Search

International audience; Delays appear in the dynamics of many systems due to non-vanishing reaction times of control systems. In biochemical systems, long sequences of repeated steps, especially in biopolymerization processes, can be modeled by delays. However, modelling systems with delays is often complicated by physical constraints, such as the requirement that solutions representing concentrations of chemical species remain positive. In this work, we consider a model for a detoxifying enzyme whose synthesis is controlled by its substrate. The model includes bindingsite clearance delays, caused by the time required for an RNA polymerase or ribosome to clear its binding site before another such machine can bind. The existence of a positive equilibrium and the positivity and boundedness of solutions of the corresponding delay-differential equations are proven. In addition, the stability of the model is studied using the "small-gain" theorem.

In this work, stable H ∞ controller design problem for linear time-invariant single-input single-output plants, which have infinitely many open-right-half-plane zeros and possibly infinitely many right-half-plane poles, is considered. Interpolation-based approach is developed to solve the problem. The approach requires to construct a unit in H ∞ satisfying certain interpolation conditions at the open-right-half-plane zeros of the plant and H ∞ norm constraint. In order to construct such a unit function, first, construction of an H ∞ function satisfying interpolation conditions at given countably many distinct points in the open-right-half plane is presented. Then, by using the upper bound on the H ∞ norm of the constructed function and the small-gain theorem, first, sufficient condition is presented for the solution of the sensitivity minimization problem by a stable controller and design methodology for such a controller is presented. Then, stable H ∞ controller design approach for the considered class of plants is presented under certain assumptions. A numerical example is given to verify the presented results.

The strong stabilization problem, i.e., the problem of designing a real stable stabilizing controller, is considered for multi-input multi-output infinite-dimensional real linear systems. The considered systems may have infinitely many poles and zeros in the open right-half-plane, as well as on the imaginary axis. It is shown that the well-known parity interlacing property (pip) for real-rational systems is also a necessary condition in the most general case as long as the plant has coprime factorizations over H ∞ , which is a necessary condition for stabilizability. Furthermore, it is shown that pip is also sufficient under some additional mild assumptions.

WOS: 000391588600005, Sometimes a network of dynamical systems shows a form of incomplete synchronization, characterized by synchronization of some but not all of its component systems. This type of incomplete synchronization is called partial synchronization or cluster synchronization. Partial synchronization is associated with the existence of partial synchronization manifolds, which are linear invariant subspaces of C, the state space of the network of systems. We focus on partial synchronization manifolds in networks of identical systems, characterized by linear diffusive coupling described by a weighted graph, and allowing for time-delay in the coupling. We present equivalent existence criteria for partial synchronization manifolds in terms of invariant spaces, the block-structure of a reordered adjacency matrix, and the solvability of a Sylvester equation. We emphasize decomposable networks, according to the rational dependency structure of the coupling weights, and according to the delay values, respectively. It is obvious that if the existence conditions for partial synchronization manifolds are satisfied for all subnetworks simultaneously, they hold for the original network, yet the converse result is not always true, as we shall illustrate with an example. Furthermore, as our main results, we show that if the decomposition is according to the weights and the basis weights are rationally independent numbers, or if the decomposition is according to different delay values, then finding a partial synchronization manifold for the original network is equivalent to finding common partial synchronization manifolds for the subnetworks, i.e., restricting to the analysis of the subnetworks does not impose any conservatism, which simplifies the analysis significantly. For the case of decomposable networks according to the weights, with rationally independent basis weights, we provide a fourth existence criterion for partial synchronization manifolds in terms of a balanced coloring of an associated multigraph. In addition, we briefly describe publicly available software for detecting partial synchronization manifolds. Our equivalent existence criteria, which depend on the network and delay structure but not on the dynamics of the systems at the nodes, are sufficient for the presence of a partial synchronization manifold. We show that under a mild assumption on the systems at the nodes, namely, left-invertibility, these conditions are necessary as well. In all criteria it turns out that the distinction between noninvasive and invasive delayed coupling is important, i.e., whether or not a coupling term between two systems vanishes whenever the latter are synchronized., Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Office (IAP P6-DYSCO); OPTEC, the Optimization in Engineering Center of KU Leuven; Research Foundation-Flanders (FWO) [G.0712.11N]; Odysseus grant from FWO, This research was supported by the Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Office (IAP P6-DYSCO), by OPTEC, the Optimization in Engineering Center of KU Leuven, and by project G.0712.11N of the Research Foundation-Flanders (FWO). The third author was supported by an Odysseus grant from FWO.

