Murat Arcak

Passivity as a Design Tool for Group Coordination

We pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set. To stabilize this set we study a class of feedback rules that are implementable with local information available to each member. When the information flow between neighboring members is bidirectional, we show that the closed-loop system exhibits a special interconnection structure which inherits the passivity properties of its components. By exploiting this structure we develop a passivity-based design framework, which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature. The passivity approach offers additional design flexibility compared to these special cases, and systematically constructs a Lurie-type Lyapunov function for the closed-loop system. We further study the robustness of these feedback laws in the presence of a time-varying communication topology, and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense.

Nonlinear observers: a circle criterion design and robustness analysis

Globally convergent observers are designed for a class of systems with monotonic nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a time-varying multivariable sector nonlinearity. Using LMI software, observer gain matrices are computed to satisfy the circle criterion and, hence, to drive the observer error to zero. In output-feedback design, the observer is combined with control laws that ensure input-to-state stability with respect to the observer error. Robustness to unmodeled dynamics is achieved with a small-gain assignment design, as illustrated on a jet engine compressor example.

A unifying passivity framework for network flow control

Network flow control regulates the traffic between sources and links based on congestion, and plays a critical role in ensuring satisfactory performance. In recent studies, global stability has been shown for several flow control schemes. By using a passivity approach, this paper presents a unifying framework which encompasses these stability results as special cases. In addition, the new approach significantly expands the current classes of stable flow controllers by augmenting the source and link update laws with passive dynamic systems. This generality offers the possibility of optimizing the controllers, for example, to improve robustness in stability and performance with respect to time delays, unmodeled flows, and capacity variation.

Observer design for systems with multivariable monotone nonlinearities

Globally convergent observers are designed for a class of systems with multivariable nonlinearities. The approach is to represent the observer error system as the feedback interconnection of a linear system and a state-dependent multivariable nonlinearity. We first extend an earlier design (Automatica 37 (12) (2001) 1923) to multivariable nonlinearities, satisfying an analog of the scalar nondecreasing property. Next, we exploit the structure of the nonlinearity to relax the positive real restriction on the linear part of the observer error system. This relaxed design renders the feasibility conditions less restrictive, and widens the applicability of the observer, as illustrated with examples. Finally, output nonlinearities are studied and the design is extended to be adaptive in the presence of unknown parameters.

Passivity-based designs for synchronized path-following

We consider a formation control system where individual systems are controlled by a path-following design and the path variables are to be synchronized. We first show a passivity property for the path following system and, next, combine this with a passivity-based synchronization algorithm developed in Arcak, M. (2006), The passivity approach expands the classes of synchronization schemes available to the designer. This generality offers the possibility to optimize controllers to, e.g., improve robustness and performance. Two designs are developed in the proposed passivity framework: The first employs the path error information in the synchronization loop, while the second only uses synchronization errors. A sampled-data design, where the path variables are updated in discrete-time and the path following controllers are updated in continuous time, is also developed

Synchronization of Interconnected Systems With Applications to Biochemical Networks: An Input-Output Approach

This paper provides synchronization conditions for networks of nonlinear systems. The components of the network (referred to as “compartments” in this paper) are made up of an identical interconnection of subsystems, each represented as an operator in an extended L2 space and referred to as a “species.” The compartments are, in turn, coupled through a diffusion-like term among the respective species. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of the network. The paper also explores results for state-space models, as well as biochemical applications. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling. The theory is illustrated by providing synchronization conditions for networks of Goodwin oscillators.

A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation

We study observer design for sampled-data nonlinear systems using two approaches: (i) the observer is designed via an approximate discrete-time model of the plant; (ii) the observer is designed based on the continuous-time plant model and then discretized for sampled-data implementation (emulation). We investigate under what conditions, and in what sense, these designs achieve convergence for the unknown exact discrete-time model. We present examples which show that designs that violate our conditions may indeed lead to instability when implemented on the exact model.

A Unifying Integral ISS Framework for Stability of Nonlinear Cascades

We analyze nonlinear cascades in which the driven subsystem is integral input-to-state stable (ISS), and we characterize the admissible integral ISS gains for stability. This characterization makes use of the convergence speed of the driving subsystem and allows a larger class of gain functions when the convergence is faster. We show that our integral ISS gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem. The result is used to develop a new observer-based backstepping design in which the growth of the nonlinear damping terms is reduced.

Rigid body attitude coordination without inertial frame information

We study a motion coordination problem where the objective is to achieve identical orientation and synchronous rotation for a group of rigid bodies. Unlike existing designs which assume that the inertial frame is available to each agent, we develop a passivity-based design which relies only on relative attitude information with respect to neighboring agents. The desired equilibria, where all the rigid bodies possess the same attitude and rotate at a desired angular velocity, are shown to be locally asymptotically stable and a manifold of undesired equilibria may exist. We then consider the situation where the reference angular velocity is available only to the leader, and propose a distributed adaptive controller with which the other agents reconstruct this reference angular velocity.

Adaptive design for reference velocity recovery in motion coordination

We study a coordination problem where the objective is to steer a group of agents to a formation that translates with a prescribed reference velocity. Unlike existing designs which assume that the reference velocity information is available to each agent, we consider the situation where this information is available only to a leader. We then develop an adaptive design with which the other agents reconstruct the reference velocity and recover the desired formation. This design relies only on relative distance information with respect to neighbouring agents and, thus, can be implemented in a decentralized fashion.