Negative Imaginary Research Articles

On positive realness and negative imaginariness of uncertain discrete-time state-space symmetric systems

ABSTRACTThis paper provides an analysis of uncertain discrete-time negative imaginary and positive real systems with state-space symmetry. First, a necessary and sufficient condition is provided for a minimal state-space symmetric system to be discrete-time positive real or discrete-time negative imaginary. Then, based on the discrete-time symmetric negative imaginary and positive real lemmas, bounds on symmetrically structured perturbations that maintain the negative imaginariness and positive realness of a discrete-time state-space symmetric system are derived. Finally, an RL network example is provided to illustrate the main results of the paper.

Fuzzy Self-Tuning of Strictly Negative-Imaginary Controllers for Trajectory Tracking of a Quadcopter Unmanned Aerial Vehicle

Robustness in the face of uncertainties is an integral part of designing a real-time control system. Based on negative imaginary (NI) systems theory, we design robust and adaptive control systems for accurate trajectory tracking of a quadcopter aerial vehicle. Considering the challenging dynamics of unmanned aerial vehicles, we employ knowledge-based fuzzy inference systems (FIS) to facilitate automatic tuning in our SNI controllers, leading to the development of adaptive SNI control systems. Unlike fixed-gain controllers that have no ability to adapt to the variations in environmental conditions or changes in the dynamics of the plant, our adaptive SNI controllers are able to perform self-tuning to constantly update their parameters. The concept of adaptive autopilots will enhance the ability of the closed-loop control systems to accommodate large uncertainties. To demonstrate their efficacy, we design and implement our adaptive SNI controllers in the three-position control loops of the AR.Drone quadcopter after conducting extensive computer simulations. We also perform a rigorous comparative study with respect to the performance of fixed-gain SNI controllers, fixed-gain NI systems, in addition to model-predictive-control systems, and proportional integral derivative (PID) control systems as our benchmarks. To complete the study, we conduct a stability analysis based on Kharitonov's Theorem.

Robust output consensus of homogeneous multi-agent systems with negative imaginary dynamics

This paper addresses the robust output consensus problem for multiple homogeneous negative imaginary (NI) systems subject to L2 external disturbances and model uncertainties. The principal aim of this paper is to extend and overcome limitations in previous studies which had required restrictive assumptions to be imposed and thereby derive necessary and sufficient conditions for robust output consensus of networked NI systems when such assumptions do not hold. In doing so, the recently published robust NI stability results are utilised. An advantage of the derived conditions is that they depend on the dc and infinite frequency gains of the systems as well as the network graph but not on the precise dynamics of the systems. Furthermore, it is shown that the derived conditions specialise to the earlier results in literature either by imposing the same assumptions or by imposing different ones which had not been known previously. It is also shown that the derived conditions simplify in the single-input single-output case and an intuition about these results is provided. Finally, examples are given which demonstrate the effectiveness of the proposed results over earlier results when earlier assumptions are not met.

Necessary and Sufficient Conditions on Negative Imaginariness for Interval SISO Transfer Functions and Their Interconnection

This article is concerned with the study of the negative imaginariness of interval transfer functions. Necessary and sufficient conditions for (strictly) negative imaginary interval transfer functions subject to parameter uncertainty are first derived based on Kharitonov's theorem. Then, certain properties of the interconnection system with negative imaginary interval transfer functions are analyzed. Finally, an example is provided to illustrate the main results of the article.

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.

Adaptive Second-Order Strictly Negative Imaginary Controllers Based on the Interval Type-2 Fuzzy Self-Tuning Systems for a Hovering Quadrotor With Uncertainties

We introduce a new adaptive control technique based on the second-order strictly negative imaginary (SNI) controller, coupled with the interval type-2 fuzzy self-tuning mechanism. We demonstrate the performance of our controllers in the three control loops of the AR.Drone quadrotor to achieve stable and balanced hover performance in flights with a variable Center-of-Gravity (CoG). To facilitate an automatic tuning mechanicsm in our SNI controller, we employ the knowledge-based interval Type-2 Takagi-Sugeno fuzzy system, known for its ability to accommodate the footprint-of-uncertainties (FoU). The proposed adaptive control law is implemented to compensate the dynamic variations in the quadrotor's CoG when the mass of the payload is constantly varied. The robustness and the effectiveness of the proposed control technique are highlighted not only using extensive computer simulations, but also through numerous real-time flight tests. Besides, we also conduct the stability analysis based on the SNI theorem.

