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Some Observations About Dissipative Properties for Dynamical Systems Under Change of Variables
Abstract
This note gives results about the preservation of some dissipative properties of systems under a change of variables. In the textbooks it is not mentioned explicitly the relationship between the equations associated with the dynamics of a system and the selected Lyapunov function to establish its stability property, when a change of coordinates is used. Based on the fact that Lyapunov stability is preserved under these changes of variables, it is shown that various forms of dissipativity can be preserved. In addition, we will show that the input-state stability (ISS), integral input-to-state stability (iISS) and input/output to state stability (IOSS) can be preserved under this class of transformation. Some examples are given to show these results. Keywords: Preservation of dissipativity, preservation of input-state stability, preservation of integral input-to-state stability, preservation of input/output to state stability, change of coordinates.
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A complete account is given of the theory of so-called dissipative dynamical systems. The concept of dissipativeness is defined as a general input-output property which includes, as notable special cases, passivity and other properties related to finite-gain. The aim is to treat input-output and state properties side-by-side with emphasis on exploring connections between them. The key connection is that a dissipative system in general possesses a set of energy-like functions of the state. The properties of these functions are studied in some detail. It is demonstrated that this connection represents a direct generalization of the well-known Kalman-Yakubovich lemma to arbitrary dynamical systems. Applications to stability theory and passive system synthesis are briefly discussed for non-linear systems.
We show that the well-known Lyapunov sufficient condition for “input-to-state stability” (ISS) is also necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided.
We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of separation principles for input-output-to-state stability -- characterizations of the property in terms of weaker stability notions. When applied to the closely related notion of integral ISS, these characterizations yield analogous results. 1
In this paper, the authors present a unifying framework for design
of controllers that suppress low-frequency oscillations in power
systems. The proposed methodology is based on the physical notion of
dissipativity, and on its system-theoretic counterparts. They illustrate
the capabilities of this framework in four application areas: power
system stabilizers (PSSs); supplementary control for high-voltage DC
(HVDC) lines; auxiliary control for static VAr systems (SVCs); and
regulation of thyristor-controlled series capacitors (TCSCs). All case
studies involve a standard benchmark case of a multi-machine power
system, and all controllers use only locally available information
This paper is the first in a two-part sequence which aims to state rigorously the energy-based concepts which are fundamental to nonlinear network theory, passivity and losslessness, and to clarify the way they enter the input-output and the state-space versions of the subject. In this part we examine the conflicting definitions of passivity which exist in the literature and demonstrate the contradictions between them with several examples. We propose a particular definition of passivity which avoids these contradictions by eliminating the dependency on a state of "zero stored energy," and we show that it has the appropriate properties of representation independence and closure. We apply it to several specific classes of n -ports and derive equivalent passivity criteria. The exact conditions are given under which this definition is equivalent to one based on an internal energy function, and we use the concept of an internal energy function to provide a canonical network realization for a class of passive systems.
The notion of input to state stability (iss) is now recognized as a central concept in nonlinear systems analysis. It provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite L 2 gains. It plays a central role in recursive design, coprime factorizations, controllers for non-minimum phase systems, and many other areas. In this paper, a newer notion, that of integral input to state stability (iiss), is studied. The notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states, in that sense generalizing the linear "H 2 " theory. It allows to quantify sensitivity even in the presence of certain forms of nonlinear resonance. We obtain here several necessary and sufficient characterizations of the iiss property, expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations. These characterizations serve to show that integral input to state stabi...
In this paper we address the problem of supression of low frequency oscillations in power systems. These oscillations appear in strongly interconnected networks because of load and topology changes, and they may cause loss of synchronism and generator tripping. We propose the utilisation of passivation techniques to design power system stabilizers for the synchronous generators. The generator to be controlled is described by a standard lagrange model, with three forcing terms: the mechanical torque coming from the turbine, the terminal voltage of the network and the field voltage, which is our control variable. In view of the significant differences between the mechanical and the electrical time scales, the first signal can be treated as a constant disturbance. The terminal voltage may be viewed as the output of an operator, -defined by the remaining part of the network-, which is in feedback interconnection with the generator. Our basic assumption is that the network is always absorbing energy from the generator, whence the interconnection subsystem (as viewed from the generator) is passsive. The control objective is then to close a loop around the field voltage so as to passivize the generator system. We characterize, in terms of a simple linear matrix inequality, a class of linear state-feedback controllers which achieve this objective.
The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.
This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.
This paper studies the dissipativity properties of full order dynamic models of synchronous generators. It is shown that, under widely accepted assumptions, these models satisfy a balance between the internal storage of a generalized energy and a suitably defined power supply associated to the stator and excitation circuits. This property has particular relevance since it allows to incorporate the detailed model of synchronous machines to classical energy functions used to analyze power system stability. The dissipation inequality implies also that the linear model around the equilibrium point meet a convex condition in the frequency domain, able to be exploited in the stability analysis of interconnected systems. The impact of excitation control and resistive losses on these properties is studied through a numerical example.
It is demonstrated how a number of widely used tools for stability
analysis can be conveniently unified and generalized using integral
quadratic constraints (IQCs). The approach has emerged under a
combination of influences from the western and Russian traditions of
control theory. An IQC based stability theorem is presented, which
covers classical small gain conditions with anti-causal multipliers, but
gains flexibility by avoiding extended spaces and truncated
signals
We present a new variant of the input-to-state stability (ISS) property which is based on using a one-dimensional dynamical system for building the class 𝒦L function for the decay estimate and for describing the influence of the perturbation. We show the relation to the original ISS formulation and describe characterizations by means of suitable Lyapunov functions. As applications, we derive quantitative results on stability margins for nonlinear systems and a quantitative version of a small gain theorem for nonlinear systems.
This paper introduces a unified approach to robustness analysis with respect to nonlinearities, time-variations and uncertain parameters. From an original idea by Yakubovich, the approach has been developed under a combination of influences from the western and russian traditions of control theory. It is shown how a complex system can be described by using certain integral quadratic constraints (IQC's), derived for its elementary components. A stability theorem for systems described by IQC's is presented, that covers classical passivity/dissipativity arguments, but simplifies the use of multipliers and the treatment of causality. The paper is divided into two parts. Part I presents the basic ideas for stability analysis, refering to a simple example. A systematic computational approach is described and relations to other methods of stability analysis are discussed. Last, but not least, it contains a summarizing list of IQC's for important types of system components, that exist in variou...
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