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Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems
 IEEE Transactions on Automatic Control
, 1998
"... . This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the cir ..."
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Cited by 259 (4 self)
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. This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach. Keywords. Piecewise linear systems, Lyapunov stability, linear matrix inequalities. 1. Introduction Construction of Lyapunov functions is one of the most fundamental problems in systems theory. The most direct application is stability analysis, but analogous problems appear more or less implicitly also in performance analysis, controller synthesis and system identification. Consequently, methods for constructing Lyapunov functions for general nonlinear systems is of great theoretical and practical interest. The objective of this paper is to develop a uniform and compu...
Flocking in Fixed and Switching Networks
, 2003
"... The work of this paper is inspired by the flocking phenomenon observed in Reynolds (1987). We introduce a class of local control laws for a group of mobile agents that result in: (i) global alignment of their velocity vectors, (ii) convergence of their speeds to a common one, (iii) collision avoidan ..."
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Cited by 192 (10 self)
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The work of this paper is inspired by the flocking phenomenon observed in Reynolds (1987). We introduce a class of local control laws for a group of mobile agents that result in: (i) global alignment of their velocity vectors, (ii) convergence of their speeds to a common one, (iii) collision avoidance, and (iv) minimization of the agents artificial potential energy. These are made possible through local control action by exploiting the algebraic graph theoretic properties of the underlying interconnection graph. Algebraic connectivity a#ects the performance and robustness properties of the overall closed loop system. We show how the stability of the flocking motion of the group is directly associated with the connectivity properties of the interconnection network and is robust to arbitrary switching of the network topology.
Tracking control of nonlinear systems using sliding surfaces with application to robot manipulators
 International Journal of Control
, 1983
"... We develop a methodology of feedback control to achieve accurate tracking in a class of nonlinear, timevarying systems in the presence of disturbances and parameter variations. The methodology uses in its idealized form piecewise continuous feedback control, resulting in the state trajectory &apos ..."
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Cited by 157 (1 self)
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We develop a methodology of feedback control to achieve accurate tracking in a class of nonlinear, timevarying systems in the presence of disturbances and parameter variations. The methodology uses in its idealized form piecewise continuous feedback control, resulting in the state trajectory 'sliding ' along a timevarying sliding surface in the state space. This idealized control law achieves perfect tracking; however, nonidealities in its implementation result in the generation of an undesirable high frequency component in the state trajectory. To rectify this, we show how continuous control laws may be used to approximate the discontinuous control law to obtain robust tracking to within a prescribed accuracy and decrease the extent of high frequency signal. The method is applied to the control of a twolink manipulator handling variable loads in a flexible manufacturing system environment.
RigidBody Dynamics With Friction and Impact,”
 SIAM Rev.,
, 2000
"... Abstract. Rigidbody dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeli ..."
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Cited by 137 (1 self)
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Abstract. Rigidbody dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeling rigidbody dynamics with friction is difficult due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law. This is particularly crucial for handling situations with large coefficients of friction, which can result in paradoxical results known at least since Painlevé [C. R. Acad. Sci. Paris, 121 (1895), pp. 112115]. This single example has been a counterexample and cause of controversy ever since, and only recently have there been rigorous mathematical results that show the existence of solutions to his example. The new mathematical developments in rigidbody dynamics have come from several sources: "sweeping processes" and the measure differential inclusions of Moreau in the 1970s and 1980s, the variational inequality approaches of Duvaut and J.L. Lions in the 1970s, and the use of complementarity problems to formulate frictional contact problems by Lötstedt in the early 1980s. However, it wasn't until much more recently that these tools were finally able to produce rigorous results about rigidbody dynamics with Coulomb friction and impulses. Key words. rigidbody dynamics, Coulomb friction, contact mechanics, measuredifferential inclu sions, complementarity problems AMS subject classifications. Primary, 70E55; Secondary, 70F40, 74M PII. S0036144599360110 Rigid Bodies and Friction. Rigid bodies are bodies that cannot deform. They can translate and rotate, but they cannot change their shape. From the outset this must be understood as an approximation to reality, since no bodies are perfectly rigid. However, for a vast number of applications in robotics, manufacturing, biomechanics (such as studying how people walk), and granular materials, this is an excellent approximation. It is also convenient, since it does not require solving large, complex systems of partial differential equations, which is generally difficult to do both analytically and computationally. To see the difference, consider the problem of a bouncing ball. The rigidbody model will assume that the ball does not deform while in flight and that contacts with the ground are instantaneous, at least while the ball is not rolling. On the other hand, a full elastic model will model not only the contacts and the resulting deformation of the entire ball while in contact, but also the elastic oscillations of the ball while it is in flight. Apart from the computational complexity of all this, the analysis of even linearly elastic bodies in contact with a *
Asymptotic Controllability Implies Feedback Stabilization
 IEEE Trans. Autom. Control
, 1999
"... It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a controlLyapunov function, iteratively sending trajectories into smaller and smal ..."
