Retrospect and prospect on integration of gain and phase for feedback stability analysis

Di Zhao·Li Qiu

Received:26 September 2022/Accepted:27 September 2022/Published online:7 November 2022

©The Author(s),under exclusive licence to South China University of Technology and Academy of Mathematics and Systems Science,Chinese Academy of Sciences 2022

Abstract In this paper, we review existing approaches to integrating small gain and small phase analysis for feedback stability of dynamical systems,and give a brief outlook for possible future directions in exploring this topic.Small gain analysis has been very successful and popular in control theory since 1960s,while the small phase analysis for multiple-input-multiple-output systems has not been well understood until recently and is now gradually taking shape.Nevertheless,there have been attempts to analyzing feedback stability via the integration of gain and phase information over decades, including the combination of small gain with positive realness as well as that with negative imaginariness.Such combinations can be subsumed into a recently proposed framework for gain-phase integration,which brings in new geometrical methods and also sheds new lights on several future directions.

Keywords Small gain·Small phase·Passivity·Positive real·Negative imaginary·Davis-Wielandt shell

1 Introduction

For single-input-single-output (SISO) linear time-invariant(LTI)systems,a successful control theory blending gain and phase as two equally important interrelated pillars has been popular since 1950s[1].In particular,the well-known Bode gain and phase plots [1] of a loop transfer function give a significant amount of useful information about the closedloop stability and performance.For multiple-input-multipleoutput(MIMO)LTI systems,networked systems,and even nonlinear systems, the control theory sees the thriving of the gain concept since 1960s [2], with particular impacts on H∞control[3],robust and optimal control[4],nonlinear systems analysis[5],and so on.However,the notion of phase for MIMO systems, which should be paired with the gain,has been sought after for decades [6–8] and is not clearly defined until recently[9–14],let along playing a fair role as thegaininsystemsanalysis.Moreimportantly,duetothelack of clear definition and understanding of phase, the control theory blending gain and phase for MIMO and nonlinear systems has not been well understood.

The concept of gain for matrices or MIMO systems has been well established and intensively studied by taking advantages of matrix singular values and induced norms.Recently, with the research developments of matrix phases[9,10], a phase theory has been proposed and is gradually taking shape for MIMO LTI systems [11–13] as well as nonlinear dynamical systems [14]. The phase concept subsumes the well-known notions on input-output properties of dynamical systems,namely,positive realness[15]and negative imaginariness [16], which can be regarded as special qualitative descriptions of systems with aπ-spread phase sector.

Closed-loopstabilityisoneofthemostfundamentalissues in systems and control theory.A small gain theorem basically tells that a closed-loop system is stable if the loop gain is less than 1[2],which applies to various systems from SISO LTI cases to MIMO nonlinear cases. In parallel, as suggested by the Nyquist stability criterion, closed-loop stability of SISO LTI systems can also be guaranteed by restricting the“loop phase”to be within the range(−π,π).Recently,such a condition has been extended to MIMO LTI systems [11]and nonlinear dynamical systems [14]. However, in many real-world problems,it is conservative or even unrealistic to carry out stability analysis with pure gain or phase information,e.g.,mechanical systems with collocated actuators and speed and position sensors[16,17].Therefore,an integration of small gain and small phase analysis is warranted,and the review and future directions of such integration are the main focus of this paper.

In the rest of this paper, we first review the existing approaches integrating gain and phase for feedback stability in Section 2,then introduce several interesting future directions in Section 3,and finally conclude the paper in Section 4.

2 Integration of gain and phase

The gain and phase have been elegantly combined for SISO LTI systems,via the study of Bode plots and Nyquist stability criterion.Whereas,due to the limitation by the unclear definition of system phase,there have been only a few attempts on integrating the gain and phase-type information of MIMO and/or nonlinear systems[8,18–20].

As positive realness and negative imaginariness can be viewed as qualitative description of system phase,combining these properties with small gain naturally gives stability criteria involving gain and phase-type information.For example,[18] introduces several ways for frequency-wise combination of passivity and small gain property of systems in the loop, [19] proposes a combined small gain and passivity approach to robust controller design,[20]considers systems in the loop share complementary mixed small gain,passivity and negative-imaginariness properties, and at a higher standpoint the dissipativity theory [21] can also be treated as an implicit combination of small gain and passivity as it can recover both by utilizing suitable supply rates.Moreover,[8]establishes multi-variable gain-phase and sensitivity relations, extending the classical Bode integral relations to multi-variable systems.

Based on the recently proposed notion of phases for MIMO LTI systems [11], some new attempts on integration of the gain and phase have been adopted from various aspects as in[22].A new framework has been primarily built up for the purpose of analysis and synthesis of feedback systems using combined small gain and small phase approaches.Starting with a stability condition facilitating the idea that small phase condition is adopted at low-frequency ranges while the small gain is adopted at the high,[22]considers also frequency-wise integration of gain and phase via simultaneous gain-phase constraints as well as a higher-dimensional geometric approach using the Davis-Wielandt(DW)shell.

3 Some new trends in the integrated theory

In this section,we introduce two interesting directions for the integrated gain and phase control theory.One is about robust feedback stability conditions against sector uncertainty,and the other is on further exploration of geometric methods via the Davis-Wielandt shells.

A central problem for the gain-based robust control theory is the disk uncertainty problem(i.e.,robust stability against all uncertainties with a uniform gain constraint, similar to a disk on the complex plane)and the multi-disk uncertainty problem(i.e.,robust stability against uncertainties from multiple sources with distinct gain constraints). In particular,solving a general multi-disk problem has been shown to be NP difficult[4],while a special multi-disk problem with cascaded two-port structures has been analytically resolved in[23].On the other hand,phase uncertainty corresponds to an angular region on the complex plane,and those systems satisfying simultaneous gain and phase constraints are therefore within a sector-shaped uncertainty set.Robust control problems can be formulated in terms of the sector uncertainty.Study on such sector uncertainty problems(or sectored-disk problems) has just start up, and a necessary and sufficient robust feedback stability condition against general sector uncertainty is,to the best of the authors’knowledge,an open problem. In short, among various robust control problems,the disk problem has been well resolved,the multi-disk problem has existed for some time but been solved only for special cases,and the sectored-disk problem is new and open.

Existing and commonly adopted methods for integrated gain and phase stability analysis are mainly based on algebraic manipulations,e.g.,convex combination,set intersection and so on.A geometric method has been proposed in[22]with emphasis on horizontal and vertical projections of the 3D object—Davis-Wielandt shell.As the shell highly fuses gain and phase as one object,it naturally provides,in addition to simply projected shapes,much more perspectives in utilizing the combined gain-phase information.For instance,the intersections,unions and cutting surfaces via inclined planes of DW shells are all worth investigation,serving the purpose of gain-phase integration. Moreover, connections between the Davis-Wielandt shell and the sector uncertainty problems may also be worth investigation, as partially revealed by some mathematical studies on the shell(see,e.g.,[24]).

4 Conclusion

In this paper, we have briefly introduced the existing approaches to integrating gain and phase information of MIMO and/or nonlinear dynamical systems for the analysis of feedback stability,including the combined small gain and positive realness(or passivity),the combined small gain and negative imaginariness,and a recently proposed framework with combined small gain and small phase analysis.Not only the framework has provided stability conditions via integrating gain and phase,it also brings in some related new notions and interesting future directions.