Robust Control of Flexible Motion Systems: A Literature Study

Transcription

1 Robust Control of Flexible Motion Systems: A Literature Study S.L.H. Verhoeven DCT Report APT Supervisors: Dr. ir. J.J.M. van Helvoort Dr. ir. M.M.J. van de Wal Ir. T.A.E. Oomen Prof.ir. O.H. Bosgra Philips Applied Technologies Mechatronics Program Drives and Control Group Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Technology Group Eindhoven, January 2009

2

3 Summary This literature study focusses on robust control of flexible motion systems. Traditionally, motion systems are designed such that the frequency of the dominant flexible dynamics is high compared to the required bandwidth. As many independent single-input single-output controllers as degrees-of-freedom are then used to control the rigid body modes of the system, where both the feedback and feedforward controller design is based on the input/output behaviour of the plant. However, increased throughput requirements lead to lighter motion systems, causing the dominant flexible dynamics to shift towards the required bandwidth. As a consequence, actuator forces will deform the body relative to the tensionless equilibrium. The traditional geometric relation between measurement information on the one hand and desired position information on the other hand is then no longer valid. The actual system performance may thus be limited by internal deformations that are not represented well in the input/output model. This is the essence of beyond-rigid-body control. The first part of this literature study gives an overview of the theory behind H -optimisation and µ-synthesis. These norm-based controller design techniques are considered relevant for beyond rigid body control, due to a variety of reasons. First, these techniques allow for an explicit distinction between performance variables and measured variables. Second, they are able to explicitly account for system uncertainty. Information of high-frequency dynamics is not accurately available and putting a lot of control effort into controlling these dynamics is undesired. Third, the control problem in H -optimisation and µ-synthesis is solved in a uniform way, regardless of the number of inputs and outputs. Therefore, it is easier to deal with - possibly non-square - plants with many actuators and sensors. The second part discusses literature on actively controlling the internal dynamics of a body. A common approach is the explicitly control a set of modes, while neglecting the other modes. In literature, it is shown that this method often works well for relatively simple systems, e.g., beams or thin plates, but it is believed that for more complex systems the application of modal control is less straightforward and may not work. One of the reasons for this, is the so called spillover effect, which is the effect of the neglected modes on the closed-loop system. By adding extra actuators (over-actuation) and sensors (over-sensing) to a flexible motion system, it is possible to explicitly control the flexible modes. Several control structures exist in which over-actuation (and over-sensing) can be applied. It can be used for either feedforward, feedback, or feedback and feedforward simultaneously. Which control structure leads to the best result depends on the system, the actuator/sensor configuration, the performance measure and the external disturbances. In order to achieve the best performance the work of the control engineer should therefore not be limited to merely controller design, but should also include the placement of the actuators and sensors. i

4

5 Abbreviations ARE BRB DOF DGKF DISO EMC FEM FRF GM HCARE HCARE IMSC IC ILC IO LFT LHP LPV LQG MIMO MM NP NS PM RHP RP RS SISO SSV TFM Algebraic Riccati Equation Beyond Rigid Body Degree-Of-Freedom Doyle, Glover, Khargonekar, Francis Double-Input Single-Output Efficient Modal Control Finite Element Model Frequency Response Function Gain Margin H Controller Algebraic Riccati Equation H Filter Algebraic Riccati Equation Independent Modal Space Control Integrated Circuit Iterative Learning Control Input/Output Linear Fractional Transformation Left Half Plane Linear Parameter Varying Linear Quadratic Gaussian Multi-Input Multi-Output Modulus Margin Nominal Performance Nominal Stability Phase Margin Right Half Plane Robust Performance Robust Stabiilty Single-Input Single-Output Structured Singular Value Transfer Function Matrix iii

