一新適應性方法之重複控制器設計

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Department of Mechanical Engineering College of Engineering

National Taiwan University Doctoral Dissertation

一新適應性方法之重複控制器設計 A New Adaptive Approach to the

Repetitive Controller Design

鍾智賢

Chih-Hsien Chung

指導教授：陳明新博士 Advisor: Min-Shin Chen, Ph.D.

中華民國 99 年 7 月

July, 2010

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誌謝

光陰似箭，時光荏苒，六年來一邊工作一邊進修博士學位的生涯已告一段落，這期間苦樂酸甜參半其中。在完成論文的這一刻，內心充滿喜悅與感恩，原來，辛苦得來的果實是如此甜美。

首先，感謝我的恩師陳明新教授，於研究期間悉心指導。質樸的陳老師總是在我研究中遇到困境時，耐心的與學生討論解決問題。老師展現謙謙君子及學者風範，深深銘記在我們的心中，感謝恩師在學習及論文研究過程中的悉心指正，

讓論文的研究更周延。感謝口試委員中央大學黃衍任教授、本校傅立成教授、顏家鈺教授與王富正教授對論文的剴切斧正的建議，使得本論文更臻完備。

回顧這六年來，慶幸自己何其幸運，遇到涵養豐富的師長及和一群可愛的學弟們。非常感謝溫文儒雅的王富正教授提供研究室給我學習，讓我有一個優質研究的環境。謝謝系統整合控制實驗室學弟們，這六年來的幫忙，特別是偉儁及敏峯在你們前後共四年間的熱心幫忙外，尚提供不少動漫，使得用餐心情特別愉悅。謝謝博班同袍祈澈及學弟俊賢在我需要幫忙時，總能伸出溫暖的雙手，解決了燃眉之急。謝謝學弟奕良及逸哲於口試時的鼎力相助。感謝台科大碩班學長錟鋒、學弟銘志及柏璋於求學期間的協助與關懷。非常感謝台科大黃緒哲教授及碩班恩師黃安橋教授，多年來對學生的諸多協助與鼓勵。

感謝大漢技術院同事們多年來的幫忙，特別是經常被我麻煩做期刊文法校正工作的陳宗輝博士，還有許文政、蘇信政、鄭芳松、粘世智及室友陳建昌博士這段時間的幫忙及鼓勵，我亦要特別感謝邱福澤老師多年的照顧及溫馨接送。

感謝我的母親及家人的鼓勵與支持，讓我有繼續前進的勇氣。謝謝我的太太玉萍有你的體諒與陪伴讓我可以無後顧之憂的順利完成博士學位。謝謝小寶貝東序，因為有你，讓爹地更有信心與毅力順利完成博士學位。最後，僅將我的成就與喜悅，與各位一同分享。

99 年 8 月 22 日

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本論文提出一新的適應性前饋控制設計(AFC)技術來達成週期性追蹤和／或週期性干擾消除。與傳統的AFC相比較，主要的不同點是運用重新參數化的回歸式於適應性機制。這控制架構是包含一像干擾觀測器(DOB-like)的架構來產生一經系統濾波後的週期性訊號，和一訊號辨識器來對此訊號進行參數識別。因此，AFC 控制系統的穩定問題不再受到受控體架構的影響。本論文更進一步利用此重新參數化的技術於新的干擾估測器設計，進而獲得此新AFC設計方法的一重複控制的通式。

不同於以往的適應性設計，本方法有以下的優點：第一點是它的增益可以任意選擇而不影響控制系統的穩定度。第二點是透過重新參數化過程，它不需反轉系統模式，可以適用於極小相與非極小相系統。第三點是它顯示週期性干擾輸入點與控制設計無關。第四點是本方法可運於線性系統有存在不確定性。最後一點是此新AFC控制與基於干擾觀測器的控制有一等價關係的表示，進而提供可應用彼此相關領域之控制知識的機會，因此，AFC的適應性增益可以藉由任何線性控制方法或適應性方法更有效率的選擇，如特徵值安置，卡曼濾波器和最小平方法等。

另外，在控制設計方面，此控制技術提供工程師一非常友善及直覺的設計。

關鍵字：重複控制；適應性前饋消除；干擾觀測器；週期性干擾消除；內模式控制；週期性追蹤控制。

I

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ABSTRACT

This dissertation proposes a new technique of adaptive feedforward control (AFC) that achieves periodic tracking and/or periodic disturbance rejection. The key difference compared with conventional AFC is a new re-parameterization regression form employed in adaptive mechanism. This new control structure is a combination of the disturbance-observer-like (DOB-like) structure and the disturbance identifier, where the DOB-like output generates a periodic disturbance which is filtered by the plant model, and the disturbance identifier is to identify the unknown parameter of the filtered disturbance. Consequently, the stabilizability problem is no longer subject to the plant structure. Utilizing the re-parameterization technique, the dissertation further proposes a general form of AFC control using repetitive control.

The proposed new control has several advantages over previous designs. First, its adaptation gain can be arbitrarily chosen without upsetting the system stability. Second, through re-parameterization process, the adaptive algorithm can be applied to minimum phase as well as non-minimum phase systems without using any approximations. Third, it is shown that the desired AFC control is independent of where the disturbance enters the system. Fourth the proposed control is proved to be robust with respect to system uncertainties. Finally and most importantly, the equivalent interpretation between the disturbance observer based control and the new AFC control provides an opportunity to apply knowledge to each other field. Therefore, AFC’s adaptation gain can be efficiently chosen by any linear control methods or adaptive algorithms, such as eigenvalue assignment, Kalman filter, least-squares algorithm and so on. Besides, the control technique provides engineers with very friendly and intuitive design on the control performance.

keywords : repetitive control; adaptive feedforward cancellation; disturbance observer; periodic disturbance rejection; internal model control; periodic tracking control.

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Abstract I

List of Figures V

List of Tables IX

1 Introduction 1

1.1 Motivation . . . 1

1.2 Literature Survey . . . 2

1.3 Overview of the Dissertation . . . 6

2 Review of Internal Model Based Repetitive Control 9 2.1 Periodic Signal . . . 10

2.2 Internal Model Principle . . . 11

2.3 Time Delay Repetitive Control . . . 12

2.4 Plug-in Time Delay Repetitive Control . . . 15

3 AFC Control 19 3.1 Review of Adaptive Algorithm . . . 21

3.2 Problem Formulation . . . 22

3.3 Linear Regression Form. . . 23

3.4 Review of Adaptive Feedforward Control . . . 29

3.5 New AFC Design for Open-Loop Stable System . . . 35 III

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IV Contents

3.5.1 Gradient Based AFC . . . 35

3.5.2 LS Based AFC . . . 41

3.6 New AFC Design for Open-Loop Unstable System . . . 46

3.7 Robustness Analysis . . . 50

3.8 Adaptive Disturbance Estimation . . . 62

4 Disturbance Observer Based Control 71 4.1 Review of Disturbance Observer Based Control . . . 72

4.1.1 Disturbance Observer Based Control . . . 72

4.1.2 Unknown Input Disturbance Observer Based Control . . . 78

4.2 New Disturbance Observer Based Control . . . 82

4.3 DOB-AFC design . . . 84

4.4 Robustness Analysis . . . 88

5 Conclusions 91

Bibliography 92

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2.1 IMP structure for the periodic reference with single frequency . . . . 12

