Spectral methods in vibration suppression control

ARENBERG DOCTORAL SCHOOL Faculty of Engineering Science

CZECH TECHNICAL UNIVERSITY IN PRAGUE Faculty of Mechanical Engineering

Spectral methods in vibration suppression control systems with time delays

Dan Pilbauer Supervisors: Prof. Ing. T. Vyhlídal, Ph.D. Prof. dr. ir. W. Michiels

Dissertation presented in partial fulfillment of the requirements for the double degree of Doctor of Engineering Science: Computer Science at KU Leuven and Doctor at CTU in Prague June 2017

Spectral methods in vibration suppression control systems with time delays

Dan PILBAUER

Dissertation presented in partial fulfillment of the requirements for the double degree of Doctor of Engineering Science: Computer Science at KU Leuven and Doctor at CTU in Prague

Examination committee: Prof. Ing. V. Kučera, DrSc., dr.h.c.ab (2nd defense), chair Prof. dr. ir. Paul Van Houttec (1st defense), chair Prof. Ing. T. Vyhlídal, Ph.D.a , supervisor Prof. dr. ir. W. Michielsc , supervisor Prof. dr. ir. K. Meerbergenc Prof. dr. R. Vandebrilc (1st defense) Prof. dr. ir. W. Desmetc (1st defense) Prof. Ing. M. Schlegel, CSc.d Prof. Ing. D. Henrion, Ph.D.a e Doc. Ing. M. Hromčík, Ph.D.a Prof. Ing. P. Zítek, DrSc.a (2nd defense) Prof. Ing. A. Víteček, CSc., Dr.h.c.f (2nd defense) Prof. Ing. B. Šulc, CSc. a (2nd defense) Ing. K. Belda, Ph.D.g (2nd defense)

a CTU

Prague, Czechia Czechia c KU Leuven, Belgium d UWB, Czechia e CNRS-LAAS, France f VSB–TU Ostrava, Czechia g ACSR-UTIA, Czechia b CIIRC,

June 2017

The property rights of findings resulting from research conducted by the doctoral student belong to both institutions in equal shares, by which the exploitation of the findings shall be subject to the agreement of both institutions. The institution which takes the initiative to submit a patent application, or any other form of protection for which formalities must be observed, shall inform the competent office of the other institution of this intention in sufficient time and at the very latest at the same time as the application is submitted. Both institutions grant one another an irrevocable, worldwide, non-exclusive and free licence to use the research results obtained for any objectives related to research or education. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm, electronic or any other means without written permission from the publisher.

Preface After almost four years, the time of my doctoral studies has nearly come to an end. Although the thesis is work of an individual, the whole story is made by many people either surrounding me or supporting me, and I think that the story deserves at least a small place in the thesis. My first step in starting a PhD was motivational collaboration with Tomáš Vyhlídal which already took place during my master studies at CTU in Prague. This then resulted in my application for the position where Tomáš Vyhlídal was the supervisor. I have to mention that as I was living mostly on my own, Tomáš Vyhlídal also helped me to find a part-time job in a company doing very interesting automation, where I participated on a joint applied research project. Unfortunately, this came with a price of time spent on extra work. My short-time employment in the company was ended by the economic crises. This turned to be a positive thing as it allowed me to fully concentrate on the research work. A further step towards the successful years of my PhD studies was again supported by strong both professional and personal friendship between my supervisor Tomáš Vyhlídal and none-other than Wim Michiels. They are both leading persons in the field of time delay theory and we started a collaboration with Wim Michiels. This resulted in one of the best opportunities in my life which was joining a dual degree program between CTU Prague, Czechia and KU Leuven, Belgium. The collaboration quickly brought joint papers for international journals and presentations of the results at prestigious conferences around the world. Again, there was a price to pay and that was moving to Belgium for more than two years, which was a very difficult decision as most of my long term relationships and friendships were in Czechia. Thanks to Wim Michiels, my stay in Belgium quickly compensated my decision and I am extremely happy that this happened. I need to mention once again that the thesis would not have been made without my promoters Tomáš Vyhlídal and Wim Michiels and I hereby give my gratitude to you. My thanks also go to the other members of my examination committee, K. Meerbergen, R. Vandebril, W. Desmet, M. Schlegel, D. Henrion, M. Hromčík, P. Zítek, A. Víteček, B. Šulc, K. Belda, agreeing to read the

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PREFACE

document and providing valuable feedback on the text. I also thank Prof. Vladimír Kučera for agreeing to be the chairman during my public defense in Prague, and Prof. Paul Van Houtte to accept this task in Leuven. Working at the universities is not just about research but also teaching, helping students and not least collaborating with other researchers. I have had the pleasure to work together with, not just leading persons in the field, such as prof. Nejat Olgac from universtity in Connecticut, Martin Hromčík from University in Prague or founder of time-delay sanctum at the faculty of mechanical engineering at CTU Prague prof. Pavel Zítek , but also with young researchers Luca Fenzi, Vyacheslav Kungurtsev, Baran Alikoc, Jaroslav Bušek, Vláďa Kučera and Milan Anderle. We had at least one co-authorship with each of the mentioned people in papers in an international journal or at international conferences. Another important part of successful research, is the environment made by friends colleagues and friends. The difficult time in Belgium has been brightened by many visits of my friends Lucka, Kuba and Terka, Jarda, Michal, Nepi and Pája, Honza and Evička, Péta, Verča, Klárka, and visits of my family members; my mother Zdeňka, Honza and Ruda, and also to a great distant support from my father Václav and Marie. Many new friendships also arose over a glass of Belgian beer so thank you Richard for making my time in Leuven easier and also for many useful English lessons. Next, I have made about 50 new friends, thanks to the great spirit of a great game of rugby at Rugby Club Leuven (RCL). Thank you RCL for letting me start playing rugby with my old knees. I have enjoyed numerous injuries, heart surgery and an uncountable number of pintjes. Here, I would like to thank Jana for all her advice regarding my numerous injuries. Next to my friends outside of the walls of the university, I have had the pleasure of meeting new people inside the university. Many thanks go to my office mates Dong and Nick. We had such a great time and fun together; noon running with Nick, lunchtime sandwiches with Dong and many dinners, housewarmings, friday breakfasts and red-shirt-fridays. I also want to thank the other colleagues Petr, Przemyslav, Andreas, Simon, Pieterjan, Joris, Bert, Pieter V., Pieter L., Ward, Roel, Yaidel, Sathish, Luca, Francesco, Adrian, Libo, Deesh, Alexei for being awesome even if you were not in my office nor the same floor. Thanks also goes to my department in Prague and colleagues Milan, Jaromír, Jaroslav, Goran, Petr, Ivo. Last but surely not least, this whole thing would not have been possible in a single way without the loving care of Sandra who moved to Belgium with me despite the great job and family troubles she had had. You supported me and encouraged me from the very beginning.

Abstract Time-delays are important components of many systems from engineering, economics and the life sciences. Due to infinite dimensionality of time delay systems, analysis and computational points are challenging tasks. However, utilizing time delays in the controllers may be beneficial for stability, robustness and performance of the controlled systems. Taking this into account, the objective of this thesis is to propose novel methods for vibration control of mechanical systems utilizing time delays. In particular, two types of delay based algorithms are considered, delayed resonators for active vibration suppression and input shapers for compensating the oscillatory modes of flexible systems. A unifying factor of these two applications is the spectral domain design where the infinite dimensionality needs to be taken into account. For the topic of delayed resonator, an alternative structure utilizing a distributed time delay is proposed and thoroughly analyzed. The main benefit of the proposed distributed delay resonator is the retarded spectrum which brings improved stability properties compared to the existing resonator with a lumped delay. Next to the theoretical analysis, the viability of the novel resonator type is verified experimentally. Subsequently, the methodology is proposed to optimize distribution of the delay in order to improve the robustness in vibration suppression. For the computationally related problem of input shapers, the thesis provides a systematic approach for optimizing the delay distribution. The design is defined as a multi-objective problem balancing the contradictory requirements on robustness in oscillatory mode suppression and the shaper response time. Additionally, optimization based systematic design of a controller for architectures with inverse shaper is investigated, which is used for compensating oscillations induced by both reference and disturbance variables. Next to the spectral optimization, robust H-infinity based design is proposed and validated by simulations.

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Beknopte samenvatting Tijdsvertragingen zijn belangrijke componenten van systemen uit de ingenieurswetenschappen, de economie en de biowetenschappen. Zulke systemen behoren tot de klasse van oneindig-dimensionale systemen, wat enerzijds uitdagingen met zich meebrengt vanuit het oogpunt van de analyse en ontwikkeling van rekenkundige werktuigen. Anderzijds kan het gericht gebruik van vertragingen in regelaars gunstig zijn voor de stabiliteit, robuustheid en prestaties van het gecontroleerde systeem. Het doel van de thesis is het ontwikkelen van nieuwe methodes voor vibratiecontrole van mechanische systemen met behulp van vertragingen in de regelaars. Meer bepaald worden twee vertragingsgebaseerde technieken behandeld, namelijk de zogenaamde vertraagde resonatoren voor actieve vibratiecontrole, en ingangsvervormers voor het compenseren van oscillaties van flexibele systemen. Een unificerend element bij deze technieken is een ontwerp in het frequentiedomein, waarbij de oneindige dimensie expliciet in rekening gebracht moet worden. Wat de vertraagde resonator betreft, wordt een alternatief schema met gedistribueerde vertragingen voorgesteld en geanalyseerd. Het belangrijkste voordeel van dit nieuwe schema is dat het spectrum gelijkaardig is aan dat van de klasse van systemen met tijdsvertragingen van het zogeheten vertraagde type, wat de stabiliteitseigenschappen aanzienlijk verbetert ten opzichte van bestaande schema’s met discrete vertragingen. Naast de theoretische analyse wordt de doelmatigheid van het nieuwe type resonator experimenteel aangetoond. Vervolgens wordt een methodologie voorgesteld om de distributie van de vertragingen te optimaliseren, om de robuustheid te verbeteren bij vibratiecontrole. Wat ingangsvervormers betreft, wordt een gelijkaardige, systematische aanpak voorgesteld om de verdeling van de vertragingen te optimaliseren. Het ontwerp van de vervormers wordt daarbij vertaald naar een optimalisatieprobleem met meerdere objectieven, waarbij een balans gezocht wordt tussen de conflicterende vereisten van enerzijds robuustheid met betrekking tot de frequentie van de te onderdrukken oscillatie en anderzijds de responstijd van de vervormer. Daarenboven worden systematische technieken ontwikkeld voor het ontwerp van regelaars in schema’s die inverse ingangsvervormers bevatten, waarbij zowel de locatie van het spectrum als Honeindig normen geoptimaliseerd worden. Aan de hand van simulaties worden deze nieuwe technieken gevalideerd. v

Abstrakt Dopravní zpoždění je nedílnou součástí systémů v řadě oblastí, mezi které patří zejména inženýrství, ekonomika a biologie. Přítomnost dopravních zpoždění ve struktuře systému, která přináší nekonečné spektrum dynamických módů, výrazně komplikuje analýzu a syntézu řízení. V některých případech ale lze využít dopravních zpoždění ve struktuře algoritmů řízení s pozitivním účinkem na stabilitu, robustnost a dynamické vlastnosti systému. Těmito případy je motivován obecný cíl práce, kterým je využití zpoždění v návrhu moderních metod tlumení vibrací a oscilací mechanických systémů, konkrétně pomocí zpožděných rezonátorů a tvarovačů signálu. Metodika zpožděného rezonátoru je založena na tlumení vibrací použitím dopravního zpoždění ve zpětné vazbě aktivně řízeného absorbéru. Tvarovače signálu se zpožděním umožňují kompenzovat kmitavé módy flexibilních prvků systému. Syntéza obou těchto typově podobných problémů je založena na využití spektrálních návrhových metod, při kterých je důležité brát v potaz nekonečnou dimenzi systému se zpožděním. Pro systémy s aktivním tlumením vibrací je navržen a důkladně analyzován nový typ rezonátoru s distribuovaným zpožděním. Důležitým přínosem navrženého aktivního tlumení vibrací jsou výhodné spektrální vlastnosti retardovaného typu, díky kterým je dosaženo zlepšení z hlediska stability v porovnání s dříve využívaným soustředným zpožděním. Kromě teoretické analýzy je provedeno experimentální ověření funkčnosti navrženého rezonátoru. Následně je metoda rozšířena o optimalizaci distribuce zpoždění za účelem zlepšení robustnosti systému z hlediska tlumení vibrací. Analogický princip syntézy je následně aplikován na problematiku návrhu tvarovačů signálu s optimalizovanou distribucí zpoždění. Návrh je definován jako vícekriteriální problém, ve kterém je kladen důraz na vyvážení protichůdných požadavků na robustnost v potlačování oscilací flexibilních podsystémů a rychlost odezvy tvarovače. Práce je navíc doplněna o návrh regulátorů pro systémy řízení s inverzním tvarovačem signálu ve zpětné vazbě za účelem kompenzace oscilací buzených jak řídicí tak i poruchovou veličinou. Kromě optimalizace spektrálních vlastností systému řízení je navržena metodika optimalizace robustnosti na bázi H-nekonečno. Navržené metody syntézy jsou ověřeny na řadě simulačních příkladů.

