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Equilibrium-independent passive-short (EIPS) systems are a class of systems that satisfy a passivity-like dissipation inequality with respect to any forced equilibria with non-positive passivity indices. This paper presents a geometric approach for finding a passivizing transformation for such systems, relying on their steady-state input-output relation and the notion of projective quadratic inequalities (PQIs). We show that PQIs arise naturally from passivity-shortage characteristics of an EIPS system, and the set of their solutions can be explicitly expressed. We leverage this connection to build an input-output mapping that transforms the steady-state input-output relation to a monotone relation, and show that the same mapping passivizes the EIPS system. We show that the proposed transformation can be implemented through a combination of feedback, feed-through, post- and pre-multiplication gains. Furthermore, we consider an application of the presented passivation scheme for the analysis of networks comprised of EIPS systems. Numerous examples are provided to illustrate the theoretical findings. IEEE

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IEEE Transactions on Automatic Control

A Geometric Method for Passivation and Cooperative Control of Equilibrium-Independent Passivity-Short Systems

A methodology to stabilize synchronized, balanced, or symmetric phase patterns of unicycle-type agents on a desired common circle, while restricting their trajectories to a certain region of interest, is proposed in this note. These phase patterns are characterized by the motion of the collective centroid of the group of agents and derived by optimizing the average linear momentum of the group. Under a mild assumption on initial states of the agents, we design control laws by exploiting the concept of barrier Lyapunov function in conjunction with bounded phase potential functions. We show that the agents asymptotically stabilize to a desired phase arrangement and their trajectories remain bounded during stabilization. We obtain bounds on the different quantities of interest in the postdesign analysis and show that these bounds depend on the initial conditions and can be altered by adjusting the controller gains. We also prove convergence when the control input is saturated to a prespecified value. Finally, we provide a discussion on the application and limitation of the proposed approach and characterize the feasible initial conditions. © 2020 IEEE.

Trajectory-Constrained Collective Circular Motion with Different Phase Arrangements

This paper analyses collective motion of multi-vehicle systems in balanced or splay formation when the vehicles are equipped with heterogeneous controller gains. Balancing refers to a situation in which the positional centroid of the vehicles is stationary. The splay formation is a special case of balancing in which the vehicles are spatially distributed with equal angular separation between them. The paper proposes strategies to achieve such balanced and splay formations about a desired centroid location while allowing the vehicles to move either along straight line paths or on individual circular orbits. Feedback control laws that can tolerate heterogeneity in the controller gains, which may be caused by imperfect implementation, are derived and analyzed. It is shown that drastic failures leading to controller gains becoming zero for almost half of the vehicles in the group can be tolerated and balanced formation can still be achieved. On the other hand, splay formation can still be achieved if the controller gain is zero for at most one vehicle. Simulation examples are given to illustrate the theoretical findings. © 2019

Journal of the Franklin Institute

Robust stabilization of balanced and splay formations with heterogeneous controller gains

This paper studies circular formation control of multi-vehicle systems, modeled with unicycle dynamics, in a uniform flowfield with different temporal phase arrangements, while assuring that the trajectories of the vehicles remains bounded within a region. Using the idea of Barrier Lyapunov function in conjunction with temporal phase potential functions, we derive control laws that achieves our objectives, under a mild assumption on the initial states of the vehicles. We further obtain bounds on the various signals in the post-design analysis and show that these depends on the initial conditions and controller gains. Simulations are provided to illustrate the theoretical findings. © 2019 IEEE.

27th Mediterranean Conference on Control and Automation, MED 2019 - Proceedings

Temporal circular formation control with bounded trajectories in a uniform flowfield

This paper considers collective circular motion of multi-agent systems in which all the agents are required to traverse different circles or a common circle at a prescribed angular velocity. It is required to achieve these collective motions with the heading angles of the agents synchronized or balanced. In synchronization, the agents and their centroid have a common velocity direction, while in balancing, the movement of agents causes the location of the centroid to become stationary. The agents are initially considered to move at unit speed around individual circles at different angular velocities. It is assumed that the agents are subjected to limited communication constraints, and exchange relative information according to a time-invariant undirected graph. We present suitable feedback control laws for each of these motion coordination tasks by considering a second-order rotational dynamics of the agent. Simulations are given to illustrate the theoretical findings. © 2017 Elsevier B.V.

