Externally positive linear systems from transfer function properties

The characterisation of single-input-single-output externally positive linear systems is considered. A complete characterisation of the class of externally positive second-order and a class of underdamped third-order systems is given and connections to negative-imaginary systems are highlighted. It is shown that negative-imaginary systems have non-negative step responses, leading to a condition for external positivity based on negative imaginary systems theory. Finally, a class of externally positive systems which can be verified using the developed results but which fail a recently developed numerical test for external positivity based upon linear matrix inequalities are introduced. These results extend the class of system for which external positivity can be verified, facilitating large-scale control and less conservative absolute stability analysis

Robust Formation Control for Networked Robotic Systems Using Negative Imaginary Dynamics

This paper proposes a consensus-based formation tracking scheme for multi-robot systems utilizing the Negative Imaginary (NI) theory. The proposed scheme applies to a class of networked robotic systems that can be modelled as a group of single integrator agents with stable uncertainties connected via an undirected graph. NI/SNI property of networked agents facilitates the design of a distributed Strictly Negative Imaginary (SNI) controller to achieve the desired formation tracking. A new theoretical proof of asymptotic convergence of the formation tracking trajectories is derived based on the integral controllability of a networked SNI systems. The proposed scheme is an alternative to the conventional Lyapunov-based formation tracking schemes. It offers robustness to NI/SNI-type model uncertainties and fault-tolerance to a sudden loss of robots due to hardware/communication fault. The feasibility and usefulness of the proposed formation tracking scheme were validated by lab-based real-time hardware experiments involving miniature mobile robots

Synchronization of Diverse Agents via Phase Analysis

In this paper, the synchronization of heterogeneous agents interacting over a dynamical network is studied. The edge dynamics can model the inter-agent communications which are often heterogeneous by nature. They can also model the controllers of the agents which may be different for each agent or uniform for all the agents. Novel synchronization conditions are obtained for both cases from a phase perspective by exploiting a recently developed small phase theorem. The conditions scale well with the network and reveal the trade-off between the phases of node dynamics and edge dynamics. We also study the synchronizability problem which aims to characterize the allowable diversity of the agents for which controllers can be designed so as to achieve synchronization. The allowable diversity is captured in terms of phase conditions engaging the residue matrices of the agents at their persistent modes. Controller design algorithms are provided for the cases of agent-dependent and uniform controllers, respectively

New Results on Negative Imaginary Systems Theory with Application to Flexible Structures and Nano-Positioning

Flexible structure systems arise in many important applications such as ground and aerospace vehicles, atomic force microscopes, rotating flexible spacecraft, rotary cranes, robotics and flexible link manipulators, hard disk drives and other nano-positioning systems. In control systems design for these flexible systems, it is important to consider the effect of highly resonant modes. Such resonant modes are known to adversely affect the stability and performance of flexible structure control systems, and are often very sensitive to changes in environmental variables. These can lead to vibrational effects which limit the ability of control systems in achieving desired levels of performance. These problems are simplified to some extend by using force actuators combined with colocated measurements of velocity, position, or acceleration. Using force actuators combined with colocated measurements of velocity can be studied using positive real systems theory, which has received a great attention since 1962. Using force actuators combined with colocated measurements of position and acceleration can be studied using negative imaginary (NI) systems theory. In this thesis, we provide a generalization and development of negative imaginary systems theory to include a wider class of systems. In the generalization of NI systems theory, we provide a new negative imaginary definition that allows for flexible systems with free body motion. Also, we provide a new stability condition for a positive feedback control system where the plant is NI according to the new definition and the controller is strictly negative imaginary (SNI). This general stability result captures all previous NI stability results which have been developed. This thesis also presents analytical tools for negative imaginary systems theory, which can be useful in the practical applications of the theory. Two methods that can be used for checking the negative imaginary property for a given system are presented. Also, methods for enforcing NI dynamics on mathematical system models to satisfy an NI Property are explored. A systematic method to design controllers for NI systems with guaranteed robust stability also is presented. A practical application of control system design for a three-mirror cavity locking system is presented in the end of the thesis