2.2 IMP structure for the periodic reference with multiple frequencies . . 12

2.3 Periodic signal generator . . . 13

2.4 Time delay repetitive control system . . . 13

2.5 A system equivalent to Figure 2.4 . . . 14

2.6 Modified time delay repetitive control system . . . 14

2.7 Structure of plug-in repetitive control system . . . 15

2.8 Plug-in time delay repetitive control system . . . 16

3.1 System . . . 24

3.3 Control system with Plug-in AFC controller . . . 29

3.4 AFC control System . . . 29

3.5 AFC control . . . 31

3.6 IMP control which is equivalent to Figure 3.5 . . . 31

3.7 Trajectory of the output error e(t). . . 34

3.8 Time history of the disturbance d₁(t) and the estimate ˆd₁(t) . . . 34

3.9 AFC control system. . . 36

3.10 Time history of periodic disturbance d(t) . . . 40

3.11 Trajectory of the reference signal r(t) and the output y(t). . . 40 V

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VI List of Figures

3.12 Trajectory of the control input u(t) . . . 41

3.13 Output error e(t) . . . 43

3.14 Trajectory of the disturbance d₁(t) and estimated disturbance ˆd₁(t) . 44 3.15 Trajectory of the parameter estimation ˆθ(t) . . . 44

3.16 Output error e(t) . . . 45

3.17 Trajectory of the disturbance d₁(t) and estimated disturbance ˆd₁(t) . 46 3.18 Trajectory of the reference signal r(t) and the output y(t). . . 49

3.19 Trajectory of the control input u(t) . . . 49

3.20 AFC control system under model with uncertainty. . . 52

3.21 Feedback connection . . . 54

3.22 LTI control system which is equivalent to Figure 3.20 . . . 57

3.23 Root-locus of ΓF(s) . . . 59

3.24 Trajectory of r(t) and y(t) under model with uncertainty . . . 61

3.25 Adaptive Disturbance Estimation . . . 64

3.26 Time history of the disturbance d(t). . . 65

3.27 Trajectory of the disturbance error |d(t) − ˆd(t)| . . . 66

3.28 Trajectory of the state x(t). . . 68

3.29 Trajectory of the disturbance d(t) and the estimated ˆd(t) . . . 68

3.30 Trajectory of the norm of state estimation error kx(t) − ˆx(t)k . . . . 69

4.1 Disturbance observer based control . . . 73

4.2 Trajectory of the the output error e(t) on Case 1 . . . 76

4.3 Time history of the disturbance d1 and the estimate ˆd1 on Case 1 . . 76

4.4 Trajectory of the the output error e(t) on Case 2 . . . 77

4.5 Time history of the disturbance d₁ and the estimate ˆd₁ on Case 2 . . 77

4.6 Unknown input disturbance observer . . . 79

4.7 Trajectory of the the output error e(t) . . . 81

4.8 Trajectory of the periodic disturbance d₁ . . . 81

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4.9 DOB control system . . . 84

4.10 Trajectory of the the output error e(t) . . . 87

4.11 Trajectory of the periodic disturbance d₁ . . . 87

4.12 DOB control system under model with uncertainty . . . 89

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3.1 Closed-loop poles of ΓF(s) versus adaptation gain γ . . . 60

IX

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Introduction

1.1 Motivation

In many industrial applications, the control system is required to track or reject periodic exogenous signals (desired reference output or disturbance). Examples include periodic motion of robot manipulators [1], repeatable runout in disk drive [2], torque ripples in harmonic drives [3], periodic force disturbance in metal cutting, and so on. Clearly, these disturbances degrade the system performance. Hence, the basic requirements in control systems are that they have the ability to regulate the controlled variables to reference commands without a steady-state error against unknown and un-measurable disturbance inputs. Such control that can successfully drive the system to track or reject periodic signals is called the repetitive control.

Besides, it may be desirable to estimate the unknown periodic disturbances acting on the system. In some cases, the purpose of disturbance estimation is to monitor the performance of systems for decision making. For examples, in the manufacturing processing, one may wish to estimate the cutting torque in drilling process [4] and the cutting force in CNC machine centers [5].

Although there are several approaches to cancel periodic disturbance, these approaches may cause original closed-loop system stability affected or these designs

1

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2 1.2. Literature Survey

are excessive complexity. Therefore one of the objectives of the dissertation is to implement a simple plug-in type repetitive controller to cancel exogenous periodic signal. The other objective is to construct an adaptive disturbance/state observer to monitor the system performance for decision making. The last purpose is to construct a robust update law for repetitive control.

1.2 Literature Survey

Since many control systems are often subject to the disturbance, one of the fundamental research topics in control theory is to study the problem of disturbance rejection. The disturbances are mostly divided into un-deterministic and deterministic disturbance. In the case of un-deterministic disturbance, the robust disturbance attenuation control, such as H∞ control [6] and variable structure control [7] which are high-gain controllers, has been investigated by many researchers. Although those are common methods for the improved performance of control systems, such high- gain controllers may not be applied in a mechanical system due to the reason of mechanical resonance. In contrast, from the 70s to the present, there have been many researchers who proposed various approaches to realize effective disturbance suppression without using high gains [8]-[13]. One of these methods employs that disturbance is estimated using an observer and cancelled out, and then the control design is reduced to nominal feedback control which generates just minimal control based on disturbance free assumption. Thus the disturbance rejection problem is transformed to the disturbance estimation design.

A recent survey paper on the disturbance estimation for linear systems can be found in [14], and extension to the disturbance estimation for nonlinear systems can be found in [15] and [16]. One approach for the disturbance estimation is the use of disturbance observer [17, 18], which does not need the dynamic model of the unknown disturbance. The disturbance observer estimates the equivalent

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disturbance which is the difference between the actual plant output and the output of the nominal model. The estimate is then inversely added at the input of the plant, so as to compensate for the disturbance effect on the output. Despite the simple structure, disturbance observer based (DOB) control as an effective add-on controller [19] is successful in enhancing disturbance attenuation capability. However, this approach relies on inverse the system dynamics, and hence can not be applied to non- minimum-phase systems (systems with unstable zeros). Even for minimum-phase systems, the obtained disturbance estimate may not asymptotically converge to true disturbance due to a Q-filter in the estimation process. Besides, in some systems, such as disk drive servo, the rotational speed is usually required as increasingly as possible for improving data transfer rate, so does the frequency of the periodic disturbance, which leads to a high loop gain in the track-following servo. However, such a design may not be feasible since the increase of the Q-filter cut-off frequency may cause an undesirable increase of the control bandwidth, which is established by a feedback controller.