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Contents Abstract

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Contents

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1 State of the art and objectives 1.1 Time Delay systems . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Classification . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Description of linear control systems with time delays 1.1.3 Spectrum of linear time-delay systems . . . . . . . . . 1.1.4 Control of time delay systems . . . . . . . . . . . . . . 1.2 Vibration control methods utilizing time delays . . . . . . . . 1.2.1 Delayed resonator . . . . . . . . . . . . . . . . . . . . 1.2.2 Delay based input shapers . . . . . . . . . . . . . . . . 1.2.3 Open problems . . . . . . . . . . . . . . . . . . . . . . 1.3 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

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Delayed Resonators

1 1 3 4 4 6 7 8 12 16 18 19

23

2 Distributed Delayed Resonator: design and complete stability analysis 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries on active delayed resonators with acceleration feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The resonator with a delayed acceleration feedback . . . . . . . 2.4 DDR absorber coupled with SDOF primary structure . . . . . 2.5 Parametric design of the DDR absorber . . . . . . . . . . . . . 2.6 Experimental verification . . . . . . . . . . . . . . . . . . . . . 2.7 Delayed resonator with various measurements in feedback . . . 2.7.1 Summary on stability maps . . . . . . . . . . . . . . . .

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24 24 25 27 35 37 39 43 49

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CONTENTS

2.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Design and analysis of robust resonators with distributed delay 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Design of the robust absorber . . . . . . . . . . . . . . . . . 3.2.1 Delayed resonator background . . . . . . . . . . . . 3.2.2 Definition of the feedback law . . . . . . . . . . . . . 3.2.3 Robustness definition . . . . . . . . . . . . . . . . . 3.2.4 Stability of the resonator-system interconnection . . 3.2.5 Definition of the optimization problem . . . . . . . . 3.3 Case study examples . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Input shapers

50 53 53 54 54 54 55 56 58 60 61

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4 Design and analysis of robust input shapers with distributed delay 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Robust input shapers with piece-wise delay distribution . . . . 4.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 4.2.2 The primary objective functions . . . . . . . . . . . . . 4.2.3 Linear constraints . . . . . . . . . . . . . . . . . . . . . 4.2.4 A multi-objective problem definition . . . . . . . . . . . 4.2.5 Demonstration example . . . . . . . . . . . . . . . . . . 4.2.6 Graphic user interface for the shaper design . . . . . . . 4.3 Robust input shapers with smooth kernel functions . . . . . . . 4.3.1 Proposed novel class of input shapers . . . . . . . . . . 4.3.2 Requirements on the shaper functionality . . . . . . . . 4.3.3 Optimization problem formulation . . . . . . . . . . . . 4.3.4 Residual vibrations . . . . . . . . . . . . . . . . . . . . . 4.3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Implementation aspects . . . . . . . . . . . . . . . . . . 4.3.7 On-line computation of integral . . . . . . . . . . . . . . 4.3.8 Exponentially decaying basis functions . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 70 71 71 72 73 74 75 76 79 79 81 84 84 88 89 91 91 96

5 H infinity design of fixed-order controllers for systems precompensated by input shapers 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Synthesis of the problem . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . 5.3.2 Control objectives and associated optimization problems

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CONTENTS

5.3.3 5.4 5.5

Reformulation in standard DDAE design . . . . . . . . . . . . . . . . Case study example . . . . . . . . . . . . Conclusions and future works . . . . . . .

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form and . . . . . . . . . . . . . . . . . .

controller . . . . . . . . . . . . . . . . . .

6 Fixed order controller design for flexible systems with dynamics pre-compensated by an inverse shaper 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Properties of shapers in mutually coupled structures . 6.3 Design of the feedback control . . . . . . . . . . . . . . 6.3.1 Distributed delay input shaper . . . . . . . . . 6.3.2 Reformulation into the DDAE standard form . 6.3.3 Closed loop architecture . . . . . . . . . . . . . 6.4 Case study analysis . . . . . . . . . . . . . . . . . . . . 6.4.1 Mechanical model . . . . . . . . . . . . . . . . 6.4.2 Application of the inverse shaper . . . . . . . . 6.4.3 Controller design and simulation results . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .

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coupled . . . . . . . . . . .

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113 113 114 117 118 119 119 120 120 122 124 129

7 Contributions and conclusions 131 7.1 Contributions towards the thesis objectives . . . . . . . . . . . 132 7.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Bibliography

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Curriculum Vitae

153

List of publications

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Chapter 1

State of the art and objectives Time delay is mostly considered as an undesired element of the system. The presence of time-delays imposes fundamental limitations on stabilizability, achievable performance and robustness. Time-delay systems are also challenging from analysis and computational points of view as they form a class of infinitedimensional systems. Even in the linear time-invariant case, this is apparent as the dynamic behavior is determined by the solutions of either a non-linear eigenvalue problem or an infinite-dimensional linear eigenvalue problem, for which only recently methods are being developed. Moreover, any control design problem that involves tuning of finitely many controller parameters can be interpreted as a reduced-order control design problem or an under-actuated control problem, which are both known to be very difficult to solve. This thesis provides novel methods utilizing various versions of time delay as an alternative to a classical delay. The methods are grounded in spectral theory of time delay systems together with recent optimization tools for such systems. This chapter shows an overview of the main topics related to the thesis along with state-of-the-art. The chapter contains necessary background of the theory related to the chapters of this thesis. The chapter continues with the statement of the research objectives and concludes with the outline of the thesis and explanation of the interconnection between KU Leuven and CTU Prague as a part of a dual degree study programme.

1.1

Time Delay systems

Time delays represent an alternative for modeling, especially for control purposes. As many classical models, described by ordinary or partial differential equations, 1

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STATE OF THE ART AND OBJECTIVES

are causal, i.e. the future state is fully determined by the state at the current time, without dependence on previous states. If the model is taken under more rigorous scope, the situation describing the reality is more complex and requires more then just a present state to describe the future ones. This leads to a more general description of models by so-called delay differential equations (DDE). Many applications where time delay is present can be found for example in [45, 63, 64]. Time-delays are important components of many systems from engineering, economics and the life sciences, because of the fact that the transfer of material, energy and information is mostly not instantaneous. They appear, for example, as computation and communication lags, they model transport phenomena and heredity and they arise as feedback delays in control loops. In practice, any system involving a feedback control will involve time delays as time delays arises from a finite time required to sense information and react to it. Moreover, time delays also arise when a high dimensional system is approximated, for example systems described by partial differential equations. Inherent for time delay systems (TDS) is the memory in the sense that the state at the current time depends explicitly on the state in the past. Such systems can be described by differential equations with deviating argument which are generally called functional differential equations (FDE) [43, 45]. Some classical examples of systems with delays can be found in all areas of science. In mechanical engineering so called regenerative chatter effect, causing inaccuracy in cutting of workpiece and degrades lifespan of work tools [56,92,132]. The delay comes from the modeling of the rotating tool, where teeth cut the material with delay caused by distance in between them. A model of internal combustion engine uses mean torque prediction, where the crankshaft rotation is described by delayed differential equation where the delay originates in the engine cycle [26, 60]. There are also many examples in biology [6, 128], where modeling of population dynamics is done by time delay, where populations grow, reproduce after a certain time and die, respectively [36, 65]. Next, natural control of a level of carbon dioxide in the blood by breathing process [70, 149], dynamics of HIV [170], myelogenous leukemia [84], hormones components in blood flow [27]. Just to name a few examples. There could be many more examples from the field of biology that need time delay in modeling. An increasing demand in performance requires accurate mathematical models for simulation and control that contain time-delay elements. So far the focus in the area has been mainly on detrimental effects of delays (loss of stability, performance degradation,..). However, very recently an increased research interest concerns situations where opportunities of delays in controllers are exploited, such as vibration control, positioning, temperature control, control of platoons of cars and many others. This thesis focuses on improving existing vibration control techniques by utilizing time delays. Control design is based

TIME DELAY SYSTEMS

3

on spectral methods and related computational tools, developed recently for time delay systems. This thesis follows this direction, particularly by utilizing novel computational methods for spectrum optimization and use of delays to improve existing vibration control techniques. Time delay also shows benefits in cases where the delay is not present [9, 69, 85]. Stabilizing effect of feedback with delay is shown in [131]. Delay can be artificially introduced in supply-chain to make better decisions, so called wait-and-act method [133, 134]. Delays can be also used for stabilizing unstable periodic orbits in chaotic systems to control them [93, 127] or to predict future outputs of nonlinear systems [18]

1.1.1

Classification

Systems including time delays can be described by equations where the rate of change depends on current states but, most importantly, on previous states from the past. For instance, such equation can be x(t) ˙ = f (t, x(t), x(t − τ ))

(1.1)

with x being state variable vector of the system and τ a single delay, or a system with multiple delays x(t) ˙ = f (t, x(t), x(t − τ1 ), x(t − τ2 ), ..., x(t − τn )).

(1.2)

In (1.2) the delays are discrete (point wise). Systems can also contain distributed delay Z τ

x(t) ˙ =f

x(t − θ)dθ .

t, x(t),

(1.3)

0

The time delay can have time or state dependency x(t) ˙ = f (t, x(t), x(t − τ (t, x(t)))).

(1.4)

The mentioned types of TDS (1.1)-(1.4) represent a class of retarded type. Systems can also include dependency on derivatives, which is then a class of neutral type x(t) ˙ = f (t, x(t), x(t − τ1 ), x(t ˙ − τ2 )). (1.5) Compared to ordinary differential equations, systems being described by delayed differential equations (1.1)-(1.5) require additional initial conditions, describing the dynamics of the system over the past. For the linear time invariant case, the initial conditions are in a form h i x(t) = φ(t), t ∈ − max τi , 0 (1.6) i

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STATE OF THE ART AND OBJECTIVES

representing the trajectories of the solution in the past [78]. All mentioned types of systems involving time delay (1.1)-(1.5) create a very broad class for which it is very difficult to define general properties and therefore the thesis focuses only on systems without state/time dependent delays.

1.1.2

Description of linear control systems with time delays

In time domain, linear time invariant (LTI) systems for control engineering are mostly described in state-space representation as   ˙ = Ax(t) + Bu(t)  x(t) (1.7) y(t) = Cx(t) + Du(t),   x(0) = x0 where the A, B, C, D are real matrices of appropriate sizes. Nevertheless, when delay is present in the system the description needs to be extended to a more general description. Firstly, delay can appear on the left side of (1.7), at the derivatives x(t). ˙ Secondly delays can be present at states x(t), inputs u(t) and therefore in the output equation as well. Finally, the set of equations can be completed with algebraic equations including delays. Systems representing all the mentioned cases can be written in the form of delay-differential algebraic equations (DDAEs), also called descriptor system with delays, as ( Pm E x(t) ˙ = A0 x(t) + i=1 Ai x(t − τi ) + Bu(t), Ω (1.8) y(t) = Cx(t), where x(t) is n-dimensional generalized state variable at time t, A0 , Ai ∈ Rn×n , i = 1, ..., m are matrices with constant entries and m is positive integer, delays τi > 0 and the matrix E ∈ Rn×n is allowed to be singular. The motivation for this description in this thesis is in easy interconnection of the system with other control elements. Examples of systems, where it is advantageous to use delay-differential algebraic equations, are e.g., lossless propagation models [42] or [48] dealing with chip interconnections, or population dynamics [65]. Systems including all types of delays (e.g. input/output) can be easily transformed in the form (1.8) by introducing slack variables to eliminate delays and direct feedthrough terms in the equations [78]. One can observe that even though (1.8) look similar to a delay expression of retarded type it should be noted that neutral systems can be described in the same form. Throughout the thesis, it has to be assumed that (1.8) has index 1 [75].

1.1.3

Spectrum of linear time-delay systems

The definition of the spectra of a system is similar to the definition for systems like (1.7). Considering the general form of TDS in the form of DDAEs (1.8) the

TIME DELAY SYSTEMS

5

characteristic equation is

det M (s) = 0,

where M (s) := sE − A0 −

m X

(1.9) Ai e−sτi .

(1.10)

i=1

The roots of (1.9) are called the characteristic roots of (1.8). The exponential terms in (1.10) arise from the Laplace transform of the delay terms. The exponential terms make the characteristic function det M (s) quasi-polynomial. In general equation (1.9) is transcendental, which means that it has an infinite number of roots in the complex plane C. This is a true generalization of the eigenvalues and the characteristic equation. If Ai = 0, ∀ i = 1...m or τi = 0, ∀ i = 1...m then the DDAE reduces back to an ODE. The spectrum of (1.9) can have a different distribution. The type of spectrum distribution might not be easy to see from the DDAEs and needs to be carefully investigated. If the system is of retarded type then the following holds [78]. If there exists a sequence {sk }k≥1 of characteristic roots of (1.9) such that lim k=(sk )k → ∞

(1.11)

lim 1, the range of the frequencies for which the DR is stable is rather limited. Thus from the practical point of view, the DR with l = 1 is the best option as it provides the widest stability range, which is not bounded from above [166] - theoretically at least. It can also be read from Fig. 2.12 that the absorber damping ratio should be close to ζ = 0.1, for which the lower stability bound is minimal.

