A technique to generate (periodic or nonperiodic) oscillations systematically in first-order, con... more A technique to generate (periodic or nonperiodic) oscillations systematically in first-order, continuous-time systems via a nonlinear function of the state, delayed by a certain time d, is proposed. This technique consists in choosing a nonlinear function of the delayed state with some passivity properties, tuning a gain to ensure that all the equilibrium points of the closed-loop system be unstable, and then imposing conditions on the closed-loop system to be semipassive. We include several typical examples to illustrate the effectiveness of the proposed technique, with which we can generate a great variety of chaotic attractors. We also include a physical example built with a simple electronic circuit that, after applying the proposed technique, displays a similar behavior to the logistic map.
Abstract The nonsmooth phenomena such as backlash and Coulomb friction, often occurring in mechan... more Abstract The nonsmooth phenomena such as backlash and Coulomb friction, often occurring in mechanical systems, typically produce undesired inaccuracies, oscillations and instability thereby degrading the system performance. The present paper addresses these phenomena in a benchmark two-mass system composed of motor and load subsystems joined by an elastic shaft. Only partial measurements of the motor position are assumed to be available. The H ∞ synthesis is then developed for a class of nonsmooth systems with backlash. The effectiveness of the proposed synthesis and its robustness features in the presence of friction forces and backlash effects are supported by an experimental study made for an industrial emulator.
Abstract This paper focuses on the onset of rotating waves in a network of pendulum-like oscillat... more Abstract This paper focuses on the onset of rotating waves in a network of pendulum-like oscillators that interact via Huygens’ coupling, i.e. in oscillators that are coupled by a flexible media. Conditions for the existence and stability of rotating wave solutions in the coupled system are derived by using the Poincare method of perturbation. Additionally, the amplitude and frequency of the rotating wave solution are also analytically determined. The theoretical results are further illustrated by means of computer simulations.
Journal of Dynamic Systems, Measurement, and Control, 1982
The optimal control of a class of discrete multivariable nonlinear systems given by: xk+1 = a (xk... more The optimal control of a class of discrete multivariable nonlinear systems given by: xk+1 = a (xk) + B (xk) uk, yk = C xk, is analyzed. A closed-loop structure is obtained with the proposed performance index. The addition of numerical integrators to the output error and the design of an optimal control law for the resultant augmented system lead to a very robust control structure. The performance of this control law is evaluated by applying it to a simulated continuous culture fermentation process.
IMA Journal of Mathematical Control and Information, 1999
IMA Journal of Mathematical Control & Information (1999) 16, 23-41 Stability of discrete nonl... more IMA Journal of Mathematical Control & Information (1999) 16, 23-41 Stability of discrete nonlinear systems under nonvanishing perturbations: application to a nonlinear model-matching problem CÉSAR CRUZ-HERNÂNDEZt Electronics & Telecommunications Department, CICESE, ...
IMA Journal of Mathematical Control and Information, 2001
IMA Journal of Mathematical Control and Information (2001) 18, 479489 ... Stability robustness o... more IMA Journal of Mathematical Control and Information (2001) 18, 479489 ... Stability robustness of linearizing controllers with state estimation for discrete-time nonlinear systems ... C ESAR C RUZ -H ERNANDEZ AND J OAQUIN A LVAREZ -G ALLEGOS
Proceedings of 1994 American Control Conference - ACC '94
The dynamic behavior of a single pendulum con-trolled by a Proportional-Derivative (PD) com-pensa... more The dynamic behavior of a single pendulum con-trolled by a Proportional-Derivative (PD) com-pensator is analyzed. By using the Melnikov the-ory it is shown that the pendulum may exhibit a chaotic behavior when the tracking signal is peri-odic and the dissipation and ...
An analysis of the dynamical behavior of second-order linear plants controlled with conventional ... more An analysis of the dynamical behavior of second-order linear plants controlled with conventional controllers is presented. The control signal is passed through classical non-linearities before being applied to the plant. Existence, stability, and some bifurcations occurring in some control schemes are analyzed. Existence of periodic and homoclinic orbits is also discussed. Using the Melnikov/Smale and Genesio/Tesi methods some conditions about existence of strange invariant sets are established. Numerical and experimental results support the analysis presented.
This work is focused on a particular type of synchronization observed in coupled systems interact... more This work is focused on a particular type of synchronization observed in coupled systems interacting via unidirectional coupling, namely mixed synchronization: part of the state variables of the coupled systems achieve complete synchronization, whereas the remaining state variables exhibit anti-phase synchronization. This chapter paper presents a modified master–slave scheme, in which the master system interacts with the slave system via a second order dynamic coupling. In the analysis, the stability of the mixed synchronous solution is investigated by using the well-known master stability function approach. A classical chaotic system, namely the Lorenz system, is considered as a particular example.