The second approach for disturbance estimation applied to the unknown disturbance is generated by a known dynamic model. In this dissertation one considers the problem of rejecting periodic disturbances, whose magnitude and phase are unknown but frequency is known. In this case, the periodic disturbance model is augmented with the system model to form an expanded system. A Luenberger observer is then constructed to estimate not only the system state, but also the state of disturbance model. The disturbance estimated method was called unknown input disturbance observer [20] or Kalman disturbance observer [21]. Consequently, by using the reconstructed disturbance injecting into the plant input, disturbance rejection is accomplished. Note that, since the disturbance observer constructed by augmented system is only used to estimate the actual disturbance acting on the system, it does not control the plant. Therefore, for achieving the closed-loop system stability and performance, a normal feedback controller is still required.

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4 1.2. Literature Survey

In addition to the above methods using the disturbance estimation, repetitive control (RC) is a specialized control strategy designed for tracking a specific periodic command or rejecting a periodic disturbance. A recent survey paper on repetitive control designs can be found in [22] and [23]. These designs are roughly classified as being either internal model base or external model base. The internal model based repetitive control design is originally proposed in [24], which is based on the internal modelling principle [25]. In [24], a time delay internal model is placed inside the nominal stable feedback loop to guarantee asymptotic tracking or rejecting of the periodic signal. However, this approach may alter original closed-loop system stability and performance. Hence, it is often realized in a plug-in manner [22, 26].

The advantages of the internal model based repetitive control are that convergence is very rapid and that the controller is linear, making analysis easy. However, the internal model introduces an infinite number of open-loop poles on the stability boundary; making stabilization of the overall system difficult [27]. As a result, Hara et al. in [28] proposed a low-pass filter included in the repetitive controller to ensure closed-loop stability, that is Q-filter. However, it makes exact internal model lost and the system performance at the high frequency harmonics be sacrificed [29].

Another disadvantage of this approach is that robustness to noise and un-modelled dynamics is impaired by the time delay internal model [22]. Moon et al. in [30]

proposed another repetitive controller design method on Q-filter, which is based on Nyquist plot technique, for the system with un-model dynamics. However, even under the ideal case, it can not reject the periodic disturbances asymptotically.

When the disturbance frequencies are unknown, adaptive internal model is often used for disturbance rejection [31]-[34].

The other approach for repetitive control designs is the basis function approach, or often called adaptive feedforward cancellation (AFC) control, being a main method in the external model based repetitive control design. With this approach, the periodic exogenous signal is modelled as a linear combination of finite or

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infinite basis functions with unknown coefficients [22] [35]. An adaptive algorithm is proposed to estimate these unknown coefficients, and a feedforward control that cancels the disturbance efforts is then constructed [36][37]. The adaptive approach may be superior to the disturbance model based approach when the frequency can not be obtained but the angle can be measured by the sensor, or injected signal need to be disconnected temporarily. However, stability of the adaptive system is ensured only if the system is SPR (strictly positive real). When the system is not SPR, the adaptation gain must be constrained to be small in order to maintain stability. The equivalence between the AFC and the internal model based approach is established in [38]. A modified adaptive algorithm with an extra phase advance is proposed in [38][39] to expedite the algorithm’s convergence. In [40], Ariyur and Krsti´c start with the sensitivity method but arrive at the same scheme. However, the adaptation gain is still constrained by the stability requirement. In [3], a different adaptive algorithm is proposed, whose adaptation gain can be arbitrarily chosen without disturbing the system stability. However, this adaptive algorithm is based on inversion of the system transfer function; hence, they can be applied to minimum-phase systems only.

When the system is in a non-minimum phase, an approximation algorithm based on the zero-phase-error-tracking design may be used [22]. The other AFC approach called frequency adaptive control technique (FACT), which utilizes a collection of frequency sampling filters (FSF) to obtain the magnitude of individual frequency components of the truncated periodic signal and uses these individual components to do adaptive update again, is proposed in [2]. The feature of FACT design is able to cancel any unwanted harmonic signals without influencing the uncompensated ones but needs more computational cost and carefully chooses adaptation gain for the system stability.

The stability conditions for the general AFC controller design is analyzed in [41]

depending on the available adaptation method used in AFC design. Bayard uses LTI representations of adaptive systems with sinusoidal regressors to do stability

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6 1.3. Overview of the Dissertation

analysis. Under the plant model known exactly, he proved that adaptive algorithm using augmented error signal is completely phase-stabilized. In [42], Guo further shows that those AFC control algorithms on the time-varying frequency case are equivalent to linear time-varying compensators which is implemented by the IMP on the state space. It provides an opportunity to apply knowledge obtained from either adaptive control or linear control to the other field.

1.3 Overview of the Dissertation

Even though the repetitive control approach is very complete already, but AFC is preferred over other schemes because the AFC controller can easily freeze the parameter update when the output signal is not available during certain periods of time, and can be driven by the measuring frequency, making the control response more robust to variation in frequency. Furthermore, the adaptive implementation can adopt angular measurements directly. However, under non-minimum phase system, arbitrary update gain and controllable convergence rate, the current researches in AFC control have not obtained effective solution yet. In view of the tradeoff between system stability and disturbance rejection in the previous controller design, the goal of this dissertation is to propose a new AFC design technique to cancel exogenous periodic signal without altering the closed-loop stability. The key difference compared with conventional AFC is a new linear regression form employed in adaptive mechanism. This new control structure is similar to a typical DOB control, but the proposed AFC control uses a disturbance identifier instead of the low-pass filter Q(s) in DOB control and does not need inverse plant model to obtain disturbance estimate. Consequently, the stabilizability problem is no longer subject to the plant structure. The proposed AFC control is just one of the special cases of [41] which called augmented error algorithm. Although, both control structures are the same, the proposed AFC control relies on an adaptive identifier through

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re-parameterization process to prove that AFC control system is nominally stable.

Utilizing the re-parameterization technique, a general form of AFC control using repetitive control is proposed in advance.

The resultant new control has several advantages over previous designs. First, since the adaptation gain of the proposed AFC is independent of the state feedback gain, under exactly known plant model it can be arbitrarily chosen without affecting the system stability. It means that the proposed AFC adds into the nominal closed- loop system without affecting the performance. Second, through re-parameterization process, the adaptive algorithm can be applied to minimum phase as well as non- minimum phase systems without using any approximations. Third, the new design is only one estimation algorithm while previous indirect schemes need two estimation algorithms [2]. Fourth, this dissertation shows that the desired adaptive control remains the same no matter where the disturbance enters the system. This justifies many previous AFC designs in the literature in which the disturbance is ”assumed”

to come into the system at the input point even though in real situations it may not be the case. Finally, for promoting that the repetitive control performance has more design freedom on adaptive update law, we further propose DOB-AFC that is a general AFC form. The interpretation of AFC in terms of disturbance observer design can be implemented by any linear control methods, such as eigenvalue assignment, Kalman filter, least-squares algorithm and so on. Therefore, the control technique provides engineers with very friendly and intuitive design. Certainly, when the system model can not be exactly obtained, the control structure using LMI method will provide more robust performance.

