Negative Imaginary Property Research Articles

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.

Converse theorems for the property of negative imaginary systems

This paper deals with the converse problems about the robust stability of finite dimensional negative imaginary systems. Various converse negative imaginary theorems are established for linear time-invariant systems. It is shown that the feedback interconnection stability of a specific system with an arbitrary strictly negative imaginary system implies the negative imaginary property of the specific.

Robust stabilization of LTI negative imaginary systems using the nearest negative imaginary controller

AbstractThis paper considers the problem of robust stabilization of linear time‐invariant systems with respect to unmodelled dynamics and structure uncertainties. To that end, a methodology to find the nearest negative imaginary system for a given non‐negative imaginary system is presented first. Then, this result is employed to construct a near optimal linear quadratic Gaussian controller achieving desired performance measures. The problem is formulated using port‐Hamiltonian method and the required conditions are defined in terms of linear matrix inequalities. The technique is presented using the fast gradient method to solve the problem systematically. The designed controller satisfies a negative imaginary property and guarantees a robust feedback loop. The effectiveness of the approach is demonstrated by a simulation on a numerical example.

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.

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.

A new consensus-based formation tracking scheme for a class of robotic systems using negative imaginary property

This brief proposes a consensus-based formation tracking scheme for multi-robot systems utilising 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 networked NI/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, two-wheeled mobile robots.

Negative Imaginary Stability Result Allowing Purely Imaginary Poles in Both the Interconnected Systems

This letter extends the class of systems to which negative imaginary (NI) stability result can be (directly) applied. The existing closed-loop stability results for NI interconnections require at least one of the systems to satisfy strictly negative imaginary (SNI) properties and hence do not allow purely imaginary poles in both the systems. In contrast to this, a set of necessary and sufficient conditions is derived in this brief for the internal stability of two NI systems in a positive feedback loop with possible poles on the imaginary axis excluding the origin. This proposed result also captures all the existing closed-loop stability results of NI systems, without poles at the origin, as special cases. Illustrative numerical examples are studied to show the usefulness of this development.

Vibration control of a class of mechatronic systems using Negative Imaginary theory

This paper proposes a new controller design methodology for a class of mechatronics systems that satisfy stable and minimum-phase Negative Imaginary (NI) property. The proposed methodology relies 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 established by the internal stability result of NI theory. In this scheme, the design problem of the IMC parameter (known as the Youla parameter) has been cast as a Negative Imaginary (NI) controller synthesis problem. We propose an LMI-based IMC design approach, which can be readily solved using the commercially available SDP solver packages. A frequency-domain design approach is also presented, which depends on the solution of a simple constrained least-square problem. An in-depth simulation case study on the vibration attenuation problem of a lightweight cantilever beam is executed to demonstrate the usefulness and potential application of the proposed NI-based IMC scheme.

Negative imaginariness and positive realness analysis of state-space symmetric systems with interval uncertainty

In this article, the state-space symmetric systems with symmetrical interval uncertainty that have positive real and negative imaginary properties are studied. First, a necessary and sufficient test in view of a state matrix is derived for a state-space symmetric system to be negative imaginary, which allows having poles at the origin. Second, bounds on symmetrical interval uncertainty that guarantee the positive realness and negative imaginariness of state-space symmetric systems are provided. Finally, the main results are illustrated by a resistor–capacitor network and a numerical design example.

Velocity-Free Attitude Control of Quadrotors: A Nonlinear Negative Imaginary Approach.

In this paper, we propose a new approach to the attitude control of quadrotors, by which angular velocity measurements or a model-based observer reconstructing the angular velocity are not needed. The proposed approach is based on recent stability results obtained for nonlinear negative imaginary systems. In specific, through an inner-outer loop method, we establish the nonlinear negative imaginary property of the quadrotor rotational subsystem. Then, a strictly negative imaginary controller is synthesized using the nonlinear negative imaginary results. This guarantees the robust asymptotic stability of the attitude of the quadrotor in the face of modeling uncertainties and external disturbances. First simulation results underline the effectiveness of the proposed attitude control approach are presented.