WOS: 000327713900004, The full synchronization of coupled nonlinear oscillators has been widely studied. In this paper we investigate conditions for which partial synchronization of time-delayed diffusively coupled systems arises. The coupling configuration of the systems is described by a directed graph. As a novel quantitative result we first give necessary and sufficient conditions for the presence of forward invariant sets characterized by partially synchronous motion. These conditions can easily be checked from the eigenvalues and eigenvectors of the graph Laplacian. Second, we perform stability analysis of the synchronized equilibria in a (gain, delay) parameter space. For this analysis the coupled nonlinear systems are linearized around the synchronized equilibria and then the resulting characteristic function is factorized. By such a factorization, it is shown that the relation between the behaviour of different agents at the zero of the characteristic function depends on the structure of the eigenvectors of the weighted Laplacian matrix. By determining the structure of the solutions in the unstable manifold, combined with the characterization of invariant sets, we predict which partially synchronous regimes occur and estimate the corresponding coupling gain and delay values. We apply the obtained results to networks of coupled Hindmarsh-Rose neurons and verify the occurrence of the expected partially synchronous regimes by using a numerical simulation. We also make a comparison with an existing approach based on Lyapunov functionals., Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Oce [IAP P6-DYSCO]; OPTEC, the Optimization in Engineering Center of the K.U. Leuven; K.U. Leuven Research Council [STRT1-09/33]; Research Foundation-Flanders (FWO) [G.0712.11N], The authors would like to thank the anonymous reviewers for their useful suggestions that helped us to improve the quality of the paper. This work has been supported by the Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Oce (IAP P6-DYSCO), by OPTEC, the Optimization in Engineering Center of the K.U. Leuven, by the project STRT1-09/33 of the K.U. Leuven Research Council and the project G.0712.11N of the Research Foundation-Flanders (FWO).

International Federation of Automatic Control;(IFAC) - Technical Committee on Non-Linear;Control Systems (TC 2.3);TC 2.1 Control Design;TC 2.4 Optimal Control, 9th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2013 -- 4 September 2013 through 6 September 2013 -- Toulouse -- 99971, We present an approach which allows to accurately predict both the occurrence and type of partially synchronous regimes of delay-coupled non-linear oscillators. Unlike the conventional approach, we build on an analysis of the stability properties of the synchronized equilibrium in the (coupling gain, delay) parameter space. As partially synchronous regimes are closely related to the presence of invariant manifolds, we first present necessary and sufficient conditions for the existence of forward invariant sets. Next, from the existence of these invariant sets and from the characterization of solutions in the unstable manifold of the synchronized equilibrium, we predict which (gain,delay) parameters may result in fully/partially synchronous behavior. We illustrate the approach for a network of delay coupled Hindmarsh-Rose neurons, STRT1-09/33 Fonds Wetenschappelijk Onderzoek G.0712.11N IAP P7-DYSCO, This work has been supported by the Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Oce (IAP P7-DYSCO), by OPTEC, the Optimization in Engineering Center of the K.U.Leuven, by the project STRT1-09/33 of the K.U.Leuven Research Council and the project G.0712.11N of the Research Foundation - Flanders (FWO). 1 This work was done while the author was visiting KU Leuven.

et al.;IFAC TC 1.5. Networked Systems;IFAC TC 2.1. Control Design;IFAC TC 2.2. Linear Control Systems;IFAC TC 2.3. Non-Linear Control Systems;IFAC TC 2.5. Robust Control, 10th IFAC Workshop on Time Delay Systems, TDS-2012 -- 22 June 2012 through 24 June 2012 -- Boston, MA -- 92592, Stable controllers are required due to modelling imperfections and for better performance achievements. As is well known, a stable stabilizing controller for a given real-rational plant exists if and only if the plant staisfies the so-called parity interlacing property. In this paper, a recent result, which proves that this condition is also necessary for infinite-dimensional plants is first presented. The condition is also sufficient if the plant satisfies certain additional properties. This result is then applied to time-delay systems. Two examples are considered. For the first example it is shown that it is not possible to design a stable stabilizing controller. For the second example it is shown that there exists a stable controller which stabilizes the plant and the design of such a controller is demonstrated

The stable H ∞ controller design problem is considered for multi-input–multi-output systems with multiple input/output time-delays. An algorithm is presented to solve this problem. The algorithm makes use of the small-gain theorem and the structure of the H ∞ controller for the class of systems under consideration.

According to some recent results, use of non-causal uncertainty blocks may be advantageous in the robust controller design of systems with multiple uncertain time-delays. In this work, performance and robustness improvements obtained by utilizing such an approach are presented. The flow controller design problem for networks with multiple uncertain time-delays is considered as a case study. It is shown that higher performance levels and larger stability margins are, in general, obtained by using non-causal uncertainty blocks. A number of simulations, which illustrate the time-domain performance improvement, are also presented.

Flow controllers are needed to avoid congestion in different types of networks. The dynamics of such networks, however, can usually be represented by infinite-dimensional models, which are either based on partial-differential equations or delay-differential equations. An optimal ℋ∞ flow controller design approach has recently been proposed for networks whose dynamics involve multiple and uncertain time-varying time delays. In the present paper, performance level and stability margins of controllers designed by this approach are analyzed. It is shown that, by choosing certain controller design parameters large, stability margins can be improved; while choosing them small improves performance levels. Therefore, there is a clear trade-off between robustness and performance.

Robust controller design for a flow control problem where uncertain multiple time-varying time-delays exist is considered. Although primarily data-communication networks are considered, the presented approach can also be applied to other flow control problems and can even be extended to other control problems where uncertain multiple time-varying time-delays exist. Besides robustness, tracking and fairness requirements are also considered. To solve this problem, an ℋ∞ optimization problem is set up and solved. Unlike previous approaches, where only a suboptimal solution could be found, the present approach allows to design an optimal controller. Simulation studies are carried out in order to illustrate the time-domain performance of the designed controllers. The obtained results are also compared to the results of a suboptimal controller obtained by an earlier approach. Copyright © 2009 John Wiley & Sons, Ltd.