Impedance Passive Systems Research Articles

Strong stabilization of (almost) impedance passive systems by static output feedback

The plant to be stabilized is a system node $ \Sigma $ with generating triple $ (A,B,C) $ and transfer function $ {\bf G} $, where $ A $ generates a contraction semigroup on the Hilbert space $ X $. The control and observation operators $ B $ and $ C $ may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator $ E $ such that, if we replace $ {\bf G}(s) $ by $ {\bf G}(s)+E $, the new system $ \Sigma_E $ becomes impedance passive. An easier case is when $ {\bf G} $ is already impedance passive and a special case is when $ \Sigma $ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback $ u = - {\kappa} y+v $, where $ u $ is the input of the plant and $ {\kappa}>0 $, stabilizes $ \Sigma $, strongly or even exponentially. Here, $ y $ is the output of $ \Sigma $ and $ v $ is the new input. Our main result is that if for some $ E\in {\mathcal L}(U) $, $ \Sigma_E $ is impedance passive, and $ \Sigma $ is approximately observable or approximately controllable in infinite time, then for sufficiently small $ {\kappa} $ the closed-loop system is weakly stable. If, moreover, $ \sigma(A)\cap i {\mathbb R} $ is countable, then the closed-loop semigroup and its dual are both strongly stable.

Building systems from simple hyperbolic ones

In this article we introduce a technique that derives from the existence and uniqueness of solutions to a simple hyperbolic partial differential equation (p.d.e.) the existence and uniqueness of solutions to hyperbolic and parabolic p.d.e.’s. Among others, we show that starting with an impedance passive system associated to the undamped wave equation, we can obtain an impedance passive system associated to the heat conduction equation.

Spectral properties of pseudo-resolvents under structured perturbations

The changes in the spectrum caused by structured perturbations of pseudo-resolvents and operators on Banach spaces are considered. In particular, if a point is in the resolvent set of an operator, necessary and sufficient conditions for it to remain in the resolvent set under structured perturbations are given. The structured perturbations of an operator are specified by an operator node that has three generating operators and a characteristic function together with an admissible feedback operator. In addition, the robustness of stability under structured perturbations is analyzed. The results are applied to boundary control systems and impedance passive systems.

A complete model of a finite-dimensional impedance-passive system

We extend the classes of standard discrete- and continuous-time input/state/output matrix systems by adding reverse internal and/or external channels. The reverse internal channel permits the impulse response to contain a differentiating part, and the reverse external channel allows us to include inputs which are forced to be zero and outputs which are undetermined. The purpose of this extension is obtaining a class of state-space matrix systems that can be used to realise all right-coprime positive-real rational relations—in particular non-proper positive-real rational transfer functions can be realised. We generalise the notions of impedance and scattering passivity to extended systems. When we restrict our attention to passive systems, the new class of extended impedance-passive systems is closed under the operations of interchanging the input and the output, as well as frequency inversion and duality. We generalise two system Cayley transformations to extended systems. The first transformation that we consider is the internal Cayley transformation, which maps an impedance- or scattering-passive continuous-time system into a discrete-time approximation of the original system that is again impedance passive or scattering passive, respectively. The second transformation is the external Cayley transformation that maps a contiuous- or discrete-time impedance-passive system into a scattering-passive system with the same time axis. In our extended setting, the two Cayley transformations become bijections between the respective classes of extended passive systems.

Passive and Conservative Continuous-Time Impedance and Scattering Systems. Part I: Well-Posed Systems

Let U be a Hilbert space. By an ℒ (U)-valued positive analytic function on the open right half-plane we mean an analytic function which satisfies the condition . This function need not be proper, i.e., it need not be bounded on any right half-plane. We study the question under what conditions such a function can be realized as the transfer function of an impedance passive system. By this we mean a continuous-time state space system whose control and observation operators are not more unbounded than the (main) semigroup generator of the system, and, in addition, there is a certain energy inequality relating the absorbed energy and the internal energy. The system is (impedance) energy preserving if this energy inequality is an equality, and it is conservative if both the system and its dual are energy preserving. A typical example of an impedance conservative system is a system of hyperbolic type with collocated sensors and actuators. We give several equivalent sets of conditions which characterize when a system is impedance passive, energy preserving, or conservative. We prove that a impedance passive system is well-posed if and only if it is proper. We furthermore show that the so-called diagonal transform (which may be regarded as a slightly modified feedback transform) maps a proper impedance passive (or energy preserving or conservative) system into a (well-posed) scattering passive (or energy preserving or conservative) system. This implies that, just as in the finite-dimensional case, if we apply negative output feedback to a proper impedance passive system, then the resulting system is (energy) stable. Finally, we show that every proper positive analytic function on the right half-plane has a (essentially unique) well-posed impedance conservative realization, and it also has a minimal impedance passive realization.