A series of studies on disturbance rejection methods of control systems is orga- nized as follows. Chapter 2 reviews internal model based repetitive control, which includes dynamics model of multiple frequencies disturbance and time-delay model.

Chapter 3 firstly reviews the adaptive algorithm, formulates the problem, constructs a linear regression form through re-parameterization process, and introduces con-

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8 1.3. Overview of the Dissertation

ventional adaptive feedforward control. And then one proposes the new AFC design and the adaptive disturbance estimation. In Chapter 4, one firstly reviews disturbance observer based control, which includes disturbance observer and unknown input disturbance observer for un-deterministic disturbance and deterministic disturbance respectively. And then, based on the linear regression form in Chapter 3, we propose a new disturbance observer design, which is different from previous designed, and makes use of the equivalence between the new AFC and disturbance observer to design a general form of AFC, called DOB-AFC. Chapter 5 gives the concluding remarks.

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Review of Internal Model Based Repetitive Control

In many industrial applications [1]-[5], the control system is required to track or reject exogenous periodic signals. When the system is subjected to periodic signal input, it is well known that the repetitive controller can work well. The conventional RC is often regarded as a simple learning control because the control input is calculated using the result of preceding periods to improve the current performance.

One closely related study of repetitive control is iterative learning control (ILC) [44]

which is achieved by iteration of the control action within finite duration. The difference between RC and ILC is the setting of the initial conditions for each trial. In the ILC, the same initial condition is assumed in every trial. Hence, the iterative action is discrete and it is enough to assure not only the stability but the convergence of the error. In the repetitive control, the repetitive process is continuously because the initial conditions are set to the final conditions of the previous trial. The difference in initial-condition resetting leads to different analysis techniques and results [45].

In this chapter, an internal model based repetitive control which is a typical one will be introduced. In Section 2.1, one firstly gives a brief review of a periodic signal for easy description on the latter sections and chapters. Section 2.2 reviews an internal model principle which states that a generating system model of the ex-

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10 2.1. Periodic Signal

ogenous signal must be included in the feedback system in order to achieve perfect tracking at the steady state. Based on this internal model principle, the time delay internal model based repetitive control which includes all frequency modes of periodic signal in the closed-loop system is then introduced in Section 2.3. Finally, for keeping original feedback control performance and stability, a plug-in time delay repetitive control was presented In Section 2.4.

2.1 Periodic Signal

The objective of this dissertation is to construct a control that can reject or track an unknown periodic signal. Therefore, in the thesis, the periodic signal is assumed to satisfy the following assumptions.

Assumption A2.1. d(t) is a periodic signal (in the thesis, it is taken as disturbance) that is d(t) = d(t + T ) for some known period T .

Assumption A2.2. d(t) is continuous and has a piecewise continuous derivative.

The periodic signal has a Fourier series representation d(t) = θ_0,c+

∞

X

i=1

θ_i,ccos(ω_it) +

∞

X

i=1

θ_i,ssin(ω_it), (2.1) where ω_i = i · 2π/T is the harmonic frequency in which 2π/T is the fundamental frequency, and θ_0,c, θ_i,c and θ_i,s are constant coefficients. In practical applications, one uses a (2N + 1)-term finite series approximation for the periodic signal,

d_N(t) = θ_0,c+

N

X

i=1

θ_i,ccos(ω_it) +

N

X

i=1

θ_i,ssin(ω_it) = φ^T(t)θ_d, (2.2) where the regressor φ(t) is a bounded vector

φ(t) =^h 1 cos(ω₁t) sin(ω₁t) . . . cos(ω_Nt) sin(ω_Nt) ⁱ^T ∈ R^{2N +1}, (2.3) and θ_d contains unknown parameters

θ_d =^h θ_0,c θ_1,c θ_1,s θ_2,c θ_2,s . . . θ_N,c θ_N,s ⁱ^T ∈ R^{2N +1}. (2.4)

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The theorem below suggests that under certain conditions, the finite series approximation is a ”good” approximation of the periodic signal as long as N is large enough.

Theorem 2.1 [43] : Under Assumption A2.2, the finite series approximation d_N(t) in (2.2) converges uniformly to the true signal d(t) in (2.1) as N approaches infinity.

Because of Theorem 2.1, this dissertation will make no difference between d_N(t) and d(t) as long as N is sufficiently large. In fact, the low-pass properties of physical systems, at most a handful of harmonics needs to be considered in general. Hence, in the remainder of this thesis, one will write

d(t) = φ^T(t)θ_d. (2.5)

2.2 Internal Model Principle

After reviewing the property of the periodic signal, one goes back to the internal model principle (IMP) design. The IMP was initially proposed by Francis and Won- ham [25]. It means that the controlled output tracks a class of reference commands without a steady-state error if the generator for the references is included in the stable closed-loop system. Figure 2.1 shows the basic control structure of IMP, where P (s) is a linear time-invariant plant, C(s) is the controller, y(t) is a controlled output, e(t) is a tracking error, and r(t) is a periodic reference signal which is expressed as the following form

r(t) = θ_1,ccos(ω₁t) + θ_1,ssin(ω₁t), (2.6) in which ω₁is a known frequency and θ_1,c and θ_1,sare unknown constant coefficients.

The compensator including an internal model 1/(s²+ ω₁²) is to provide a closed-loop transmission zero to cancel unstable poles of the periodic input so that it achieves perfect tracking. The design problem is to choose the remaining transfer function C(s) so that the closed-loop transfer function is stable and has desire input-output properties.

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12 2.3. Time Delay Repetitive Control

e y

2 2

1

s Z C s( ) u

r P s( )

Figure 2.1: IMP structure for the periodic reference with single frequency

( ) y P s

2 2

n( )

n

C s s Z

u

1

2 2

1

( ) C s

s Z ¹ u

un

r e

Figure 2.2: IMP structure for the periodic reference with multiple frequencies

The advantages of this type of controller are that it is linear, making analysis easier, and that convergence is very rapid. When the periodic exogenous signals is the sum of two or more sinusoids, the method is easily extended to the cases as shown in Figure 2.2. However, the stability problem becomes more and more difficult as poles are added on the jw-axis.