6

7 Contents Summary i Abbreviations iii Contents v 1 Introduction Background Project motivation and problem formulation Outline of the Report Overview of common terms in literature Global literature overview Robust Control Introduction Benefits of advanced control General control configuration Including weights in the general control configuration Including uncertainty in the general control configuration Control problems Nominal Performance Robust Stability and Robust Performance Modelling uncertainty Robust stability Motivation for the structured singular value v

8 vi Contents Robust performance Restatement of control problems Solutions to the H optimal control problem DGKF solution to H control problem µ-synthesis Feedforward design Summary and conclusions Robust control for an ASML wafer scanner Introduction Control goal and control structure Plant modelling Performance quantification Weighting filters for scaling Weighting filters for loop shaping Weighting filters to account for power spectra Uncertainty quantification Results Summary and conclusions Vibration control of flexible structures Introduction System description Distributed parameter system Nodal models Modal models Relevance of modal analysis Modal control Independent modal space control Coupled control Spillover Example: spillover effect

9 Contents vii 4.4 Robust control for large space structures Problem formulation Modelling uncertainty and performance specification Tradeoffs between robustness and performance Control of flexible modes in the controller crossover region Summary and conclusions Control of flexible motion systems Introduction Three aspects of controller design First aspect: control structure Second aspect: actuator/sensor position Third aspect: performance definition Control of flexible motion systems without over-actuation Conventional controller design Advanced controller design Actuator/sensor placement Interpretation of transmission zeros Control of flexible motion systems with over-actuation Internal and external over-actuation Double-input single-output (DISO) Internal over-actuation Summary and conclusions Conclusions and recommendations Conclusions Recommendations A Interpretation of transmission zeros 81 Bibliography 87

10

11 Chapter 1 Introduction 1.1 Background Philips Applied Technologies has a long history of research on advanced control for high precision motion systems. Several topics, such as multivariable control, H -optimisation, µ- synthesis, LPV control, and ILC have been examined for high-precision and high-throughput stages. This research has especially been done for ASML, which is a leading company in the market for chip manufacturing machines, i.e., wafer scanners. These machines are used for the production of Integrated Circuits (ICs). ICs are produced on a silicon wafer (200 mm mm diameter) by a photolithographic process. An important mechanical component of the wafer scanner is the wafer stage that positions the silicon wafer with respect to the imaging optics. Because very fine patterns have to be produced on the wafer, a position accuracy in the order of nanometers and microradians is required. High-accuracy stages are nowadays controlled in six Degrees-Of-Freedom (DOFs): the three rigid body translations and rotations. The design of the control loops is mostly based on six actuators and sensors, to independently control the six rigid body DOFs. To keep control design simple, Single-Input Single-Output (SISO) controllers are common practice, despite the Multi-Input Multi-Output (MIMO) nature of the problem. The design of the feedback and feedforward controllers is usually based on the plant Input/Output (IO) behaviour. Based on the IO plant model, controller design aims at creating suitable (closed-loop) transfer function behaviour. However, the actual performance of a motion system is not necessarily represented well by the IO behaviour of the plant and the corresponding closed-loop transfer functions that rather represent servo performance. This situation occurs if internal plant dynamics become relevant that is not directly sensed, as for a wafer stage [36]. The actual system performance is in terms of the positioning of that part of the silicium wafer that is subject to the light exposure. The servo performance is only an approximation of this true goal, since it is based on laser interferometer data (measured at the edges of the wafer stage) that is transferred via a sensor transformation into coordinates of the area on the wafer subject to exposure. However, the sensor transformation assumes the wafer stage to be a rigid body system and hence the possible contribution of internal dynamics (flexible modes) is neglected. Because various 1

