On non-proper negative imaginary systems

doi:10.1016/j.sysconle.2015.10.008

Systems & Control Letters

Volume 88, February 2016, Pages 47-53

https://doi.org/10.1016/j.sysconle.2015.10.008 Get rights and content

Abstract

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.

Introduction

Broadly speaking, negative imaginary systems are systems whose transfer functions satisfy negative imaginary condition [1], [2], [3], [4]. This type of systems was firstly introduced in [2], and arose in many practical areas, such as lightly damped or undamped flexible structures with collocated position sensors and force actuators [1], [2], [3], [4], [5], [6], [7], DC machine systems from input voltages to shaft rotational velocities [6], [8], nanopositioning systems [9] and RLC circuit networks [10]. A main contribution in [2], [4] was that a negative imaginary lemma was proposed to test the negative imaginary properties of systems under the assumption of minimal state-space realization. Along this line of research, [11] and [12] generalized the negative imaginary lemma by removing the minimality assumption and using the algebraic Riccati equation, respectively. Another main research problem of negative imaginary systems was the internal stability of positive feedback interconnected negative imaginary systems, see [2], [4], [13]. The stability result of [2], [4] has been used in modulated–demodulated systems [14], multi-input multi-output collocated structures [15] and a string of coupled systems [16]. In addition, many efforts have been devoted to the study of negative imaginary theory and applications, see [3], [5], [6], [7], [13], [16], [17], [18] and references cited therein.

In this paper, we extend the definition of negative imaginary functions in [2], [4], [13] to the case where the functions are allowed to be non-proper. It is well known that descriptor systems play an important role in realistic systems. However, the existing definitions of negative imaginary functions are not applicable to non-proper and non-symmetric descriptor systems. The first definition of negative imaginary functions in [2] did not allow poles on the purely imaginary axis. Then, the definition was extended in [4] by allowing poles on the imaginary axis except at the origin. In [13], the authors further extended the definition by allowing poles at the origin. All definitions of negative imaginary functions in [2], [4], [13] were proposed in terms of properties on the $j ω$ -axis, and the transfer functions were limited to be proper real–rational. Although the authors of [17], [18] gave a generalized definition about negative imaginary functions which were not necessarily rational and proper, the transfer function matrices in [17], [18] were restricted to be symmetric. Compared to the definitions of negative imaginary functions in [2], [4], [13], [17], [18], our new definition proposed in this paper does not require the transfer function matrix to be symmetric and proper. This paper aims at the study of negative imaginary systems where symmetry and properness are not necessary. Inspired by the positive real conditions in [19], [20], [21], this paper provides a sufficient condition corresponding to complex variable $s$ to test the negative imaginary properties of systems.

Negative imaginary theory is related to positive real theory. Under certain technical assumptions, negative imaginary functions can be transformed into positive real functions and vice versa. A relationship between negative imaginary and positive real functions with no poles at the origin was provided in [4]. The relationship in [4] was further developed in [22] by allowing the negative imaginary functions to have a simple pole at the origin. Subsequently, the relationship in [4] was modified to the case where the transfer function matrices were symmetric [17]. Also, some transformations between negative imaginary and positive real functions were presented in [6]. In this paper, we give new relationships between (lossless) negative imaginary and (lossless) positive real functions. We consider the possibilities of having poles at the origin and infinity in the transfer functions, and remove the symmetric restrictions. To achieve this, we give a minor decomposition theory of negative imaginary transfer function matrices.

The rest of this paper is organized as follows: Section 2 introduces the new concept of (lossless) negative imaginary functions. Section 3 gives the main results of this paper. A generalized lemma is presented to provide a sufficient condition for non-proper negative imaginary functions. When the given transfer function matrix is symmetric, the generalized lemma is modified to provide a necessary and sufficient condition for negative imaginary functions. Section 4 presents relationships between (lossless) negative imaginary and (lossless) positive real functions. Section 5 concludes the paper.

