Characterising discrete-time linear systems with the “mixed” positive real and bounded real property

Abstract

In this paper, we present characterisations of linear, shift-invariant, discrete-time systems that exhibit mixtures of small gain-type properties and positive real-type behaviours in a certain manner. These “mixed” systems are already fairly well characterised in the continuous-time domain, but the widespread adoption of digital controllers makes it necessary to verify whether commonly used discretisation procedures preserve the characteristic of “mixedness”. First, we analyse the effects of classical discretisation methods on the “mixed” property using Nyquist methods. A frequency domain feedback stability result is then presented. Finally, we develop a spectral-based characterisation of “mixed” discrete-time systems which provides a practical computational test that can also be applied to the MIMO case. Several examples validate the developed theory.

Introduction

Passivity-based analysis and control [10], [25], a well-established theory for engineering systems that appears in a wide range of application areas such as circuit network theory [3], signal processing systems [46], mechanical networks [41] and robotics [30], [12], has recently generated interest from the engineering community once again due to novel results on the stability of network flow control schemes [45]. Traditionally, passivity results guarantee the stability of a feedback interconnection of two stable systems if, for instance, both the systems are passive, and one of the systems is input strictly passive with finite gain [42]. Furthermore, parallel interconnections of passive systems are passive [25], and thus passivity is useful for designing stable control systems that are modular-based. Indeed, motivated by gradient update methods for the optimisation problem of maximising a sum of utility functions (dependent on packet rates from different information sources) subject to capacity constraints in the communication links, [45] developed a framework for stabilising source and link control laws by showing that they can be replaced by dynamic systems with prescribed passivity properties. It was also pointed out that the extra generality afforded in doing so issues one with an ability to design the controllers to improve robustness in stability and performance with respect to time delays, unmodelled flows and capacity variation.

It must always be noted that passivity is a sufficient condition for stability, and problems can arise from using purely passivity-based techniques for real-world applications. For example, unmodelled dynamics can destroy assumed or nominal passivity over certain frequency bandwidths [34], [1], and meeting set passivity criteria can conflict with system performance requirements [16]. A range of work exists on investigating ways to relax passivity requirements while being able to maintain the stability properties of nominally passive control systems; for instance, [22], [23] studied in depth the concepts of finite frequency positive realness (i.e., positive realness only over a certain frequency band) and “restricted passivity”.

Building on the idea of finite frequency positive realness, “mixed” systems were recently examined in [17], [18], [19], [20]. “Mixed” system was a term coined to refer to systems that combine notions of passivity and small gain type behaviour in a certain manner; for example, a “mixed” system has small gain behaviours over frequency bands where positivity is violated. “Mixed” systems were intended to aid in the formalisation and extension of the well-known engineering notion that keeping feedback-loop gain small at high frequencies where passivity might be violated avoids destabilisation of high frequency dynamics; for instance, see [33], [5]. The stability of large-scale interconnections of “mixed” systems was considered in [20], and a spectral-based characterisation, leading to a test, for “mixed” systems in continuous-time was presented in [19].

While the study of systems with finite frequency positive realness (e.g. “mixed” systems [20], [19]; see also [47], [31]) has seen much progress over the past number of years, many basic questions remain. For instance, engineers rarely work with continuous-time systems exclusively. For simulation purposes, or for the purpose of control design, or in order to implement a controller, at some stage a discrete-time representation of the system must be considered. Thus, it is critical to establish whether discrete-time systems inherit fundamental properties of the continuous-time systems from which they are derived. System discretisation has currently become an issue of importance once again, and several papers [32], [48], [6], [11], [40], [36] have recently appeared on this topic, particularly in the switched systems community.

The main purpose of this paper is to introduce the concept of “mixedness” to discrete-time systems. Initially, we emphasise that not all discretisation procedures preserve the “mixed” property. We provide some results that show that “mixedness” is preserved by Tustin׳s and Euler׳s backward methods. However, for some applications, these methods may not be suitable. Given the potential for “mixedness” to be lost upon discretisation, we then adapt previous results for the continuous-time case to the discrete-time domain. First, stability results for feedback interconnections of “mixed” discrete-time systems are derived. Then, a computationally viable spectral-based characterisation method, for determining whether a system in discrete-time is “mixed,” is developed.

The remainder of the paper is organised as follows. In Section 2, “mixed” systems in discrete-time are defined and some preliminary work that is required later in the paper is provided. In Section 3, we motivate the need to characterise “mixed” systems in discrete-time, by observing how certain discretisation processes do not necessarily preserve the property of “mixedness”. In Section 4, a feedback stability result for “mixed” discrete-time systems, based on the Nyquist stability theorem, is presented. A spectral-based characterisation of “mixed” discrete-time systems, based on their state-space information, is derived in Section 5. This spectral-based characterisation leads to a test for determining “mixedness” in discrete-time and constitutes one of the main results of the paper. Examples of the spectral-based test are provided in Section 6, and directions for our future research are presented in Section 7.