Communications in Nonlinear Science and Numerical Simulation

Collective circular motion in synchronized and balanced formations with second-order rotational dynamics

In this brief, a non-identical gain based control design scheme is proposed to stabilise balanced collective formation in multi-vehicle systems, modelled with non-holonomic unicycle dynamics. Balanced collective formation refers to a situation in which the vehicles' motion causes their positional centroid to become stationary. This study mainly focuses on achieving stable balanced collective formation about the prescribed location of the collective centroid along with desired orientations of the vehicles. It is assumed that the vehicles are moving in a force field which is controlled externally. The proposed control algorithm consists of two controls - one of the controls, pertaining to the external force field, is directly applied to the system, and assumes the same value for all the vehicles. While the other control, which operates with heterogeneous gains, provides the steering force to the vehicles and is derived using Lyapunov analysis. Theoretical findings are validated through numerical simulations. © The Institution of Engineering and Technology 2018.

IET Control Theory and Applications

Multi-vehicle formation in a controllable force field with non-identical controller gains

In this letter, we establish a connection between cooperative control of passivity-short systems, and the regularization of a pair of dual network optimization problems. We build upon existing works that established, under a passivity requirement, an equivalence between the steady-state behavior of diffusively coupled network system and the solutions to a pair of convex-dual optimization problems. We show that when the agents are passivity-short, the resulting optimization problems are no-longer convex. By introducing a regularization term to the problem, we then establish that this corresponds to a feedback passivation of each system via an appropriately chosen linear output-feedback gain. We also obtain conditions on the regularization term such that the resultant closed system possess the so-called maximally equilibrium-independent passivity property and exhibits a solution to their network-level interactions. Finally, we illustrate theoretical results with two case studies. © 2018 IEEE.

IEEE Control Systems Letters

Regularization and Feedback Passivation in Cooperative Control of Passivity-Short Systems: A Network Optimization Perspective

This paper studies synchronization in a multi-agent system, which is defined as a situation where all the agents in a group are required to achieve a common velocity direction. The agents are assumed to be coupled through controller gains that are not necessarily identical or homogeneous, which addresses a practical scenario where the gains may vary nominally due to minor implementation errors or drastically due to major faults or errors. The paper analyzes the effect of heterogeneous gains on the common velocity direction at which the system of agents synchronizes. Conditions under which heterogeneous controller gains result in a synchronized formation are derived and it is shown that the resulting common velocity direction lies in the conic hull of the initial velocity vectors of agents. A detailed analysis of the two agents system shows that there exists a less restrictive condition on heterogeneous gains that results in synchronization. Effect of saturation is also studied for two cases when the controller gains are bounded and when the control efforts are bounded. Both all-to-all and limited communication topologies are considered. Simulations are given to support the theoretical findings. © 2017, Springer Science+Business Media Dordrecht.

Nonlinear Dynamics

Synchronization of multi-agent systems with heterogeneous controllers

This paper studies collective formations of multi-agent systems, modeled with unicycle dynamics, while admitting heterogeneity in both controller gains and speeds of the agents along with saturation on the controller outputs. This addresses a practical scenario where the speeds are usually nonidentical and the controller gains may vary due to imperfect implementation. The paper analyzes the effect of both heterogeneous controller gains and speeds simultaneously on the collective formations obtained by optimizing the average linear momentum of the group of agents. A detailed analysis of the two-agent system is given in the paper and some results on the locus of the collective centroid with varying controller gains are obtained. Effect of saturation is also studied for two cases when the controller gains are bounded and when the control efforts are bounded. Simulation examples are given to illustrate the theoretical findings. © 2017 The Franklin Institute