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Cited by 118 (12 self)
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It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a controlLyapunov function, iteratively sending trajectories into smaller and smaller neighborgoods of a desired equilibrium. A major technical problem, and one of contributions of the present paper, concerns the precise meaning of "solution" when using a discontinuous controller. I. Introduction A longstanding open question in nonlinear control theory concerns the relationship between asymptotic controllability to the origin in R n of a nonlinear system x = f(x; u) (1) by an "open loop" control u : [0; +1) ! U and the existence of a feedback control k : R n ! U which stabilizes trajectories of the system x = f(x; k(x)) (2) with respect to the origin. For the special case of linear control systems x = Ax + Bu, this relationship is well understood: asymptotic cont...
Stability criteria for switched and hybrid systems
 SIAM Review
, 2007
"... The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, an ..."
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Cited by 114 (8 self)
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The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NPhardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.
Effective Synthesis of Switching Controllers for Linear Systems
, 2000
"... In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the sys ..."
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Cited by 110 (8 self)
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In this work we suggest a novel methodology for synthesizing switching controllers for continuous and hybrid systems whose dynamics are defined by linear differential equations. We formulate the synthesis problem as finding the conditions upon which a controller should switch the behavior of the system from one "mode" to another in order to avoid a set of bad states, and propose an abstract algorithm which solves the problem by an iterative computation of reachable states. We have implemented a concrete version of the algorithm, which uses a new approximation scheme for reachability analysis of linear systems.
Stable Flocking of Mobile Agents, Part II: Dynamic Topology
 In IEEE Conference on Decision and Control
, 2003
"... This is the second of a twopart paper, investigating the stability properties of a system of multiple mobile agents with double integrator dynamics. In this second part, we allow the topology of the control interconnections between the agents in the group to vary with time. Specifically, the contro ..."
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Cited by 99 (4 self)
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This is the second of a twopart paper, investigating the stability properties of a system of multiple mobile agents with double integrator dynamics. In this second part, we allow the topology of the control interconnections between the agents in the group to vary with time. Specifically, the control law of an agent depends on the state of a set of agents that are within a certain neighborhood around it. As the agents move around, this set changes giving rise to a dynamic control interconnection topology and a switching control law. This control law consists of a a combination of attractive/repulsive and alignment forces. The former ensure collision avoidance and cohesion of the group and the latter result to all agents attaining a common heading angle, exhibiting flocking motion. Despite the use of only local information and the time varying nature of agent interaction which affects the local controllers, flocking motion can still be established, as long as connectivity in the neighboring graph is maintained.
Stability And Robustness For Hybrid Systems
, 1996
"... Stability and robustness issues for hybrid systems are considered in this paper. General stability results that are extensions of classical Lyapunov theory have recently been formulated. However, these results are in general not straightforward to apply due to the following reasons. First, a search ..."
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Cited by 84 (6 self)
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Stability and robustness issues for hybrid systems are considered in this paper. General stability results that are extensions of classical Lyapunov theory have recently been formulated. However, these results are in general not straightforward to apply due to the following reasons. First, a search for multiple Lyapunov functions must be performed. However, existing theory does not unveil how to find such functions. Secondly, if the most general stability result is applied, knowledge about the continuous trajectory is required, at least at some time instants. Because of these drawbacks stronger conditions for stability are suggested, in which case it is shown that the search for Lyapunov functions can be formulated as a linear matrix inequality (LMI) problem for hybrid systems consisting of linear subsystems. Additionally, it is shown how robustness properties can be achieved when the Lyapunov functions are given. Specifically, it is described how to determine permitted switch regions ...