Delayed velocity feedback The stability regions derived in Sec. 2.3 for distributed and classical delayed resonator can be also derived for velocity feedback. Using the same method (CTCR), stability maps are shown for distributed delay given by Z gd τd u(t) = x˙ a (t − θ)dθ, (2.50) τd 0

46

DISTRIBUTED DELAYED RESONATOR: DESIGN AND COMPLETE STABILITY ANALYSIS

Figure 2.12: Stability region of the DR (2.2) with respect to the damping ratio ζ of the absorber, the excitation frequency ω and the branch number l, derived in [166]. For a given l, the overall stability region is given as a union with the regions lying inside corresponding to larger branch numbers. for which the gain and delay is given by −1 + 2ζ 2 − 4ζ 2 ω 2 − ω 4 2(ω 2 − 1) 2 1 − ω2 τd = atan + (l − 1) π ; l ∈ N+ . ω 2ζω g¯d =

(2.51) (2.52)

Figure 2.13: Stability regions of the delay resonator with velocity feedback and distributed delay (2.50)-(2.52) with respect to ζ, ω and l. The stability region is determined as a union with the regions lying inside corresponding to the larger branch numbers. Classical delayed resonator velocity feedback is analyzed in [32]. Experiments provided in the paper show discuss effect of imperfections of the absorber

DELAYED RESONATOR WITH VARIOUS MEASUREMENTS IN FEEDBACK

47

mechanism operating properties (friction etc.). Moreover, measurement of speed is mostly not possible and needs to be calculated from position. The feedback is in the form u(t) = g x˙ a (t − τv ) (2.53) with the parameters tuned by 1p (1 − ω 2 )2 + 4ζ 2 ω 2 ) ω 1 − ω2 1 −1 tan + 2(l − 1)π τ= ω 2ζω g=

(2.54) (2.55)

The stability ranges are shown in Fig. 2.13. For l = 1, interestingly, when ω = 1 the delay τd = 0 (and gv = 2ζ) implying that the feedback simply eliminates the dissipative damping effect for non-delayed case. Moreover, the delay τd is negative for ω > 1. Thus, only higher branches of the delay values can be considered for such cases.

Figure 2.14: Stability regions of the delay resonator with velocity feedback (2.2)-(2.55) with respect to ζ, ω and l. For a given l > 1, the stability region is determined as a union with the regions lying inside corresponding to the larger branch numbers. Dashed yellow - upper applicability boundary for l = 1. The stability maps for velocity feedback with distributed delay is in Fig. 2.14. As can be seen, the distributed delay removes issues with frequencies around the natural frequency of the absorber. Besides that, the dynamics of the system is not neutral and therefore removes some additional stability issues. In both cases of the velocity feedback, branches l > 1 are necessary in order to get stable absorber for excitation frequencies ω > 1. This is different in comparison with acceleration feedback, where switching branches is not necessary in order to stay in stable regions.

48


Delayed position feedback To complete the chapter, the stability maps are also derived for position feedback. Absorber with classical delay and with relative position measurement was proposed in [89]. The position feedback with classical delay is defined as u(t) = gxa (t − τ ) and parameters are given by p g = (1 − ω 2 )2 + (2ζω)2 

τ=



2 atan  ω

1 − ω2 −

(2.56)

(2.57)

q   2 2 (1 − ω 2 ) + (2ζω)  + (l − 1) π  ; l ∈ N+ (2.58) 2ζω

Stability regions for position feedback with classical delay in Fig. 2.15 shows

Figure 2.15: Stability regions of the delay resonator with position feedback (2.56)-(2.58) with respect to ζ, ω and l. The stability region is determined as a union with the regions lying inside corresponding to the larger branch numbers. interestingly wide area of stability for branch l = 1, which covers full frequency range for ζ > 0.7 and full damping range for ω > 0.63. The stability range for higher branch number get very narrow and covers only a small area around the natural frequency of the absorber. The feedback with distributed delay applied on the position measurement is of the form Z gd τd u(t) = xa (t − θ)dθ (2.59) τd 0

DELAYED RESONATOR WITH VARIOUS MEASUREMENTS IN FEEDBACK

49

with parameters of the feedback given by −1 + 2ζ 2 − 4ζ 2 ω 2 − ω 4 4ζ 2 2ζω τd = atan + (l − 1) π ; l ∈ N+ ω ω2 − 1

g¯d =

(2.60) (2.61)

Finally, stability maps for delayed resonator with distributed delay and position

Figure 2.16: Stability regions of the delay resonator with position feedback with distributed delay (2.59)-(2.61) with respect to ζ, ω and l. The stability region is determined as a union with the regions lying inside corresponding to the larger branch numbers. measurement is shown in Fig. 2.16. The shape of the areas is very similar to the feedback without distributed delay in Fig. 2.15. However, the stability range becomes smaller for decreasing damping ratio ζ and for l > 1 the range shrinks to only natural frequency of the absorber.

2.7.1

Summary on stability maps

As shown in Sec. 2.3 and 2.7, selection of absorber feedback measurements, i.e., position, velocity or acceleration, affects the operating range of the active absorber. Also selection of the delay distribution, i.e. pointwise or distributed, change the range of the absorber’s stability. When the feedback is taken from the absorber’s acceleration measurements, the delay distribution should be chosen with respect to the parameters of the system, if the damping is low then the pointwise delay (2.46)-(2.48) has a wider range in the stability region (around the nominal input frequency ω = 1) with respect to the input exciting frequency ω, see Fig. 2.12. On the other hand, when the damping increases, the stability can be lost for the nominal frequency and hence the distributed delay (2.6)-(2.16) needs to be used.

50


When delay velocity feedback is used, there is no stable absorber with distributed delay (2.16)-(2.52) for the nominal frequency, see Fig. 2.13. For an application requiring operation around the nominal frequency, distributed delay cannot be used. Fortunately, point-wise delay (2.53)-(2.55) with branch number l = 2 can solve this problem, see Fig. 2.14, even though the delay can be relatively small and can cause implementation problems. The position feedback has similar stability regions for distributed delay (2.59)-(2.61) as for classical pointwise delay (2.56)-(2.58). Interestingly, for branch number l = 1 there is very broad range of parameters that provide stable feedback for the absorber. Roughly speaking, classical pointwise delay feedback is stable for all parameters except for the region lying under the line between points [ω = 0.6 ζ = 0] and [ω = 0 ζ = 0.7]. For the distributed delay, the limit frequency is shifted towards the nominal frequency and so the line is between points [ω = 1 ζ = 0] and [ω = 0 ζ = 0.7]. This suggests to use a classical point-wise delay for very low damped systems and for a higher damping ratio the choice can be made with respect to the application aspects (e.g. filtering properties of the distributed delay).

2.8

Conclusions

The key contributions of the chapter are the design, analysis and experimental verification of a delayed resonator with a distributed delay (DDR) in the acceleration feedback. Besides that, the analysis without experimental verification is done for velocity and position feedback with lumped delay and distributed delay as well. Similar to the classical delayed resonator with a lumped delay (DR) proposed in [86, 88] and recently analysed in [166], the parameterization of the DDR is fully analytic and its implementation is easily manageable. Even though the adjustment of the overall algorithm consisting of substitution of the lumped delay by a distributed delay may seem rather simple, it has fundamental consequences for the overall closed loop dynamics and brings the following key benefits: • The equally distributed delay performs the task of a moving average filter, which is beneficial taking into account that the signals from accelerometers are affected by measurement noise. • The resulting dynamics of the resonator as well as that of the resonatorsystem coupling exhibit retarded time delayed system characteristics in contrast to the neutral dynamics if the DR is applied. As demonstrated, the retarded character of DDR has positive consequences on the lower bound of applicable excitation frequencies, especially when the DDR absorber has higher damping. Besides that, the retarded dynamics is

CONCLUSIONS

51

considerably easier to handle compared to neutral dynamics from stability perspectives. Stability maps for various measurements with lumped and distributed delay are derived and showed. As showed, the feedback of the delayed resonator should be selected with respect to the operating ranges of the absorber’s parameters and exciting frequency range. The design procedure and the transmissibility have also been compared with the delay free PI algorithm. This seemingly simplest possible implementation of feedback from the acceleration sensor can be problematic when common high-frequency noise is involved. The PI algorithm also needs an additional low pass filter if the acceleration sensor is affected by measurement noise. Even though both the theoretical and experimental verification were performed for a case study with a single-degree-of-freedom set-up only, the methodology can be deployed on more complex systems. A multiple delay resonator and its dynamic analysis can be targeted in the further effort. Another open topic left for further investigation is the analysis of nonlinearities, such as deadzone and actuator limitations on the resonator performance. The extension of the applicable range of delayed resonators is a the task for subsequent research. Another open topic is a use of more complex delay distribution to enhance the robustness of the vibration suppression, analogously as it was done recently for distributed delay input shapers [151]. This issue will be targeted in the next chapter.

Chapter 3

Design and analysis of robust resonators with distributed delay 3.1

Introduction

This chapter is the next step in the development of the delay-based vibration absorbers. The second chapter of this thesis shows how distributed delay can significantly improve performance of the system. In Chapter 2, the delay was either lumped or linearly distributed on the delay interval. In this chapter, the delay is distributed in a such way that it becomes less sensitive against the excitation frequency. The distribution of the delay is profiled by polynomial functions which make the distribution smooth and gives more degrees of freedom in the design. The degrees of freedom are depending on the degree of the polynomial. The design is enriched by the stabilization of the overall system which leads to an objective optimization with spectral constraints. Such problems are very difficult to solve and hereto firstly the constraints are eliminated and subsequently the unconstrained optimization problem is solved. The chapter is organized as follows. Section 3.2 introduces a novel type of delayed resonator together with a definition of objectives of the design which results in an optimization problem. In Section 3.3 the numerical results with simulation examples are described. In Section 3.4 the conclusions are presented.

53

54

3.2

DESIGN AND ANALYSIS OF ROBUST RESONATORS WITH DISTRIBUTED DELAY

Design of the robust absorber

To show a parallel with the classical delayed resonator and the resonator with distributed delay the basic idea is briefly recalled in the first subsection. The introduction is then followed by the definition of the novel delayed feedback for the absorber. Then, a criterion of robustness, closed loop stability, and eventually an optimization problem are defined. The DR is parameterized by solving the optimization problem.

3.2.1

Delayed resonator background

The classical DR absorber with lumped delay, introduced in [88], has feedback given by u(t) = g¨ xa (t − τ ). (3.1) The feedback is simply tuned by two parameters g and τ . The parameters can be obtained analytically, simply by joining the resonator (3.1) with absorber ma x ¨(t)+ca x(t)+k ˙ a x(t) = u(t) and placing two dominant roots on the imaginary axis for the given input excitation frequency. Alternatively, the feedback with distributed time delay was proposed in Chapter 2 in the form Z g τ u(t) = x ¨a (t − θ)dθ. (3.2) τ 0 The feedback is also tuned only by two parameters g and τ as for the classical case (3.1). The delay is distributed equally on the time interval [−τ 0], where the slope of the distribution is given by the gain parameter g/τ . As shown, for classical DR with lumped or distributed delay only two parameters are free to be assessed, which means that placing the complex conjugate couple of roots removes two degrees of freedom given by the parameters and any further modifications are not possible. Some modifications allow to modify the feedback by adding delay-free parts as shown in [66] or a possible extension would be adding more absorbers as to get more parameters as done in [59], where multiple resonators are used to suppress multiple harmonics. Adding more absorbers is not within our scope and this chapter focuses on modification of the feedback. The novel DR feedback is an extension of the previously mentioned distributed time delay feedback (3.2). Here, the time delay is distributed with a polynomial, where the coefficients of the polynomial are used to meet multiple requirements, i.e. robustness and stability.

3.2.2

Definition of the feedback law

The remainder of the chapter is focused only on feedback from the absorber’s acceleration. From a practical point of view, acceleration is the most convenient

DESIGN OF THE ROBUST ABSORBER

55

and easiest to physically implement. The following feedback rule is considered Z τ u(t) = h¨ x(t) + g(θ)¨ x(t − θ)dθ (3.3) 0

where g(θ) =

N X

ai θi e−λθ

(3.4)

i=0

and parameter h is a coefficient of the delay-free part of the feedback. The delay free coefficient h is introduced in order to increase the total number of parameters and hence to have more flexibility in the design. The transfer function from x to u can be written as P (s) = hs2 +

N X

ai gi (s)

(3.5)

i=0

Rτ where gi (s) = 0 s2 θi e−sθ dθ. Together with delay-free parameter h we get N + 2 parameters to be assessed.