2.3 Time Delay Repetitive Control

In this section, our objective is based on the IMP to obtain a repetitive control with minimal system scheme that generates all periodic signals of period T . Based on the reason, Inoue et al. [24] originally employed a time delay system as shown in Figure 2.3 to serve as a periodic signal generator. It is readily seen that the delay element

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stores the function of the past one period and the system has infinitely many poles on the imaginary axis at jkω, where ω = 2π/T . It is therefore expected from the IMP that the asymptotic tracking property for exogenous periodic signals may be achieved by implementing the model 1/(1 − e^−sT) into the closed-loop system. A controller including this model is said to be a repetitive controller and a system with such a controller is called a repetitive control system [46] as shown in Figure 2.4. In Figure 2.4, the feedback controller C(s) is designed to stabilize the plant and has desire input-output properties.

e

sT

Figure 2.3: Periodic signal generator

C s y

r e

( )

sT P s

e

⁻

Figure 2.4: Time delay repetitive control system

Therefore, the transfer function from r to e is W_er(s) = 1 − e^−sT

1 − (1 − P (s)C(s)) e^−sT. (2.7) Consequently, s = j2kπ/T becomes the transmission zeros of W_er(s). Therefore, the system asymptotically tracks the periodic signal of a fixed period T if the closed- loop system is stable. Let us start with some simple stability analysis. An easy loop transformation converts Figure2.4 to Figure2.5[29]. Using the small gain theorem, the converted system is L₂ input/output stable if

k1 − P (s)C(s)k∞< 1. (2.8)

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14 2.3. Time Delay Repetitive Control

1−P s C s( ) ( )

e

⁻sT

(

¹⁻^e⁻^sT

)

^{r t}^{( )} ^{e t}^{( )}

Figure 2.5: A system equivalent to Figure 2.4

Although the condition is only a sufficient condition, it is actually very close to necessity since the delay e^−sT introduces a large amount of phase shift especially in the high frequency range. It is clearly seen that the above condition can never be satisfied for a strictly proper P (s)C(s). This restriction comes from the apparently unrealistic over specification of tracking in a very high frequency band. One way of handling this is to introduce a low-pass filter in front of the delay term, thereby replacing the delay element e^−sT by Q(s)e^−sT for some strictly proper stable ra- tional filter Q(s) [28]. This, named finite dimensional repetitive control, relaxes the tracking requirement in the high frequency range, thereby relaxing the stability condition. The modified repetitive control system is shown in Figure 2.6. Then the stability condition becomes

kQ(s) (1 − P (s)C(s)) k∞< 1. (2.9)

Clearly, the high frequency band condition is relaxed here compared with (2.8), and the above condition can be satisfied with strictly proper P (s)C(s). Although the stability robustness was improved, it was paid by the degradation of the steady-state tracking performance.

C s y

r e

( ) ( ) ^sT P s

Q s e⁻

Figure 2.6: Modified time delay repetitive control system

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The advantages of the time delay RC are that it is linear, making analysis easier, and that convergence is very rapid. The other advantage is that the repetitive compensator for any periodic signals is easily implemented by including the delay element. However, it alters the loop gain of the system and is not possible for selective harmonic cancellation. Besides, for the purpose of ensuring closed-loop stability, additional filtering is usually added to such schemes, but it sacrifices high frequency performance.

2.4 Plug-in Time Delay Repetitive Control

Normally, repetitive controller is realized in a plug-in manner, as shown in Figure 2.7. In Figure 2.7, the nominal controller is usually designed to stabilize the plant and reject a disturbance being across a broad frequency spectrum, and the repetitive controller is used to compensate periodic signals which have a known fundamental frequency.

r + y -

e

Repetitive Controller

++ v Nominal Controller

C(s)

Plant P(s)

Figure 2.7: Structure of plug-in repetitive control system

In this section, our purpose is to make a description of a plug-in time delay repetitive controller design. Based on the stability analysis on the time delay repetitive control which has been introduced in the previous section, we know that perfect tracking for the periodic signal including higher order harmonic signal is the unrealistic. Therefore, for making stability condition be relaxed, a Q filter scheme must

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16 2.4. Plug-in Time Delay Repetitive Control

be also considered in the plug-in manner. Under this way, the RC could be added directly into the existing closed-loop system since it did not influence internal stability and system performance very much. Such plug-in manner was presented as shown in Figure 2.8.

C(s) y

++

r + -

e P s( )

( ) ^sT Q s e

++ v

Figure 2.8: Plug-in time delay repetitive control system

According to the Figure2.8, the transfer function from r to e is W_er(s) = 1 − Q(s)e^−sT

1 + P (s)C(s) − Q(s)e^−sT. (2.10) Define a sensitivity function as

S(s) = 1

1 + P (s)C(s), (2.11)

where S(s) is a stable sensitivity function since the nominal closed-loop system is stable. Then (2.10) becomes as

W_er(s) =

1 − Q(s)e^−sTS(s)

1 − S(s)Q(s)e^−sT . (2.12)

Note that the above equation shows the transmission zeros of W_er(s) are no longer s = j2kπ/T , but in the lower frequency range, they are still very approximate.

It means that one must sacrifice the system performance at the high frequency harmonics in order to ensure closed-loop stability.

Let us start with some simple stability analysis. By an appropriate operating, Figure 2.8 can also be expressed equivalently as Figure 2.9. Therefore it exists a

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( ) ( ) S s Q s

e

sT

¹^{Q s e}^{( )} ^sT

^{S s r t}^{( ) ( )} ^{e t}^{( )}

Figure 2.9: A system equivalent to Figure 2.8 stabilizing repetitive control when the following condition is satisfied,

kQ(s)S(s)k∞< 1. (2.13)

Clearly, when (2.13) is satisfied, the plug-in repetitive control system is stable.

Compared with time delay repetitive control, the main advantage of the design method is that it employs a plug-in manner to reduce changing nominal closed-loop system stability.

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AFC Control

As everyone knows, an internal model based repetitive control may cause large phase shift to make original controlled system change into narrow bandwidth, and hence makes the original system have poor transient response and be stabilized difficultly.

An external model design where the model is adjusted adaptively to match the actual external signal and placed outside of the basic feedback loop, was then set up to take care of the problems. Among the external model based repetitive control designs, adaptive feedforward control (AFC) is a main method. In AFC design, it assumes that the unknown disturbance consists of the sum of sinusoids of known frequencies as the equation (2.1). The Fourier coefficients of the periodic disturbance with known frequency will be estimated adaptively by an adaptive algorithm in real- time. Since the output signal of repetitive controller is as being injected from outside of the feedback loop, it is more like feedforward and therefore is expected not to alter original closed-loop system stability and performance very much. This also implies that both repetitive controller and feedback controller designs are mutually independent.

Since synthesis of conventional repetitive control systems involves trade-off between robust stability and system performances, an optimized design method which can address the problem systematically is difficult to obtain. As a result, the goal

19

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20

of this chapter is to find an ideal control that achieves asymptotic tracking of the periodic reference r(t) regardless of the periodic disturbance d(t). Under this con- sideration, a modified AFC control will be presented. The key difference compared with conventional AFC is a new re-parameterization regression form employed in adaptive mechanism. Consequently, the stabilizability problem is no longer subject to the plant structure.

