Negative Imaginary Research Articles

Necessary and Sufficient Conditions for State Feedback Equivalence to Negative Imaginary Systems

Necessary and Sufficient Conditions for State Feedback Equivalence to Negative Imaginary Systems

On negative imaginariness and [formula omitted] control of descriptor systems with state–space symmetry

This paper studies negative imaginariness and the static output feedback H∞ control synthesis of descriptor systems based on the state–space symmetric model. First, under the assumption that the transfer function matrix is proper, necessary and sufficient conditions to determine negative imaginary properties are established for descriptor systems with state–space symmetry. Then, based on state–space symmetry, a sufficient condition is derived for the design of static output feedback controller, such that the resulting closed-loop system satisfies negative imaginary and H∞ performance. Finally, two circuit networks and a numerical example are provided to validate the main results of the paper.

A dynamic controller synthesis methodology for negative imaginary systems using the internal model control principle

This paper proposes a new controller design methodology for stable and minimum-phase Negative Imaginary (NI) systems relying on the classical Internal Model Control (IMC) principle. The closed-loop stability of the proposed scheme depends only on the DC loop gain, which is theoretically justified by the feedback stability results of the NI theory. The main objective is to design the Youla parameter of an IMC scheme, which has been cast as a Negative Imaginary (NI) controller synthesis problem. Two different methodologies have been proposed. A frequency-domain IMC design approach is first presented, which depends on solving a constrained, linear, least-square estimation problem. Then, an LMI-based methodology is developed, which can be solved by the commercially available SDP solver packages. An in-depth simulation case study on the vibration attenuation problem of a lightweight cantilever beam (a potential application of the NI theory) was carried out to demonstrate the usefulness of the NI-based IMC design methodology. Finally, the simulation results were experimentally validated on a custom-made vibration suppressor to confirm the feasibility of the proposed scheme.

Synthesizing Negative-Imaginary Systems With Closed-Loop H∞-Performance Via Static Output Feedback Control

Abstract This paper develops an algorithm for synthesizing negative imaginary (NI) systems with a specified H∞-norm bound by exploiting the convex relaxation technique in the static output-feedback (SOF) control design framework. This scheme improves the robustness objective since the present framework can handle both NI and H∞-norm-bounded unstructured uncertainty of the system. The non-strict inequality involved in the NI system description that creates a major difficulty in the SOF control design, can efficiently be tackled by the developed convex relaxation-based iterative algorithm. The advantages of the algorithm are shown using several examples and the results are compared with the existing works.

Discrete-time Negative Imaginary Systems from ZOH Sampling

A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case.

On the Stability of Networked Nonlinear Negative Imaginary Systems with Applications to Electrical Power Systems

In this paper, we propose employing battery-based feedback control and nonlinear negative imaginary (NI) systems theory to reduce the need for such expansion. By formulating a novel Luré-Postnikov-like Lyapunov function, stability results are presented for the feedback interconnection of two single nonlinear NI systems, while output feedback consensus results are established for the feedback interconnection of two networked nonlinear NI systems based on a network topology. This theoretical framework underpins our design of battery-based control in power transmission systems. We demonstrate that the power grid can be gradually transitioned into the proposed NI systems, one transmission line at a time.

Properties of interconnected negative imaginary systems and extension to formation‐containment control of networked multi‐UAV systems with experimental validation results

AbstractThis paper extends the properties of a positive feedback interconnection of two negative imaginary (NI) systems to multi‐agent NI systems and proposes a new formation‐containment control methodology relying on the characteristic loci technique. Inspired by recent applications of NI and passivity‐based control techniques in the multi‐agent systems (MAS) domain, a new formation‐tracking and containment control scheme is developed for a class of networked multi‐UAV systems. The proposed scheme offers a two‐stage and two‐loop control configuration where the inner loop uses a cascaded PID controller to ensure stable hovering of the UAVs, and the outer loop deploys a distributed “mixed” SNI and strictly passive controller to achieve the formation‐containment objectives. This scheme works with a dynamic output feedback control strategy; hence, it offers advantages when the full‐state measurement is not possible. In contrast to the well‐known Lyapunov theory‐based cooperative control schemes, the present one exploits the characteristic loci technique to prove the formation‐tracking and containment phenomena theoretically. The paper also provides experimental validation results on a fleet of Crazyflie 2.1 nano quadcopters.

A negative imaginary lemma and state-feedback negative imaginary synthesis for commensurate fractional-order systems

This article focuses on negative imaginariness of commensurate fractional-order linear time-invariant systems. Enlightened by the generalized Kalman–Yakubovic–Popov lemma, minimal state-space realization based necessary and sufficient conditions are developed to characterize negative imaginariness of commensurate fractional-order systems for both 0<α<1 and 1<α<2 cases. Meanwhile, the problem of negative-imaginary state-feedback controller synthesis for fractional-order systems is addressed. A fractional-order viscoelastic system, a fractional-order RLC circuit network and a numerical example are used as illustrative examples to demonstrate the main developed theory.