12 2 Introduction forces (actuation, disturbances, gravity) act on a body with finite stiffness, the body will exhibit internal deformations relative to the tensionless equilibrium. The traditional geometric relation between measurement information on the one hand and desired position information on the other hand is then no longer valid. A similar reasoning applies to the actuator side, where internal deformations refute the validity of the traditional actuator transformation that is derived on the basis of a rigid-body plant model. The actual system performance may thus be limited by modal deformations that are not represented in the IO model used for feedback and feedforward controller design. 1.2 Project motivation and problem formulation Because of the fierce competition in the IC market, it is desirable to put more and smaller transistors on a single IC and to increase the throughput of the wafer scanner. ASML is therefore faced with the industrial challenge to build bigger and lighter stages, while at the same time the requirements on the servo error and throughput become ever demanding. With this tendency, it will eventually become necessary to much more rigorously address the presence of flexible modes in the control system design. Explicitly taking into account the internal plant dynamics (besides the usual IO behaviour) in the controller design provides opportunities to improve the actual system performance of mechatronic stages. At this moment, studying the following control design freedom is considered worthwhile in the context of beyond-rigid-body (BRB) control : Over-actuation: The usage of more actuators than free rigid body DOFs, to enable the possibility to actively control the internal flexible modes, instead of the present - rather passive - approach of limiting the undesired effect of these modes as much as possible. Over-sensing: The usage of more sensors than rigid body DOFs, to explicitly sense the internal flexible modes or to at least improve the observability of such modes and hence to enable the possibility to improve the controller design in the face of flexible modes. Vibration control of structures: Using over-actuation and over-sensing to explicitly control the flexible modes of a motion system is in some way equivalent to controlling a system without rigid body modes, e.g., a structure. Since a lot of research has already been done in that field, it is believed that valuable information can be obtained from it. Explicit distinction between sensed vs. performance variables: Exploiting the explicit distinction that can be made in the controller design between the variables that are sensed (y) and the variables that represent the actual system performance (z). As discussed above, for wafer stages z could be in terms of the positioning of the spot on the wafer subject to exposure, while y would rather be in terms of the spots on the edge of the wafer measured by the interferometer, see Figure 1.1. The internal dynamics between these performance and sensed locations may cause a relevant deviation of the servo accuracy in y vs. the exposure accuracy in z. The so-called generalised

13 1.3 Outline of the Report 3 standard plant set-up, which is depicted in Figure 1.2 and is also used in norm-based controller design, can be used to make an explicit distinction between y and z in the controller design, thereby providing the potential to design more effective controllers. P z z y K u P Figure 1.1: Standard feedback configuration with an explicit distinction between sensed and performance variables. Considering the control design freedom listed above, the goal of this literature study can be divided into two parts: G Examine norm-based controller design and its applicability to an ASML wafer scanner. Examine the control of flexible modes in structures and in flexible motion systems. w z G u y K Figure 1.2: General control configuration. 1.3 Outline of the Report The report is organised as follows. Chapter 2 discusses H -optimisation and µ-synthesis. Both techniques can be used to design MIMO controllers for a wide variety of system. It is expected that these norm-based controller designs lead to better results for the problem at hand than classical loop shaping methods due to a variety of reasons. For example, due to the explicit distinction between the performance and sensed variables and because of the systematic way of dealing with a large number of inputs and outputs.