On the dissipativity property of negative imaginary systems

This paper extends the concept of dissipativity to a new class of dynamical systems known as negative imaginary systems. The paper also introduce a new definition for nonlinear negative imaginary systems to extend the existent definition. Two different quadratic dynamic supply rates are introduced to allow for extending the dissipativity concept to cover the negative imaginary systems. One of which is a differential operator and the other is an integral operator. Both supply rates are used in order to formulate the class of negative imaginary systems as dissipative systems. This extension allows for wider class of dynamical systems to be considered in the negative imaginary framework. We also show how the new definition extends the negative imaginary system to analyze a class of higher order evolutionary dynamics. In particular, we show that the second order replicator dynamics satisfy the negative imaginary property and hence we can conclude convergence with certain class of games. Also, a nonlinear negative imaginary lemma based on the above definitions is derived.

Passivity Analysis of Replicator Dynamics and Its Variations

In this article, we focus on studying the passivity properties of different versions of replicator dynamics (RD). The RD represents a deterministic monotone nonlinear dynamic, which allows incorporation of the distribution of population types through a fitness function. We use tools for control theory, in particular, the passivity theory, to study the stability of the RD when it is in action with evolutionary games. The passivity theory allows us to identify the class of evolutionary games in which stability with RD is guaranteed. We show that several variations of the first order RD satisfy the standard loseless passivity property. In contrary, the second-order RD does not satisfy the standard passivity property; however, it satisfies a similar dissipativity property known as negative imaginary property.

Strictly negative imaginary state feedback control with a prescribed degree of stability

This paper presents conditions for the synthesis of a strictly negative imaginary closed-loop system with a prescribed degree of stability under the assumption of full state feedback. A perturbation method is used to ensure the closed-loop system has both the strict negative imaginary property and a prescribed degree of stability. This approach involves the real Schur decomposition of a matrix followed by the solution to two Lyapunov equations. Also, we present and clarify the perturbation properties of strictly negative imaginary systems.

Sensor blending and Control allocation for non-square linear systems to achieve negative imaginary dynamics

This paper deals with the design of static pre- and post- compensators to transform stable, non-square LTI systems into the class of strongly strict negative imaginary systems. The pre-compensator plays the role of a control allocator while the post-compensator does sensor blending in order to make a non-square system square along with satisfying the strongly strict negative imaginary property. A specific structure of the post-compensator is also given that guarantees a feasible solution of the LMI conditions when applied to systems with number of outputs greater than or equal to number of inputs. The proposed pre- and post- compensators can also stabilize a non-square plant in closed-loop upon satisfying a particular DC-gain condition and furthermore, they can be utilized to develop a simple constant input tracking framework for non-square systems. The LMI-based design methodology offers a numerically tractable solution framework and hence the easy implementation of the proposed scheme in practical applications. Illustrative examples are provided throughout the paper to demonstrate the usefulness of the proposed results in widening the scope of the negative imaginary theory to non-square LTI systems (e.g. safety-critical systems having redundant sensors and actuators).

Forward action to make time-delay systems positive-real or negative-imaginary

Positive-real and negative-imaginary systems are characterized by properties which make them particularly important in modeling physical devices, such as large-scale flexible structures or electric networks. This paper focuses on a strategy developed in order to ensure the positive-real or negative-imaginary property to systems with external time-delays. A forward action is taken into consideration in order to make systems with time-delay positive-real or negative-imaginary, guaranteeing also closed-loop stability performance. Numerical examples are included.

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.

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.

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.

Guest Editorial

Guest Editorial

Output feedback negative imaginary synthesis under structural constraints

The negative imaginary property is a property that many practical systems exhibit. This paper is concerned with the negative imaginary synthesis problem for linear time-invariant systems by output feedback control. Sufficient conditions are developed for the design of static output feedback controllers, dynamic output feedback controllers and observer-based feedback controllers. Based on the design conditions, a numerical algorithm is suggested to find the desired controllers. Structural constraints can be imposed on the controllers to reflect the practical system constraints. Also, the separation principle is shown to be valid for the observer-based design. Finally, three numerical examples are presented to illustrate the efficiency of the developed theory.