3.2.3

Robustness definition

Introducing the feedback (3.3) into the absorber equation (1.19), throughput of the resonator may now be defined as R(s) − P (s) T (ω, τ ) = Q(s) m (jω)2 + c jω + k − h(jω)2 − (jω)2 PN a g (ω) a a a i=0 i i = , (3.6) ca jω + ka where P (s) is defined by Eq. (3.5) and R(s), Q(s) is defined in (1.24). The denominator Q(s) normalizes the function (3.6) for limω→0 T (ω, τ ) = 1. Eq. (3.6) can be interpreted as the ratio between input and output force of the primary structure. If the classical delayed resonator (3.1) or resonator with distributed delay (3.2) is applied then this function goes to zero for the nominal frequency. For other frequencies this function tends to increase rapidly and therefore, if there is an uncertainty or multiple frequencies on the input signal, the quality of the absorption can decrease. Hereby we want to reduce this influence. For a given frequency range I = [ωmin , ωmax ], around the nominal frequency of the exciting force, we want to minimize an average of the frequencies that passes through this bandwidth. By discretizing points on this area as ωk ,

56


equidistantly or with Chebyshev points, the robustness criterion, an average of T (ω, τ ), can be expressed as r(p, τ ) =

Nω 1 X 2 T (ωk , τ ) Nω

(3.7)

k=1

where p = [h a0 ... can be expressed as where

aN ] is vector of gains to be assessed. The function (3.6) T (ω, τ ) = |l(ω, τ )p + b(ω, τ )|

l(ω, τ ) = [−h b(jω) =

(3.8) 2

− aN gN (jω)] ca(jω) jω+ka

− a0 g0 (jω) ...

ma (jω)2 +ca (jω)+ka ca jω+ka

|T (ω, τ )|2 = pT l(ω, τ )∗ l(ω, τ )p + 2< ¯bl(ω, τ ) + |b|2

(3.9) (3.10)

and the robustness measure takes the form

where

r(p, τ ) = pT H(τ )p + Bp,

(3.11)

PNω H = N1ω i=0 l(ωi , τ )∗ l(ωi , τ ), PNω ¯ 2 B = Nω i=0 < bl(ω, τ ) .

(3.12)

Note that function (3.11) is a quadratic form in parameters p if τ is fixed.

3.2.4

Stability of the resonator-system interconnection

A general approach to approximate the feedback (3.3) by stable LTI systems is described in [79], see also the references therein. The feedback (3.3) can be rewritten into the form Z τ Cs eAs θ Bs y(t − θ)dθ, (3.13) u(t) = hy(t) + 0

where

 −λ    As =   ...   0

1 −λ

0 .. . ..

.

··· .. . .. . −λ

···

 0 ..  .    ∈ RNp +1×Np +1 , 0   1  −λ

(3.14)

and Bs = 0!a0

1!a1

...

N !aNp

|

, Cs = 0

...

0

1 .

(3.15)


57

The corresponding transfer function can then also be expressed as Gs (s) = h + Cs (sI − As )−1 I − e−τ (sI−As ) Bs . This suggest a realization and implementation by a dynamic system z(t) ˙ = As z(t) + Bs y(t) − eτ As Bs y(t − τ ), u(t) = Cs z(t) + hy(t).

(3.16)

(3.17)

Given a number λ > 0 the matrix As (3.14) is Hurwitz and the solution (3.17) is stable. Also notice that the expression (3.17) is still linear in the parameters p. The absorber and primary structure coupled together can be described as x(t) ˙ = Ax(t) + B1 u(t) + B2 f (t), (3.18) y(t) = C x(t) ˙ = CAx(t) + CB1 u(t) + CB2 f (t), where output is acceleration and the derivatives of the states are in the output equation. f (t) is a harmonic force exciting the primary structure, and matrices A, B1 , B2 correspond to a linear model of a mechanical system (1.20)–(1.21) translated into a state space representation with a state vector x = [xa x˙ a xp x˙ p ]T . Matrix C defines the measured acceleration of the absorber C = [0 1 0 0]. The system (3.18) and the feedback (3.13) create together a System of Delay Differential Algebraic Equations (DDAE)  ˙ = As z(t) + Bs y(t) − e−As τ Bs y(t − τ ),   z(t)  x(t) ˙ = Ax(t) + Bu(t), . (3.19) u(t) = Cs z(t) + hy(t),    y(t) = CAx(t) + CB1 u(t) which can be rewritten in a    I As 0   I ˙ =   ξ(t)  Cs 0  0 0 | {z } | E

compact form as    0 0 Bs 0 0 0 −e−As τ Bs   A B 0 0  ξ(t−τ )  ξ(t)+0 0 0 0 0 0  0 −I h 0 CA CB1 −I 0 0 0 0 {z } | {z } A0

A1

(3.20) where ξ = [z x u y]T is a new state vector. With the new matrix notation the overall system can be written as ˙ = A0 ξ(t) + A1 ξ(t − τ ). E ξ(t)

(3.21)

58


For the resulting system, it is required that all poles of the system lie in the left half of the complex plane, i.e., having spectral abscissa α < 0, where the spectral abscissa is defined as

with

3.2.5

α(p) := sup( 0, min fk (p) = r(p) + γk (3.26) 0 α ≤ 0, x where γ is a penalty coefficient. In each iteration k of the method, the penalty coefficient γk is increased (e.g. by a factor of 10) until the solution is feasible. Thus, in each iteration the unconstrained problem is solved and the result is used as an initial guess for the next iteration. Solutions of the successive unconstrained problems eventually converge to the solution of the original constrained problem. The second term in (3.26) is a non-smooth, non-convex function and it is thus not possible to use standard solvers for quadratic programming problems. Here, an alternative approach based on the hybrid algorithm for non-smooth optimization (HANSO) [67] together with the software provided by the research team of W. Michiels [75] is used for the given purpose. The software only requires the objective function (3.26) and its derivative. The derivative of the objective function (3.26) is defined as ( ∂α(p,s(p)) df ∂r(p) α>0 ∂p = +γ , (3.27) dp ∂p 0 α≤0


59

where the first part is simply ∂r(p) = 2Hp + B, ∂p and for the second part it results in   u∗ ∂M ∂α ∂h w    = − u∗ ∂M   ∂h w   ∂s    ∗ ∂M    u w    ∂a0 ∂α    = − ∂M 1 ∂α ∂a0 u∗ ∂s w = − ∗ ∂M = < ..    ∂p u    ∂s w .         ∗ ∂M   ∂α = − u ∂aN w   ∂aN

(3.28)

 u∗ ∂M ∂h w    u∗ ∂M    ∂a0 w    , ..   .    ∗ ∂M u ∂aN w 

(3.29)

u∗ ∂M ∂s w

where w and v are the left and right eigenvector corresponding to the characteristic function of the system

with

M (s, p)v = 0, w∗ M (s, p) = 0,

(3.30)

M (s) := (sI − A0 (p) − A1 (p)e−sτ ).

(3.31)

Note on selection of the constant λ and time delay τ In the proposed method a length of the delay τ is selected as a fixed value. The length of the time delay τ could be considered as an additional design parameter, but it would increase the complexity of the optimization problem and also the time needed to solve the problem. As the parameters of the system (1.20)–(1.21) and the range of exciting frequencies are known, a sufficiently good value of the delay can be estimated. The delay is suggested to be at least one period of the excitation frequency T =c

2π , c > 1. ω

(3.32)

The choice of λ, once again, induces a trade-off. For large λ, the additional dynamics are very fast and do not affect the overall control system’s performance, but since the kernel basis functions (3.4) have a rapid decay, it might become more difficult to find parameter values for the shaper satisfying the design requirements, affecting the performance. The parameter λ is then selected as λ=

ω 1 = , 2π τ

(3.33)

which gives a good compromise between a position of −λ poles of the system (3.21), which are safely far away from the imaginary axis, and a speed of decay.

60


3.3

Case study examples

Six examples are provided to show the functionality of the proposed method. The selected parameters of the system are given in Table 3.1. The optimization parameters are defined in Table 3.2. The first example is selected to optimize the robustness function (3.6) in the neighborhood of the natural frequency of the absorber. The next examples have always one different parameter from the previous one (shown in bold). Each example is accompanied by two figures, the first with the spectra of the absorber itself and the spectra of the system coupled with the absorber. The second figure shows the robustness function (3.6) of the absorber (1.19) with feedback (3.3) and also the robustness function of the overall system (1.20)-(1.21). The figures of the robustness function are always compared with classical DR resonator. In addition, one extra example (#7) shows a system which would be unstable with classical delayed resonator (3.1) but is stable with the robust delayed resonator (3.3). Table 3.1: Parameters of the system ma 0.223kg

ca 1.273kgs−1

ka 350N m−1

mp 1.52kg

cp 10.11kgs−1

kp 1960N m−1

Table 3.2: Optimization parameters Case # 1 2 3 4 5 6

Nom. excit. freq. ωf 39.62 39.62 39.62 39.62 35 45

delay length 4 ωπf 4 ωπf 8 ωπf 8 ωπ 8 ωπf 8 ωπf

# of coefficients 4 6 6 6 6 6

Rel. Region Iω [0.98; 1.02]ωf [0.98; 1.02]ωf [0.98; 1.02]ωf [0.95;1.05]ωf [0.98; 1.02]ωf [0.98; 1.02]ωf

7

30

8 ωπf

6

[0.98; 1.02]ωf

The first example shows the case, where the delay and number of coefficients selected is relatively low. The resulting absorber has good performance for the nominal exciting frequency but has a comparable sensitivity with respect to the classical delayed resonator, see Fig. 3.1 and 3.7. Increasing the number of coefficients, as shown in the second case (Fig. 3.2 and 3.8), doesn’t bring substantial improvements whereas it is so for the third case. If we look on the spectra in Fig. 3.9, the absorber has two eigenvalues around the exciting frequency. This results in an improved robustness function as shown in the Fig. 3.3, where the shape of the curve is flat in the selected region.

CONCLUSIONS

61

Next, the fourth example shows the case, where the region in which robustness is minimized, is extended. This results in placing two eigenvalues of the absorber further from the nominal exciting frequency and hence extending the flatness of robustness curve on a wider interval, see Fig. 3.4 and 3.10 for more details. The last two examples (#5 and #6) show the performance in frequencies below and above natural frequency of the absorber (1.19). As shown in Fig. 3.11 and 3.12, the stability constraint (3.24) is active when the optimum is reached resulting in an eigenvalue at the stability boundary. The resulting system is still stable but close to the border of instability. This chapter can be further extended towards more restrictive constraints on the shape of the spectra, e.g. the constraint (3.24) could be shifted to the left half plane. On the top of the previous results, this section provides one extra case #7, which has Fig. 3.13 enhanced by spectra of the system coupled with the classical delayed resonator. The exciting frequency is selected far below natural frequency of the absorber. As shown, the system with classical delayed feedback (3.1) is not stable, whereas with the proposed robust delayed resonator (3.3) the resulting system is stable. Additionally, time domain simulations are showed in Fig. 3.14 and 3.15. The simulations are shown for example # 3. First Fig. 3.14 shows a simulation where the exciting input is a combination of three harmonic forces f (t) = A1 sin(ω1 t) + A2 sin(ω2 t) + A3 sin(ω3 t), where ω1 = 39.1, ω2 = 39.62, ω3 = 40.2 and A1 = A2 = A3 = 10N . As depicted, the classical delayed resonator removes one nominal harmonic (ω2 = 39.62) force but the two other frequencies remain in the output signal. The robust delayed resonator results in a much smaller amplitude, almost negligible. The next three figures 3.15 show results where only one of the three harmonic forces is applied. The simulations shows excellent results in comparison with classical delayed resonator except for nominal frequency. For the nominal frequency the classical delayed resonator has precise pole placement whereas the robust delayed resonator has small misalignment which is the price for increased robustness for other frequencies.

3.4

Conclusions

As the main contribution, a new type of delayed resonator with a delay distributed with a polynomial function is presented along with the method for the design of the parameters. The main benefit of the proposed feedback is to design an absorber with smaller sensitivity to the excitation frequency. The method also includes a constraint, securing stability of the overall system. The stability is checked by including a constraint in the optimization problem. The constraint is later translated into an unconstrained penalty method. The spectral constraint makes the optimization problem non-smooth and such problem has to

62


be solved with specialized algorithms. The presented results show a successful implementation using HANSO [94]. The effectiveness of the proposed method is presented at six examples. The examples show increased performance in comparison with classical delayed resonator and also show successful results in case where the classical approach would lead to an unstable system. The proposed method improves previous work as follows: • more robust feedback, e.g., an increased insensitivity against uncertainty of the input frequency; • the approach imposes stability of the overall system in the design of the feedback law; • the freedom in the design allows to tailor the feedback for various systems with various requirements on functionality; • the value of the delay can be selected, which can be beneficial when the feedback is used in real system. The length of the delay must be long enough with respect to the sampling period. Subsequent work will involve an extension of the optimization problem in order to increase the response speed of the system to a change of exciting frequency. As shown, the optimization satisfies the stability requirements but the resulting system might have slow response as the overall system has eigenvalues close to the imaginary axis. Finally, the simulation results will be verified on laboratory set-up.