In the beginning of this chapter, one first gives a brief review of adaptive algorithm in order to describe it easily at the latter sections. In Section 3.2, the problem formulation in which we study will be discussed. In Section 3.3, a new re-parameterization regression form is proposed to employ in adaptive mechanism.

Section3.4 reviews conventional adaptive feedforward control (AFC). After reviewing conventional AFC, a new AFC design, which is based on the linear regression form of Section 3.3, is proposed to be independent of feedback control design. Fur- thermore, regarding that open-loop system is stable whether or not, the proposed control design takes different kinds of strategies in Section 3.5 and Section 3.6, respectively. The robustness of the proposed AFC with respect to un-modelled dynamics is studied in Section 3.7. Finally, Section 3.8 introduces an adaptive disturbance estimation algorithm for situations when it is desirable to track or monitor the unknown disturbance.

Note:

In Chapter 3 and 4, notations in the time domain and frequency domain may be mixed in one expression; for example, y(t) = W (s)u(t), where W (s) is the transfer function from u(t) to y(t).

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3.1 Review of Adaptive Algorithm

The adaptive algorithm is usually used in situations where one wishes to estimate an unknown constant vector θ ∈ R^p, which characterizes either a signal or a dynamic system. The first step of the estimation process is to obtain, through a re-parameterization procedure, a linear regression form in θ,

w(t) = φ^T(t)θ, (3.1)

where w(t) ∈ R is an available signal, φ(t) ∈ R^p is a known bounded regressor, and θ ∈ R^p is the unknown constant vector to be estimated. Let ˆθ(t) be an estimate of θ. Based on the above linear regression form, there are two different kinds of identifier structures. One is the gradient algorithm, the other is the least-squares (LS) algorithm. The gradient algorithm suggests the following update law for ˆθ(t),

˙ˆθ(t) = γφ(t)(w(t) − φ^T(t)ˆθ(t)), (3.2)

with a positive adaptation gain γ > 0, and an arbitrary initial guess ˆθ(0). Note that the regressor φ(t) in the linear regression form (3.1) needs to be uniformly bounded for the gradient algorithm (3.2). If one denotes the estimation error ˜θ(t) = θ − ˆθ(t), the update law (3.2) results in a linear error dynamics

θ(t) = −γ φ(t)φ˙˜ ^T(t)˜θ(t). (3.3)

Theorem 3.1 [47] : If the regressor vector φ(t) is persistently exciting in the sense that for some finite interval length δ, the following matrix is positive definite for all t > 0,

Z t+δ t

φ(τ )φ^T(τ )dτ > 0,

then the error dynamics (3.3) is exponentially stable, and ˆθ(t) in (3.2) converges to θ exponentially.

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22 3.2. Problem Formulation

Based on the linear regression form (3.1), the LS algorithm suggests the following update law for ˆθ(t),

˙ˆθ(t) = γΩ(t)φ(t)w(t) − φ^Tθ(t)ˆ , (3.4) Ω(t) = −γ˙ −ηΩ(t) + Ω(t)φ(t)φ^T(t)Ω(t), (3.5)

where the adaptation gain γ > 0 is the design parameter which can be arbitrary chosen, the matrix Ω ∈ R^pxp is called covariance matrix and acts in the update law of ˆθ as a time-varying directional adaptation gain, and η > 0 being a forgetting factor prevents that Ω becomes arbitrarily small in some directions. The initial condition of the matrix Ω must be Ω(0) > 0. From the textbook [47], one knows that it has the result which is similar to Theorem 3.1, that is, if the regressor φ(t) is persistently exciting, then the matrix Ω(t) in (3.5) is positive definite and ˆθ(t) in (3.4) converges to θ exponentially.

3.2 Problem Formulation

After reviewing the property of adaptive algorithm, one considers a linear time invariant (LTI) system subject to an unknown periodic disturbance:

˙x(t) = Ax(t) + Bu(t) + Gd(t), (3.6)

y(t) = Cx(t) + J d(t),

where x(t) ∈ Rⁿ is the state vector, u(t) ∈ R is the control input, y(t) ∈ R is the system output, d(t) ∈ R is an unknown periodic disturbance, and A ∈ R^n×n, B ∈ Rⁿ, G ∈ Rⁿ, C ∈ R^1×n, and J ∈ R are known constant matrices. Note that the formulation of this thesis allows the disturbance d(t) to enter the system at any place. The disturbance can enter the system at the input point (G = B and J = 0), at the output point (G = 0 and J 6= 0) [37], or at any place in the system. One contribution of this dissertation is that the proposed control law remains the same no

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matter where the disturbance comes into the system; in other words, the proposed control law is independent of the matrices G and J .

Our objective in repetitive control design is to construct a control u(t) for the system (3.6) that can drive the system output y(t) to asymptotically track a periodic reference r(t) despite the existence of unknown periodic disturbance d(t).

The disturbance d(t) and the reference r(t) are assumed to satisfy the following assumptions.

Assumption A3.1. d(t) and r(t) are of the same period; that is, d(t) = d(t + T ) and r(t) = r(t + T ) for some known period T .

Assumption A3.1 is only for easiness of presentation, the disturbance d(t) and reference r(t) are assumed to have the same period. The proposed control can be easily modified to allow d(t) and r(t) to have different periods.

Assumption A3.2. d(t) and r(t) are both continuous and have a piecewise continuous derivative.

Since the periodic signals d(t) and r(t) satisfy Assumption A3.1 and A3.2, the periodic disturbance thus has a Fourier series representation in (2.2), and the periodic reference signal also has has a finite series approximation

r(t) = φ^T(t)θr, θr ∈ R^{2N +1}, (3.7)

where the harmonic regressor φ(t) was defined in (2.3), and θ_r is the unknown constant vector to be estimated.

3.3 Linear Regression Form

Since this thesis will adopt the AFC approach to deal with the repetitive control design problem, one thus needs transform the state space system (3.6) to the following

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24 3.3. Linear Regression Form

y -

u +

+ d

1( ) P s

2( ) P s

r

+ e

Figure 3.1: System input-output description,

y(t) = P₁(s)u(t) + P₂(s)d(t), (3.8) where P₁(s) and P₂(s) are all stable transfer functions. The objective of repetitive control design is to construct a control u(t) for the system (3.8) that can drive the system output y(t) to asymptotically track a periodic reference r(t).

Thus, setting a tracking error as

e(t) = y(t) − r(t), (3.9)

and substituting (3.8) into (3.9), it becomes

e(t) = P₁(s)u(t) + P₂(s)d(t) − r(t). (3.10) Figure 3.1 shows the system block diagram. Re-arrange (3.10) into

e(t) − P₁(s)u(t) = P₂(s)d(t) − r(t). (3.11) It is important to note that on the right hand side of the above equation is still the periodic signal as a result of the periodic signal d(t) passing into stable filter P₂(s).

