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Rank properties of Poincaré maps for hybrid systems with applications to bipedal walking
 in Hybrid Systems: Computation and Control, pp.151–160
, 2010
"... The equivalence of the stability of periodic orbits with the stability of fixed points of a Poincare ́ map is a wellknown fact for smooth dynamical systems. In particular, the eigenvalues of the linearization of a Poincare ́ map can be used to determine the stability of periodic orbits. The main o ..."
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The equivalence of the stability of periodic orbits with the stability of fixed points of a Poincare ́ map is a wellknown fact for smooth dynamical systems. In particular, the eigenvalues of the linearization of a Poincare ́ map can be used to determine the stability of periodic orbits. The main objective of this paper is to study the properties of Poincare ́ maps for hybrid systems as they relate to the stability of hybrid periodic orbits. The main result is that the properties of Poincare ́ maps for hybrid systems are fundamentally different from those for smooth systems, especially with respect to the linearization of the Poincare ́ map and its eigenvalues. In particular, the linearization of any Poincare ́ map for a smooth dynamical system will have one trivial eigenvalue equal to 1 that does not affect the stability of the orbit. For hybrid systems, the trivial eigenvalues are equal to 0 and the number of trivial eigenvalues is bounded above by dimensionality differences between the different discrete domains of the hybrid system and the rank of the reset maps. Specifically, if n is the minimum dimension of the domains of the hybrid system, then the Poincare ́ map on a domain of dimension m ≥ n results in at least m − n + 1 trivial 0 eigenvalues, with the remaining eigenvalues determining the stability of the hybrid periodic orbit. These results will be demonstrated on a nontrivial multidomain hybrid system: a planar bipedal robot with knees.
An Iterative Approach to Calculating Dynamic ATC
 Proc. of Bulk Power System Dynamics and Control IV  Restructuring
, 1998
"... Abstract — Stability limits place restrictions on the available transfer capability (ATC) of power systems. Calculation of these limits is therefore very important, but has traditionally been quite difficult. This paper proposes an iterative algorithm for determining parameter values which result in ..."
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Abstract — Stability limits place restrictions on the available transfer capability (ATC) of power systems. Calculation of these limits is therefore very important, but has traditionally been quite difficult. This paper proposes an iterative algorithm for determining parameter values which result in marginal stability of a system. (A system is marginally stable for a particular disturbance if the postdisturbance trajectory lies on the stability boundary.) A knowledge of the critical parameter values allows the dynamic ATC to be determined. The algorithm is based on the GaussNewton solution of a nonlinear leastsquares problem. This solution process uses trajectory sensitivities. I.
Estimating Wind Turbine Parameters and Quantifying Their Effects on Dynamic Behavior
"... IEEE does not in any way imply IEEE endorsement of any of Power Systems ..."
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IEEE does not in any way imply IEEE endorsement of any of Power Systems
Stability analysis and control of nonlinear phenomena in boost converter using modelbased Takagi–Sugeno fuzzy approach
 IEEE Trans. Circuits Syst. I, Regul. Pap
, 2010
"... Abstract—The application of a novel TakagiSugeno fuzzy modelbased approach to prohibit the onset of subharmonic instabilities in dcdc power electronic converters is presented in this paper. The use of a modelbased fuzzy approach derived from an average mathematical model to control the nonlinear ..."
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Abstract—The application of a novel TakagiSugeno fuzzy modelbased approach to prohibit the onset of subharmonic instabilities in dcdc power electronic converters is presented in this paper. The use of a modelbased fuzzy approach derived from an average mathematical model to control the nonlinearities in power electronic converters has been reported in the literature, but this is known to act as a lowpass filter thus ignoring all nonlinear phenomena occurring at converter clock frequency. This paper shows how converter fastscale instabilities can be captured by extending the TS fuzzy modeling concept to nonsmooth dynamical systems by combining the TS fuzzy modeling technique with nonsmooth Lyapunov stability theory. The new method is applied to the current mode controlled boost converter to demonstrate how the stability analysis can be directly applied by formularizing the stability conditions as numerical problem using Linear Matrix Inequalities (LMIs). Based on this methodology, a new type of switching fuzzy controller is proposed. The resulting control scheme is able to maintain the stable periodone behavior of the converter over a wide range of operating conditions while improving the transient response of the circuit. Index Terms—TakagiSugeno fuzzy approach, nonsmooth Lyapunov theory, Linear Matrix Inequality, DCDC Converter.
Detection and stabilization of hybrid periodic orbits of passive running robots
 In Mechatronics and Robotics
, 2004
"... Abstract — In this paper a new algorithm to detect hybrid periodic orbits of autonomous hybrid dynamical system is developed. Conventional Newton algorithm is modified so that it suits to the analysis of Poincare ́ return map of hybrid dynamical systems that include multiple phases (modes) and discr ..."
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Abstract — In this paper a new algorithm to detect hybrid periodic orbits of autonomous hybrid dynamical system is developed. Conventional Newton algorithm is modified so that it suits to the analysis of Poincare ́ return map of hybrid dynamical systems that include multiple phases (modes) and discrete jumps. Then, the algorithm is applied to a specific example; planar onelegged robot model having a springy leg and a compliant hip joint. With the algorithm, passive running gaits of the onelegged robot are automatically detected for various parameter sets and initial conditions. The analysis of the characteristic multiplier of the return map revealed the stability and the bifurcation of the passive running gaits. Two kinds of controllers that achieve orbital stabilization are presented. A similarity is found between the detection algorithm and the stabilizing controller. The algorithm can be applied to any kinds of the robots (e.g. walking robot). I.
