incrementally port-Hamiltonian systems are ubiquitous in physical systems modeling, as already illustrated by the following examples.

An appealing class of maximal cyclically monotone port-Hamiltonian systems is defined as follows. Consider any Hamiltonian $H:\mathcal{X}\to\mathbb{R}$, and any convex function $K:\mathcal{X}^{*}\times\mathcal{U}\to\mathbb{R}$. Then the system

is a maximal cyclically monotone port-Hamiltonian system with maximal cyclically monotone relation $\mathcal{M}=\mbox{graph }(\partial K)$. Special case occurs if the convex function $K(e,u)$ is of the form

with $P$ a convex function of $e$, and $B$ an $n\times m$ matrix. This yields the restricted system class

A physical example of the form (14) is the following.

An example of a maximal cyclically monotone port-Hamiltonian system of the form (14) that is not stemming from physical systems modeling, but instead from optimization, is the following.

An extended class of maximal monotone port-Hamiltonian systems is defined as

where $J$ is a skew-symmetric matrix, and $P$ a convex function as above. It will follow from Proposition 5.4 that if $J\neq 0$ then the underlying maximal monotone relation is not derivable from a convex function, and the system is not cyclically monotone port-Hamiltonian anymore. An example within this class is the following.

A cornerstone of port-Hamiltonian systems theory is the fact that the power-conserving interconnection of port-Hamiltonian systems defines again a port-Hamiltonian system. This in turn is based on the fact that the composition of Dirac structures is again a Dirac structure. In this section we will show that the same property holds for incrementally port-Hamiltonian systems. This follows from the corresponding compositionality property of (maximal) (cyclically) monotone relations.

Let us start by considering two monotone relations $\mathcal{M}_{a}\subset{\mathcal{F}}_{a}\times{\mathcal{F}}\times{\mathcal{E}}_{a}\times{\mathcal{E}}$ and $\mathcal{M}_{b}\subset{\mathcal{F}}_{b}\times{\mathcal{F}}\times{\mathcal{E}}_{b}\times{\mathcal{E}}$. Define the composition of $\mathcal{M}_{a}$ and $\mathcal{M}_{b}$, denoted as $\mathcal{M}_{a}\overset{\scriptscriptstyle\mathcal{F}\times\mathcal{E}}{\leftrightarrows}\mathcal{M}_{b}$, as before, by

Thus the composition of $\mathcal{M}_{a}$ and $\mathcal{M}_{b}$ is obtained by imposing the interconnection constraints

on the vectors $(f_{a},f_{1},e_{a},e_{1})\in\mathcal{M}_{a}$ and $(f_{b},f_{2},e_{b},e_{2})\in\mathcal{M}_{b}$ and looking at the resulting vectors $(f_{a},f_{b},e_{a},e_{b})\in{\mathcal{F}}_{a}\times{\mathcal{F}}_{b}\times{\mathcal{E}}_{a}\times{\mathcal{E}}_{b}$.

Whenever interconnection flow and effort spaces $\mathcal{F}$ and $\mathcal{E}$ are clear from the context, we will simply write $\mathcal{M}_{a}\leftrightarrows\mathcal{M}_{b}$. The following result is straightforward.

Also the composition of two maximal monotone relations turns out to be maximal monotone; provided certain (mild) regularity conditions are met. To elaborate on this, we first introduce some nomenclature and review some known facts about maximal monotone relations.

For a set $S\in{\mathcal{F}}$, $\operatorname{cl}S$ denotes its closure. The relative interior of a convex set $C\subseteq{\mathcal{F}}$ is denoted by $\operatorname{rint}C$. A set $S\subseteq{\mathcal{F}}$ is said to be nearly convex if there exists a convex set $C\subseteq{\mathcal{F}}$ such that $C\subseteq S\subseteq\operatorname{cl}C$. For a nearly convex set $S$, in general, there can be multiple convex sets $C$ satisfying $C\subseteq S\subseteq\operatorname{cl}C$. For any such set $C$, however, we have that $\operatorname{cl}C=\operatorname{cl}S$. As such, $\operatorname{cl}S$ is convex if $S$ is nearly convex. Based on this observation, one can extend the notion of relative interior to nearly convex sets by defining $\operatorname{rint}S=\operatorname{rint}(\operatorname{cl}S)$.

Let $S\subseteq{\mathcal{F}}_{1}\times{\mathcal{F}}_{2}\times{\mathcal{E}}_{1}\times{\mathcal{E}}_{2}$. The projection of $S$ on ${\mathcal{F}}_{1}\times{\mathcal{F}}_{2}$, denoted by $\Pi(S,{\mathcal{F}}_{1}\times{\mathcal{F}}_{2})$, is defined as

We define projections of $S$ on ${\mathcal{E}}_{1}\times{\mathcal{E}}_{2}$, ${\mathcal{F}}_{i}\times{\mathcal{E}}_{j}$, ${\mathcal{F}}_{i}$, and ${\mathcal{E}}_{j}$ in a similar fashion.

Let $S\subseteq\mathcal{F}\times\mathcal{G}$ be a nearly convex set. Then, both $\Pi(S,\mathcal{F})$ and $\Pi(S,\mathcal{G})$ are nearly convex sets. Furthermore, one can show that

Let $\mathcal{M}\subset{\mathcal{F}}\times{\mathcal{E}}$ be a maximal monotone relation. Then, the projections $\Pi(\mathcal{M},{\mathcal{F}})$ and $\Pi(\mathcal{M},{\mathcal{E}})$ are nearly convex sets [17, Thm. 12.41].

Let $L:\mathcal{G}\rightarrow\mathcal{H}$ be a linear map and $L^{*}:\mathcal{H}^{*}\rightarrow\mathcal{G}^{*}$ denote its adjoint. For maximal monotone relations $\mathcal{M}\subseteq\mathcal{H}\times\mathcal{H}^{*}$ and $\mathcal{N}\subseteq\mathcal{G}\times\mathcal{G}^{*}$, define $\mathcal{M}_{L}\subseteq\mathcal{G}\times\mathcal{G}^{*}$ and ${}_{L}\mathcal{N}\subseteq\mathcal{H}\times\mathcal{H}^{*}$ by

From [17, Thm. 12.43], we know that $\mathcal{M}_{L}$ is maximal monotone if

and ${}_{L}\mathcal{N}$ is maximal monotone if

Furthermore, if $\mathcal{M}$ is generated by $\phi:\mathcal{H}\rightarrow(-\infty,+\infty]$ and

then $\mathcal{M}_{L}$ is generated by $\phi\circ L$ given by $h\mapsto\phi(Lh)$. Dually, if $\mathcal{N}$ is generated by $\psi:\mathcal{G}\rightarrow(-\infty,+\infty]$ and

then ${}_{L}\mathcal{N}$ is generated by the function $(\psi^{\star}\circ L^{*})^{\star}$.