Therefore, we guess that it has the following representation

P2(s)d(t) − r(t) = P1(s)d1(t), (3.12) where d₁(t) is a periodic signal with the period T . To prove the existence of such d₁(t) in (3.12), one needs the following assumption.

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Assumption A3.3. P₁(jω_k) 6= 0 for k = 0, 1, . . . , N where ω_k = k · 2π/T , T is the period of both d(t) and r(t), and P1(s) is the stable transfer function.

Remark 3.1 : Assumption A3.3 requires that the transfer function P₁(s) has no system zero at s = jω_k. The reason is obvious: if P₁(s) has a system zero at s = jω_k, due to the zero gain of P₁(s) at s = jω_k, the equivalent periodic signal d₁ in (3.12) can not generate the k’th harmonic sinusoidal at the output point of P₁(s). This assumption can be waived if the disturbance enters the system at the input point (P₁(s) = P₂(s)), and there is no tracking mission (r(t) = 0). However, as long as there is a periodic tracking mission (r(t) 6= 0), or the disturbance enters the system ”not” at the input point, Assumption 3.3, which has been neglected by most previous literature, is necessary.

Lemma 3.2 : Given stable transfer function P₁(s) and P₂(s), and periodic reference r(t) in (3.7) and periodic disturbance d(t) in (2.5), if Assumption A3.3 holds, there exists an equivalent disturbance d₁(t) ,

d₁(t) = φ^T(t)θ, (3.13)

satisfying equation (3.12), with φ(t) ∈ R^{2N +1} as in (2.3), and θ ∈ R^{2N +1} some constant vector.

Proof: Since the steady state output of a stable system P₁(s) subject to a sinusoid input is also a sinusoid but with different amplitude and phase, all subsequent analysis will assume that the filter output has reached a steady-state condition.

Therefore, one has, for sufficiently large t,

P₁(s)[φ(t)] =







P₁(j0)

|P₁(jω₁)| cos(ω₁t +⁶ P₁(jω₁))

|P₁(jω₁)| sin(ω₁t +⁶ P₁(jω₁)) ...

|P₁(jω_N)| cos(ω_Nt +⁶ P₁(jω_N))

|P1(jωN)| sin(ωNt +⁶ P1(jωN))







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=







p_0,c

p1,ccos(ω1t) − p1,ssin(ω1t) p_1,scos(ω₁t) + p_1,csin(ω₁t)

...

p_N,ccos(ω_Nt) − p_N,ssin(ω_Nt) p_N,scos(ω_Nt) + p_N,csin(ω_Nt)







= M₁φ(t). (3.14)

where

pk,c = |P1(jωk)| cos(⁶ P1(jωk)), p_k,s = |P₁(jω_k)| sin(⁶ P₁(jω_k)),

p_0,c = P₁(j0), (3.15)

and M₁ is a square matrix, M₁ = diag p_0,c,

"

p_1,c −p_1,s p_1,s p_1,c

#

, · · · ,

"

p_N,c −p_N,s p_N,s p_N,c

#!

. (3.16)

Since M₁ is block diagonal, its determinant is

|M₁| = p_0,c

N

Y

k=1

(p²_k,c+ p²_k,s)

= P₁(j0)

N

Y

k=1

|P₁(jω_k)|². (3.17) From Assumption A3.3, one concludes that |M1| 6= 0; hence, M1 is invertible. Note also that

P₁(s)[φ^T(t)] = {P₁(s)[φ(t)]}^T

= [M₁φ(t)]^T

= φ^T(t)M₁^T. (3.18)

Substituting (2.5) and (3.7) into (3.12), and using a relationship similar to (3.14) on P₂(s), equation (3.12) becomes

P₂(s)[φ^T(t)θ_d] − φ^T(t)θ_r = φ^T(t)[M₂^Tθ_d− θ_r]

= P₁(s)[d₁(t)]. (3.19)

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It is straightforward to check, using (3.14), that d₁(t) = φ^T(t)θ with θ = M₁^−T[M₂^Tθ_d− θr] satisfies the above equation. This proved the existence of d1(t). End of proof.

After identifying the existence of the equivalent periodic disturbance d₁(t), substituting (3.12) into (3.11), one has

e(t) − P₁(s)u(t) = P₁(s)d₁(t). (3.20)

Therefore, based on the equivalence of system, the system block in Figure 3.1 can be simplified to Figure 3.2. Note that equation is the same as those previous AFC

1( ) P s d1

Figure 3.2: A system equivalent to Figure 3.1

systems when there is no tracking mission (r(t) = 0, e(t) = y(t)). The equation shows that an ideal control, which achieves asymptotic tracking of the periodic reference r(t) regardless of the periodic disturbance d(t), is u = −d₁. Therefore, the control design problem becomes the problem of estimating d₁(t). According to (3.13) in Lemma 3.2, the estimation of the periodic disturbance d₁(t) is further reduced to the estimation of the unknown constant vector θ in (3.13). In order to estimate θ, one needs a linear regression form in θ. This can be obtained by substituting (3.13) into (3.20),

e(t) − P₁(s)u(t) = P₁(s)[φ^T(t)θ] = P₁(s)[φ^T(t)]θ. (3.21)

The above equation is in fact a linear regression form, but the regressor P1(s)φ(t) increases many computational cost and analysis difficulties. In order to have a fine expression in the linear regression form (3.21), one further utilizes LTI system property, that is, when the periodic signal, d₁(t), enters into an LTI system, P₁(s),

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the system can be characterized by a superposition or sum of the zero-input response,

(t), and the zero-state response, ψ^T₁(t)θ. Therefore, (3.21) is rewritten as

e(t) − P₁(s)u(t) = ψ^T₁(t)θ + (t), (3.22)

where (t) is the exponentially decaying term since the plant model is stable, and the regressor ψ₁(t) is equivalent to P₁(s)φ(t) arriving in the steady state, that is ψ₁(t) = P₁(s)φ(t)|_t→∞, and thus it has

ψ₁(t) = M₁φ(t), (3.23)

in which M₁ is the nonsingular block matrix as defined in (3.16). The regressor ψ₁(t) is bounded since both M₁ and φ(t), defined in (3.16) and (2.3) respectively, are bounded. After some transient times, the exponentially decaying term (t) in equation (3.22) approaches to zero. Therefore, we will first neglect the presence of the (t) but latter show in Section3.7it does not affect the property of the identifier.

In this case, the linear regression form in (3.22) is represented as

e(t) − P₁(s)u(t) = ψ₁^T(t)θ. (3.24)

Note that equation (3.24) shows that the linear regression form remains the same no matter where the disturbance enters the system. Therefore, if the system satisfies Assumption A3.3, one can assume that the periodic disturbance enter the system at input point.

Remark 3.2 : The key step in deriving the linear regression form (3.24) is to take the constant vector θ out of the square bracket of (3.21) after P1(s). Without this step, one must resort to model reference adaptive control scheme to estimate θ, as is done in many previous AFC designs, which have to enforce the minimum-phase assumption of P₁(s) or small adaptation gain assumption.