Integral Control of Stable Negative-Imaginary Systems Preceded by Hysteresis Nonlinearity

In this note, constant reference tracking using integral control for negative-imaginary (NI) systems preceded by hysteresis nonlinearity is addressed. The hysteresis nonlinearity is considered to be slope-restricted, though no specific modeling framework is exploited. It is effectively shown that integral tracking controllers are guaranteed to exist when a stable NI system with sign-definite dc-gain is preceded by such hysteresis nonlinearities. When the dc-gain is negative-definite, under positive feedback configuration with positive slope-restricted hysteresis, it is established that the integral controller gain can be any positive diagonal matrix. The development of the proposed model-independent inversion-free scheme is carried out in general state-space setting and is applicable for multiple-input multiple-output systems with decoupled hysteresis nonlinearities. Effectiveness of the results is illustrated with examples.

Distributed robust stabilization of networked multiagent systems with strict negative imaginary uncertainties

SummaryThis paper deals with the distributed robust stabilization problem for networked multiagent systems with strict negative imaginary (SNI) uncertainties. Communication among agents in the network is modelled by an undirected graph with at least one self‐loop. A protocol based on relative state measurements of neighbouring agents and absolute state measurements of a subset of agents is considered. This paper shows how to design the protocol parameters such that the uncertain closed‐loop networked multiagent system is robustly stable against any SNI uncertainty within a certain set for various different network topologies. Tools from negative imaginary (NI) theory are used as an aid to simplify the problem and synthesise the protocol parameters. We show that a state, input, and output transformation preserves the NI property of the network. Consequently, a necessary and sufficient condition for the transfer function matrix of the nominal closed‐loop networked system to be NI and satisfy a DC gain condition is that multiple reduced‐order equivalent systems be NI and satisfy a DC gain condition simultaneously. Based on the reduced‐order systems, we derive sufficient conditions in an LMI framework which ensure the existence of a protocol satisfying the desired objectives. A numerical example is given to confirm the effectivenesses of the proposed results.

Controller synthesis for negative imaginary systems using nonlinear optimisation and H 2 performance measure

ABSTRACT Negative imaginary (NI) systems theory has attracted considerable attention in the area of robust control of highly resonant flexible structures systems and nano-positioning systems, where the NI property is naturally satisfied. In this paper, we present a control synthesis methodology for NI systems based on nonlinear optimisation techniques. In the presented method, a parametrised library of strictly negative imaginary (SNI) controllers is created and used in a standard numerical nonlinear optimisation setup. The SNI controller library contains generic controllers that are widely used in the control highly resonant flexible structures such as positive position feedback (PPF) and integral resonant control (IRC). Sequential quadratic programming (SQP) techniques are used in the numerical optimisation problem. The synthesised controller satisfies the SNI property as well as optimising performance. As an application of these results, an example of controlling a Euler–Bernoulli beam with piezoelectric actuator and sensor is presented.

On positive realness, negative imaginariness, and [formula omitted] control of state-space symmetric systems

This paper studies positive realness and negative imaginariness of state-space symmetric systems. First, a necessary and sufficient condition is provided for a state-space symmetric system to be, respectively, positive real and negative imaginary. Then an explicit formulation for the infimum of H∞ norm under a mixed H∞ and negative imaginary performance and an explicit solution for the associated optimal static output feedback control gain are developed based on symmetric negative imaginary theorem. Finally, three examples are provided to illustrate the main results of the paper.

Bilinear Transformation for Discrete-Time Positive Real and Negative Imaginary Systems

This paper studies the connection between discrete-time and continuous-time negative imaginary systems. First, we analyze differences between two statements that are claimed to provide equivalent conditions for systems to be discrete-time positive real. Our conclusion is that one is equivalent to the definition of discrete-time positive real transfer matrices, the other is not. Second, by means of the bilinear transformation, a connection between discrete-time and continuous-time negative imaginary transfer matrices is established. Third, the concept of discrete-time lossless negative imaginary systems is introduced, and a discrete-time lossless negative imaginary lemma is developed to characterize the lossless negative imaginary properties in terms of minimal state-space realization. Some properties of discrete-time lossless negative imaginary transfer matrices are also studied. Several numerical examples illustrate the developed theory.

H2 model reduction for negative imaginary systems

This paper studies the model reduction problem for negative imaginary (NI) systems. For a given high-order system that is stable and NI, our goal is to approximate it by a low-order NI system so that the norm of the approximation error system is minimised. By using the Galerkin projection, the model reduction problem is formulated as a minimisation problem over the Stiefel manifold. The first-order necessary condition is derived for the construction of a local optimal reduced-order system. A gradient descent algorithm is provided to solve the first-order necessary condition. The resulting reduced-order system preserves the stability and the NI structure of the original system. Finally, two examples are presented to demonstrate the effectiveness of the proposed model reduction method.