14 4 Introduction In Chapter 3 it is shown how H -optimisation and µ-synthesis can be used to create MIMO controllers for ASML wafer stages. In [50] it is shown that these types of controllers can be regarded as feasible successors for the standard SISO controllers that are currently used. Chapter 4 focusses on vibration control in flexible structures. Flexible structures are in this report regarded as systems without rigid body modes. Although flexible structures are very different from wafer stages and traditional motion systems, i.e., no rigid body modes, similarities occur in the form of closely spaced flexible modes and the uncertainty in the high-frequency modes. Chapter 5 focusses on vibration control in motion systems. The concept of using more actuators and sensors than rigid body modes is introduced, which boils down to adding more actuators and sensors to explicitly control the flexible modes. The benefit of over-actuation and over-sensing is shown for a simple free-free beam system. In Chapter 6 the main results and conclusions are summarised and recommendations are given for further research. 1.4 Overview of common terms in literature In literature, different terms are often used to describe similar things. In this section, a short overview is given of some common terms that are considered relevant in the context of BRB control. Flexible mode or vibration mode: A flexible mode refers to a periodic motion that is physically possible in the absence of any external influence and in which the elastic displacement w(p,t) at position p and time t all move in unison, i.e., all displacements pass through zero simultaneously and they all attain their maxima simultaneously, see [28]. In this report the term flexible mode is used. Rigid bode mode: Similar as a flexible mode, but instead of a periodic motion it describe a direction of displacement without flexible deformation. Hence, the corresponding natural frequency is zero. Structural analysts often ignore this mode, because there is no deformation involved. However, for the control engineer this mode is crucial, since it allows the structure or system to be moved or track a command. Mode Shape: Modes shapes can refer to flexible modes and rigid body modes. System: It is quite difficult - or maybe impossible - to give one good description of a system since a system can be almost anything. In this literature study and in the context of BRB control, the term system is used to describe a physical product with unalterable properties. For example, a wafer stage, a car, or a two-mass-spring system. In other contexts, the term system can mean different things. Structure or flexible structure: A structure is a type of system in which rigid body modes are not considered relevant for control purposes or do not exist. To avoid confusion. the term flexible structure should then ideally be used to describe a structure that has no rigid body modes. However, in literature both terms are used in parallel.

15 1.5 Global literature overview 5 Flexible system: A system in which flexible modes are present and not negligible under normal operation. In theory, all systems have a finite stiffness and can therefore be regarded flexible. However, not all systems contain flexible modes that are relevant during normal operation. Flexible motion system: A motion system in which the flexible modes are relevant during normal operation. Intelligent structure: This term often refers to (large) structures in which control is applied. For example, a building that is able to withstand earthquakes. Active vibration control: Control effort aimed at controlling the flexible modes in a system. Rigid body modes are not present or are not considered relevant. Hence, the term structural vibration control is also used. Passive vibration control has the same goal as active vibration control, but the goal is basically achieved by modifying the structure, e.g., vibration isolation, or adding local springs and masses, instead of using actuators and sensors, see [46]. Vibration damping: Often the control law in active vibration control focusses on damping the flexible modes and this is referred to as vibration damping. Another possibility is to compensate for a flexible mode. 1.5 Global literature overview In this section, a compact overview is given of the literature that - at this moment - is considered relevant in the context of norm-based control and controlling flexible systems. Since it is impossible to give a complete overview, only the most well known literature sources are used. Not all literature listed below is used for this literature study and the literature is divided into four groups: Robust control. A lot of books and papers have been written about classical and robust control theory. Two well known books are the book of Skogestad and Postlethwaite [48] and the book of Zhou et al. [54]. The former focusses on practical feedback control and the latter more on system theory. Modal control. Two books about the modal system description are [18, 35, 40]. Although these books emphasise the advantages of the modal system description, some drawbacks of the modal description exist and are presented in a paper of Hughes, see [28]. The concept of modal control is also covered in [18, 40], but a lot more literature is available, see, e.g., [8, 34, 47]. In [47] the concept of Efficient Modal Control is introduced, i.e., using displacement or energy content of each mode as weight to determine the feedback control force. Robust control of flexible modes. A lot of literature is available on robust control of flexible structures, due to the difficulties in accurate modelling of flexible structures. A comprehensive and recent overview of the application of H -Optimisation and µ- synthesis in controlling flexible modes is given in [29]. Also, a tutorial is presented for