63

3

0.3

2

0.2

T (ω)

T(ω)

CONCLUSIONS

1

0.1 0

0

39 0

10

20

30

40

50

39.5

40

ω

ω 1

0.06

A(ω)

A(ω)

0.04

0.5

0.02

0

0

39

0

10

20

30

40

50

39.5

40

ω

ω

3

0.3

2

0.2

T (ω)

T(ω)

Figure 3.1: Case # 1: blue - robust DR (3.3), red- lumped delay DR (3.1). TopThe robustness function of the absorber (3.6), Bottom- robustness function of the overall system (1.20)-(1.21)

1

0.1 0

0

39 0

10

20

30

40

50

39.5

40

ω

ω 1

0.06

A(ω)

A(ω)

0.04

0.5

0.02

0

0

39

0

10

20

30

40

50

39.5

40

ω

ω



3

0.3

2

0.2

T (ω)

T(ω)

64

1

0.1 0

0

39 0

10

20

30

40

39.5

50

40

ω

ω 1

0.06

A(ω)

A(ω)

0.04

0.5

0.02

0

0

39

0

10

20

30

40

39.5

50

40

ω

ω


1

2

T (ω)

T(ω)

3

0.5

1 0 0

38 0

10

20

30

40

39

50

40

41

40

41

ω

ω 1

0.15

A(ω)

A(ω)

0.1

0.5

0.05

0

0

38

0

10

20

30

40

50

39

ω

ω


CONCLUSIONS

65

0.4

2

T (ω)

T(ω)

3

0.2

1 0 0

34.5 0

5

10

15

20

25

30

35

40

45

50

35

35.5

ω

ω 1

0.06

A(ω)

A(ω)

0.04

0.5

0.02

0

0

34.5

0

5

10

15

20

25

30

35

40

45

50

35

35.5

ω

ω

3

0.3

2

0.2

T (ω)

T(ω)


1

0.1 0

0

44.5 0

10

20

30

40

50

60

45

45.5

ω

ω 1

0.06

A(ω)

A(ω)

0.04

0.5

0.02

0

0

44.5

0

10

20

30

40

50

60

45

45.5

ω

ω


66


200

200 Coupled system RDR Absorber70 Spectral abscissa

150

Coupled system RDR Absorber70 Spectral abscissa

150

60

60 50

40

100

ℑ(s)

ℑ(s)

100

ℑ(s)

ℑ(s)

50

30

30

50

50 20

20

10

0 -20

-10

0

0 -1

10

10

0 0

1

-20

ℜ(s)

ℜ(s)

-10

0

10

0

Figure 3.8: Spectra, case # 2

200

200 Coupled system RDR Absorber70 Spectral abscissa

150

60

60

40

100

ℑ(s)

ℑ(s)

50

ℑ(s)

50

100

30

40 30

50

50 20

20

10

0 -20

-10

0

0 -1

10

10

0 0

1

-20

ℜ(s)

ℜ(s)

-10

0

10

0 -1

0


200

200

70

90


Coupled system RDR 80 Absorber Spectral abscissa

150

70

150 50

60

100

ℑ(s)

ℑ(s)

ℑ(s)

40

100

30

50

1

ℜ(s)

ℜ(s)


50 40 30

50

20

20

10

10

0 -20

1

ℜ(s)


150

ℑ(s)

0 -1

ℜ(s)


ℑ(s)

40

0 -10

0

ℜ(s)

10

0 -1

0

ℜ(s)


1

-20

-10

0

10

0 -1

0

ℜ(s)

ℜ(s)


1

CONCLUSIONS

67

200

60 Coupled system RDR Absorber RDR Coupled system DR Spectral abscissa

50

150 40

ℑ(s)

ℑ(s)

100

30

20

50

10

0 0

-20

-10

ℜ(s)

0

-1

10

-0.5

0

0.5

1

ℜ(s)

Figure 3.13: Spectrum of the system where the systems coupled with the classical DR (black dots) is unstable in comparison with the the system coupled with robust DR (green dots), which is stable.

0.015 system without feedback system with the classical DR system with the robust DR

0.01

xp [m]

0.005

0

-0.005

-0.01

-0.015 0

5

10

15

20

25

30

time[s]

Figure 3.14: Simulation of the system (case # 3 from Table 3.2) excited by a combination of three harmonic frequencies

68


6

×10 -3 system without feedback system with the classical DR system with the robust DR

4

xp [m]

2

0

-2

-4

-6 0

5

10

15

20

25

30

time[s] 6

×10 -3 system without feedback system with the classical DR system with the robust DR

4

xp [m]

2

0

-2

-4

-6 0 6

5

10

15

20

25

30

time[s]

×10 -3

system without feedback system with the classical DR system with the robust DR

4

xp [m]

2

0

-2

-4

-6 0

5

10

15

20

25

30

time[s]

Figure 3.15: Simulation of the system (case # 3 from Table 3.2) excited by a harmonic force. The system is excited by a harmonic force with a frequency above (top), at exactly (middle) and below (bottom) the natural frequency of the absorber

Part II

Input shapers

69

Chapter 4

Design and analysis of robust input shapers with distributed delay 4.1

Introduction

The presented work is a further step in a systematic research on involving distributed delays in the structure of the shapers by the teams of thesis supervisors. The first results were presented in [159, 160], where the lumped delay in the ZV shaper structure was substituted by an equally distributed delay. Subsequently, more complex delay distributions were considered in [162]. Next to the smoothening effect at the signal accommodation part, the retarded characteristics of the shaper spectrum can be considered as an implementation benefit, particularly, if the shaper is implemented within a closed loop system [154, 158]. In the recent work [151], an optimization based technique, the constrained linear least squares method in particular, was applied to the shaper design. The work was motivated by design algorithms for digital signal shapers [112, 144], see also a recent work [37] (Chapter 6) on optimization based design of multi-mode shapers with lumped delay. The delay distribution in [151] consisted of a discrete series of equally distributed time delays and the optimization objective was to achieve enhanced robustness of the shaper. The main objective of this chapter is to propose a novel class of shapers, characterized by a piece-wise distributed delay, and to present on optimization approach for the corresponding shapers tuning. This will be done by optimizing the coefficients of the shaper. Besides, compared to [151], additional options and constraints on the shaper properties and robustness will be considered providing

70

ROBUST INPUT SHAPERS WITH PIECE-WISE DELAY DISTRIBUTION

71

the end-user extensive freedom in defining requirements on the optimized shaper performance. Subsequently, a class of shapers is analyzed where the delay distribution is governed by a smooth polynomial kernel function. A similar optimization approach as for the shapers with piece-wise distributed delay is presented for the corresponding shapers tuning. This will be done by optimizing the shape of the smooth polynomial kernel function. Compared to the piece-wise distributed delay technique, additional options and constraints on the shaper properties will be considered providing the end-user even more freedom in the shaper design. Let us remark that the presented work is also related to work by Cole, et al. [23–25], where the shaper was considered and designed as a general finite impulse response (FIR) filter. The FIR filters are designed to operate on an arbitrary command input signal to ensure a finite settling time, prescribed roll-off rate and the filter frequency response. Analogously to the proposed shapers with distributed delays, the designed FIR filters produce an input smoothing effect. Next to the fact that a completely different methodology has been involved in this work, utilizing time delay system theory in combination with optimization methods, the main contribution of this chapter with respect to the work by Cole, et al., consists of taking into account robustness criteria in terms of residual vibrations as well as further structural issues on the shaper design. Besides, the implementation by a time delay system with discrete delay is considered, next to the implementation by the convolution integral discretization, often applied for FIR filters. The chapter is based on [100, 103], of which the doctoral candidate is the main author. The chapter is organized as follows. Firstly, robust input shapers with piecewise distributed delay are introduced. The section is followed by the formulation of the optimization problem for tuning the shaper, and a procedure that solves this optimization problem. The section is completed with a demonstration on several examples. Secondly, the piece-wise distributed delay is substituted with a smooth polynomial kernel function. Also for this case, the problem is firstly defined and then solved by the proposed method. The section is completed with many examples, comparisons and implementation comments.

4.2 4.2.1

Robust input shapers with piece-wise delay distribution Problem formulation

Considering the robust shaper form that was proposed in [151], i.e. given by (1.34)-(1.35), which can be reformulated as G(s, T ) = A +

PNp

k=0

ak e−sτk . s

(4.1)

72

DESIGN AND ANALYSIS OF ROBUST INPUT SHAPERS WITH DISTRIBUTED DELAY

The parameters of (4.1) are the gain of the non-delayed shaper part A, the gains ak and delays τk (k = 0, 1, ..., N ) determining the overall distribution of the delay. The optimization procedure which follows, considers that the delays are equidistantly distributed over the interval [0, T ], i.e. τm = τm−1 + ∆τ, m = 1, ..., Np with τ0 = 0 and ∆τ = NTp . The two criteria to optimize over the free parameters A, ak , k = 0, ..., N are the residual function (4.34) and the action time T . Additionally, a set of (partly optional) constraints is to be taken into the consideration as addressed below.

4.2.2

The primary objective functions

In minimizing the residual vibration function over the interval covering the p vicinity corresponding to the oscillatory mode sˆz = −ωζ − jω 1 − ζ 2 to be compensated, the uncertainties in both, the natural frequency ω and the damping ratio ζ are taken into account. Assume that ω ∈ I1 , ζ ∈ I2 , where I1 = [ωmin , ωmax ], I2 = [ζmin , ζmax ],

(4.2)

with nominal frequency ωnom and nominal damping ζnom assumed to be the midpoints of these intervals. To handle the uncertainty in ω and ζ a grid of Nω and Nζ Chebyshev points is defined on I1 and I2 . The Chebyshev points are more efficient for polynomial approximation, see [135]. The points are defined as (k − 1)π ωmax + ωmin 2

ωk =

ζl =

ζmax + ζmin 2

−

−

ωmax − ωmin 2

ζmax − ζmin 2

cos

cos

,

Nω − 1 (k − 1)π Nζ − 1

.

(4.3)

with k = 1, ..., Nω and l = 1, ..., Nζ . Define the vector of gains to be assessed as | x = A a0 a1 · · · aNp , (4.4) then the robustness criterion can be expressed by f (x, T ) =

Nζ Nω X 1 X V (ζl , ωk )2 . Nω Nζ k=1 l=1

The shaper transfer function (4.1) can be expressed as p G −ωζ − jω 1 − ζ 2 = L(ζ, ω, T )x with

L(ζ, ω, T ) = 1 g0 (¯ s, T ) · · · gNp (¯ s, T ) ,


where gk = in

e−sτk s

, k = 0..Np and s¯ = −ωζ −jω

V (ζ, ω)2

p

73

1 − ζ 2 . Consequently, it results

= x| L(ζ, ω, T )L(ζ, ω, T ) xe2ζωT = x| 1 and determining whether the parameter A is excluded from the optimisation and fixed on a specific value or is left as a decision variable. • Step 2 - Define nominal values of the flexible structure, see Fig. 1.2, i.e. nominal frequency ωnom and nominal damping ratio ζnom . For the given nominal parameters, the zeros of the shapers can be placed exactly on the nominal pole or not. Then the uncertainty intervals (4.2) over which the residual function is to be minimized are to be determined. • Step 3 - Finally, the Pareto front is obtained by solving the problem (4.13) with respect to the parameter α. The end user can then select a compromise between the two objective functions, the response time T and average residual vibration on the parameters domain (ω, ξ) ∈ I1 × I2 , just by adjusting the parameter α.

4.2.5

Demonstration example

Consider the oscillatory mode given by the nominal frequency ωnom = 10 and damping ratio ζnom = 0.01. The residual vibration function is to be minimized

76


0.7 0.6

√ xT Hx

0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

T/Tnom

Figure 4.1: Pareto frontier generated for different number of coefficients N = 5(blue), 10(red), 15(yellow) with marked points for α = 0.78 over the the intervals I1 = [0.85, 1.15]ωnom , I2 = [0.85, 1.15]ζnom , covered by Nζ = 10, Nω = 10 discretisation points to obtain the mesh (4.3). Performing the optimization procedure over the full parameter set consisting of the parameters A and ak , k = 0, ..., N p and taking into account the Equality constraint II and the Inequality constraints I and II, the whole Pareto front is given in Fig. 4.1 for Np = 5, 10 and 15. It can nicely be observed that increasing the number of coefficients provides more degrees of freedom for the optimization and enhances the diversity of the solutions. Next, for the specific value of α = 0.78 the comparison of the residual vibration function (4.34), the shaper step responses and the spectral properties are given in Figs. 4.2, 4.3 and 4.4 is shown. As can be seen, increasing the number of points from Np = 5 to Np = 10 led to considerable improvement in the residual vibration function over the target region. On the other hand, an increase from Np = 10 to Np = 15 had a negligible effect on the shaper robustness. In the step response in Fig. 4.3 a considerable difference between the case with Np = 5 and the cases obtained for Np = 10, 15 - an increase of Np provides enhanced variability of speed can also be observed. Finally, in Fig. 4.4, it can seen that as required all the obtained shapers provide the retarded character of the spectrum.