Notation: $R^{m \times n}$ and $C^{m \times n}$ denote the sets of $m \times n$ real and complex matrices, respectively. $Re [.]$ and $Im [.]$ denote the real and imaginary parts of complex numbers, respectively. $A^{T}$ , $\bar{A}$ and $A^{*}$ denote the transpose, the complex conjugate and the complex conjugate transpose of a complex matrix $A$ , respectively. $H (A)$ denotes the Hermitian part of matrix $A$ . $λ_{\max}$ and $λ_{\min}$ denote the maximum and minimum eigenvalues for a square complex matrix with only real eigenvalues, respectively.

Section snippets

Non-proper negative imaginary transfer functions

A new definition of negative imaginary transfer functions is proposed in this section and some preliminaries which will be used later are introduced. First, we recall the definition of positive real transfer functions in [19], [20].

Definition 1

[19]

A square transfer function matrix $F (s)$ is positive real if

(1)
all elements of $F (s)$ are analytic in $Re [s] > 0$ ;
(2)
$F (s)$ is real for real positive $s$ ;
(3)
$F^{*} (s) + F (s) \geq 0$ for $Re [s] > 0$ .

Assume that $F (s)$ is a rational transfer function matrix. Conditions 1 and 2 in the above definition are

Negative imaginary test

This section presents the main results of this paper. We propose two generalized lemmas, which are analogous to the definition of positive real functions in [19], [20], [21], to test the non-proper negative imaginary properties of systems in terms of complex variable $s$ .

The following lemma provides a sufficient condition for real–rational transfer function matrices to be negative imaginary, where poles at the origin and infinity are allowed in the transfer functions.

Lemma 5

Let $G (s)$ be a square

Relations between positive real and negative imaginary functions

According to Definition 2, Lemma 2 and the proofs of Lemma 5, Lemma 7, the negative imaginary function $G (s)$ can be decomposed as the form $G (s) = G_{0} (s) + s A_{1} + s^{2} A_{2} + Σ_{i = 3}^{q} s^{i} A_{i},$ where $A_{1} + A_{1}^{T} \leq 0$ , $A_{2} = A_{2}^{*} \leq 0$ , $A_{i} = 0$ , $i = 3, \dots, q$ and $G_{0} (s)$ is proper. Continue to conceive a partial fraction expansion of $G_{0} (s)$ as (3), which is similar to the minor decomposition of positive real transfer functions in [19, p216]. The minor decomposition in (3) is a subsequent grouping together of terms with poles in the left

Conclusions

This paper has studied the non-proper negative imaginary properties of square real–rational transfer functions. The new definition of negative imaginary functions that might be non-proper and non-symmetric has been proposed. The generalized negative imaginary lemmas for negative imaginary systems with poles at the origin and infinity have been presented in terms of $s$ . The relationships between (lossless) negative imaginary and (lossless) positive real transfer functions have been studied. The

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^☆: This work was supported by National Natural Science Foundation of China under Grant 61374026, Program for New Century Excellent Talents in University 11-0880 and Fundamental Research Funds for the Central Universities WK2100100013.

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On non-proper negative imaginary systems☆

Abstract

Introduction

Section snippets

Non-proper negative imaginary transfer functions

[19]

Negative imaginary test

Relations between positive real and negative imaginary functions

Conclusions

Automatica

Systems Control Lett.

Systems Control Lett.

Mechatronics

Automatica

Feedback control of negative-imaginary systems

IEEE Control Syst.

Stability robustness of a feedback interconnection of systems with negative imaginary frequency response

IEEE Trans. Automat. Control

Finite frequency negative imaginary systems

IEEE Trans. Automat. Control

A negative imaginary lemma and the stability of interconnections of linear negative imaginary systems

IEEE Trans. Automat. Control

Robust performance analysis for uncertain negative-imaginary systems

Internat. J. Robust Nonlinear Control