Regulation and controlled synchronization for complex dynamical systems

Abstract

SUMMARYIn this paper we investigate the problem of controlled synchronization as a regulator problem. In controlled synchronization one is given autonomous transmitter dynamics and controlled receiver dynamics. The question is to "nd a (output) feedback controller that achieves matching between transmitter and controlled receiver. Several variants of the problem where the standard solvability assumptions for the regulator problem are not met turn out to have a solution. Simulations on two standard synchronization examples are also included.

A dynamical control view on synchronization

“…For instance, this is true for Lur'e systems, with a transmitter system of the form and receiver dynamics i i = A2 + c p (~~) + BU It follows that provided the pair (A, C) is detectable, as well as (AT, BT) is detectable (or equivalently, (A, B) is stabilizable) then there exists a (linear) dynamic output feedback of the form (50) such that the two systems asymptotically synchronize. For further details and insight in the controlled synchronization problem the reader is referred to (Huijberts et al 2000b). …”

A dynamical control view on synchronization

“…For instance, this is true for Lur'e systems, with a transmitter system of the form and receiver dynamics i i = A2 + c p (~~) + BU It follows that provided the pair (A, C) is detectable, as well as (AT, BT) is detectable (or equivalently, (A, B) is stabilizable) then there exists a (linear) dynamic output feedback of the form (50) such that the two systems asymptotically synchronize. For further details and insight in the controlled synchronization problem the reader is referred to (Huijberts et al 2000b). …”

A global asymptotic synchronization problem via internal model approach

“…Specifically, we will show how the problem under consideration can be reformulated as a robust output regulation problem with the Duffing system as an uncertain plant subject to a nonlinear exosystem. It should be noted that using the output regulation theory to handle the synchronization problem has been proposed in [8]. However, the approach of [8] utilizes the feedforward control which cannot deal with the model uncertainty.…”

“…It should be noted that using the output regulation theory to handle the synchronization problem has been proposed in [8]. However, the approach of [8] utilizes the feedforward control which cannot deal with the model uncertainty. Here we will use the internal model approach which is capable of handling the model uncertainty in the Duffing system.…”

Output synchronization control of ship replenishment operations: Theory and experiments

“…The synchronization phenomenon was early reported by Huygens (1673) who observed that a pair of pendulum clocks hanging from a lightweight beam oscillated with the same frequency. In the last century, synchronization has received attention in the Russian scientific community where it was observed in balanced and unbalanced rotors and vibro-exciters (Blekhman, 1988), and has in the last decade attracted interest from researchers within physics, dynamical systems, circuit theory, and more lately control theory (see for instance Brunt, 1998;Huijberts, Nijmeijer, & Willems, 2000). A coordinated synchronization scheme applied to robot control was presented in Rodriguez- Angeles and Nijmeijer (2001), and in Nijmeijer and Rodriguez-Angeles (2003) both coordinated and cooperative synchronization schemes were treated.…”

A dynamical control view on synchronization

“…For instance, this is true for Lur'e systems, with a transmitter system of the form and receiver dynamics i i = A2 + c p (~~) + BU It follows that provided the pair (A, C) is detectable, as well as (AT, BT) is detectable (or equivalently, (A, B) is stabilizable) then there exists a (linear) dynamic output feedback of the form (50) such that the two systems asymptotically synchronize. For further details and insight in the controlled synchronization problem the reader is referred to (Huijberts et al 2000b). …”

A global asymptotic synchronization problem via internal model approach

“…Specifically, we will show how the problem under consideration can be reformulated as a robust output regulation problem with the Duffing system as an uncertain plant subject to a nonlinear exosystem. It should be noted that using the output regulation theory to handle the synchronization problem has been proposed in [8]. However, the approach of [8] utilizes the feedforward control which cannot deal with the model uncertainty.…”

“…It should be noted that using the output regulation theory to handle the synchronization problem has been proposed in [8]. However, the approach of [8] utilizes the feedforward control which cannot deal with the model uncertainty. Here we will use the internal model approach which is capable of handling the model uncertainty in the Duffing system.…”

Output synchronization control of ship replenishment operations: Theory and experiments

“…The synchronization phenomenon was early reported by Huygens (1673) who observed that a pair of pendulum clocks hanging from a lightweight beam oscillated with the same frequency. In the last century, synchronization has received attention in the Russian scientific community where it was observed in balanced and unbalanced rotors and vibro-exciters (Blekhman, 1988), and has in the last decade attracted interest from researchers within physics, dynamical systems, circuit theory, and more lately control theory (see for instance Brunt, 1998;Huijberts, Nijmeijer, & Willems, 2000). A coordinated synchronization scheme applied to robot control was presented in Rodriguez- Angeles and Nijmeijer (2001), and in Nijmeijer and Rodriguez-Angeles (2003) both coordinated and cooperative synchronization schemes were treated.…”

A dynamical control view on synchronization

“…For instance, this is true for Lur'e systems, with a transmitter system of the form and receiver dynamics i i = A2 + c p (~~) + BU It follows that provided the pair (A, C) is detectable, as well as (AT, BT) is detectable (or equivalently, (A, B) is stabilizable) then there exists a (linear) dynamic output feedback of the form (50) such that the two systems asymptotically synchronize. For further details and insight in the controlled synchronization problem the reader is referred to (Huijberts et al 2000b). …”

A global asymptotic synchronization problem via internal model approach

“…Specifically, we will show how the problem under consideration can be reformulated as a robust output regulation problem with the Duffing system as an uncertain plant subject to a nonlinear exosystem. It should be noted that using the output regulation theory to handle the synchronization problem has been proposed in [8]. However, the approach of [8] utilizes the feedforward control which cannot deal with the model uncertainty.…”

“…It should be noted that using the output regulation theory to handle the synchronization problem has been proposed in [8]. However, the approach of [8] utilizes the feedforward control which cannot deal with the model uncertainty. Here we will use the internal model approach which is capable of handling the model uncertainty in the Duffing system.…”

Output synchronization control of ship replenishment operations: Theory and experiments

“…The synchronization phenomenon was early reported by Huygens (1673) who observed that a pair of pendulum clocks hanging from a lightweight beam oscillated with the same frequency. In the last century, synchronization has received attention in the Russian scientific community where it was observed in balanced and unbalanced rotors and vibro-exciters (Blekhman, 1988), and has in the last decade attracted interest from researchers within physics, dynamical systems, circuit theory, and more lately control theory (see for instance Brunt, 1998;Huijberts, Nijmeijer, & Willems, 2000). A coordinated synchronization scheme applied to robot control was presented in Rodriguez- Angeles and Nijmeijer (2001), and in Nijmeijer and Rodriguez-Angeles (2003) both coordinated and cooperative synchronization schemes were treated.…”