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Almost global convergence to practical synchronization in the generalized Kuramoto model on networks over the n-sphere Markdahl, Johan ; Proverbio, Daniele ; et al in Communications Physics (2021), 4 From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we ... [more ▼] From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we generalise to networks over n-dimensional spheres. We show that, for almost all initial conditions, the sphere model converges to a set with small diameter if the model parameters satisfy a given bound. Moreover, for even n, a special case of the generalized model can achieve phase synchronization with nonidentical frequency parameters. These results contrast with the standard n = 1 Kuramoto model, which is multistable (i.e., has multiple equilibria), and converges to phase synchronization only if the frequency parameters are identical. Hence, this paper shows that the generalized network Kuramoto models for n ≥ 2 displays more coherent and predictable behavior than the standard n = 1 model, a desirable property both in flocks of animals and for robot control. [less ▲] Detailed reference viewed: 50 (0 UL)High-Dimensional Kuramoto Models on Stiefel Manifolds Synchronize Complex Networks Almost Globally Markdahl, Johan ; ; Goncalves, Jorge in Automatica (2020) Detailed reference viewed: 73 (3 UL)Robust synchronization of heterogeneous robot swarms on the sphere Markdahl, Johan ; Proverbio, Daniele ; Goncalves, Jorge in 2020 59th IEEE Conference on Decision and Control (CDC) (2020) Synchronization on the sphere is important to certain control applications in swarm robotics. Of recent interest is the Lohe model, which generalizes the Kuramoto model from the circle to the sphere. The ... [more ▼] Synchronization on the sphere is important to certain control applications in swarm robotics. Of recent interest is the Lohe model, which generalizes the Kuramoto model from the circle to the sphere. The Lohe model is mainly studied in mathematical physics as a toy model of quantum synchronization. The model makes few assumptions, wherefore it is well-suited to represent a swarm. Previous work on this model has focused on the cases of complete and acyclic networks or the homogeneous case where all oscillator frequencies are equal. This paper concerns the case of heterogeneous oscillators connected by a non-trivial network. We show that any undesired equilibrium is exponentially unstable if the frequencies satisfy a given bound. This property can also be interpreted as a robustness result for small model perturbations of the homogeneous case with zero frequencies. As such, the Lohe model is a good choice for control applications in swarm robotics. [less ▲] Detailed reference viewed: 19 (0 UL)A multifactorial evaluation framework for gene regulatory network reconstruction Mombaerts, Laurent ; Aalto, Atte ; Markdahl, Johan et al in Foundations of Systems Biology in Engineering (2019) In the past years, many computational methods have been developed to infer the structure of gene regulatory networks from time series data. However, the applicability and accuracy presumptions of such ... [more ▼] In the past years, many computational methods have been developed to infer the structure of gene regulatory networks from time series data. However, the applicability and accuracy presumptions of such algorithms remain unclear due to experimental heterogeneity. This paper assesses the performance of recent and successful network inference strategies under a novel, multifactorial evaluation framework in order to highlight pragmatic tradeoffs in experimental design. The effects of data quantity and systems perturbations are addressed, thereby formulating guidelines for efficient resource management. Realistic data were generated from six widely used benchmark models of rhythmic and nonrhythmic gene regulatory systems with random perturbations mimicking the effect of gene knock-out or chemical treatments. Then, time series data of increasing lengths were provided to five state-of-the-art network inference algorithms representing distinctive mathematical paradigms. The performances of such network reconstruction methodologies are uncovered under various experimental conditions. We report that the algorithms do not benefit equally from data increments. Furthermore, at least for the studied rhythmic system, it is more profitable for network inference strategies to be run on long time series rather than short time series with multiple perturbations. By contrast, for the non-rhythmic systems, increasing the number of perturbation experiments yielded better results than increasing the sampling frequency. We expect that future benchmark and algorithm design would integrate such multifactorial considerations to promote their widespread and conscientious usage. [less ▲] Detailed reference viewed: 96 (8 UL)Decentralized cooperative tracking subject to motion constraints ; Markdahl, Johan ; et al in Automatica (2018), 96 This paper addresses the formation control problem, where three agents are tasked with moving an object cooperatively along a desired trajectory while also adjusting its posture to some desired attitudes ... [more ▼] This paper addresses the formation control problem, where three