16 6 Introduction designing H -based controllers for a smart plate, i.e., a plate equipped with actuators and sensors. Most of the literature discussed in [29] is quite similar. It mainly concerns the creation of SISO or MIMO H -based controllers with collocated actuators and sensors, for controlling a flexible beam, plate, or antenna-like structure. The only differences occur in the combined work of Halim and Moheimani [23, 24, 35], where a spatial performance norm is minimised, i.e., performance is required at an infinitely large set of points. The work done in [43] is also different, because actuators and sensors are used in a non-collocated setting. Unfortunately, no motivation is given for this choice. In [32], the performance of a H -based controller is compared to traditional velocity feedback and LQG control. The main conclusion is that simple velocity feedback outperforms the H -based controller. This is opposite to the results in [43], where the H -based controller is superior. A possible explanation for this big difference is that the H control problem is not formulated well. In [11], a method is proposed to design control laws based on H -optimisation, for flexible structures with closely spaced modes and collocated actuators and sensors. Moreover, the solution presented avoids calculation of the Algebraic Riccati Equations, see Chapter 2, so an explicit solution for the controller is obtained. In [21, 26, 27], robustness is achieved in a different way. Only parametric uncertainty is considered and stability of the closed loop system, including the parametric uncertainty, is proven by using Lyapunov stability theory. In [27], the topic is active robust shape control of flexible structures and the authors propose a method to control the shape of the structure under the influence of disturbances. For example, maintaining a certain optimal wing cross section during flights. Controlling the shape of a structure can not be done without controlling the flexible modes of a structure. Hence, it is considered relevant. In the early 90 s a lot of work has been done by G.J. Balas, see [1, 3 7]. In this work the flexible modes of a structure, called the Caltech experimental flexible structure, are suppressed by using µ-synthesis. Actuators and sensors are used in a non-collocated setting and the structure has closely spaced flexible modes and uncertainty in the higher frequency modes. At Eindhoven University of Technology and Philips, research has been done on controlling flexible modes in motion systems. Recent work that is considered relevant for this literature study is the work done by M. Schneiders, see [44 46] and J.W. van Wingerden [53]. Both authors discuss the use of extra sensors and actuators to explicitly control the flexible modes in a motion system. Actuator and sensor selection. The problem of choosing a good location for the actuators and sensors (possibly non-collocated) in a flexible motion system is briefly introduced in some of the literature about robust control of flexible modes, see, e.g., [43, 44, 53]. The topic of actuator/sensor selection is investigated more thoroughly in [22, 25, 31, 42, 44, 52].

17 Chapter 2 Robust Control 2.1 Introduction As mentioned in the introduction, wafer stages are currently controlled by six SISO controllers, despite the MIMO nature of the problem. This chapter first briefly discusses H -optimisation and µ-synthesis as a type of MIMO control. In literature, H -optimisation and µ-synthesis are extensively studied in various areas of engineering. For more details the reader is advised to study [37, 48, 50, 51]. A more comprehensive overview is given by [14, 54, 55]. 2.2 Benefits of advanced control SISO feedback controllers are usually designed using manual loop shaping. Based on Bode diagrams and Nyquist plots of the open-loop transfer function, parameters are tuned such that properties like BandWidth (BW), Phase Margin (PM), and Gain Margin (GM) are met. Often, this results in a series connection of low-order filters, like integrators, lead-lag filters, notches, and low-pass filters. This conventional design has some disadvantages: Loop shaping is usually performed in an open-loop setting. The open-loop gain should be large at low frequencies to meet the performance requirements (reference tracking and disturbance rejection) and small at high frequencies, in order to not amplify measurement noise. In between, the open loop gain is approximately one. The point where the open-loop gain crosses the 0 [db] line from above for the first time, is defined as the BW of the system. 1 At the bandwidth the phase should be large enough (phase margin) to be stable. However, it is more natural to do loop shaping in a closed loop fashion, since in the end it is the closed-loop performance that counts. It is not guaranteed that the best controller is found by manually shaping the openloop, since this is subject to the experience of the control engineer. This drawback 1 This is the definition of BW that is used in this report. In literature this point is often referred to as crossover frequency and the bandwidth is defined in closed-loop; either in the sensitivity, S, or complementary sensitivity, T. 7