4.2.6

Graphic user interface for the shaper design

The shaper design procedure proposed and demonstrated above has been implemented as a Matlab tool, including the graphic user interface (GUI) shown in Fig. 4.5, which consists of four parts:


77

1

0.8

0.6

V

I1

0.4

0.2

0 0

5

10

15

20

ω

Figure 4.2: Residual vibration function of the shapers for given α = 0.78 for a different number of coefficients N = 5(blue), 10(red), 15(yellow)

1

u

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

t

Figure 4.3: Step response of the shapers with a different number of coefficients N = 5(blue), 10(red), 15(yellow)

78


1000 30

ℑ(s)

800

20 10

600

ℑ(s)

0 -0.4

-0.2

0

ℜ(s)

400

200

0 -8

-6

-4

-2

0

ℜ(s)

Figure 4.4: Zeros of the shapers for a given α = 0.78 and number of coefficients N = 5(blue), 10(red), 15(yellow) with cropped and zoomed area around target p 2 mode s = −ωnom (ζnom + j 1 − ζnom ) (black cross) • The shaper parameterization window to define the structural and required performance properties of the shaper. After defining all the parameters, the optimisation task is initiated by pressing the button Draw pareto front. • The Pareto front window serves for the objective function visualization and for determining the user-defined shaper performance point by the slider below the graph. • The shaper’s attributes window corresponding to the selected point on the Pareto front toolbar. The button Show is to demonstrates the shaper properties and the button Export is to save the results to the Matlab workspace. • The shaper’s characteristics window with the step response, impulse response and residual vibration function for the selected point of the Pareto front. A user-friendly, MATLAB based, interface for the shaper design is available from http://twr.cs.kuleuven.be/research/software/delay-control/shaper-design/

ROBUST INPUT SHAPERS WITH SMOOTH KERNEL FUNCTIONS

79

Figure 4.5: Graphic user interface for the design of the shaper

4.3

Robust input shapers with smooth kernel functions

This section propose a novel class of input shapers where the piece-wise distributed delay (4.1) is substituted with a smooth polynomial function.

4.3.1

Proposed novel class of input shapers

Considering a class of input shapers of the form u(t) = Aw(t) + (1 − A)

Z

T

w(t − η)dh(η),

(4.14)

0

described by u(t) = Aw(t) +

Z

T

g(θ)w(t − θ)dθ,

(4.15)

0

where w(t) ∈ R, u(t) ∈ R are input and output, respectively, the smooth function g is the general kernel function of the distributed delay, T > 0 is the maximum time-delay and A ∈ [0, 1] is the static gain. Note that the Eq. (4.15) has a similar form as delayed feedback for delayed resonator (3.3), proposed in Chapter 3. The difference lies in the application. Here, the polynomial distribution of the delay is used for feedforward control and in Sec. 3 the delay

80


is used in feedback control. The later requires stability whereas here the stability is not necessary at this moment. The transfer function of the shaper is given by Z T G(s) = A + g(θ)e−sθ dθ. (4.16) 0

The kernel function can be expressed as a combination of various types of basis functions (e.g. splines, exponentials, polynomials,...). Since implementation and realization constraints need to be taken into an account when designing the kernel function, this work is based on the kernel function g chosen as the polynomial Np X g(θ) = ai θ i , (4.17) i=0

where ai ∈ R, i = 0, 1, ..., Np , are the coefficients and Np is the selected degree of the polynomial. Thus, the transfer function (4.16) can be rewritten into a form Np X G(s) = A + ai gi (s), (4.18) i=0

where the functions gi (s) are given by gi (s) =

Z

T i −sθ

θe 0

dθ =

i! −

Pi

i! i−j −T s e j=0 (i−j)! (T s) , i+1 s

(4.19)

and can be interpreted as so called “moments” of e−sθ . The impulse response of the input shaper (4.15) and (4.17) is given by Aδ(t) + g(t), t ∈ [0, T ], h(t) = 0, t ≥ T, with δ denoting the Dirac impulse, and the step response by Rt A + 0 g(θ)dθ, t ∈ [0, T ], s(t) = 1, t ≥ T. The spectrum of zeros is given as the solution of the equation G(s) = 0. Multiplying both sides of the equation by sNp +1 , the following equation is obtained   Np i X X i! (4.20) AsNp +1 + ai i!sNp −i − T i−j sNp −j e−T s  = 0, (i − j)! i=0 j=0 which has the same distribution of zeros as G(s), except for additional zeros in the s-plane origin.


81

The inverse application of the shaper in a feedback interconnection requires A > 0. It is easy to see that for A > 0, the quasi-polynomial at the left-hand side of (4.20) is retarded, because of the fact that the s power corresponding to the exponential terms ranges from 0 (for i = Np ) to Np (for i = 0), i.e. it is lower than the quasi-polynomial order Np + 1. This is a desirable property since in such a feedback configuration the shaper zeros are turned into poles, see [154, 158]. For A = 0, which is the case considered as well in what follows, the spectrum can be either retarded or neutral, depending on the values of the parameters ai , i = 1..Np and T . It needs to be stressed that in the feed-forward shapersystem interconnection the retarded character of the spectrum is not required and shapers with neutral spectrum of zeros can be well applied too.

4.3.2

Requirements on the shaper functionality

This section outlines various design requirements for input shapers that can be taken into account in their design by optimization methods. Most of them will be directly expressed in terms of linear equality or inequality constraints on the shaper parameters A, a0 , . . . , aNp . Others will be expressed by the non-negativity of the polynomial on an interval. Linear equality constraints The first linear constraint stems from placing zeros of the shaper at the expected position of the oscillatory mode to be compensated, p 2 sˆn = −ζnom ωnom ± j 1 − ζnom ωnom , where ωnom and ζnom are the nominal frequency and damping. requirements corresponds to G(ˆ sn ) = 0 ⇒ A +

Np X

ai gi (ˆ sz ) = 0,

These

(4.21)

i=0

which can be turned into two real equations for the case of a complex zero, < {G(ˆ sz )} = 0,

(4.22)

= {G(ˆ sz )} = 0.

(4.23)

It is also possible to place more zeros. However, each additional zero decreases the number of degrees of freedom. In order to arrive at a feasible solution, this may lead to a significant increase of the time delay of the shaper.

82


The second equality constraint comes from the basic requirement on input shaping to have a static gain equal to one, which leads to G(0) = 1 ⇒ A +

Np X

ai gi (0) = 1.

(4.24)

i=0

The additional linear equality constraint that might be required corresponds to the requirement of continuity of both the step response and its derivatives at times t = 0 or (and) t = T . At t = 0 it is expressed by A = 0, g(0) = a0 = 0,

(4.25)

and for t = T by g(T ) =

Np X

ai T i = 0.

(4.26)

i=0

Linear inequality constraints The first constraints come from the basic requirement on the delay free part of the shaper 0 ≤ A ≤ 1. The next fundamental requirement that need to be considered is the nondecreasing step response, or equivalently, the non-negative impulse response, which can be formulated as g(α) ≥ 0, ∀α ∈ [0, T ].

(4.27)

Condition (4.27) is a semi-infinite polynomial inequality (requirement to be satisfied for a continuum of α values). The problem can be solved via Pólya’s relaxations, which will be addressed in the following subsection. As demonstrated and motivated by Singh [117], limiting the jerk, defined as the 1st derivative of impulse response, can help to increase durability of actuators in the control systems. Analogously, limiting the jounce - the 2nd derivative of the impulse signal can also be imposed. For example as demonstrated in [30], [83], limiting jerk and jounce reduces CNC machining vibrations and increases product quality. These quantities have also been taken into account in planning a trajectory of a manipulating robot in [15] or [71]. The constraints on the jerk and jounce are described by |g(α)| ˙ ≤ J1 , ∀α ∈ [0, T ], (4.28) |¨ g (α)| ≤ J2 where J1,2 are chosen limits on Jerk and Jounce, respectively. The inequality constraints (4.28) are also semi-infinite. They can be transformed to the


following polynomial inequalities,  g(α) ˙ + J1 ≥ 0    J1 − g(α) ˙ ≥0 , ∀α ∈ [0, T ], g ¨ (α) + J ≥ 0  1   J1 − g¨(α) ≥ 0

83

(4.29)

and included via Pólya’s relaxation technique, which is addressed next. Pólya’s relaxation When applying Pólya’s relaxation to the polynomial inequality p(α) ≥ 0, ∀α ∈ [0, T ],

(4.30)

with p denoting a polynomial of degree Np , the following steps need to be followed [13], where the first two reformulate the polynomial p to a homogeneous polynomial over a unit simplex: 1. Rescale the interval for α to [0, 1] by introducing a new variable θ = θ ∈ [0, 1].

α T

⇒

2. Set θ = θ1 , introduce the additional variable θ2 ∈ R, and homogenize the polynomial p(T θ1 ) by multiplying single monomials with powers of (θ1 + θ2 ), until all monomials have the same degree N (larger than or equal to Np ). The non-negativity requirement (4.30) on p is now equivalent with the non-negativity requirement of the corresponding homogeneous multivariate polynomial pN (θ1 , θ2 ) for all θ1 and θ2 satisfying θ1 ≥ 0, θ2 ≥ 0, θ1 + θ2 = 1,

(4.31)

i.e., for all (θ1 θ2 ) belonging to the unit simplex in R2 . 3. Compute the coefficients of the multi-variate polynomial pN (θ1 , θ2 ). 4. Sufficient conditions for (4.30) are obtained by requiring that all coefficients of pN be non-negative. 5. If necessary, increase Pólya’s relaxation degree N and repeat from step (2) on. It can be shown that the gap between sufficient and necessary conditions tends to zero by increasing Pólya’s relaxation degree, see, e.g., [47]. Example. Considering requirement (4.27) of a non-decreasing step response on interval [0, T ], for the case where Np = 2, i.e., g(α) := a0 + a1 α + a2 α2 ≥ 0, ∀α ∈ [0, T ].

(4.32)

84


With θ =

α T,

condition (4.32) is equivalent to a0 + a1 T θ + a2 T 2 θ2 ≥ 0, ∀θ ∈ [0, 1].

(4.33)

This in turn is equivalent to a0 (θ1 + θ2 )2 + a1 T (θ1 + θ2 )θ1 + a2 T 2 θ12 ≥ 0, ∀θ1 ≥ 0, θ2 ≥ 0 : θ1 + θ2 = 1, i.e., a positivity constraint of a homogeneous multivariate polynomial over the unit simplex. Working out this expression in monomials gives θ12 (a0 + a1 T + a2 T 2 ) + (2a0 + a1 T )θ1 θ2 + a0 θ22 ≥ 0, ∀θ1 ≥ 0, θ2 ≥ 0 : θ1 + θ2 = 1. Hence, sufficient conditions for (4.32) are given by the linear inequalities a0 + a1 T + a2 T 2 ≥ 0, 2a0 + a1 T ≥ 0, a0 ≥ 0. This is called Pólya’s relaxation of degree 2. Tighter conditions can be obtained by increasing the degree of the relaxation, at the price of an increase in the number of inequalities. For Pólya’s relaxation of order 3, one writes the constraint as a0 (θ1 + θ2 )3 + a1 T (θ1 + θ2 )2 θ1 + a2 T 2 (θ1 + θ2 )θ12 ≥ 0, ∀θ1 ≥, θ2 ≥ 0 : θ1 + θ2 = 1. Working out this expressions and requiring the coefficients of all (third-order) monomials to be non-negative results in a0 + a1 T + a2 T ≥ 0, 3a0 + 2a1 T + 3a2 T 2 ≥ 0, 3a0 + a1 T ≥ 0, a0 ≥ 0.

4.3.3

Optimization problem formulation

The procedure for the design of the shaper parameters A, a0 , · · · , aNp and T is presented. The procedure is based on a multi-objective optimization problem yielding a trade-off between a fast response time of the shaper and a robustness requirement, expressed in terms of residual vibrations. The optimization problem is constrained by selected requirements specified in Section 4.3.2.