Feedback Stability of Generalized Positive Real and Negative Imaginary Systems

This paper derives necessary and sufficient conditions for robust stability of a feedback interconnection of linear time-invariant (LTI) systems that exhibit positive realness on finite nonzero frequencies (excluding pole locations) through a weighting function, i.e. a multiplier. This class of LTI systems importantly includes the well-studied negative imaginary systems without poles at the origin as a subset. Under the assumption that the instantaneous gain of the loop transfer function is zero, it is shown that the aforementioned feedback interconnection is stable if and only if the static (a.k.a. DC) loop gain is less than unity. This condition is identical to the one that guarantees feedback stability of negative imaginary systems, but is applicable to a much wider class of systems beyond negative imaginariness. Our proof is based entirely on the multivariable Nyquist stability criterion — it does not make use of realisations of the transfer functions. Comparisons with integral quadratic constraint based robust stability results are also made.

H ∞ negative imaginary static output feedback controller for low frequency networked control systems

In this paper, an H ∞ negative imaginary static output feedback (SOF) controller is designed for a low frequency (LF) networked control system (NCS) with time-delays. A difference operator is introduced to deal with the time-delays in a feedback configuration via integral quadratic constraints (IQCs). Both H ∞ performance and negative imaginariness are analysed via LF IQCs and a generalised Kalman–Yakubovich–Popov (GKYP) lemma. A two-stage algorithm is presented to obtain the H ∞ negative imaginary SOF controller for the LF NCS with time-delays. Simulation results are given to show the effectiveness and superiority of the H ∞ negative imaginary SOF controller.

Robust Fuzzy Q-Learning-Based Strictly Negative Imaginary Tracking Controllers for the Uncertain Quadrotor Systems.

Quadrotors are one of the popular unmanned aerial vehicles (UAVs) due to their versatility and simple design. However, the tuning of gains for quadrotor flight controllers can be laborious, and accurately stable control of trajectories can be difficult to maintain under exogenous disturbances and uncertain system parameters. This article introduces a novel robust adaptive control synthesis methodology for a quadrotor robot's attitude and altitude stabilization. The proposed method is based on the fuzzy reinforcement learning and strictly negative imaginary (SNI) property. The first stage of our control approach is to transform a nonlinear quadrotor system into an equivalent negative-imaginary (NI) linear model by means of the feedback linearization (FL) technique. The second phase is to design a control scheme that adapts online the SNI controller gains via fuzzy Q-learning. The performance of the designed controller is compared with that of a fixed-gain SNI controller, a fuzzy-SNI controller, and a conventional PID controller in a series of numerical simulations. Furthermore, the proofs for the stability of the proposed controller and the adaptive laws are provided using the NI theorem.

Finite, fixed and prescribed-time stability and stabilization of nonlinear negative imaginary systems

One of the most challenging problems in the negative imaginary (NI) systems framework is to achieve stability, particularly rated stability for the interconnected systems as it uses the positive-feedback connection, which may have a destabilizing effect. In this paper, rated convergence properties are introduced in the framework of nonlinear negative imaginary (NNI) systems theory. New NI definitions for a general nonlinear system are provided using dissipative notions and after that the relation between these NI definitions and rated stability is investigated. Further, several positive-feedback interconnections are discussed to establish finite-time, fixed-time and prescribed-time NI properties and their respective stabilities about origin under certain assumptions. Moreover, a methodology of finite-time, fixed-time and prescribed-time NI based control that renders a NNI system, finite-time, fixed-time and prescribed-time stable about origin inspite of the positive-feedback is developed. Finally, encouraging academic and real-world examples are reported to test the validity of the theoretical results.

Characterization of Input–Output Negative Imaginary Systems in a Dissipative Framework

In this article, we define the notion of stable input–output negative imaginary (IONI) systems. This new class captures and unifies all the existing stable subclasses of negative imaginary (NI) systems and is capable of distinguishing between the strict subclasses (e.g., strongly strictly negative imaginary, output strictly negative imaginary (OSNI), input strictly negative imaginary, etc.) in the literature. In addition to a frequency-domain definition, the proposed IONI class has been characterized in a time-domain dissipative framework in terms of a new quadratic supply rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w(u,\bar{u},\dot{\bar{y}})$</tex-math></inline-formula> . This supply rate consists of the system’s input ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula> ), an auxiliary input ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bar{u}$</tex-math></inline-formula> ) that is a filtered version of the system’s input, and the time-derivative of an auxiliary output of the system ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\dot{\bar{y}}$</tex-math></inline-formula> ). This supply rate corrects earlier supply rate attempts in the literature, which were only expressed in terms of the input ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula> ) and the time-derivative of the system’s output ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\dot{y}$</tex-math></inline-formula> ). In this article, IONI systems are proved to be a class of dissipative systems with respect to the proposed supply rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w(u,\bar{u},\dot{\bar{y}})$</tex-math></inline-formula> . Subsequently, an equivalent frequency-dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(Q(\omega), S(\omega), R(\omega))$</tex-math></inline-formula> dissipative supply rate is also proposed for IONI systems. These findings reveal the connections between the NI property and classical dissipativity in both the time domain and frequency domain. We also provide linear matrix inequality (LMI) tests on the state-space matrices to check whether a system belongs to the IONI class or any of its important subclasses. Finally, the derived results are specialized for OSNI systems since such systems exhibit interesting closed-loop stability properties when connected, in a positive feedback loop, to NI systems without poles at the origin. Several illustrative numerical examples are provided to make the results intuitive and useful.