It is shown how a very simple procedure of time reversal can solve, in a much easier way than oth... more It is shown how a very simple procedure of time reversal can solve, in a much easier way than other techniques, an important problem arising in some control strategies of two conventional DC-to-DC switched power converters used to generate an alternate voltage signal: the boost and the buck-boost circuits. This problem is related to the non-minimum phase nature of the capacitor voltage normally used as output in these devices, which compels t6control indirectly this variable by controlling the inductor current. A stable model that generates an adequate reference for this signal is proposed, which can be used in-line or in batch mode. The procedure is illustrated by numerical simulations using a sliding -mode controller proposed elsewhere.
In this paper, the onset of mixed synchronization in a triplet of mechanical oscillators interact... more In this paper, the onset of mixed synchronization in a triplet of mechanical oscillators interacting via Huygens’ coupling, i.e. a suspended rigid bar, is investigated. The term mixed synchronization refers to the case where two oscillators synchronize inphase, while the third oscillator synchronizes in anti-phase with respect to the other two oscillators. Sufficient conditions for the onset of mixed synchronization are derived by using the Poincaré method and the obtained analytic results are complemented with numerical simulations. Ultimately, it is demonstrated that two synchronous modes can be observed in the coupled system at the same time namely, in-phase and anti-phase synchronization.
This paper presents a strategy for detecting anomalies—behavior that is different from nominal—in... more This paper presents a strategy for detecting anomalies—behavior that is different from nominal—in systems with periodic outputs. Different from classical detectors available in the literature which require the computation of a residual, the detection algorithms proposed here are based on the concept of the Poincaré map and on the notion of cyclic group. The performance of the detectors is illustrated by numerical simulations and validated by experiments on an actuated mass-spring-damper oscillator.
Abstract In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-... more Abstract In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-equilibrium point on a discontinuity surface is analyzed. The discontinuous system is piecewise linear and approximated to a non-smooth continuous system. The discontinuous term is represented by a sign function that is replaced by a saturation function with high slope. Some of the conditions that determine the chaotic behavior of the approximate system are formally established. Besides, the convergence of its chaotic solutions to those of the discontinuous system is shown. Several bifurcation diagrams of both systems show the similarity of their dynamical behavior in a wide parameter range, and particularly for a parameter region determined from the application of the Melnikov technique to non-smooth systems, where a chaotic behavior can be displayed.
Abstract The existence of Smale horseshoes into the dynamics of a two degree of freedom (2-DOF) r... more Abstract The existence of Smale horseshoes into the dynamics of a two degree of freedom (2-DOF) robot manipulator driven by a PD controller is proved. The PD controller is considered as a non-Hamiltonian perturbation of an undriven double pendulum, then a technique developed by Holmes and Marsden (1982). which uses a combination of a reduction scheme and the method of Melnikov with an energy balance argument, is applied.
A technique to generate (periodic or nonperiodic) oscillations systematically in first-order, con... more A technique to generate (periodic or nonperiodic) oscillations systematically in first-order, continuous-time systems via a nonlinear function of the state, delayed by a certain time d, is proposed. This technique consists in choosing a nonlinear function of the delayed state with some passivity properties, tuning a gain to ensure that all the equilibrium points of the closed-loop system be unstable, and then imposing conditions on the closed-loop system to be semipassive. We include several typical examples to illustrate the effectiveness of the proposed technique, with which we can generate a great variety of chaotic attractors. We also include a physical example built with a simple electronic circuit that, after applying the proposed technique, displays a similar behavior to the logistic map.
Abstract The nonsmooth phenomena such as backlash and Coulomb friction, often occurring in mechan... more Abstract The nonsmooth phenomena such as backlash and Coulomb friction, often occurring in mechanical systems, typically produce undesired inaccuracies, oscillations and instability thereby degrading the system performance. The present paper addresses these phenomena in a benchmark two-mass system composed of motor and load subsystems joined by an elastic shaft. Only partial measurements of the motor position are assumed to be available. The H ∞ synthesis is then developed for a class of nonsmooth systems with backlash. The effectiveness of the proposed synthesis and its robustness features in the presence of friction forces and backlash effects are supported by an experimental study made for an industrial emulator.
Abstract This paper focuses on the onset of rotating waves in a network of pendulum-like oscillat... more Abstract This paper focuses on the onset of rotating waves in a network of pendulum-like oscillators that interact via Huygens’ coupling, i.e. in oscillators that are coupled by a flexible media. Conditions for the existence and stability of rotating wave solutions in the coupled system are derived by using the Poincare method of perturbation. Additionally, the amplitude and frequency of the rotating wave solution are also analytically determined. The theoretical results are further illustrated by means of computer simulations.
Journal of Dynamic Systems, Measurement, and Control, 1982
The optimal control of a class of discrete multivariable nonlinear systems given by: xk+1 = a (xk... more The optimal control of a class of discrete multivariable nonlinear systems given by: xk+1 = a (xk) + B (xk) uk, yk = C xk, is analyzed. A closed-loop structure is obtained with the proposed performance index. The addition of numerical integrators to the output error and the design of an optimal control law for the resultant augmented system lead to a very robust control structure. The performance of this control law is evaluated by applying it to a simulated continuous culture fermentation process.