4.3.4

Residual vibrations

The design procedure takes into account robustness of the shaper, in the sense that it is less insensitive with respect to changes of the suppressed system’s parameters. Our robustness criterion is expressed in terms of the residual


85

vibrations introduced by [122], which can be expressed in terms of the transfer function (4.16), as shown in [151] p V (ζ, ω) = G −ωζ − jω 1 − ζ 2 eζωT , (4.34) and which takes into account uncertainty not only in the nominal frequency (as it is usual), but also in the damping of the vibration to be suppressed. Formula (4.34) expresses the amplitude of thepresidual vibration at time t = T . More precisely, let s = −ωζ − jω 1 − ζ 2 be the pole to be compensated by a shaper zero and assume that ω ∈ I, ζ ∈ I2 , where I1 = [ωmin , ωmax ], I2 = [ζmin , ζmax ],

(4.35)

with a nominal frequency ωnom and a nominal damping ζnom assumed to be the midpoints of these intervals. To handle the uncertainty in ω and ζ, a grid of Nω and Nζ Chebyshev points is defined, which are more efficient for polynomial approximation [135], on I1 and I2 ωmax + ωmin ωmax − ωmin (k − 1)π ωk = − cos ; k = 1, ..., Nω , 2 2 Nω − 1 (4.36) (k − 1)π Nζ − 1 Define the vector of gains to be assessed x = A a0 robustness criterion can be expressed by ζl =

ζmax + ζmin 2

−

ζmax − ζmin 2

f (x, T ) =

cos

; l = 1, ..., Nζ . (4.37) a1

···

aNp

|

, the

Nζ Nω X 1 X V (ζl , ωk )2 . Nω Nζ k=1 l=1

The shaper transfer function (4.16) can be expressed as p G −ωζ − jω 1 − ζ 2 = L(ζ, ω, T )x with h i p p L(ζ, ω, T ) = 1 g0 −ωζ − jω 1 − ζ 2 · · · gNp −ωζ − jω 1 − ζ 2 . Consequently we have V (ζ, ω)2

= x| L(ζ, ω, T )L(ζ, ω, T ) xe2ζωT = x| 0 a scaling parameter, is given by 2 3 Np +1 T . A, a0 ρ, a1 ρ , a2 ρ , . . . , aNp ρ , ρ The user interface shown for the previous case with piece-wise distributed delay 4.5 helps users to define requirements on the shaper. Similarly, a userfriendly, MATLAB based, interface for the design of the shapers with a smooth polynomial kernel is available from http://twr.cs.kuleuven.be/research/software/delay-control/shaper-design/

88


1 α∈ [0.999,1]

0.9 0.8 non-feasible area

√ xT Hx

0.7 0.6 0.5 0.4

α= 0.998

0.3

α= 0.989

0.2

α= 0.907 α= 0.680

0.1

α= 0.014

0 0

0.5

1

1.5

2

2.5

T/Tnom

Figure 4.6: The Pareto front(green curve) of (4.42) for the first example in Section 4.3.5. The red circles represent the solutions of (4.41) for certain values of α.

4.3.5

Results

This section provides six design examples of shaper (4.14). The shapers are tuned for an oscillatory mode defined by the damping ratio ζnom = 0.01 and natural frequency ωnom = 1. The uncertainty intervals are given by I1 = [0.85, 1.15] ωnom and I2 = [0.85, 1.15] ζnom , and Nω = 10, Nζ = 10 number of points are used in the discretization of the omega and ζ interval, respectively. The first three examples illustrate the design of robust shapers with different constraints. The next three examples show a comparison with the ZV shaper. The selected constraints for the examples are indicated in Table 4.1. The parameters of the designed shapers are shown in Table 4.2, where the two objectives, i.e., the response time and average residual vibrations, are shown as well. Fig. 4.6 shows the complete Pareto front of multi-objective optimization problem (4.42) for example #2 with constraints listed in Table 4.1. As expected, a faster response time (horizontal axis) is obtained by increasing α, at the price of an increase of the average residual vibrations (vertical axis) on the given intervals (4.35). To obtain comparable results, the parameter α for examples #1-3 is selected as α = 0.0501. A comparison of step and impulse response is in Fig. 4.7. As expected,


89

a smoother response leads to a (slight) increase in the response time. Step response of the interconnection with system with defined oscillatory mode is depicted in Fig. 4.8. Figure has highlighted 5% range of oscillations with black dotted lines. As can be seen, the example #3(yellow) has no oscillations after the shaper response time t > T , because one of the constraints is to place an exact zero. Examples #1 and #2 have residual vibrations after the response time, but in a very small range, which corresponds with Pareto front charts. Next, Fig. 4.9 shows residual vibrations with respect to frequency. The interval of frequencies, where the shapers are optimized, is indicated with black dashed lines around the nominal frequency ωnom . Shapers are also optimized in the neighbourhood of the nominal damping ratio ζnom . Residual vibrations with respect to ζ and ω around the nominal values are shown in Fig 4.10. Nominal values are indicated with black dashed lines. The dependency on ζ is much small than the one on ω and almost not noticeable. A good conception about the dynamics of the shaper is provided in Fig. 4.11. The spectra are computed by the QPmR algorithm [168]. As can be seen, the spectrum of zeros with A > 1 in #1 is truly retarded, as derived in the Subsection 2.2. Interestingly, the spectrum corresponding to #3 is also retarded despite A = 0, whereas the spectrum for the case #2 is neutral. In a detailed view, the distribution of the zeros of the shapers corresponds to Fig. 4.9, where a decrease of residual vibrations is seen on frequencies around the zeros in the complex plane. Examples #4-6 are shown to provide a comparison with the classical ZV shaper, which is described by (4.14) for N = 1 and where the parameters (A0 , A1 and τ1 ) are determined by placing a zero at the oscillatory mode (ζnom = 0.01, ωnom = 1). To have comparative results, parameter α in (4.41) is selected as α = 1, so the optimization objective insists on a response time as fast as possible. The suppression of the undesired oscillation is then imposed by placing an exact zero at sˆn . Example #4 shows only a 10% slower response time then the ZV shaper and the responses are comparable. Hence, if the response time is the main criterion, the proposed shaper mimics a ZV shaper. The next two examples have more requirements on smoothness, and the response time increases accordingly. As in the previous three examples, the step response is in Fig. 4.12 with a comparison with the classical ZV shaper and the step response for shaper interconnected with oscillatory system is shown in Fig. 4.13. The residual vibrations in Fig. 4.14 have almost the same shape for ω < 1. Examples with longer response time T have lower residual vibrations on higher frequencies.

4.3.6

Implementation aspects

This section propose two ways on how to implement the shaper (4.15). The first method is based on an on-line computation of the integral. The second

90

#


placing a zero on ωnom

1 2 3 4 5 6

constraint on initial step at t = 0 (A = 0)

zero derivative of step response at t = 0

zero derivative of step response at t = T

× ×

×

× ×

×

×

× × × ×

α 0.680 0.680 0.680 1 1 1

× ×

Table 4.1: Optional constraints selected for every example are indicated by ×. In all cases a polynomial of degree 7 is used. All examples include constraints of a non-decreasing step response imposed via a Pólya’s relaxation of degree 10, and of a steady state gain equal to one 1

0.2 0.15

0.6

h(t)

s(t)

0.8

0.4

0.1 0.05

0.2

0

0 0

2

4

6

8

0

10

2

4

6

8

10

t

t (a) Step responses

(b) Impulse responses

Figure 4.7: Step and impulse responses of the shaper examples #1(red),#2(blue) and #3(yellow) approach consists of realizing the shaper by a dynamic system. In order to avoid an unstable realization the basis functions gi for the second case are modified. # 1 2 3 4 5 6

T 2.65π 3.02π 2.81π 1.11π 1.27π 1.67π

√

x| Hx 0.036 0.051 0.048 0.18 0.17 0.16

A 0.129 0 0 0.489 0.455 0

a0 1.006e-7 0 0.175 3.039e-5 6.709e-8 0

a1 -9.954e-8 0.173 -0.178 -4.788e-5 2.812e-6 1.253

a2 7.898e-2 -0.146 8.062e-2 2.151e-5 0.181 -1.624

a3 -4.245e-2 6.268e-2 3.668e-3 4.822e-2 -0.317 0.893

a4 1.150e-2 -1.286 -6.623e-3 -7.648e-2 0.239 -0.264

a5 -1.824e-3 1.234e-3 1.117e-3 5.017e-2 -9.702e-2 4.316e-2

a6 1.515e-4 -4.658e-5 -6.343e-5 -1.602e-2 2.234e-2 -3.304e-3

a7 -4.881e-6 1.850e-7 7.584e-7 2.169e-3 -2.187e-3 5.908e-5

Table 4.2: The optimized coefficients of the polynomial (4.17). The numbers in the first column correspond with example numbers. The second column is the response time of the shaper. The third column is the average of the residual vibrations.


91

1.2 1

y(t)

0.8 0.6 0.4 0.2 0 0

5

10

15

t Figure 4.8: Step response of the shaper interconnected with a second order oscillatory system with output y for examples #1(red),#2(blue) and #3(yellow). The dot-dashed line is a system without shaper

4.3.7

On-line computation of integral

This method doesn’t require further modifications of the shaper equation but requires the on-line evaluation of the integral in every discrete step time, which may lead to high memory and computation costs. This methods may also lead to instability when the numerical integration method is not selected appropriately (see [81] and the references therein). The shaper can be implemented as X y(t) ≈ Au(t) + wk g(θk )u(t − θk ), (4.43) where θk and wk are nodes and weights of the quadrature formula.

4.3.8

Exponentially decaying basis functions

The shaper described by (4.15) and (4.17) can be rewritten into the form y(t) = Au(t) +

Z

T

CeAθ Bu(t − θ)dθ, 0

(4.44)

92


Figure 4.9: Residual vibrations of the shaper examples #1(red),#2(blue) and #3(yellow). The graph is plotted for nominal damping ratio ζnom . The cropped zoom view shows details around nominal frequency ωnom .

Figure 4.10: Contour plots of the residual vibrations of the shaper for examples #1-#3, with respect to ζ and ω.


93

Figure 4.11: The zeros of the shaper with parameters given in Table 4.2. The spectra correspond to examples #1(red, circles), #2(blue, squares) and #3(yellow, diamonds) where

and

 0    A=  ...   0 B = 0!a0

1 0

1!a1

0 .. .

..

.

···

 ··· 0 . .. . ..    ∈ RNp +1×Np +1 , .. . 0  0 1 0 |

1 .

(4.46)

The corresponding transfer function can then also be expressed as G(s) = A + C(sI − A)−1 I − e−T (sI−A) B.

(4.47)

...

N !aNp

,C = 0

...

0

(4.45)

94


1 0.8

s(t)

0.6 0.4 0.2 0 0

1

2

3

4

5

6

t Figure 4.12: Step response of input shaper examples #4(brown),#5(green) and #6(purple). For comparison, the ZV shaper step response shown with black dashed line

1.2 1

y(t)

0.8 0.6 0.4 0.2 0 0

1

2

3

4

5

6

t Figure 4.13: Step response of the shaper interconnected with a second order oscillatory system with output y for examples #4(brown),#5(green) and #6(purple).


95

1

V (ζ, ω)

0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

ω Figure 4.14: Residual vibrations of the shaper examples #4(brown),#5(green) and #6(purple). The black dashed line corresponds to the ZV shaper. This suggest a realization and implementation by the dynamical system z(t) ˙ = Az(t) + Bu(t) − eT A Bu(t − T ), (4.48) y(t) = Cz(t) + Au(t). Unfortunately, the system (4.48) is always unstable, because the matrix A has a multiple non-semisimple eigenvalue at zero. In (4.47), however, the eigenvalues of A are removable singularities. A general approach to approximate FIR filters by stable LTI systems is described in [79] and the references therein. In what follows a solution which does not involve an approximation is presented. The instability problem can be resolved by modifying the basis functions gi . Given a number λ > 0, one can also consider a delay kernel of the form g(θ) =

Np X

ai θi e−λθ .

i=0

Note that the expression is still linear in ai so all results presented before regarding the tuning of the parameters A, a0 , . . . , aNp to compensate a zero, to have a non-negative step response, etc., can be easily adapted. In the implementation there is one difference. Namely, we have g(θ) = CeAθ B

96


1

0.2 0.15

0.6

h(t)

s(t)

0.8

0.4

0.1 0.05

0.2 0

0 0

2

4

6

8

10

0

2

4

6

8

10

t (b) Impulse responses

t (a) Step responses

Figure 4.15: Step and impulse responses of the shaper examples with λ = 0 (red) and λ = 0.4 (blue) with B and C as before but with  −λ 1 0  .  −λ . .  .. A=  ... .   0 ···

··· .. . .. . −λ

 0 ..  .    ∈ RNp +1×Np +1 0   1  −λ

(4.49)

Now A is a Hurwitz matrix for λ > 0, thus, the implementation using differential equation (4.48) is already stable. The choice of λ, once again, induces a trade-off. For large λ, the additional dynamics are very fast and do not affect the overall control system’s performance, but since the kernel basis functions have a rapid decay, it might become more difficult to find parameter values for the shaper satisfying the design requirements, affecting the performance. Example. Example #2 is now revisited and compared the design for λ = 0 and λ = 0.4. A comparison of step and impulse response is shown in Fig. 4.15, where imposing λ > 0 introduces a longer response time and a slight modification of the shape of the responses, see Fig. 4.16. The residual vibrations, see Fig. 4.16, are slightly different in the optimized interval with lower average residual vibrations.

