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See detailL2-gain of double integrators with saturation nonlinearity
Goncalves, Jorge UL

in Proceedings of the 15th IFAC World Congress (2002)

This paper uses quadratic surface Lyapunov functions to efficiently check if a double integrator in feedback with a saturation nonlinearity has L2-gain less than gamma > 0. We show that for many of such ... [more ▼]

This paper uses quadratic surface Lyapunov functions to efficiently check if a double integrator in feedback with a saturation nonlinearity has L2-gain less than gamma > 0. We show that for many of such systems, the L2-gain is non-conservative in the sense that they are approximately equal to the low erbound obtained by replacing the saturation with a constant gain of 1. These results allow the use of classical analysis tools like mu -analysis or IQCs to analyze systems with double integrators and saturations, including servo systems like some mechanical systems, satellites, hard-disks, CD players, etc. [less ▲]

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See detailQuadratic Surface Lyapunov Functions in the Analysis of Feedback Systems with Double Integrators and Saturations
Goncalves, Jorge UL

in Proceedings of the 10th IEEE Mediterranean Conference on Control and Automation (2002)

Many systems like servo systems, satellites, harddisks, and CD players, can be modeled as linear systems with a single integrator and a saturation. Many times, such systems are controlled with a PI ... [more ▼]

Many systems like servo systems, satellites, harddisks, and CD players, can be modeled as linear systems with a single integrator and a saturation. Many times, such systems are controlled with a PI controller resulting in a feedback interconnection with a double integrator and a saturation. In this paper, we propose a loop transformation that results in bounded operators so that classical analysis tools like mu􀀀-analysis or IQCs can be applied. In order to show boundedness of all operators, we use quadratic surface Lyapunov functions to efficiently check if a double integrator in feedback with a saturation nonlinearity has L2 􀀀-gain less than gamma > 0 􀀀. We show that for many of such systems, the L2 􀀀-gain is non-conservative in the sense that this is approximately equal to the lower bound obtained by replacing the saturation with a constant gain of . [less ▲]

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See detailGlobal analysis of piecewise linear systems using impact maps and quadratic surface Lyapunov functions
Goncalves, Jorge UL; Megretski, A.; Dahleh, M. A.

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 ▲]

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See detailQuadratic Surface Lyapunov Functions in Global Stability Analysis of Saturation Systems
Goncalves, Jorge UL

in Proceedings of the American Control Conference (2001)

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See detailGlobal stability analysis of on/off systems
Goncalves, Jorge UL

in Proceedings of the 39th IEEE Conference on Decision and Control (2000)

This paper considers quadratic surface Lyapunov functions in the study of global stability analysis of on/off systems (OFS), including those OFS with unstable nonlinearity sectors. In previous work ... [more ▼]

This paper considers quadratic surface Lyapunov functions in the study of global stability analysis of on/off systems (OFS), including those OFS with unstable nonlinearity sectors. In previous work, quadratic surface Lyapunov functions were successfully applied to prove global asymptotic stability of limit cycles of relay feedback systems. In this work, we show that these ideas can be used to prove global asymptotic stability of equilibrium points of piecewise linear systems (PLS). We present conditions in the form of LMI that, when satisfied, guarantee global asymptotic stability of an equilibrium point. A large number of examples was successfully proven globally stable. These include systems with an unstable affine linear subsystem, systems of relative degree larger than one and of high dimension, and systems with unstable nonlinearity sectors, for which all classical fail to analyze. In fact, existence of an example with a globally stable equilibrium point that could not be successfully analyzed with this new methodology is still an open problem. This work opens the door to the possibility that more general PLS can be systematically globally analyzed using quadratic surface Lyapunov functions. [less ▲]

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See detailGlobal stability of relay feedback systems
Goncalves, Jorge UL; Megretski, A.; Dahleh, M. A.

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 ▲]

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See detailSemi-global analysis of relay feedback systems
Goncalves, Jorge UL; Megretski, A.; Dahleh, M. A.

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 ▲]

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See detailLinear smoothing over arbitrary polytopes
Francis, Amrita; Natarajan, Sundararajan; Ortiz-Bernardin, Alejandro et al

Scientific Conference (n.d.)

The conventional constant strain smoothing technique yields less accurate solutions that other techniques such as the conventional polygonal finite element method [1, 2]. In this work, we propose a linear ... [more ▼]

The conventional constant strain smoothing technique yields less accurate solutions that other techniques such as the conventional polygonal finite element method [1, 2]. In this work, we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex poly- topes. The method relies on sub-division of the polytope into simplical subcells; however instead of using a constant smoothing function, we employ a linear smoothing function over each subcell. This gives a new definition for the strain to compute the stiffness matrix. The convergence properties and accuracy of the proposed scheme are discussed by solving few benchmark problems. Numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the optimal convergence rate as in traditional quadrilateral and hexahedral finite elements. The accuracy is also improved, and all the methods tested pass the patch test to machine precision. [less ▲]

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