Absolute stability analysis for negative-imaginary systems (original) (raw)
Related papers
On LTI output strictly negative-imaginary systems
Systems & Control Letters, 2017
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 negativeimaginary 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.
IEEE Transactions on Automatic Control, 2011
In this letter, an analytical framework is proposed to examine stability of two stable, linear time invariant systems interconnected in positive feedback where the systems have "mixed" properties of negative-imaginary and small-gain. Using the notion of dissipativity, the interconnection of systems is guaranteed to be finite-gain stable under the condition that the dc loop gain is contractive. This work builds on Griggs, et al. [14] and exploits a new set of frequency dependent triplets that was introduced in above reference to "mix" two unconditional stability statements, i.e., small-gain and passivity. Unlike the above reference the present work explores the important question of how a conditional stability statement as needed when two negative-imaginary systems are connected in a feedback loop can be "mixed" with an unconditional stability statement as needed when two contractive systems are connected in a feedback loop. The usefulness of the proposed analytical framework is demonstrated via a numerical example.
On Discrete-Time Output Negative Imaginary Systems
IEEE Control Systems Letters, 2022
This letter introduces the notion of linear Discrete-time Output Negative Imaginary (D-ONI) systems. The D-ONI class is defined in the z-domain and it includes the systems having poles on the unit circle. The proposed definition involves a real parameter δ ≥ 0, which indicates the strictness properties. δ > 0 specifies the strict subset, Discrete-time Output Strictly Negative Imaginary (D-OSNI), within the stable D-ONI class. Interestingly, the new D-ONI class captures the existing D-NI systems while restricted to discrete-time LTI systems having a real, rational and proper transfer function. However, the D-OSNI systems are not identical to the existing strictly D-NI (D-SNI) subset. Instead, these two subsets intersect each other. An LMIbased state-space characterisation is derived to check the strict/non-strict D-ONI properties of a given system relying on the value of δ. The paper also establishes the connections between the discrete-time Passive and discretetime NI systems. Finally, a closed-loop stability result is proposed for a positive feedback interconnection of two D-ONI systems without poles at z = −1 and z = +1.
Stability Analysis and Vibration Control of a Class of Negative Imaginary Systems
Jurnal Teknologi, 2015
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.
Modelling and control of nonlinear negative imaginary systems
International Journal of Dynamics and Control
This article investigates mathematical modelling and control of a nonlinear negative imaginary system using an illustrative benchmark physical example of a quadrotor dynamic model. A generalized methodology based on Euler–Lagrange equation is applied to obtain nonlinear negative imaginary dynamic model for the quadrotor. In this method, the Kronecker product is employed to formulate the Coriolis matrix, which is then used to construct a mathematical model of a quadrotor. This article further presents nonlinear negative imaginary systems theory-based analysis and synthesis framework to find the control solution for quadrotor’s attitude stability problem while hovering. The multi-loop control scheme is applied to the quadrotor that directly uses the Euler angles (angular positions) instead of angular velocity measurements. Numerical simulation results of this paper show that the investigated control strategy ensures the asymptotic stabilization of quadrotor attitude system model in th...
A Strongly Strict Negative-Imaginary Lemma for
2014
Abstract. A state-space characterization is given for strongly strict negative-imaginary systems. It facilitates both robust analysis and synthesis methods for interconnected negative-imaginary systems. Numerical advantages are achieved by avoiding a non-convex rank constraint, a non-strict inequality condition and a minimality assumption present in previous literature. Key words: negative-imaginary systems, positive-real, non-minimal realization Notation. Notation is standard. R and RH ∞ denote the set of all proper realrational and proper real-rational stable transfer function matrices, respectively. R and C are the fields of real and complex numbers, respectively. The superscript (·) n×m denotes an operator with m columns and n rows. Re[s] and Im[s] denote the real and imaginary part of a complex number s ∈ C, respectively. Furthermore, let A ∗ be the complex conjugate transpose of matrix A, and let det(A) be the determinant of a square matrix A. 1. Introduction. Negative-imagina...
Robust performance analysis for uncertain negative‐imaginary systems
2012
Negative-imaginary systems are important in engineering practice as this class of systems appears quite often in practical problems, for example, lightly damped flexible structures with collocated position sensors and force actuators. In this paper, an analytical framework for robust performance of uncertain negativeimaginary systems is proposed. The results are obtained by transforming negative-imaginary systems into a bounded-real framework via the positive-real property. This paper deals with all the significant technical difficulties that appear due to the transformation and the punctured j-axis frequency condition of negative-imaginary systems. The problem is equivalently cast into a structured singular value condition that gives a quantitative performance test for this class of systems. This result also gives an analytical framework for robust stability when the perturbations are mixture of bounded-real and negative-imaginary uncertainties. A numerical example is presented to show the usefulness of the proposed methods. Copyright
Output strictly negative imaginary systems and its connections to dissipativity theory
2019 IEEE 58th Conference on Decision and Control (CDC), 2019
This paper generalises the notion of output strictly negative imaginary systems and provides a complete characterisation both in frequency domain and time domain. The paper also reveals the missing link between the negative imaginary theory and dissipativity. A new time domain supply rate is introduced to characterise the class of output strictly negative imaginary systems that consists of input to the system, the derivative of an auxiliary output of the system and a real parameter δ > 0. Further, in addition to the output strictly negative imaginary systems, all stable negative imaginary systems are shown to be dissipative with respect to the same supply rate with δ = 0. An equivalence is also established between the output strictly negative imaginary systems property and time domain dissipativity of this class of systems with respect to the proposed supply rate and a specific positive definite storage function. Several numerical examples are studied to elucidate the essence of the theoretical developments.
On state-space characterization for strict negative-imaginariness of LTI systems
2011
Negative-imaginary systems appear quite often in engineering applications, for example, in flexible structures with collocated position sensors and force actuators, in electrical circuits, in system biology, etc. In this paper, a strongly strict negative-imaginary lemma is proposed to ensure the strict negative-imaginary property of an LTI system. This result will facilitate both robustness analysis and controller synthesis for interconnected negative-imaginary systems. In the proposed characterization, numerical advantages are achieved by avoiding a minimality assumption, a non-convex rank constraint and a non-strict inequality condition present in previous literature. Two numerical examples are provided to illustrate the effectiveness of the proposed results.
Solution to the positive real control problem for linear time-invariant systems
IEEE Transactions on Automatic Control, 1994
In this paper we study the problem of synthesizing an internally stabilizing linear time-invariant controller for a linear time-invariant plant such that a given closed-loop transfer function is extended strictly positive real. Necessary and sufficiency conditions for the existence of a controller are obtained. State-space formulas for the controller design are given in terms of solutions to algebraic Riccati equations (or inequalities). The order of the constructed controller does not exceed that of the plant. Recent work by Safonov and Chiang 1251 has pointed out some advantages of a positive real formulation of the p synthesis problem. Molander and Willems 1181 gave some sufficient conditions for the solvability of the positive real synthesis problem for the state-feedback case. Safonov et al. [26] approached the positive real control problem using the Cayley transform to Manuscript