agents are tasked with moving an object cooperatively along a desired trajectory while also adjusting its posture to some desired attitudes, i.e. position and attitude tracking. Two decentralized control laws based on locally available information are proposed. The first control law maintains constant inter-agent distances over time, i.e. the formation of agents moves as a single rigid-body. The second control law relaxes this constraint by only maintaining similarity of the agent formation as a polygon in Euclidean space. [less ▲] Detailed reference viewed: 153 (1 UL)Dynamic controllers for column synchronization of rotation matrices: a QR-factorization approach Thunberg, Johan ; Markdahl, Johan ; Goncalves, Jorge in Automatica (2018), 93 In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the ... [more ▼] In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the corresponding k columns of the relative rotations between the agents are assumed to be available for the control design. When one specific column is considered, the problem is equivalent to synchronization on the (d-1)-dimensional unit sphere and when all the columns are considered, the problem is equivalent to synchronization on SO(d). We design dynamic control laws for these synchronization problems. The control laws are based on the introduction of auxiliary variables in combination with a QR-factorization approach. The benefit of this QR-factorization approach is that we can decouple the dynamics for the $k$ columns from the remaining d-k ones. Under the control scheme, the closed loop system achieves almost global convergence to synchronization for quasi-strong interaction graph topologies. [less ▲] Detailed reference viewed: 165 (2 UL)Almost Global Consensus on the n-Sphere Markdahl, Johan ; Thunberg, Johan ; Goncalves, Jorge in IEEE Transactions on Automatic Control (2018), 63(6), 1664-1675 This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve in continuous time on the n-dimensional unit ... [more ▼] This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve in continuous time on the n-dimensional unit sphere or n-sphere. For any connected, undirected graph and all n 2 N\{1}, each protocol in said class is shown to yield almost global consensus. The feedback laws are negative gradients of Lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol. This convergence result sheds new light on the general problem of consensus on Riemannian manifolds; the n-sphere for n 2 N\{1} differs from the circle and SO(3) where the corresponding protocols fail to generate almost global consensus. Moreover, we derive a novel consensus protocol on SO(3) by combining two almost globally convergent protocols on the n-sphere for n in {1, 2}. Theoretical and simulation results suggest that the combined protocol yields almost global consensus on SO(3). [less ▲] Detailed reference viewed: 135 (7 UL)Towards almost global synchronization on the Stiefel manifold Markdahl, Johan ; ; Goncalves, Jorge in Proceedings of the 57th IEEE Conference on Decision and Control (2018) Detailed reference viewed: 31 (2 UL)A lifting method for analyzing distributed synchronization on the unit sphere Thunberg, Johan ; Markdahl, Johan ; et al in Automatica (2018) This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted ... [more ▼] This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted to the (d+1)-dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence. [less ▲] Detailed reference viewed: 100 (1 UL)Experimental design trade-offs for gene regulatory network inference: an in silico study of the yeast Saccharomyces cerevisiae cell cycle Markdahl, Johan ; Colombo, Nicolo ; Thunberg, Johan et al in Proceedings of the 56th IEEE Conference on Decision and Control (2017, December) Time-series of high throughput gene sequencing data intended for gene regulatory network (GRN) inference are often short due to the high costs of sampling cell systems. Moreover, experimentalists lack a ... [more ▼] Time-series of high throughput gene sequencing data intended for gene regulatory network (GRN) inference are often short due to the high costs of sampling cell systems. Moreover, experimentalists lack a set of quantitative guidelines that prescribe the minimal number of samples required to infer a reliable GRN model. We study the temporal resolution of data vs.quality of GRN inference in order to ultimately overcome this deficit. The evolution of a Markovian jump process model for the Ras/cAMP/PKA pathway of proteins and metabolites in the G1 phase of the Saccharomyces cerevisiae cell cycle is sampled at a number of different rates. For each time-series we infer a linear regression model of the GRN using the LASSO method. The inferred network topology is evaluated in terms of the area under the precision-recall curve (AUPR). By plotting the AUPR against the number of samples, we show that the trade-off has a, roughly speaking, sigmoid shape. An optimal number of samples corresponds to values on the ridge of the sigmoid. [less ▲] Detailed reference viewed: 135 (10 UL)Intrinsic reduced attitude formation with ring inter-agent graph ; Markdahl, Johan ; et al in Automatica (2017), 85 This paper investigates the reduced attitude formation control problem for a group of rigid-body agents using feedback based on relative