Computing Descent Direction of MTL Robustness for NonLinear Systems
"... Abstract—The automatic analysis of transient properties of nonlinear dynamical systems is a challenging problem. The problem is even more challenging when complex statespace and timing requirements must be satisfied by the system. Such complex requirements can be captured by Metric Temporal Logic ( ..."
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Abstract—The automatic analysis of transient properties of nonlinear dynamical systems is a challenging problem. The problem is even more challenging when complex statespace and timing requirements must be satisfied by the system. Such complex requirements can be captured by Metric Temporal Logic (MTL) specifications. The problem of finding system behaviors that do not satisfy an MTL specification is referred to as MTL falsification. This paper presents an approach for improving stochastic MTL falsification methods by performing local search in the set of initial conditions. In particular, MTL robustness quantifies how correct or wrong is a system trajectory with respect to an MTL specification. Positive values indicate satisfaction of the property while negative values indicate falsification. A stochastic falsification method attempts to minimize the system’s robustness with respect to the MTL property. Given some arbitrary initial state, this paper presents a method to compute a descent direction in the set of initial conditions, such that the new system trajectory gets closer to the unsafe set of behaviors. This technique can be iterated in order to converge to a local minimum of the robustness landscape. The paper demonstrates the applicability of the method on some challenging nonlinear systems from the literature. I.
Limitinduced stable limit cycles in power systems
 In Proceedings of the IEEE Saint Petesburg PowerTech
, 2005
"... Abstract — Heavily loaded power systems are susceptible to Hopf bifurcations, and consequent oscillatory instability. The onset of instability can be predicted by small disturbance (eigenvalue) analysis, but the ensuring behaviour may depend strongly on nonlinearities within the system. In particula ..."
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Abstract — Heavily loaded power systems are susceptible to Hopf bifurcations, and consequent oscillatory instability. The onset of instability can be predicted by small disturbance (eigenvalue) analysis, but the ensuring behaviour may depend strongly on nonlinearities within the system. In particular, physical limits place bounds on the divergent behaviour of states. This paper explores the situation where generator fieldvoltage limits capture behaviour, giving rise to a stable (though nonsmooth) limit cycle. It is shown that shooting methods can be adapted to solve for such nonsmooth limitinduced limit cycles. By continuing branches of limitinduced and smooth limit cycles, the paper established the coexistence of equilibria, smooth and nonsmooth limit cycles. Furthermore, it is shown that when branches of limitinduced and smooth limit cycles merge, the limit cycles are annihilated at a grazing bifurcation. Index Terms — Limit cycles, piecewise smooth dynamics, hybrid systems, shooting methods, Hopf bifurcations, continuation methods. I.
Dynamics and stability issues of a singleinductor dualswitching DCDC converter
 IEEE Trans. Circ. Syst. I 2010
"... Abstract—A singleinductor twoinput twooutput power electronic dc–dc converter can be used to regulate two generally nonsymmetric positive and negative outputs by means of a pulsewidth modulation with a double voltage feedback. This paper studies the dynamic behavior of this system. First, the o ..."
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Abstract—A singleinductor twoinput twooutput power electronic dc–dc converter can be used to regulate two generally nonsymmetric positive and negative outputs by means of a pulsewidth modulation with a double voltage feedback. This paper studies the dynamic behavior of this system. First, the operation modes and the steadystate properties of the converter are addressed, and, then, a stability analysis that includes both the power stage and control parameters is carried out. Different bifurcations are determined from the averaged model and from the discretetime model. The Routh–Hurwitz criterion is used to obtain the stability regions of the averaged (slowscale) dynamics in the design parameter space, and a discretetime approach is used to obtain more accurate results and to detect possible (fastscale) subharmonic oscillations. Experimental measurements were taken from a system prototype to confirm the analytical results and numerical simulations. Some possible nonsmooth bifurcations due to the change in the switching patterns are also illustrated. Index Terms—Bifurcations, dc–dc converters, dual output, nonlinear dynamics, power electronics, single inductor, stability analysis. I.
Nonuniqueness in reverse time of hybrid systems
 in Hybrid Systems: Computation & Control, M. Morari and
, 2005
"... Abstract. Under standard Lipschitz conditions, trajectories of systems described by ordinary differential equations are well defined in both forward and reverse time. (The flow map is invertible.) However for hybrid systems, uniqueness of trajectories in forward time does not guarantee flowmap inve ..."
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Abstract. Under standard Lipschitz conditions, trajectories of systems described by ordinary differential equations are well defined in both forward and reverse time. (The flow map is invertible.) However for hybrid systems, uniqueness of trajectories in forward time does not guarantee flowmap invertibility, allowing nonuniqueness in reverse time. The paper establishes a necessary and sufficient condition that governs invertibility through events. It is shown that this condition is equivalent to requiring reversetime trajectories to transversally encounter event triggering hypersurfaces. This analysis motivates a homotopy algorithm that traces a onemanifold of initial conditions that give rise to trajectories which all reach a common point at the same time. 1
Inverse Problems in Power Systems
"... Abstract — Large disturbances in power systems often initiate complex interactions between continuous dynamics and discrete events. Such behaviour can be modeled in a systematic way by a set of differentialalgebraic equations, modified to incorporate impulse (state reset) action and constraint swit ..."
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Abstract — Large disturbances in power systems often initiate complex interactions between continuous dynamics and discrete events. Such behaviour can be modeled in a systematic way by a set of differentialalgebraic equations, modified to incorporate impulse (state reset) action and constraint switching. The paper presents a practical objectoriented approach to implementing the DAIS model. The systematic nature of the DAIS model enables efficient computation of trajectory sensitivities, which in turn facilitate algorithms for solving inverse problems. The paper outlines a number of inverse problems, including parameter uncertainty, parameter estimation, boundary value problems, bordercollision bifurcations, locating critically stable trajectories, and optimal control.