Negative Imaginary Systems Theory Research Articles

Applying negative imaginary systems theory to non-square systems with polytopic uncertainty

This paper aims to apply Negative Imaginary (NI) systems theory to non-square LTI systems, for example, flexible structure systems with redundant sensors and actuators. The paper proposes static pre- and post-compensation schemes to transform stable fat (i.e. no. of inputs more than the no. of outputs) and tall (i.e. no. of outputs more than the no. of inputs) LTI plants into the class of Strongly Strict Negative Imaginary (SSNI) systems, a subset of the Strictly Negative Imaginary (SNI) systems. The proposed compensators can also stabilise a non-square or a non-NI square plant in a positive feedback loop. Moreover, the compensators can be utilised to design a simple constant reference tracking scheme exploiting the integral controllability propoerty of SSNI systems. Subsequently, observer-based pre- and post-compensators are developed to handle the cases when some of the system states are not available for direct measurement. Then, the compensation schemes are extended to stable, non-square or non-NI square systems with polytopic uncertainty. Finally, the post-compensation technique is applied to stabilise a class of tall/square uncertain LTI plants preceded by a slope-restricted nonlinearity.

Multi-vehicle formation control and obstacle avoidance using negative-imaginary systems theory

This paper presents a compelling combination of the two negative-imaginary (NI) control systems: a consensus-based formation control framework for a hybrid multi-vehicle system and a new dynamic obstacle detection and avoidance algorithm. The incorporation of the two techniques permits robots to self-detect collisions and then self-create a safe path to overcome any unexpected obstacles while traveling to the destination in a given formation. Moreover, owing to the distributed and dynamic architecture, our NI obstacle avoidance method has better adaptability to the change of environment conditions or obstacle/robot dynamics. Also, a rigorous comparative study on the performance of our obstacle avoidance method with respect to that of the artificial potential field function and switching formation-based NI obstacle avoidance algorithms is performed as a benchmark. Finally, simulation and experimental results are then implemented to demonstrate the effectiveness of the proposed theories.

Switching time-invariant formation control of a collaborative multi-agent system using negative imaginary systems theory

The movement of cooperative robots in a densely cluttered environment may be impossible if the formation type is invariant. Hence, we investigate a novel method of switching time-invariant formation control for a group of heterogeneous autonomous vehicles, which may include Unmanned Ground Vehicles (UGV) and Unmanned Aerial Vehicles (UAV). We have extended a Negative Imaginary (NI) consensus control approach to switch the formation shape of the robots whilst only using the relative distances between agents, in combination with those between agents and obstacles. All agents can automatically create a new safe formation to overcome obstacles, then restore the prototype formation once the obstacles are cleared. We address the oscillation and local minima problems, generated by existing algorithms, using a newly developed circle fitting approach. We also highlight the advantages of our proposed algorithms over the kinematic controller and the conventional Artificial Potential Field (APF) controller from both theoretical and experimental standpoints. Simulation and experimental results are then analyzed to validate the feasibility of our proposed approach.

A Survey of Riccati Equation Results in Negative Imaginary Systems Theory and Quantum Control Theory

This paper presents a survey of some new applications of algebraic Riccati equations. In particular, the paper surveys some recent results on the use of algebraic Riccati equations in testing whether a system is negative imaginary and in synthesizing state feedback controllers which make the closed loop system negative imaginary. The paper also surveys the use of Riccati equation methods in the control of quantum linear systems including coherent H∞ control.

Foundations of Not Necessarily Rational Negative Imaginary Systems Theory: Relations Between Classes of Negative Imaginary and Positive Real Systems

In this technical note we lay the foundations of a not necessarily rational negative imaginary systems theory and its relations with positive real systems theory. In analogy with the theory of positive real functions, in our general framework negative imaginary systems are defined in terms of a domain of analyticity of the transfer function and of a sign condition that must be satisfied in such domain. In this way, we do not require to restrict the attention to systems with a rational transfer function. In this work, we also define various grades of negative imaginary systems and aim to provide a unitary view of the different notions that have appeared so far in the literature within the framework of positive real and in the more recent theory of negative imaginary systems, and to show how these notions are characterized and linked to each other.

LMI based robust state feedback control of negative-imaginary systems

The importance of robust feedback control for uncertain negative imaginary system lies in its practical relevance and engineering applications, for example, lightly damped flexible structures with collocated position sensors and force actuators. This article investigates an LMI based robust static state internally stabilizing feedback controller of the negative imaginary system in presence of strictly negative imaginary uncertainties. The paper first presents some basic theory of negative imaginary systems and then, a necessary and sufficient condition is established for the internal stability of the positive feedback interconnections of negative imaginary systems. The paper provides a systematic controller synthesis procedure based on negative imaginary lemma to achieve robustness in the feedback control of negative imaginary systems. To show the effectiveness of controller synthesis method, one design example is considered and a static state-feedback controller is constructed to force the plant to be stable. The simulation results of this article demonstrate the usefulness of the design method and we believe that feedback control of the negative imaginary system can benefit control engineering practitioners in a number of ways.

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures With Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented.

Modulated–demodulated control: Q control of an AFM microcantilever

We outline the application of modulated–demodulated control to the quality (Q) factor control of an atomic force microscope microcantilever. We review the modulated–demodulated control technique, emphasize its linear time invariant nature and develop state space representations of the controller for design and analysis. The modulated–demodulated controller can be configured as both positive position feedback (PPF) and resonant controllers, which are effective in the control of negative imaginary systems. Negative imaginary systems theory has important application in the control of collocated mechanical systems and we briefly summarize the key relevant results. A high-frequency, tunable modulated–demodulated controller, designed specifically for MHz operation, was developed for experimental validation. The modulated–demodulated controller enables the use of a low-bandwidth baseband controller in the configuration of a high-bandwidth controller, thus simplifying the implementation of high-bandwidth controllers. We outline the controller characterization and demonstrate closed-loop control of a Bruker DMASP microcantilever. We also present AFM images highlighting the improvements in scan speed and image quality achieved as a result of Q control. Modulated–demodulated control appears well suited to the control of high-frequency resonant dynamics. In addition to high-speed atomic force microscopy, we believe this control technique may find applications in high-frequency microelectromechanical systems (MEMS).

Some new results in the theory of negative imaginary systems with symmetric transfer matrix function

This note represents a first attempt to provide a definition and characterisation of negative imaginary systems for not necessarily rational transfer functions via a sign condition expressed in the entire domain of analyticity, along the same lines of the classic definition of positive real systems. Under the standing assumption of symmetric transfer functions, we then derive a necessary and sufficient condition that characterises negative imaginary transfer functions in terms of a matrix sign condition restricted to the imaginary axis, once again following the same line of argument of the standard positive real case. Using this definition, even transfer functions with a pole at the origin with double multiplicity, as well as with a possibly negative relative degree, can be negative imaginary.

Infinite-Dimensional Negative Imaginary Systems

We consider second order infinite-dimensional systems with force control and collocated position measurement interconnected with finite-dimensional controllers of the same type. We show that under assumptions that generalize those in the finite-dimensional case (the theory of negative imaginary systems), asymptotic stability of the closed-loop system can be concluded, but that the closed-loop system may not be exponentially stable nor input-output stable.

Feedback Control of Negative-Imaginary Systems

This paper investigates the robustness of positive-position feedback control of flexible structures with colocated force actuators and position sensors. In particular, the theory of negative-imaginary systems is used to reveal the robustness properties of multi-input, multi-output (MIMO) positive-position feedback controllers and related types of controllers for flexible structures. The negative-imaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise input-output dynamics.