Negative Imaginary Systems Research Articles

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.

Closed-loop stability analysis of discrete-time negative imaginary systems

We derive necessary and sufficient conditions for stability analysis of a positive feedback interconnection of a discrete-time negative imaginary system and a discrete-time strictly negative imaginary system. General stability analysis results for continuous-time negative imaginary systems connected in positive feedback have recently been proposed. Those recent results extend previous theorems by removing restrictive assumptions on the infinite frequency gains imposed in the earlier literature and by extending the class of negative imaginary systems for which the results are applicable to include systems with free body dynamics (i.e., poles at the origin). Here, we present the discrete-time counterparts of the aforementioned recently developed results which specialise to simple and easy-to-check conditions under specific assumptions. Last, we illustrate some of the results by several examples.

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.

Stability analysis and controller design for Lur'e system with hysteresis nonlinearities: a negative-imaginary theory based approach

ABSTRACTIn this paper, absolute stability condition is investigated when a hysteresis nonlinearity is connected to a linear subsystem via positive feedback in Lur'e system framework. In particular, the nonlinearity is considered to belong to the time-invariant slope-restricted counter-clockwise hysteresis class. An absolute stability criterion is proposed in terms of the negative-imaginary system properties. In effect, the stability condition requires the linear subsystem to belong to the strongly strict negative-imaginary (SSNI) system class along with satisfying a matrix condition involving the DC-gain of the linear subsystem and the slope-upper-bounds of the nonlinearities. Compliance to the conditions guarantees asymptotic convergence of state-trajectories to an equilibrium set. Invoking the stability result and exploiting the relaxed minimality assumption of SSNI systems, a static state-feedback synthesis method is proposed to ensure absolute stability for such hysteretic systems. Tractable conditions in the form of linear matrix inequalities can be solved to obtain a stabilising state-feedback gain matrix. Finally, a numerical example of multi-input–multi-output system is presented to illustrate the usefulness of the proposed results.

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.

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.

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.

Properties and stability analysis of discrete-time negative imaginary systems

This paper is concerned with discrete-time negative imaginary (DT-NI) functions. First, a new definition of DT-NI functions is introduced. Then, by means of the relations between discrete-time positive real and DT-NI functions, two different versions of DT-NI lemmas are established to characterize the DT-NI properties based on state-space realizations. Also, a necessary and sufficient condition is presented to guarantee the internal stability of positive feedback interconnected DT-NI systems. Meanwhile, some other properties of DT-NI functions are studied. Several numerical examples are presented to illustrate the main results of this paper. Compared to the previous results, our results remove the symmetric assumption in rational case.

An observer-based control scheme using negative-imaginary theory

This paper presents a full-order state observer-based scheme that transforms a causal, LTI, minimally realized system into a strongly strict negative-imaginary system by defining an auxiliary output based on the observed states. The auxiliary output is used for closed-loop control with positive feedback invoking the internal stability condition of negative-imaginary theory. A set of LMI conditions is derived that determines the value of a design parameter μ required for the proposed scheme to transform a given system with a minimal state-space realization into a strongly strict negative-imaginary system. The proposed scheme is applicable for both stable/unstable and square/non-square systems. In this paper, the framework is further explored for robustness analysis of uncertain systems and numerical examples are given to elucidate effectiveness of the proposed results.

Discrete-time negative imaginary systems

In this paper we introduce the notion of a discrete-time negative imaginary system and we investigate its relations with discrete-time positive real system theory. In the framework presented here, discrete-time negative imaginary systems are defined in terms of a sign condition that must be satisfied in a domain of analyticity of the transfer function, in analogy with the case of discrete-time positive real functions, as well as analogously to the continuous-time case. This means in particular that we do not need to restrict our notions and definitions to systems with rational transfer functions. We also provide a discrete-time counterpart of the different notions that have appeared so far in the literature within the framework of strictly positive real and in the more recent theory of strictly negative imaginary systems, and to show how these notions are characterized and linked to each other. Stability analysis results for the feedback interconnection of discrete-time negative imaginary systems are also derived.

