Mark Verwoerd

Global Phase-Locking in Finite Populations of Phase-Coupled Oscillators

We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling.

D-Stability and Delay-Independent Stability of Homogeneous Cooperative Systems

We introduce a nonlinear definition of D-stability, extending the usual concept for positive linear time-invariant systems. We show that globally asymptotically stable, cooperative systems, homogeneous of any order with respect to arbitrary dilation maps are D-stable. We also prove a strong stability result for delayed cooperative homogeneous systems. Finally, we show that both of these results also hold for planar cooperative systems without the restriction of homogeneity.

On Computing the Critical Coupling Coefficient for the Kuramoto Model on a Complete Bipartite Graph

We extend recent results on the existence of global phase-locked states (GPLS) in the Kuramoto model on a complete graph to the case of a complete bipartite graph. In particular, we prove that, for the Kuramoto model on a complete bipartite graph, the value of the critical coupling coefficient, i.e., the smallest coupling coefficient for which a GPLS can exist, can be determined by solving a system of two nonlinear equations that do not depend on the coupling coefficient itself. We show that said system of equations can be solved using an efficient algorithm described in the paper.

The role of control and system theory in systems biology

The use of new technology and mathematics to study the systems of nature is one of the most significant scientific trends of the century. Driven by the need for more precise scientific understanding, advances in automated measurement are providing rich new sources of biological and physiological data. These data provide information to create mathematical models of increasing sophistication and realism—models that can emulate biological and physiological systems with sufficient accuracy to advance our understanding of living systems and disease mechanisms. New measurement and modelling methods set the stage for control and systems theory to play their role in seeking out the mechanisms and principles that regulate life. It is of inestimable importance for the future of control as a discipline that this role is performed in the correct manner. If we handle the area wisely then living systems will present a seemingly boundless range of important new problems—just as physical and engineering systems have done in previous centuries. But there is a crucial difficulty. Faced with a bewildering array of choices in an unfamiliar area, how does a researcher select a worthwhile and fruitful problem? This article is an attempt to help by offering a control-oriented guide to the labyrinthine world of biology/physiology and its control research opportunities.

Observations on the stability properties of cooperative systems

We extend two fundamental properties of positive linear time-invariant (LTI) systems to homogeneous cooperative systems. Specifically, we demonstrate that such systems are D-stable, meaning that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneous cooperative system is globally asymptotically stable (GAS) for all non-negative delays if and only if the system is GAS for zero delay.

On the use of noncausal LTI operators in iterative learning control

This paper demonstrates the use of noncausal operators in iterative learning control (ILC). First, it is shown that for a particular class of plants (having unstable zeros), perfect tracking can only be achieved by using noncausal operators. Then it is shown that with any converging algorithm (both causal and noncausal) we can associate a particular feedback controller. For causal algorithms this controller can be shown to be internally stabilizing. In the noncausal case, however, the associated controller is found to be generally destabilizing which implies that the existing notion of an equivalent controller for causal ILC does not extend to noncausal ILC.

On admissible pairs and equivalent feedback-Youla parameterization in iterative learning control

This paper revisits a well-known synthesis problem in iterative learning control, where the objective is to optimize a performance criterion over a class of causal iterations. The approach taken here adopts an infinite-time setting and looks at limit behavior. The first part of the paper considers iterations without current-cycle-feedback (CCF) term. A notion of admissibility is introduced to distinguish between pairs of operators that define a robustly converging iteration and pairs that do not. The set of admissible pairs is partitioned into disjoint equivalence classes. Different members of an equivalence class are shown to correspond to different realizations of a (stabilizing) feedback controller. Conversely, every stabilizing controller is shown to allow for a (non-unique) factorization in terms of admissible pairs. Class representatives are introduced to remove redundancy. The smaller set of representative pairs is shown to have a trivial parameterization that coincides with the Youla parameterization of all stabilizing controllers (stable plant case). The second part of the paper considers the general family of CCF-iterations. Results derived in the non-CCF case carry over, with the exception that the set of equivalent controllers now forms but a subset of all stabilizing controllers. Necessary and sufficient conditions for full generalization are given.

On equivalence classes in iterative learning control

This paper advocates a new approach to study the relation between causal iterative learning control (ILC) and conventional feedback control. Central to this approach is the introduction of the set of admissible pairs (of operators) defined with respect to a family of iterations. Considered are two problem settings: standard ILC, which does not include a current cycle feedback (CCF) term and CCF-ILC, which does. By defining an equivalence relation on the set of admissible pairs, it is shown that in the standard ILC problem there exists a bijective map between the induced equivalence classes and the set of all stabilizing controllers. This yields the well-known Youla parameterization as a corollary. These results do not extend in full generality to the case of CCF-ILC; though gain every admissible pair defines a stabilizing equivalent controller, the converse is no longer true in general.

l 1 -optimal robust iterative learning controller design

In this paper we consider the robust iterative learning control (ILC) design problem for SISO discrete-time linear plants subject to unknown, bounded disturbances. Using the supervector formulation of ILC, we apply a Youla parameterization to pose a MIMO l1-optimal control problem. The problem is analyzed for three situations: (1) the case of arbitrary ILC controllers that use current iteration tracking error (CITE), but without explicit integrating action in iteration, (2) the case of arbitrary ILC controllers with CITE and with explicit integrating action in iteration, and (3) the case of ILC controllers without CITE but that force an integral action in iteration. Analysis of these cases shows that the best ILC controller for this problem when using a non-CITE ILC algorithm is a standard Arimoto-style update law, with the learning gain chosen to be the system inverse. Further, such an algorithm will always be worse than a CITE-based algorithm. It is also found that a trade-off exists between asymptotic tracking of reference trajectories and rejection of unknown-bounded disturbances and that ILC does not help alleviate this trade-off. Finally, the analysis reinforces results in the literature noting that for SISO discrete-time linear systems, first-order ILC algorithms can always do as well as higher-order ILC algorithms.

A Convergence Result for the Kuramoto Model with All-to-All Coupling

We prove a convergence result for the standard Kuramoto model with all-to-all coupling. Specifically, we show that the critical coupling strength associated with the onset of completely phase-locked behavior converges in probability as the number of oscillators tends to infinity.