18 8 Robust Control could be resolved by formulating the control problem as an optimisation problem with a guaranteed global optimum. If the control problem becomes more complicated, loop shaping becomes hard or almost impossible. For instance, if MIMO plants with strong interaction are considered, if the number of actuators and sensors increases, or if there are performance requirements on multiple closed-loop transfer functions. There is no straightforward manner to account for modelling errors and uncertainty in the plant model. High-frequency roll-off can be used to achieve some robustness against high-frequency resonance modes and unmodelled dynamics, but a more sophisticated approach to robust controller design is desirable. The measured and regulated variables need to be the same. The performance objectives need thus be stated in terms of variables that can be measured. The disadvantages listed above are, at least in theory, resolved by controller design using H -optimisation and µ-synthesis. 2 Both topics are briefly described in the remainder of this chapter. A disadvantage of these more advanced controller design techniques is the need for a plant model, whereas for conventional controller design a measured Frequency Response Function (FRF) is sufficient. In addition, using six individual SISO controllers 3 is more transparent, and hence simpler. Control engineers are more familiar with manual loop shaping and terms like phase-, gain- and Modulus Margin (MM). 2.3 General control configuration Most advanced controller design techniques, like H -optimisation and µ-synthesis, make use of the general control configuration as depicted in Figure 2.1. Herein, G is the generalised standard plant 4 and K is the controller to be designed. The regulated variables, i.e., the variables to be kept small, are stacked in the vector z. Typical signals that are often included in z are control actions and servo errors. The measured output signals that are used as controller input are collected in vector y, which implies that the control objective does not have to be stated in terms of measured signals. The vector w contains the exogenous inputs, e.g., disturbances, sensor noise, but also reference trajectories. The controller output signals are stacked in vector u. Note y and u do not have to be of the same size, i.e., the controller does not have to be a square matrix. The generalised standard plant can be represented as: [ z y ] [ G11 G = 12 G 21 G 22 ][ w u ], (2.1) or as a state-space representation: 2 The term synthesis us used rather that design to stress that it is a more formalised approach. 3 This is also called multiloop SISO control 4 Sometimes G is referred to as standard plant or augmented (standard) plant. In this report the term standard plant refers to the physical standard plant, i.e., the plant without weighting filters.

19 2.3 General control configuration 9 w z G u y K Figure 2.1: General control configuration. G : A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 The Linear Time Invariant (LTI) controller is described by:. (2.2) with state-space realisation: u = Ky, (2.3) [ AK B K : K C K D K ]. (2.4) Including weights in the general control configuration Besides a physical plant model, the generalised standard plant can also contain weighting filters that represent performance objectives, as depicted in Figure 2.2. Here, w p and z p denote the weighted exogenous input and output signals of G, respectively, and w p = V p w p and z p = Wp 1 z p denote the physical plant variables. 5 If, for example, z includes the error signals, it may be desirable to penalise errors in the low-frequency region Including uncertainty in the general control configuration In a similar fashion uncertainty can be included in the general control configuration, as depicted in Figure 2.3. Here w u and w u = V u w u denote the scaled and unscaled output from the uncertainty block u, respectively, and z u and z u = Wu 1 z u denote the scaled and unscaled input to the uncertainty block. By explicitly representing model uncertainty, it can be ensured that the resulting controller performs well in case of plant variations, i.e., that the controller is robust. This is further explained later in this chapter. By comparing 5 The weighting function normally also includes a scaling factor, but for this report it is assumed that the physical plant is scaled properly.

20 10 Robust Control w p V p w p z p z p W p Ḡ G u K y M Figure 2.2: General control configuration with performance weights. Figure 2.2 and Figure 2.3 it can be seen that the setup is similar, except that the exogenous variables are separated in two groups: variables related to performance (subscript p) and uncertainty-related signals (subscript u). The closed-loop system can then be written as z = Mw: [ zu z p ] [ M11 M = 12 M 21 M 22 ][ wu w p ]. (2.5) w u u z u V u w u z u W u w p V p Ḡ w p z p z p W p u K y G M Figure 2.3: General control configuration with performance weights and model uncertainty. Remark 2.1 In case of vector valued signals w p, w u, z p, and z u, V p, V u, W p, and W u become