On LTI output strictly negative-imaginary systems

This paper deals with the notion of output strictly negative-imaginary systems. A definition is given for the class of output strictly negative-imaginary systems and a lemma is proposed to test the necessary and sufficient conditions required to satisfy output strictly negative-imaginary system properties. A set theoretic relationship is established between the existing class of strictly negative-imaginary systems and the newly defined output strictly negative-imaginary systems class. A stability analysis result for interconnected systems with positive feedback is presented while one of the systems is negative-imaginary and the other one is output strictly negative-imaginary, contrary to the existing results where at least one of the systems in the interconnection belongs to the strictly negative-imaginary class. Several numerical examples have been studied to demonstrate the proposed results.

H∞ model reduction for negative imaginary systems

ABSTRACTThis paper is concerned with the H∞ model reduction for negative imaginary (NI) systems. For a given linear time-invariant system that is stable and NI, our goal is to find a stable reduced-order NI system satisfying a pre-specified H∞ approximation error bound. Sufficient conditions in terms of matrix inequalities are derived for the existence and construction of an H∞ reduced-order NI system. Iterative algorithms are provided to solve the matrix inequalities and to minimise the H∞ approximation error. Finally, a numerical example is used to demonstrate the effectiveness of the proposed model reduction method.

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.

Controller synthesis for negative imaginary systems: a data driven approach

The negative imaginary (NI) property occurs in many important applications. For instance, flexible structure systems with collocated force actuators and position sensors can be modelled as negative imaginary systems. In this study, a data-driven controller synthesis methodology for NI systems is presented. In this approach, measured frequency response data of the plant is used to construct the controller frequency response at every frequency by minimising a cost function. Then, this controller response is used to identify the controller transfer function using system identification methods.

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NEGATIVE IMAGINARY THEOREM WITH AN APPLICATION TO ROBUST CONTROL OF A CRANE SYSTEM

This paper presents an integral sliding mode (ISM) control for a case of negative imaginary (NI) systems. A gantry crane system (GCS) is considered in this work. ISM is a nonlinear control method introducing significant properties of precision, robustness, stress-free tuning and implementation. The GCS model considered in this work is derived based on the x direction and sway motion of the payload. The GCS is a negative imaginary (NI) system with a single pole at the origin. ISM consist of two blocks; the inner block made up of a pole placement controller (NI controller), designed using linear matrix inequality for robustness and outer block made up of sliding mode control to reject disturbances. The ISM is designed to control position tracking and anti-swing payload motion. The robustness of the control scheme is tested with an input disturbance of a sine wave signal. The simulation results show the effectiveness of the control scheme.

Absolute stability analysis for negative-imaginary systems

This paper deals with absolute stability of a Lur’e system with positive feedback where the linear subsystem exhibits negative-imaginary frequency response and the nonlinearity connected in feedback is time-invariant, memoryless and slope-restricted. The proposed absolute stability criterion requires the linear subsystem to belong to the strongly strict negative-imaginary class. Along with that, positive definiteness of a symmetric matrix needs to be ensured, where the symmetric matrix is obtained by subtracting the dc-gain matrix of the linear subsystem from a strictly positive diagonal matrix with elements indicating the reciprocal of the maximum slope bounds of the nonlinearities. The stability criterion is proved using a Lur’e–Postnikov-type Lyapunov function. Numerical examples are presented to demonstrate the proposed results.

On non-proper negative imaginary systems

This paper studies non-proper negative imaginary systems. First, the concept of negative imaginary transfer functions that may have poles at the origin and infinity is introduced. Then, a generalized lemma is presented to provide a sufficient condition to characterize the non-proper negative imaginary properties of systems. The generalized lemma is given in terms of complex variable s in the quarter domain of analyticity. Compared to previous results, our result removes the symmetric restriction. Also, a new relationship is established between (lossless) negative imaginary and (lossless) positive real transfer function matrices by using a minor decomposition. The results in this paper give us the possibility to address non-proper and non-symmetric descriptor systems with negative imaginary frequency response. Several examples are presented to illustrate the results.

STABILITY ANALYSIS AND VIBRATION CONTROL OF A CLASS OF NEGATIVE IMAGINARY SYSTEMS

This paper presents stability analysis and vibration control of a class of negative imaginary systems. A flexible manipulator that moves in a horizontal plane is considered and is modelled using the finite element method. The system with two poles at the origin is shown to possess negative imaginary properties. Subsequently, an integral resonant controller (IRC) which is a strictly negative imaginary controller is designed for the position and vibration control of the system. Using the IRC, the closed-loop system is observed to be internally stable and simuation results show that satisfactory hub angle response is achieved. Furthermore, vibration magnitudes at the resonance modes are suppressed by 48 dB.