attitude information. Under both undirected and directed cycle ... [more ▼] This paper investigates the reduced attitude formation control problem for a group of rigid-body agents using feedback based on relative attitude information. Under both undirected and directed cycle graph topologies, it is shown that reversing the sign of a classic consensus protocol yields asymptotical convergence to formations whose shape depends on the parity of the group size. Specifically, in the case of even parity the reduced attitudes converge asymptotically to a pair of antipodal points and distribute equidistantly on a great circle in the case of odd parity. Moreover, when the inter-agent graph is an undirected ring, the desired formation is shown to be achieved from almost all initial states. [less ▲] Detailed reference viewed: 82 (0 UL)A geodesic feedback law to decouple the full and reduced attitude Markdahl, Johan ; ; et al in System and Control Letters (2017), 102 This paper presents a novel approach to the problem of almost global attitude stabilization. The reduced attitude is steered along a geodesic path on the n−1-sphere. Meanwhile, the full attitude is ... [more ▼] This paper presents a novel approach to the problem of almost global attitude stabilization. The reduced attitude is steered along a geodesic path on the n−1-sphere. Meanwhile, the full attitude is stabilized on SO(n). This action, essentially two maneuvers in sequel, is fused into one smooth motion. Our algorithm is useful in applications where stabilization of the reduced attitude takes precedence over stabilization of the full attitude. A two parameter feedback gain affords further trade-offs between the full and reduced attitude convergence speed. The closed loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the transient dynamics. The exact solutions also help us to characterize the asymptotic behavior of the system such as establishing the region of attraction by straightforward evaluation of limits. The geometric flavor of these ideas is illustrated by a numerical example. [less ▲] Detailed reference viewed: 99 (0 UL)Global converegence properties of a consensus protocol on the n-sphere Markdahl, Johan ; Goncalves, Jorge in Proceedings of the 55th IEEE Conference on Decision and Control (2016, December) This paper provides a novel analysis of the global convergence properties of a well-known consensus protocol for multi-agent systems that evolve in continuous time on the n-sphere. The feedback is ... [more ▼] This paper provides a novel analysis of the global convergence properties of a well-known consensus protocol for multi-agent systems that evolve in continuous time on the n-sphere. The feedback is intrinsic to the n-sphere, i.e., it does not rely on the use of local coordinates obtained through a parametrization. It is shown that, for any connected undirected graph topology and all n>1, the consensus protocol yields convergence that is akin to almost global consensus in a weak sense. Simulation results suggest that actual almost global consensus holds. This result is of interest in the context of consensus on Riemannian manifolds since it differs from what is known with regard to the 1-sphere and SO(3) where more advanced intrinsic consensus protocols are required in order to generate equivalent results. [less ▲] Detailed reference viewed: 181 (12 UL)Global and invariant aspects of consensus on the n-sphere Markdahl, Johan ; ; et al in Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems (2016, July) This paper concerns two aspects of the multi- agent consensus problem on the n-sphere. Firstly, it proves that a standard consensus protocol, in a certain sense, yields asymptotical stability on a global ... [more ▼] This paper concerns two aspects of the multi- agent consensus problem on the n-sphere. Firstly, it proves that a standard consensus protocol, in a certain sense, yields asymptotical stability on a global level for a nontrivial class of graph topologies. Secondly, it provides a novel consensus protocol that leaves the centroid of agent states in Rn+1 projected back to the sphere invariant. It hence becomes possible to determine the consensus point as a function of the initial states. Much of the stability analysis has an intuitive geometric appeal since it is based on the symmetries of the n-sphere rather than generic Lyapunov theory. [less ▲] Detailed reference viewed: 105 (7 UL)Exact solutions to a class of feedback systems on SO(n) Markdahl, Johan ; in Automatica (2016), 63 This paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on View the MathML source. The closed-loop systems are solved ... [more ▼] This paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on View the MathML source. The closed-loop systems are solved exactly for the rotation matrices as explicit functions of time, the initial conditions, and the gain parameters of the control laws. The exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix. Applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance. Knowledge of the future trajectory of the states can also be used as an alternative to the zero-order hold in systems where the attitude is sampled at discrete time instances. [less ▲] Detailed reference viewed: 97 (3 UL) |
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