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Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions Goncalves, Jorge ; ; in IEEE Transactions on Automatic Control (2003), 48(12), 2089-2106 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 ... [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. [less ▲] Detailed reference viewed: 99 (1 UL)Global stability of relay feedback systems Goncalves, Jorge ; ; in IEEE Transactions on Automatic Control (2001), 46(4), 550--562 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 ... [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. [less ▲] Detailed reference viewed: 55 (0 UL)Global analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions Goncalves, Jorge ; ; in Proceedings of the European Control Conference (ECC) 2001 (2001) 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 ... [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. [less ▲] Detailed reference viewed: 43 (0 UL)Global stability of relay feedback systems Goncalves, Jorge ; ; in Proceedings of the IEEE American Control Conference (2000 ACC) (2000) 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 ... [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 non-existent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that guarantee global asymptotic stability of a limit cycle induced by a relay with hysteresis in feedback with an LTI stable system. The analysis is based on finding global quadratic Lyapunov functions for a Poincare map associated with the RFS. We found that a typical Poincare 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. The search for globally quadratic Lyapunov functions is then done by solving a set of LMIs. Most examples of RFS analyzed by the authors were proven globally stable. Systems analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. This leads us to believe that quadratic stability of associated Poincare maps is common in RFS. [less ▲] Detailed reference viewed: 53 (0 UL)Semi-global analysis of relay feedback systems Goncalves, Jorge ; ; in Proceedings of the 37th IEEE Conference on Decision and Control (1998) 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 ... [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 limit cycle as time goes to infinity. However, tools to characterize this neighborhood do not exist. In this work, we present conditions, in the form of linear matrix inequalities (LMI), that guarantee the stability of a limit cycle in a reasonably large set around it. These results differ from previous local results as they take into account the high order terms of the Poincare map. [less ▲] Detailed reference viewed: 59 (0 UL) |
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