On decentralized integral controllability of stable negative-imaginary systems and some related extensions

This paper exploits the decentralized integral controllability (DIC) of stable negative-imaginary (NI) systems with non-singular DC-gain matrix to develop a constant input tracking framework with failure tolerance property in the event of sensor and/or actuator faults. By virtue of the DIC property of NI systems, it is shown that a positive feedback interconnection of two NI systems, of which one is a decentralized integrator system with non-negative gains and the other one is a stable NI system having negative definite DC-gain matrix, is internally stable. This is in contrast to the existing results where, for internal stability, at least one of the systems needs to be strictly negative-imaginary (SNI). Further it is shown that, in a positive feedback interconnection, stable NI systems with negative definite DC-gain matrix exhibit integral controllability (IC) property for any positive value of the integral gain k, which is a distinct property of stable NI systems having negative definite DC-gain matrix since the existing notion of IC for LTI systems offers only a finite range of k. Using the DIC property and the eigenvalue loci technique, some new results and research directions for the NI systems have also been presented in this paper.

Robust stability conditions for feedback interconnections of distributed-parameter negative imaginary systems

Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems.

Negative Imaginary Approached High Performance Robust Resonant Controller Design for Single-Phase Islanded Microgrid and Its Voltage Observation on Different Load Condition

This paper presents the design of a high performance robust resonant controller for the islanded single-phase microgrid operation on different loads conditions. The design of the controller is done using the results of Negative Imaginary approach. The performance of the proposed controller has been found much effective to track the instantaneous reference grid voltage. The simulation work has been done with the help of MATLAB/SimPower System toolbox. This shows that the proposed controller provides effective control of voltage against the uncertain load conditions.

Feedback Stability of Negative Imaginary Systems

This paper extends the robust feedback stability theorem of negative imaginary systems by removing restrictive assumptions on the instantaneous gains of the systems that were imposed in the earlier literature, and it further generalizes this robust analysis result into the case that allows negative imaginary systems to have poles at the origin. In doing so, we extend the class of negative imaginary systems for which this robust stability theorem is applicable. We also show that this new generalized necessary and sufficient result specializes to the earlier theorems under the same assumptions. We additionally prove that the previously known dc gain condition is not only necessary and sufficient for robust feedback stability under the earlier specified instantaneous gain assumptions, but is also necessary and sufficient for robust feedback stability under new, different and equally simple assumptions. The general robust feedback stability theorem for negative imaginary systems with free body dynamics (i.e., poles at the origin) derived in this paper also specializes to the case that is only applicable for the negative imaginary system without poles at the origin. Since the results for negative imaginary systems with free body dynamics developed in this paper depend on the existence of a matrix $\Psi$ with certain properties, we also propose a systematic construction of this matrix $\Psi$ and show that construction of one such $\Psi$ is sufficient for determining the feedback stability of the closed-loop system. Finally, examples are used to demonstrate the applicability of the results.

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.

A Negative Imaginary Theory-based Controller Synthesis for Vibration Control of a Piezoelectric Tube Scanner

In this paper, a data-driven controller is synthesised for systems with negative imaginary (NI) properties such as the piezoelectric tube scanner (PTS) of an atomic force microscopy (AFM). The scanning speed of an AFM is limited by the dynamic behaviours of its scanning unit. The controller aims to damp vibrations, increase the closed-loop bandwidth and track reference signals with high accuracy. The measured frequency response of the PTS is used to calculate the controller frequency response at each frequency by minimising a cost function. The data-driven control optimisation problem is constrained by NI properties to guarantee that the controller is stable and identifiable. The designed controller is implemented on the AFM using a dSPACE ds1103 real-time prototyping system. Experimental results show that the controller is able to significantly damp the resonances in the scanner’s lateral axes, provide a high-bandwidth closed-loop system and allow reference signal tracking.

Lyapunov-based Stability of Feedback Interconnections of Negative Imaginary Systems

Feedback control systems using sensors and actuators such as piezoelectric sensors and actuators, micro-electro-mechanical systems (MEMS) sensors and opto-mechanical sensors, are allowing new advances in designing such high precision technologies. The negative imaginary control systems framework allows for robust control design for such high precision systems in the face of uncertainties due to unmodelled dynamics. The stability of the feedback interconnection of negative imaginary systems has been well established in the literature. However, the proofs of stability feedback interconnection which are used in some previous papers have a shortcoming due to a matrix inevitability issue. In this paper, we provide a new and correct Lyapunov-based proof of one such result and show that the result is still true.

Data Driven Controller Design for Positioning Control of an AFM Scanner

In this paper, a data driven controller is designed for the positioning control of a piezoelectric tube scanner (PTS) used in an atomic force microscope (AFM). A single-input single-output (SISO) model-based data driven controller is synthesized by using a mixed negative imaginary (NI) and small gain (SG) approach. The controller is implemented on an AFM and gives significant damping and tracking performance, with 16.64 dB and 19.33 dB damping at the resonance frequency of the X and Y-axes of the PTS, respectively. Moreover, this type of controller design is an effective approach to give intuition about how the controller frequency response should, depending on the design constraints being applied, which ensures optimized performance in vibration control.