Sufficient conditions for passivity and stability of interconnections of hybrid systems using sums of storage functions (original) (raw)
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Passivity Analysis and Passification of Discrete-Time Hybrid Systems
IEEE Transactions on Automatic Control, 2008
For discrete-time hybrid systems in piecewise affine or piecewise polynomial (PWP) form, this note proposes sufficient passivity analysis and synthesis criteria based on the computation of piecewise quadratic or PWP storage functions. By exploiting linear matrix inequality techniques and sum of squares decomposition methods, passivity analysis and synthesis of passifying controllers can be carried out through standard semidefinite programming packages, providing a tool particularly important for stability of interconnected heterogenous dynamical systems.
Nonlinear Analysis: Hybrid Systems
2009
In this paper, we develop dissipativity theory for discontinuous dynamical systems. Specifically, using set-valued supply rate maps and set-valued connective supply rate maps consisting of locally Lebesgue integrable supply rates and connective supply rates, respectively, and set-valued storage maps consisting of piecewise continuous storage functions, dissipativity properties for discontinuous dynamical systems are presented. Furthermore, extended Kalman–Yakubovich–Popov set-valued conditions, in terms of the discontinuous system dynamics, characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions are derived. Finally, these results are used to develop feedback interconnection stability results for discontinuous dynamical systems by appropriately combining the set-valued storage maps for the forward and feedback systems. © 2009 Elsevier Ltd. All rights reserved.
IET Control Theory & Applications, 2010
The stability and control of systems possessing passivity violations is considered. The authors seek to exploit the finite gain characteristics of a plant over a range in which a passive mapping no longer exists while implementing a similar hybrid passive and finite gain controller. Using the dissipative systems framework the authors define a hybrid system: one which possesses a passive map, and finite gain characteristics when the passive map is destroyed. The definition of a hybrid system utilises a switching parameter to break the system into passive and finite gain regions. It is shown that this switching parameter is equivalent to an ideal lowpass filter and can be approximated by a Butterworth filter. The stability of two hybrid systems within a negative feedback interconnection is also considered. A hybrid passivity and finite gain stability theorem is developed using both Lyapunov and input -output techniques, which yield equivalent results. Sufficient conditions for the closed-loop system to be stable are presented, which resemble an amalgamation of the traditional passivity and small-gain theorems.
A notion of passivity for hybrid systems
Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228), 2001
We propose a notion of passivity for hybrid systems. Our work is motivated by problems in haptics and teleoperation where several computer controlled mechanical systems are connected through a communication channel. To account for time delays and to better react to user actions it is desirable to design controllers that can switch between different operating modes. Each of the interacting systems can be therefore naturally modeled as a hybrid system. A traditional passivity definition requires that a storage function exists that is common to all operating modes. We show that stability of the system can be guaranteed even if different storage function are found for each of the modes, provided appropriate conditions are satisfied when the system switches.
Passivity analysis of discrete-time hybrid systems using piecewise polynomial storage functions
2005
This paper proposes some sufficient criteria based on the computation of polynomial and piecewise polynomial storage functions for checking passivity of discrete-time hybrid systems in piecewise affine or piecewise polynomial form. The computation of such storage functions is performed by means of convex optimization techniques via the sum of squares decomposition of multivariate polynomials.
Stability analysis of interconnected implicit systems based on passivity
Proceedings of the 15th IFAC World Congress, 2002, 2002
This paper considers the internal stability of an interconnection of two implicit systems based on passivity theory. The passivity for implicit systems is defined as a generalization of the passivity in the traditional input-output framework. We derive a necessary and sufficient condition for the passivity of an implicit system in terms of a constrained linear matrix inequality (CLMI). Based on this CLMI condition, we derive a sufficient condition for the internal stability of an interconnection of two implicit systems.
Passivity-based Stability of Interconnection Structures
Lecture Notes in Control and Information Sciences, 2008
In the 1970s, Vidyasagar developed an approach to the study of stability of interconnected systems. This paper revisits this approach and shows how it allows one to interpret, and considerably extend, a classical condition used in mathematical biology.
Passivity of Cascaded Systems Based on The Analysis of Passivity Indices
Passivity index is defined in terms of an excess or shortage of passivity, and it has been introduced in order to extend the passivity-based stability conditions to the more general cases for both passive and non-passive systems. In this report, we revisit the secant criterion literature results from the perspective of passivity indices. While most of the passivity-based stability results in literature focus on studying the feedback interconnection of passive or non-passive systems, our results focus on the study of cascaded interconnection. In this report, we show how to use the secant criterion to quantify the excess/shortage of passivity for cascaded system which includes both passive and non-passive systems. We further show that under certain conditions, the cascaded interconnection can be directly stabilized via output feedback.
2011
Motivated by applications of systems interacting with their environments, we study the design of passivitybased controllers for a class of hybrid systems. Classical and hybrid-specific notions of passivity along with detectability and solution conditions are linked to asymptotic stability. These results are used to design passivity-based controllers following classical passivity theory. An application, pertaining to a point mass physically interacting with the environment, illustrates the definitions and the results obtained throughout this work. • 0-input pre-asymptotically stable if it is 0-input stable and 0-input pre-attractive.
On stability in hybrid systems
1998
Asymptotic stability: A definition of stability wherein small changes in initial condition are diminished as time progresses. Autonomous system: A system whose dynamics are not explicitly dependent on time or external inputs. Continuous dynamics: Equations governing the behavior of variables that continuously change value as a function of time. Continuous variable: A variable whose values lie in a continuum, such as the real numbers. Controlled system: A system whose dynamics is explicitly dependent on external inputs, which usually may be specified by a designer. Discrete dynamics: Equations governing the behavior of variables that are immediately reset to new values over time. Discrete variable: A variable whose values take on only a countable number of values. Dynamical system: A set of equations describing how a set of variables change values over time. Equilibrium point: A point that remains unchanged as a function of the dynamics of a system; if the system is placed there exactly, it will remain there forever. Global asymptotic stability: A definition of stability wherein any perturbations in initial condition (no matter how large) are diminished as time progresses. Hybrid system: A dynamical system possessing discrete and continuous variables and dynamics. Impulse: A discontinuous change in a continuous variable's value. Jacobian matrix: A matrix of partial derivatives relating changes in one vector of variables to changes in another, functionally dependent vector of variables. Lyapunov function: An energy function associated with a dynamical system that decreases over time. Lyapunov stability: A definition of stability based on the notion that small changes in initial condition lead to small changes in system behavior. Multiple Lyapunov functions: A stability technique for hybrid systems in which a decreasing energy function is associated with each discrete state. Sampled-data system: A dynamical system wherein continuous variables are updated at discrete, usually equally spaced, instants of time. Schur: A matrix whose eigenvalues all have magnitude less than one. State space: The space in which the variables of interest of a dynamical system take their values. Switched system: A hybrid system formed by switching among different dynamical equations governing the system's behavior.