21 2.4 Control problems 11 matrices. Since the off-diagonal components of these matrices are difficult to interpret, the weighting matrices are often chosen to be diagonal. 2.4 Control problems Various goals can be pursued in the controller design. The following four control problems are distinguished in [48]: Nominal Stability (NS): The closed-loop system is stable in the absence of model uncertainty. NS is always required. NS is further explained in [48]. Nominal Performance (NP): The closed-loop system is stable and it achieves the performance specifications in the absence of model uncertainty. NP implies NS. Robust Stability (RS): The closed-loop system is stable in the presence of a certain class of model uncertainties. RS implies NS Robust Performance (RP): The closed-loop system is stable and it achieves the performance specifications in the presence of model uncertainty. RP implies NS, NP, and RS. Several definitions of stability exist in literature. Here Definition 4.4 of [48] is used: Definition 2.1 A system is (internally) stable if none of its components contain hidden unstable modes and the injection of bounded external signals at any place in the system results in bounded output signals measured anywhere in the system. Here a signal u(t) is defined to be bounded if there exists a constant c such that u(t) < c for all t. This type of stability is also referred to as Bounded-Input Bounded-Output (BIBO) stability. The word internally stresses that it is not sufficient to have a stable response from one particular input to another particular output, but require bounded signals measured at any place in the system. A continuous time linear time-invariant system ẋ = Ax + Bu is stable if and only if all the poles p i are in the open Left Half Plane (LHP); that is, Re(p i ) = Reλ i (A) < 0, i. A system matrix with such a property is called Hurwitz. A system is unstable is it has any poles in the open Right Half Plane (RHP); that is Re(p i ) = Reλ i (A) > 0. The imaginary axis (jω-axis) is thus the stability boundary between a stable and unstable response. Poles on the jω-axis, like integrators and pure harmonic oscillators (s = ±jω), are unstable by Definition 2.1 given above. For example, consider a pure integrator, a constant input c o leads an unbounded output c o t. However, if stability is judged based on the response of an initial condition, different conclusions can be drawn for poles on the jω-axis. In [17], a system is stable if initial conditions decay to zero and unstable they diverge. If the system has non-repeated jω-axis poles, it is said to be neutrally stable. For example, a single integrator results in a constant output and a pure harmonic oscillator results in an oscillating response without damping. If the systems has repeated poles on the jω-axis, it is unstable. For example, a double integrator (mass floating in space). A non-zero initial velocity results in an unbounded position.

22 12 Robust Control In the next two sections the different control problems are further elaborated. Section 2.5 discusses nominal stability and nominal performance, and Section 2.6 discusses robust stability and robust performance. 2.5 Nominal Performance Consider again the control problem without model uncertainty as depicted in Figure 2.1 and Figure 2.2. If the generalised standard plant G is closed with controller K, the generalised closed-loop system M results: with the partitioning G as follows: M = F l (G,K) := G 11 + G 12 K(I G 22 K) 1 G 21, (2.6) [ zp y ] [ G11 G = 12 G 21 G 22 ][ wp u ]. (2.7) The expressing F l (G,K) in (2.6) is called a lower Linear Fractional Transformation (LFT) and can be read as close G by K. The closed-loop system M in (2.6) also contains the weighting filters V p and W p. The physical closed-loop system M can be defined in a similar way: M = F l (Ḡ,K) := Ḡ11 + Ḡ12K(I Ḡ22K) 1 Ḡ 21. (2.8) Ideally, the effect of w p on z p should be zero. Imagine, M being the sensitivity S, which is the case if w p is the reference trajectory and z p the tracking error. In an ideal situation M = 0 at all frequencies (perfect regulation), which is not possible for realistic control problems ( Waterbed effect ). Instead, the goal is to make S small at certain frequencies. To indicate in which frequencies it is important to make S small, the weighting filters V p and W p can be used. The controller design problem is then restated to making M = W p MVp small. Suppose that M is a SISO system, like the sensitivity S. The gain M(jω) is then a natural measure of smallness over the frequency domain. To come up with a scalar measure for smallness, the H -norm could be used. For an asymptotically stable SISO system this norm is defined as: M(s) := sup M(jω), (2.9) ω with sup denoting the supremum. The M(s) -norm thus denotes the maximum value of the SISO transfer function M over all frequencies. So, by making proper choices for the weighting filters, the control problem amounts to designing a controller K such that the M -norm is bounded by a given value γ, which is usually set to 1. Controller design aimed at minimising the H -norm of a suitable closed-loop system is called H optimisation. Other norms than the H -norm can also be used, but certain properties of the H -norm, like the