IMA Journal of Mathematical Control and Information, 1999
IMA Journal of Mathematical Control & Information (1999) 16, 23-41 Stability of discrete nonl... more IMA Journal of Mathematical Control & Information (1999) 16, 23-41 Stability of discrete nonlinear systems under nonvanishing perturbations: application to a nonlinear model-matching problem CÉSAR CRUZ-HERNÂNDEZt Electronics & Telecommunications Department, CICESE, ...
IMA Journal of Mathematical Control and Information, 2001
IMA Journal of Mathematical Control and Information (2001) 18, 479489 ... Stability robustness o... more IMA Journal of Mathematical Control and Information (2001) 18, 479489 ... Stability robustness of linearizing controllers with state estimation for discrete-time nonlinear systems ... C ESAR C RUZ -H ERNANDEZ AND J OAQUIN A LVAREZ -G ALLEGOS
Proceedings of 1994 American Control Conference - ACC '94
The dynamic behavior of a single pendulum con-trolled by a Proportional-Derivative (PD) com-pensa... more The dynamic behavior of a single pendulum con-trolled by a Proportional-Derivative (PD) com-pensator is analyzed. By using the Melnikov the-ory it is shown that the pendulum may exhibit a chaotic behavior when the tracking signal is peri-odic and the dissipation and ...
An analysis of the dynamical behavior of second-order linear plants controlled with conventional ... more An analysis of the dynamical behavior of second-order linear plants controlled with conventional controllers is presented. The control signal is passed through classical non-linearities before being applied to the plant. Existence, stability, and some bifurcations occurring in some control schemes are analyzed. Existence of periodic and homoclinic orbits is also discussed. Using the Melnikov/Smale and Genesio/Tesi methods some conditions about existence of strange invariant sets are established. Numerical and experimental results support the analysis presented.
This work is focused on a particular type of synchronization observed in coupled systems interact... more This work is focused on a particular type of synchronization observed in coupled systems interacting via unidirectional coupling, namely mixed synchronization: part of the state variables of the coupled systems achieve complete synchronization, whereas the remaining state variables exhibit anti-phase synchronization. This chapter paper presents a modified master–slave scheme, in which the master system interacts with the slave system via a second order dynamic coupling. In the analysis, the stability of the mixed synchronous solution is investigated by using the well-known master stability function approach. A classical chaotic system, namely the Lorenz system, is considered as a particular example.
It is shown how a very simple procedure of time reversal can solve, in a much easier way than oth... more It is shown how a very simple procedure of time reversal can solve, in a much easier way than other techniques, an important problem arising in some control strategies of two conventional DC-to-DC switched power converters used to generate an alternate voltage signal: the boost and the buck-boost circuits. This problem is related to the non-minimum phase nature of the capacitor voltage normally used as output in these devices, which compels t6control indirectly this variable by controlling the inductor current. A stable model that generates an adequate reference for this signal is proposed, which can be used in-line or in batch mode. The procedure is illustrated by numerical simulations using a sliding -mode controller proposed elsewhere.
In this paper, the onset of mixed synchronization in a triplet of mechanical oscillators interact... more In this paper, the onset of mixed synchronization in a triplet of mechanical oscillators interacting via Huygens’ coupling, i.e. a suspended rigid bar, is investigated. The term mixed synchronization refers to the case where two oscillators synchronize inphase, while the third oscillator synchronizes in anti-phase with respect to the other two oscillators. Sufficient conditions for the onset of mixed synchronization are derived by using the Poincaré method and the obtained analytic results are complemented with numerical simulations. Ultimately, it is demonstrated that two synchronous modes can be observed in the coupled system at the same time namely, in-phase and anti-phase synchronization.
This paper presents a strategy for detecting anomalies—behavior that is different from nominal—in... more This paper presents a strategy for detecting anomalies—behavior that is different from nominal—in systems with periodic outputs. Different from classical detectors available in the literature which require the computation of a residual, the detection algorithms proposed here are based on the concept of the Poincaré map and on the notion of cyclic group. The performance of the detectors is illustrated by numerical simulations and validated by experiments on an actuated mass-spring-damper oscillator.
Abstract In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-... more Abstract In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-equilibrium point on a discontinuity surface is analyzed. The discontinuous system is piecewise linear and approximated to a non-smooth continuous system. The discontinuous term is represented by a sign function that is replaced by a saturation function with high slope. Some of the conditions that determine the chaotic behavior of the approximate system are formally established. Besides, the convergence of its chaotic solutions to those of the discontinuous system is shown. Several bifurcation diagrams of both systems show the similarity of their dynamical behavior in a wide parameter range, and particularly for a parameter region determined from the application of the Melnikov technique to non-smooth systems, where a chaotic behavior can be displayed.
Abstract The existence of Smale horseshoes into the dynamics of a two degree of freedom (2-DOF) r... more Abstract The existence of Smale horseshoes into the dynamics of a two degree of freedom (2-DOF) robot manipulator driven by a PD controller is proved. The PD controller is considered as a non-Hamiltonian perturbation of an undriven double pendulum, then a technique developed by Holmes and Marsden (1982). which uses a combination of a reduction scheme and the method of Melnikov with an energy balance argument, is applied.
Uploads