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3.4 Review of Adaptive Feedforward Control

Most of the AFC control systems are implemented in a plug-in manner as shown in Figure 3.3. In Figure 3.3, d, r, u, u_f, v, e and y are, respectively, the exogenous periodic disturbance, the reference input, the control input, the nominal feedback control, the feedforward control, the output error and the system output, C(s) is the feedback controller and P₁(s) is the transfer function of the plant. For the ease to analysis, it usually assumes that the control objective is to achieve the periodic tracking mission and the system plant P1 is stable, and hence has uf = 0.

Consequently, according to Lemma 3.2, the control structure is simplified as Figure 3.4.

C s u

d

uf y r

e v

1( ) P s

Figure 3.3: Control system with Plug-in AFC controller

u P s₁( ) e

d1

Figure 3.4: AFC control System

In Figure 3.4, the system output error is

e(t) = P₁(s) (u(t) + d₁(t)) , (3.25)

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30 3.4. Review of Adaptive Feedforward Control

where the periodic disturbance d₁ has an expression form like (3.13). For being convenient to state, one assumes that d1 is a single-tone harmonic signal

d1(t) = θ1,ccos(ω1t) + θ1,ssin(ω1t), (3.26) in which ω₁is a known frequency and θ_1,c and θ_1,sare unknown constant coefficients.

Certainly, extended compensation for many sinusoids is straightforward. By adding the negative of disturbance at all time, the disturbance can be easily cancelled at the input of the plant. Hence, the feedforward control u(t) is suggested as the negative of disturbance estimation. Consequently, the feedforward control has

u(t) = −φ^T(t)ˆθ(t). (3.27)

Substituting (3.13) and (3.27) into (3.25), the plant output error is rewritten as e(t) = P₁(s)[φ^T(t)θ − φ^T(t)ˆθ(t)]. (3.28) The problem is how to find an adjustment mechanism so that the parameter estimation ˆθ(t) converges to the nominal value θ and further the disturbance is cancelled exactly. Since this expression is similar to conventional model reference identifiers structure [47], the parameter vector ˆθ(t) has a possible update law which called the pseudo-gradient algorithm, that is

˙ˆθ(t) = γφ(t)e(t), (3.29)

where γ > 0 is an adaptation gain. The AFC control diagram is shown in the Figure 3.5. According to the adaptive theory [47], if P₁(s) is a strictly positive real (SPR), the system output e(t) will converge to zero as t → ∞. As a result of the SPR condition, stabilizing controller is only guaranteed on few physical systems.

Furthermore, based on the Laplace transform analysis, Messner and Bodson in [38] obtained an equivalent LTI representation. The resulting continuous-time transfer function from e(t) to u(t) is

u(t)

e(t) = −γ s

s²+ ω₁². (3.30)

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∫

cosω1t

∫

sinω1t

−1

γ

u +

+ e

+ + dˆ¹

d1

1( ) P s

AFC Controller ˆ1,

θ c

ˆ1,

θ s

Figure 3.5: AFC control

Although it is derived from frequency domain, it can be derived from time domain more easily. Firstly, substituting the integration of (3.29) into (3.27), the control u can be written as

u(t) = −φ^T(t)

Z t 0

γφ(τ )e(τ )dτ

= −γ

Z t 0

cos(ω1(t − τ ))e(τ )dτ, (3.31) where the term of integral at the last equation expresses a convolution integral, that is cos(ω₁t) ∗ e(t). Finally, taking the Laplace transform on cos(ω₁t), one can immediately obtain the result of (3.30).

0 ₂ ₂ P s₁( )

1

s s

J Z

u e

d1

Figure 3.6: IMP control which is equivalent to Figure 3.5

It is obvious that the AFC scheme of Figure3.5 and the IMP scheme of Figure

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32 3.4. Review of Adaptive Feedforward Control

3.6 are functionally equivalence in principle. The result allows LTI analysis techniques to be used in adaptive system. Therefore, from the IMP controller design, the SPR condition can be relaxed if the adaptation gain γ is carefully chosen. Besides, Messner and Bodson also stated that the phase difference between the input disturbance and the measurement output might deteriorate the convergence property of the AFC system. The cause of the phase lag is the conventional AFC design without considering the plant model. In order to compensate for the phase lag, they add a phase shift α₁ into the regressor of (3.29). The modified regressor then becomes

φα(t) =^h cos(ω₁t + α₁) sin(ω₁t + α₁) ⁱ^T , and the update law (3.29) becomes

˙ˆθ(t) = γφ_α(t)e(t).

Note that the feedforward control is still set as (3.27). Consequently, using similar operating process in (3.31), the resulting continuous-time transfer function from e(t) to u(t) becomes

u(t)

e(t) = −γs cos(α1) + ω sin(α1) s²+ ω₁² .

To achieve the fastest elimination of the periodic disturbance at low adaptive gain, Bodson and co-workers [39] suggested that the optimal regressor phase advance α₁ is the phase of the plant at the disturbance frequency.

The main advantages of the AFC have the following items. First, it can selec- tively remove harmonics from the frequency spectrum. Second, it is not necessary to acquire an exact plant model. Third, when the output signal is not available during certain periods of time, the AFC controller can simply freeze the parameter updates. On the contrary, the internal model based controller is not robust to this variation. Forth, it can be driven by the measuring frequency, making the control response more robust to variation in frequency. Finally, the adaptive implementation can adopt angular measurements directly. It needn’t require the frequency

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to be computed from the angular measurements. However, the system must be SPR or the adaptation gain be small in order to maintain stability. Besides, from IMP equivalence perspective, when the controller introduces a number of sinusoidal signals, the stability problem becomes more and more difficult.

Example 3.1 : Consider an open-loop stable system with the input-output description as in (3.25), where the plant transfer function P₁(s) is

P₁(s) = (s + 3)(s + 5) (s + 2)(s + 4)(s + 6), and the periodic disturbance d1 is

d₁(t) = 6 cos(ωt) + cos(2ωt) + 0.5 sin(3ωt),

in which the frequency ω = 1.

In the adaptive estimation algorithm (3.29), one sets the adaptation gain γ = 5, and the regressor

φ(t) =^h 1 cos(ωt) sin(ωt) cos(2ωt) sin(2ωt) cos(3ωt) sin(3ωt) ⁱ^T , Figure 3.7 shows the output error asymptotically converges to zero and the root mean square error at 70s ≤ t ≤ 100s is 7.1387 × 10⁻⁸. Figure 3.8 shows the time history of the disturbance, where the true disturbance d₁(t) and the disturbance estimation ˆd₁(t) are shown by dotted line and solid line respectively. It shows the estimate asymptotically converges to the true disturbance. Therefore, the simulation verifies successfully that the AFC design has good performance under SPR system.