4.4

Conclusions

This section showed two main contributions. First an extension of the design of the input shaper with a piece-wise equally distributed time delay. In comparison

CONCLUSIONS

97

1

V (ζ, ω)

0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

ω Figure 4.16: Residual vibrations of the shaper designed with λ = 0 (red) and λ = 0.4 (blue) with previous work, the main advantage of the proposed method lies in the definition of the multi-objective constrained problem. Presuming higher number of degrees of freedom, the optimization allows to design a robust and/or fast shaper, where the final attributes rely on a decision maker. Then, for the given requirements, linear (in)equality constraints are defined. Some of the constraints, such as a non-decreasing step response and unity gain are necessary and demands on placing zeros to compensate specific modes of the flexible structure or initial gain are the optional constraints. The presented examples show the applicability of the proposed method and the effect of the different number of coefficients which is mainly beneficial but also brings difficulty for the case where the shaper is applied in the feedback. Secondly, a general approach for designing distributed delay shapers based on constrained optimization is presented. The main benefit compared to the previous work consists of relaxing the delay distribution from a fixed to a continuous (polynomial) shape along with a systematic design procedure grounded in convex optimization. The degrees of freedom allow a number of additional design options have been introduced, such as limitation of the jerk (analogously to [117]) and jounce. Also continuity at the start and the end of the action can be prescribed. Let us remark that compared to the command smoothers considered in [121], when utilizing this option, the designed shaper/smoother may provide both command smoothing and full compensation of the mode. Additionally, next to the possibility to assess the robustness with respect to the frequency as it is common, the robustness with respect to the

98


damping of the mode to be suppressed can be included as requirement in the shaper design. The novelty is also in the option to include the overall action length of the shaper in the optimization procedure, next to the robustness of the mode suppression. The presented work also provides an alternative to the design of FIR shapers in [24]. Next to utilizing a different approach grounded in time delay system’s theory, the presented work provides a much wider range of design options, as mentioned above. The additional contributory aspect is the possible implementation of the shaper (time delay FIR filter) as a dynamical system with discrete delay (4.48).

Chapter 5

H infinity design of fixed-order controllers for systems pre-compensated by input shapers 5.1

Introduction

The use of feedforward control to compensate undesired oscillations is a widely used technique, which was firstly proposed in 1957 by Smith and well-know as the posicast filter [124]. As shown in Fig 1.2, the reference command w(t) for the nominal system P (here P0 ) controlled by a feedback controller K (here C) is shaped by the input shaper S in such a way that the target oscillatory mode of the attached flexible structure F is not excited. Subsequently, the research focused on more robust versions, such as the zero-vibration (ZV), zero-vibration-derivative (ZVD) and extra insensitive (EI) shapers, introduced in [122]. Extensions to multi-mode shapers able to eliminate two or more modes simultaneously were proposed in [138], [141], and the corresponding discrete time versions in [22]. Feedforward methods are able to handle a change of the reference input but they are incapable to react on external disturbances on the system. The interconnection of the shaper within the feedback loop can provide disturbance rejection, as shown by Smith in [124], proposing a basic scheme with a compensator and the shaper in the feedback. Later on, a direct augmentation of the standard loop with a pre-tuned input shaper was proposed in [54] followed by a stability analysis using the root locus method in [52] and [130]. The 99

100

H INFINITY DESIGN OF FIXED-ORDER CONTROLLERS FOR SYSTEMS WITH INPUT SHAPERS

scheme, with the shaper in the loop between the controller and the system, turned out to be ineffective when the vibrations are caused by disturbance variables and, as shown in [53], including the shaper is advantageous only in elimination of the sensor disturbances but incapable to suppress the effect of actuator disturbances. Recently, a novel architecture where an inverse shaper is included in a feedback loop, has been proposed in [157] (see also [156] for the multi-mode version). This architecture provides filtering of the undesired frequency for the reference input but also for input and output disturbances. Those properties are not achievable by a standalone controller. However, as shown in [101], the controller can be designed to mimic properties of the inverse shaper but only for specific inputs and not all of them. Additionally, the controller has some other constraints which limit its usage. For more details see [101]. By inversion of the shaper and the feedback interconnection, the shaper’s zeros are turned into poles, and the overall closed loop would be a delay system of neutral type, characterized by an undesirable spectrum location and a sensitivity of stability with respect to infinitesimal delay perturbations, and none of the classical shapers with lumped delay are applicable in this context. Instead, new types of shapers based on distributed delays need to be applied, [100, 152, 161, 163], provide retarded dynamics of the closed loop system. The crucial task for the systems with inverse shaper in the feedback is the controller design. Even though the dynamics of both the system and the controller are considered as delay free and finite-dimensional, the overall closedloop dynamics become infinite dimensional due to inclusion of the inverse shaper containing delays. The design is however simplified by the fact that the delays are fixed and known. As pointed out in [157], the classical frequency-response design methods are convenient for the given task. Based on the analysis of conceptually simpler single degree of freedom applications in [157], it has been demonstrated that as long as the controller is designed as sufficiently fast and as long as it features good gain and phase margins in combination with the system, inclusion of the inverse shaper dynamics in the closed loop does not bring critical closed-loop stability concerns. In this chapter, next to addressing more complex multiple degrees of freedom (MDOF) systems, a robust control design method is proposed for the systeminverse shaper interconnection. It is based on minimizing the H-infinity norm of the closed loop weighted mixed sensitivity functions. Let us remark that it is a subsequent work to [106], where a static feedback controller was considered for the given task. In this chapter, a more complex dynamic feedback controller is considered providing us more flexibility in the loop shaping in order to increase robustness against external noise and model uncertainties. Besides, a conceptually different method based on minimizing the spectral abscissa was

PRELIMINARIES

101

used in [106], which can easily handle the requirement on the prompt actions of the controller, but lacks the ability to predefine a requirement on a robust controller setting. The mixed-sensitivity control design approach was proposed in [171]. Further reading can be found, e.g., [29, 34], for an approach based on Riccatti equation and LMIs. A direct application to the problem under consideration is not possible for two reasons. First, due to the presence of the shaper, the closed-loop system is an infinite-dimensional time-delay system. Second, as the design is aimed on finite-dimensional controllers, which are easy to implement, the design problem can be seen as a reduced order control design problem, known to be difficult to solve. In view of these properties, the algorithm and software is employed for fixed-order H-infinity synthesis proposed in [38] and [40]. Note that this chapter is an extension of the research paper [104], where the doctoral candidate is the leading author. Since the algorithm of [38] relies on an asymptotically stabilizing initial controller, our design consists of two stages. First, a stabilizing controller is designed by minimizing the spectral abscissa, as proposed in [75, 145]. Once the stabilizing controller is obtained, the H-infinity norm criterion, stemming from the mixed-sensitivity problem formulation, is optimized. As a by product of the overall control design procedure illustrates the power of the delay differential algebraic equations (DDAE) modeling framework: the system components (uncontrolled system, inverse shaper, controller), connections between components, and the weights for the H-infinity design, can be directly combined in one system of DDAEs, which are supported by both the algorithm of [75] and the one of [38]. The chapter is structured as follows. In Sec. 5.2, the architecture with the inverse shaper in feedback together with implementation aspects, such as extra eigenvalues in the spectra, is examined. Sec. 5.3 is dedicated to a rigorous examination of the design of a fixed-order controller for the system with an inverse shaper, where difficulties arise from the infinite spectrum of the closed loop system. The design of the controller is divided into two subsections, where the first part suggest stabilization of the closed loop with the inverse shaper and the second part shows a procedure for the design of the H-infinity based fixed-order controller for the infinite order system. Sec. 5.4 demonstrates a case study example and the chapter concludes with final remarks in Sec. 5.5.

5.2

Preliminaries

A closed loop architecture with the inverse shaper proposed in [157], is shown in Fig. 5.1. For the purposes of this chapter, the notation different from Fig. 1.3. Consider a nominal system with an input u and the output x has the strictly proper transfer function P0 (s), and Flexible structure features the transfer function F (s), with y being the system output and y = d + x being its input, d

102


is an unmeasurable disturbance. The feedback position controller for Nominal system has a transfer function K(s). The inverse shaper transfer function is 1 given by S(s) . d w

m

u

x

+ + P0 K − Controller Nominal system

y

yF F Flexible structure

v

S −1 Inverse shaper Figure 5.1: Inverse shaper in the feedback architecture The purpose of including the inverse shaper to the feedback loop is to project its filtering (mode compensation) properties, to the transfer functions of the closed loop system Tyw (s) =

K(s)P0 (s)S(s) K(s)P0 (s) 1 F (s) = S(s) + K(s)P (s) F (s), 1 + K(s)P0 (s) S(s) 0

Tyd (s) =

1 S(s) 1 F (s) = S(s) + K(s)P (s) F (s). 1 + K(s)P0 (s) S(s) 0

(5.1) (5.2)

If the shaper S(s) is designed to compensate poles r1,2 of the flexible structure by its zero s1,2 = r1,2 , this zero-pole compensation will project to the transfer functions (5.1) and (5.2), because S(s) appears in their numerators. A general description of a delay based input shaper is as follows, Z ϑ v(t) = aw(t) + (1 − a) w(t − µ)dh(µ), (5.3) 0

where w and v are the shaper input and output, respectively, a ∈ R+ , a < 1 is the gain parameter, and the distribution of the delays is prescribed by the non-decreasing function h(µ), with length ϑ. Considering that overall delay consists of a series of lumped and equally distributed delay of the lengths τ , delayed-zero-vibration(DZV) shaper is obtained (see [161] for more details). The transfer function of the shaper is S(s) = a + (1 − a)

1 − e−sT −sτ e . Ts

(5.4)

The time domain interpretation of the equation can be seen in Fig. 5.2 (top). The bottom of Fig. 5.2 shows the frequency response of the shaper, where

PRELIMINARIES

103

the frequency to suppress (ω0 = 0.1rad/s) is observable as drop in the output amplitude in the ω0 neighbourhood. Thus, the input shaper performs the task of a notch filter. Its key advantage compared to this classical filtration technique is the monotonously increasing step response (or nonnegative impulse response). This results in an energetically convenient solution to perform the task. Performing the same task with a classical filter would be very difficult and has many limitations as described in [101]. The shaper is parameterized by the lengths of the distributed delay T , the lumped delay τ and by the gain a, as proposed in [163].

v(t)

1

a 0 0

τ

τ+T

t

amplitude (dB)

0

-10

-20

-30

-40 0.001

0.01

0.1=ω 0

1

10

100

frequcncy (rad/s)

Figure 5.2: Top: step response of the shaper (5.4); Bottom: magnitude frequency response of the shaper (5.4) by

From an application point of view, the inversion of the shaper is described v(t) =

1 a

(xn (t) − (1 − a)r(t)) ,

with the input x2 and the output v, where Z 1 t−τ v(η)dη. r(t) = τ t−(T +τ )

(5.5)

(5.6)

104


In our controller design procedure is implemented dynamically in the form r(t) ˙ =

1 (v(t − τ ) − v(t − (T + τ ))). T

(5.7)

This transformation results in additional dynamics, characterized by the introduction of an eigenvalue at zero.

5.3 5.3.1

Synthesis of the problem Problem statement

Consider system P0 , shown in Fig. 5.1, which can be described by x(t) ˙ = Ax(t) + Bu(t), P0 y(t) = Cx(t) + Du(t),

(5.8)

where x(t) ∈ Rn is the vector of system states, u(t) ∈ R being the control input and capital letters are real-valued matrices of appropriate dimensions. Apparently (see Fig. 5.1), the subsystem F is considered as decoupled from P0 . The additional information on the system F is its oscillatory mode given by the natural frequency ω0 and the damping ratio p ζ, determining the complex conjugate couple of poles r1,2 = −ζω0 − jω0 1 − ζ 2 which will be used for the shaper design. The inverse shaper within the feedback will filter the given modes from both the reference w and disturbances d. However, the inverse shaper introduces time delays into the closed loop system and therefore the system becomes infinite dimensional. If the system has undesired dynamics or is even not stable a controller is an essential part of the scheme although design of the controller is more challenging. First, a fixed-order controller is proposed. Secondly, the controller is designed with a two step approach, consisting of stabilization and robust H-infinity design for time-delay system.

5.3.2

Control objectives and associated optimization problems

The original system (5.8) without feedback is of finite dimension. By including the inverse shaper (5.3)-(5.7) in the feedback, the system becomes infinite dimensional and, therefore, infinitely many eigenvalues are introduced into the system. Considering that the nominal system (5.8) can be unstable, a fixed-order dynamic feedback controller is proposed in order to stabilize and optimize the system in the form x˙ K (t) = AK xK (t) + BK m(t), K: (5.9) u(t) = CK xK (t) + DK m(t),

SYNTHESIS OF THE PROBLEM

105

where xK (t) ∈ Rn is the vector of the controller states, m(t) ∈ R being the controller input and the capital letters are real-valued matrices of appropriate dimensions. As mentioned above, the design of the controller has multiple steps regarding to different control objectives. Compensation of the flexible mode Compensation of the oscillatory modes of the flexible structure is automatically achieved by interconnection of the inverse shaper in the feedback with properly tuned parameters. Stabilization The H-infinity optimization requires requires initial controller to be stabilizing. Stabilization is achieved by minimizing the spectral abscissa of the closed-loop, leading to the optimization problem min c(K),

(5.10)

K

where the spectral abscissa is defined as c(K) := sup {

Spectral methods in vibration suppression control

Spectral methods in vibration suppression control

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