Stabilization of collective formations with speed and controller gain heterogeneity and saturation

This paper proposes a heterogeneous gains based controller design methodology to stabilize a particular type of collective motion in a multi-agent system where the heading angles of the agents are in balanced formation. Balancing refers to the situation in which the movement of agents causes the position of their centroid to become stationary. Our interest, in this paper, is to achieve balanced formation about a desired location of the centroid while allowing the agents to move either along straight line paths or around individual circular orbits. For this purpose, we derive feedback control laws that operate with heterogeneous control gains, and are more practical compared to the homogeneous gains based controls existing in the literature. We also show that if the heterogeneous control gains are zero for almost half of the agents of the group, it is possible to achieve balanced formation at an additional advantage of reduced computational complexity of the proposed control law. Simulations are given to illustrate the theoretical findings. © 2017 IEEE.

2017 Indian Control Conference, ICC 2017 - Proceedings

A heterogeneous control gain approach to achieve a desired collective centroid by a formation of agents

In this paper, we study the problem of a formation of agents trying to achieve a desired stationary or moving collective centroid. The agents are assumed to be moving in a force field which is controlled externally. The stabilization of the collective centroid to a fixed desired location results in a balanced formation of the agents about that point. Similarly, the centroid of the system of agents may be required to move along a certain given trajectory. For this, the centroid of the formation must converge to the desired trajectory. To solve this problem, we propose an all-to-all coupled planar motion model that explicitly incorporates an additional control pertaining to the external force field. Simulation results are presented to support the theoretical findings. © 2016 IEEE.

2016 Indian Control Conference, ICC 2016 - Proceedings

Achieving a desired collective centroid by a formation of agents moving in a controllable force field

This paper proposes a design methodology to stabilize collective circular motion of a group of N-identical agents moving at unit speed around individual circles of different radii and different centers. The collective circular motion studied in this paper is characterized by the clockwise rotation of all agents around a common circle of desired radius as well as center, which is fixed. Our interest is to achieve those collective circular motions in which the phases of the agents are arranged either in synchronized, in balanced or in splay formation. In synchronized formation, the agents and their centroid move in a common direction while in balanced formation, the movement of the agents ensures a fixed location of the centroid. The splay state is a special case of balanced formation, in which the phases are separated by multiples of 2π/N. We derive the feedback controls and prove the asymptotic stability of the desired collective circular motion by using Lyapunov theory and the LaSalle's Invariance principle. © 2015 American Automatic Control Council.

Proceedings of the American Control Conference

Stabilization of collective motion in synchronized, balanced and splay phase arrangements on a desired circle

In this paper, we study the collective motion of a group of N(> 2) identical agents trying to achieve a circular formation centered at a desired location, which is fixed. A circular formation is characterized by the motion of all agents around the same circle in the same direction. To solve this problem, we propose a planar motion model that incorporates two control inputs. One of the control inputs is chosen independently and the other control input is decided by using the composite Lyapunov function. We show that the desired location of the center of this circular formation, which is fixed, is obtained by directing the centroid of the group of agents to that desired location. This leads to a collective formation of all the agents, known as balanced circular formation. The theoretical findings are supported by simulations. © 2014 IFAC.

IFAC Proceedings Volumes (IFAC-PapersOnline)

Stabilization of balanced circular motion about a desired center

In this paper, we study the collective motion of individually controlled planar particles when they are coupled through heterogeneous controller gains. Two types of collective formations, synchronization and balancing, are described and analyzed under the influence of these heterogeneous controller gains. These formations are characterized by the motion of the centroid of the group of particles. In synchronized formation, the particles and their centroid move in a common direction, while in balanced formation the movement of particles possess a fixed location of the centroid. We show that, by selecting suitable controller gains, these formations can be controlled significantly to obtain not only a desired direction of motion but also a desired location of the centroid. We present the results for N-particles in synchronized formation, while in balanced formation our analysis is confined to two and three particles. © 2013 AACC American Automatic Control Council.