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Papers by jorge goncalves

This paper presents an entirely new constructive global analysis methodology for a class of hybri... more This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS.

This paper starts by presenting local stability conditions for limit cycles of piecewise linear s... more This paper starts by presenting local stability conditions for limit cycles of piecewise linear systems (PLS), based on analyzing the linear part of Poincaré maps. Local stability guarantees the existence of an asymptotically stable neighborhood around the limit cycle. However, tools to characterize such neighborhood do not exist. This work gives conditions in the form of LMIs that guarantee asymptotic stability of PLS in a reasonably large region around a limit cycle, based on recent results on impact maps and surface Lyapunov functions (SuLF). These are exemplified with a biological application: a 4 th -order neural oscillator, also used in many robotics applications like, for example, juggling and locomotion.

IEEE Transactions on Automatic Control, 2005

Page 1. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 11, NOVEMBER 2005 1877 The system (A... more Page 1. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 11, NOVEMBER 2005 1877 The system (A; B; C) has vector relative degree f1; 2g with invariant zero equal to 0 and, thus, satisfies H2 and moreover ...

IEEE Transactions on Automatic Control, 2003

In this paper we develop an entirely new constructive global analysis methodology for a class of ... more In this paper we develop an entirely new constructive global analysis methodology for a class of hybrid systems known as Piecewise Linear Systems (PLS). This methodology consists in inferring global properties of PLS solely by studying their behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface, by constructing quadratic Lyapunov functions on switching surfaces. We found that an impact map induced by an LTI flow between two switching surfaces can be represented as a linear transformation analytically parameterized by a scalar function of the state. This representation of impact maps allows the search for quadratic surface Lyapunov functions to be done by simply solving a set of LMIs. Global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS can this way be efficiently checked. These new results were successfully applied to certain classes of PLS. Although this analysis methodology yields only sufficient criteria of stability, it has shown to be very successful in globally analyzing a large number of examples with a locally stable limit cycle or equilibrium point. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied.

IEEE Transactions on Automatic Control, 2001

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditi... more For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. This paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincaré maps associated with RFS. These results are based on the discovery that a typical Poincaré map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions. for the Spring of 1993. He has held consulting positions with several companies in the U.S. and abroad. He is the coauthor (with Ignacio Diaz-Bobillo) of the book Control of Uncertain Systems: A Linear Programming Approach (Englewood Cliffs, NJ: Prentice-Hall, 1995, and the coauthor (with Nicola Elia) of the book Computational Methods for Controller Design (New York: Springer-Verlag, 1998). His interests include robust control and identification, the development of computational methods for linear and nonlinear controller design, and applications of feedback control in several disciplines including material manufacturing and modeling of biological systems. Dr. Dahleh was the recipient of the Ralph Budd Award in 1987 for the best thesis at Rice University, the George Axelby Outstanding Paper Award (paper coauthored with

This paper presents semi-global sufficient stability conditions of limit cycles for relay feedbac... more This paper presents semi-global sufficient stability conditions of limit cycles for relay feedback systems. Local stability conditions exist. These are based on analyzing the linear part of the Poincare map. We know that when a certain limit cycle satisfies those local conditions, a neighborhood around the limit cycle exists such that any trajectory starting in this neighborhood converges to the

IEEE Transactions on Automatic Control, 2005

IEEE Transactions on Automatic Control, 2003

IEEE Transactions on Automatic Control, 2001