A test for determining systems with “mixed” small gain and passivity properties

Abstract

“Mixedness” is a property that captures elements of the notions of passivity and small gain. In the frequency domain, a linear, time-invariant system is called “mixed” if, over some frequency bands, it is strictly passive and, over the remaining frequencies, it has a gain of less than one; there exist no frequencies over which the system has neither of the notions of these properties associated with it. In this paper, a test is developed for determining whether or not a linear, time-invariant system is “mixed”.

Introduction

Two important results in the input–output stability theory literature are the small gain and the passivity theorems. The small gain theorem states that if the product of the gains of two stable systems, interconnected via a negative feedback loop, is less than one then the interconnection is stable [1], [2], [3], [4]. The passivity theorem guarantees stability of the negative feedback interconnection if, for instance, both of the systems are passive and one of them is input strictly passive with finite gain [1], [2], [3], [5]. Circumstances in which the small gain or passivity properties fail to adequately describe a system in question suggest that alternative assumptions may need to be placed on the systems in the interconnection such that stability might be determined.

High frequency dynamics which might destroy the passivity properties of an otherwise passive system lead us to an example of the notion of a “mixed” system provided that, at those destructive high frequency dynamics, the system has a small gain. More generally, in the linear, time-invariant (LTI) case, a system is called “mixed” if, over some frequency bands, it has the property of being input and output strictly “passive” and, over the remaining frequencies, it has a “gain” of less than one; there exist no frequencies over which the system has neither of these property notions associated with it. “Mixed” systems were initially discussed in [6], [7] (and in the nonlinear case in the time domain in [8], [9]) in a dissipative systems’ framework [10], [11].

The objective of this paper is to present a necessary and sufficient test for determining whether or not a multi-input, multi-output (MIMO) LTI system is “mixed”. The procedure involves the computation of two Hamiltonian matrices, one associated with any potentially passive aspects of the system and the other associated with the notion of system small gain. The examination of the spectral characteristics of these Hamiltonian matrices, which are constructed from state-space data, leads to the elimination of an element of frequency-dependency from the test. The purely imaginary eigenvalues of the Hamiltonian matrices correspond exactly to the frequencies at which zero eigenvalues of certain transfer function matrices typically associated with system passivity and system small gain occur. Testing the sign definiteness of these transfer function matrices at a single frequency point on either side of the frequencies which give rise to the zero eigenvalues yields whether or not the system is “mixed”. Spectral conditions for positive realness of transfer function matrices are discussed in [12] and, for more general frequency domain inequalities, in [13].

The paper is divided into the following sections. The notion of a “mixed” system is defined in Section 2. In Section 3, system state-space descriptions are utilised to compute two Hamiltonian matrices and derive associated results which are required for the “mixedness” test described in Section 4. Examples are provided in Section 5.

The results of this paper are concerned with LTI systems viewed in the frequency domain. $\mathcal{R}$ denotes the set of proper real rational transfer function matrices. For a transfer function matrix $G\left(s\right)\in \mathcal{R}$, $G^{\sim }\left(s\right)$ is defined to mean $G^T(-s)$ and $G^{\ast }\left(j\omega \right)≔G^{\sim }\left(j\omega \right)$. ${\mathcal{L}}_{\infty }$ is a Banach space of matrix- (or scalar-) valued functions that are essentially bounded on $j\mathbb{R}$. The Hardy space, ${\mathcal{H}}_{\infty }$, is the closed subspace of ${\mathcal{L}}_{\infty }$ with functions that are analytic and bounded in the open right-half plane. In other words, ${\mathcal{H}}_{\infty }$ is the space of transfer functions of stable, LTI, continuous-time systems. ${\mathcal{RH}}_{\infty }$ denotes the subspace of ${\mathcal{H}}_{\infty }$ whose transfer function matrices are proper and real rational. The notation $A\in {\mathcal{RH}}_{\infty }^{m\times n}$ will be used to indicate such matrices with $m$ rows and $n$ columns.