Nonlinear Negative Imaginary Systems with Switching*

In this paper, we extend nonlinear negative imaginary (NI) systems theory to switched systems. Switched nonlinear NI systems and switched nonlinear output strictly negative imaginary (OSNI) systems are defined. We show that the interconnection of two switched nonlinear NI systems is still switched nonlinear NI. The interconnection of a switched nonlinear NI system and a switched nonlinear OSNI system is asymptotically stable under some assumptions. This stability result is then illustrated using a numerical example.

Control of negative imaginary systems exploiting a dissipative characterization

This paper utilizes a dissipative framework to construct a set of stabilizing controllers in positive feedback for LTI negative imaginary (NI) systems with no pole at origin. The technique relies on identifying a dissipative characterization of such NI systems. Contrary to the existing literature, the set of controllers presented in this paper may not belong to the NI systems class and can be designed having only a rough estimate of the DC gain of the plant. Apart from extending the class of stabilizing controllers beyond the NI family, the proposed framework also generalizes a few existing NI domain results. It is shown that the integral resonant control (IRC) scheme in MIMO setting can be captured as a special case of the developed result. Integral controllability of a class of NI systems having negative definite DC gain directly follows from the proposed stability result of this paper. Further generalization in terms of the set of integral gain matrix is also demonstrated. Effectiveness of the proposed results are illustrated through numerical examples.

A frequency-domain approach to nonlinear negative imaginary systems analysis

In this study, we extend the theory of negative imaginary (NI) systems to a nonlinear framework using a frequency-domain approach. The extended notion is completely characterized via a finite-frequency integration over a “kernel function” on energy-bounded input and output signal pairs. The notion is closely related to and carefully contrasted with the well-studied extension of negative imaginariness — the theory of counterclockwise dynamics. Conditions for feedback stability of the proposed nonlinear NI systems are then developed based on the technique of integral quadratic constraints. Examples and simulations on feedback interconnections of typical nonlinear systems are provided to demonstrate the effectiveness.

Static output feedback negative imaginary controller design with H 2 performance

In this paper, the problem of designing static output feedback (SOF) negative imaginary controller has been studied. Because the constraints brought by negative imaginary (NI) property and performance are a set of bilinear matrix inequalities (BMIs), designing an SOF NI controller with performance is a non-trivial problem. To overcome the difficulty of solving BMIs, a linearisation-based method is proposed. First, a necessary and sufficient condition is established, where the constraints of NI property and performance are reformulated as similar forms. Second, inspired by the semidefinite programming, the derived condition is converted to the difference between two convex functions. Then a linearised iterative algorithm with initialisation process is provided to compute the desired SOF NI controller. Finally, two numerical examples are presented to illustrate the proposed results.

Robust output consensus of networked negative imaginary systems via an integral quadratic constraint approach

This paper deals with the robust output consensus problem of networked homogeneous and heterogeneous negative imaginary (NI) systems under external disturbances via the integral quadratic constraint (IQC) approach. Applying the IQC-based NI stability result and properties of the network graph, a necessary and sufficient IQC-based condition is given for robust output consensus of networked homogeneous NI systems without poles at the origin. This condition depends on the corresponding complementary IQCs at zero and infinite frequencies as well as all eigenvalues of the Laplacian matrix associated with the network graph, and it covers the dc loop gain condition in the earlier literature by constructing appropriate IQC multipliers. Moreover, the robust output consensus can be preserved along a path by IQC-based conditions. Subsequently, the robust output consensus problem is also considered for homogeneous NI systems having poles at the origin by adding an assumption related to the largest eigenvalue of the Laplacian matrix. Besides, sufficient IQC-based conditions for the heterogeneous case are described by different IQC multipliers. Finally, some examples are shown to illustrate the main results.

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.

Improved voltage tracking of autonomous microgrid technology using a combined resonant controller with lead-lag compensator adopting negative imaginary theorem

Growing application of distributed generation units at remote places has led to the evolution of microgrid (MG) technology. When an MG system functions independently, i.e., in autonomous mode, unpredictable loads and uncertainties emerge throughout the system. To obtain stable and flexible operation of an autonomous MG, a rigid control mechanism is needed. In this paper, a robust high-performance controller is introduced to improve the performance of voltage tracking of an MG system and to eliminate stability problems. A combination of a resonant controller and a lead-lag compensator in a positive position feedback path is designed, one which obeys the negative imaginary (NI) theorem, for both single-phase and three-phase autonomous MG systems. The controller has excellent tracking performance. This is investigated through considering various uncertainties with different load dynamics. The feasibility and effectiveness of the controller are also determined with a comparative analysis with some well-known controllers, such as linear quadratic regulator, model predictive and NI approached resonant controllers. This confirms the superiority of the designed controller.