Sufficient conditions for passivity and stability of interconnections of hybrid systems using sums of storage functions
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Building from recent results on passivity for a class of hybrid systems, we investigate the properties of negative feedback interconnections of such systems. We establish links between the passivity properties of the individual subsystems and passivity, stability, and asymptotic stability of their interconnection. As a main difference to the continuous time counterpart, it is found that the sum of the two storage functions of two individual hybrid subsystems may not be a storage function for their interconnection. This issue motivates exploring additional sufficient conditions that guarantee that passivity and stability of the interconnected system hold using the individual storage functions. Throughout the paper, an application and examples illustrate the definitions and the results obtained.Keywords:
Passivity
The paper discusses the preservation of u-average passivity throughout suitable interconnection. The concept of power preserving connection is introduced. It is instrumental to ensure u-average passivity of the interconnected system with respect to new external controls.
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Passivity and dissipativity are energy-like concepts, widely used in control design, that capture the "energy" consumption of a dynamical system and therefore relate closely to the physical world. Passivity indices of a system are measures of its passivity margins and represent shortage and excess of passivity in a system. With the aid of passivity indices, one can measure how passive a system is, or how far from passivity it is. Passivity indices extend all the analysis and design methods based on passivity to nonpassive systems as well. One of the advantages of using passivity is its tight relationship to stability. Another is its compositionality, which, together with its generality, makes it possible to use passivity in a wide range of complex control systems. In the present entry, an overview of dissipativity and passivity is given. Passivity indices of a system and their relation to stability are defined, and methods to find the indices are presented.
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Building from recent results on passivity for a class of hybrid systems, we investigate the properties of negative feedback interconnections of such systems. We establish links between the passivity properties of the individual subsystems and passivity, stability, and asymptotic stability of their interconnection. As a main difference to the continuous time counterpart, it is found that the sum of the two storage functions of two individual hybrid subsystems may not be a storage function for their interconnection. This issue motivates exploring additional sufficient conditions that guarantee that passivity and stability of the interconnected system hold using the individual storage functions. Throughout the paper, an application and examples illustrate the definitions and the results obtained.
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This paper presents an analytical framework to examine the unconditional stability of two stable, linear timeinvariant systems in a positive feedback interconnection where one system has "mixed" negative-imaginary and passivity properties and other system has "mixed" negative-imaginary and negative-passivity properties. The examination of the stability of above mentioned interconnection is done by using a Nyquist criteria, and it is shown that the positive feedback interconnection between such two systems is guaranteed to be finite-gain stable. A numerical example is presented in the paper to demonstrate the usefulness of the proposed analytical framework.
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Passivity and dissipativity are energy-like concepts, widely used in control design, that capture the "energy" consumption of a dynamical system and therefore relate closely to the physical world. Passivity indices of a system are measures of its passivity margins and represent shortage and excess of passivity in a system. With the aid of passivity indices, one can measure how passive a system is, or how far from passivity it is. Passivity indices extend all the analysis and design methods based on passivity to nonpassive systems as well. One of the advantages of using passivity is its tight relationship to stability. Another is its compositionality, which, together with its generality, makes it possible to use passivity in a wide range of complex control systems. In the present entry, an overview of dissipativity and passivity is given. Passivity indices of a system and their relation to stability are defined, and methods to find the indices are presented.
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Generality
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We investigate the stability of a class of dynamical systems that switch among a given finite family of hybrid systems. We propose sufficient conditions tailored to this particular type of hybrid systems which guarantee the uniform global pre-asymptotic stability (UGpAS) of a given closed set. We first assume this set to be UGpAS for each system of the family. A slow switching condition is then presented to maintain this property for the overall system. We introduce for this purpose the concept of hybrid dwell time which characterizes the length of the hybrid time intervals between two successive switching instants.
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This paper focuses on the notions of QSR dissipativity and passivity in hybrid systems. The work presented in this paper can mainly be divided into two parts- (i) feedback and parallel interconnections and, (ii) networks. First, it is shown that the negative feedback interconnection of two QSR dissipative hybrid systems is QSR dissipative. Also, passivity is preserved when passive hybrid systems are connected in parallel. It is also shown that QSR dissipativity and hence passivity are closely related to the stability of hybrid systems. In the second part of this work, passivity of networked hybrid input-output automata, in presence of network delays is studied. The issues of both the constant and time varying delays is addressed using a wave variable transform based approach.
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