While port-Hamiltonian descriptor systems are known to be stable and passive, they may not be asymptotically stable or strictly passive. Necessary and sufficient conditions are presented when these properties as well as the regularity and the index one property can be achieved via state feedback while preserving the port-Hamiltonian structure.
Delay-dependent passivity and stability of linear port-Hamiltonian systems
ENSAM, Hassan II University of Casabalanca（卡萨布兰卡哈桑二世大学ENSAM）；University of Wuppertal（伍珀塔尔大学）；Fraunhofer IEG, Fraunhofer Research Institution for Energy Infrastructures and Geotechnologies IEG（弗劳恩霍夫能源基础设施和岩土技术研究所IEG）
A consequence of the Second Law of thermodynamics is that no thermodynamic system with a single heat source at constant temperature can convert heat into mechanical work in a recurrent manner. First we note that this is equivalent to cyclo-passivity at the mechanical port of the thermodynamic system, while the temperature at the thermal port of the system is kept constant. This leads to the general system-theoretic question which systems with two power ports have similar behavior: when is a system cyclo-passive at one of its ports, while the output variable at the other port (such as the tempe…
The present work address the problem of energy shaping for stochastic port-Hamiltonian system. Energy shaping is a powerful technique that allows to systematically find feedback law to shape the Hamiltonian of a controlled system so that, under a general passivity condition, it converges or tracks a desired configuration. Energy shaping has been recently generalized to consider stochastic port-Hamiltonian system. Nonetheless the resulting theory presents several limitation in the application so that relevant examples, such as the additive noise case, are immediately ruled out from the possible…
The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity, KYP inequalities and a~finite available storage is investigated.
The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is de…
Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.
When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum satisfies the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure there always exists a nearby port-Hamiltonian descriptor system with exac…
For stochastic systems with nonvanishing noise, i.e., at the desired state the noise port does not vanish, it is impossible to achieve the global stability of the desired state in the sense of probability. This bad property also leads to the loss of stochastic passivity at the desired state if a radially unbounded Lyapunov function is expected as the storage function. To characterize a certain (globally) stable behavior for such a class of systems, the stochastic asymptotic weak stability is proposed in this paper which suggests the transition measure of the state to be convergent and the ergo…
We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems subject to control and terminal state constraints. To this end, after reducing the problem to an ODE with feed-through term, we derive an input-state turnpike towards a subspace for optimal control of generalized port-Hamiltonian ordinary differential equations. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. By means of the port-Hamiltonian structure, we…
We characterize stable differential-algebraic equations (DAEs) using a generalized Lyapunov inequality. The solution of this inequality is then used to rewrite stable DAEs as dissipative Hamiltonian (dH) DAEs on the subspace where the solutions evolve. Conversely, we give sufficient conditions guaranteeing stability of dH DAEs. Further, for stabilizable descriptor systems we construct solutions of generalized algebraic Bernoulli equations which can then be used to rewrite these systems as pH descriptor systems. Furthermore, we show how to describe the stable and stabilizable systems using Dira…
The present work is a successor of [Ilchmann, Kirchhoff 2022] on generic controllability and of [Ilchmann, Kirchhoff 2023] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive new results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.
We develop optimization-based structure-preserving model order reduction (MOR) methods for port-Hamiltonian (pH) descriptor systems of differentiation index one. Descriptor systems in pH form permit energy-based modeling and intuitive coupling of physical systems across different physical domains, scales, and accuracies. This makes pH models well-suited building-blocks for component-wise modeling of large system networks. In this context, it is often necessary to preserve the pH structure during MOR. We discuss current projection-based and structure-preserving MOR algorithms for pH systems and…
We study port-Hamiltonian systems with two external ports, and the strategies and limitations for conversion of energy from one port into the other. It turns out that, apart from the cyclo-passivity of port-Hamiltonian systems, this is related to the internal connection structure of port-Hamiltonian systems. A source of motivation for energy conversion is provided by thermodynamics, in particular the Carnot theory of conversion of thermal into mechanical energy. This is extended to general port-Hamiltonian systems which are satisfying structural conditions on their topology; thus generalizing …
We discuss the problem of robust representations of stable and passive transfer functions in particular coordinate systems, and focus in particular on the so-called port-Hamiltonian representations. Such representations are typically far from unique and the degrees of freedom are related to the solution set of the so-called Kalman-Yakubovich-Popov linear matrix inequality (LMI). In this paper we analyze robustness measures for the different possible representations and relate it to quality functions defined in terms of the eigenvalues of the matrix associated with the LMI. In particular, we lo…
It is a universal phenomenon that the state and input of the continuous stirred tank reactor (CSTR) systems are both disturbed. This paper proposes a (state, input)-disturbed port-Hamiltonian framework that can be used to model and further designs a stochastic passivity based controller to asymptotically stabilize in probability the (state, input)-disturbed CSTR (sidCSTR) systems. The opposite entropy function and the availability function are selected as the Hamiltonian for the model and control purposes, respectively. Furthermore, the proposed (state, input)-disturbed port-Hamiltonian model …
The relationships between port-Hamiltonian systems modeling and the notion of monotonicity are explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to maximal cyclically monotone relations, together with their generating functions. This gives rise to new classes of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cy…
In this work we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots (FJRs) in the port-Hamiltonian (pH) framework. The proposed design method, called virtual contraction based control (v-CBC), combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system…
We present a new fixed-order H-infinity controller design method for potentially large-scale port-Hamiltonian (pH) plants. Our method computes controllers that are also pH (and thus passive) such that the resulting closed-loop systems is again passive, which ensures closed-loop stability simply from the structure of the plant and controller matrices. In this way, we can avoid computationally expensive eigenvalue computations that would otherwise be necessary. In combination with a sample-based objective function which allows us to avoid multiple evaluations of the H-infinity norm (which is typ…
In these lectures notes, we review our recent works addressing various problems of finding the nearest stable system to an unstable one. After the introduction, we provide some preliminary background, namely, defining Port-Hamiltonian systems and dissipative Hamiltonian systems and their properties, briefly discussing matrix factorizations, and describing the optimization methods that we will use in these notes. In the third chapter, we present our approach to tackle the distance to stability for standard continuous linear time invariant (LTI) systems. The main idea is to rely on the character…