23 2.5 Nominal Performance 13 sub-multiplicative property 6 turn out to be very useful to incorporate uncertainty models that are discussed later [37, 48]. In general, M is not a SISO system but a MIMO system. The main difference between the two is the presence of directions in the latter. The gain of M therefore depends on the particular direction of w p. To deal with the directionality, the Singular Value Decomposition (SVD) in introduced. Consider a, possibly complex, l m matrix M, which can also be frequency dependent. The matrix M can be factorised as follows: M = Y ΣU, (2.10) where { } stands for the complex conjugate transpose. The matrices Y and U are orthonormal matrices of size l l and m m, respectively. The l m matrix Σ contains a diagonal matrix Σ 1 of real non-negative singular values σ i, arranged in descending order: Σ = [ Σ1 0 ] if l m or: Σ = [ Σ 1 0 ] if l m, (2.11) where: Σ 1 = diag(σ 1,σ 2,...,σ k ), with: k = min(l,m). (2.12) If w p is aligned with the ith column of U (this is called the input direction), z p will be in the direction of the ith column of Y (output direction) and amplified by a gain σ i. Matrices U and Y thus contain the information about directionality, whilst the matrix Σ contains the information about the gains. The largest gain is achieved when w p is aligned with the first column of U, which corresponds to the maximum singular value σ 1. The maximum singular value σ 1 is usually denoted by σ. For MIMO system, the H -norm of (2.9) can thus be adjusted to a more general form: M(s) := sup ω σ(m(jω)). (2.13) Because the H -norm only looks at the maximum singular value, the norm is often interpreted as a worst-case gain. The following definition of the H -norm also illustrates this character: z p (t) 2 M(s) = sup. (2.14) w p(t) =0 w p (t) 2 Here, w p and z p are the input and output signals of M and 2 denotes the L 2 -norm of a signal that equals the square root of the energy of a signal. The H -norm is thus the maximum amplification of energy of the input signal w p. From (2.14) one can easily understand that the H -norm can only be defined for systems that are asymptotically stable, since z p goes to infinity when the system is unstable. 6 This is also called Schwartz inequality : GH G H

24 14 Robust Control 2.6 Robust Stability and Robust Performance To assess robust stability and robust performance, uncertainty models can be included in the control configuration as depicted in Figure 2.3. Controllers resulting from NP design also exhibit some robustness, since they have a certain phase- and gain margin. However, these properties may not be sufficient and to account for uncertainties explicitly, uncertainty models are incorporated in the controller design/analysis. In this section measures for robust stability and robust performance are given, but first it is explained how uncertainty can be included in the general control configuration Modelling uncertainty Uncertainties are differences between the actual plant and the plant model. Various sources of uncertainty exist. If a plant model is known, e.g., in state space format, there is always uncertainty in the parameters. This is called parametric uncertainty and it is not discussed further in this report. Another kind of uncertainty is dynamic uncertainty, which can arise due to various sources: Model simplification: To design a controller the plant model should be kept relatively simple. Therefore, high-frequency modes and non-linearities are often neglected. Production tolerances: Plants that are the same in theory, are not the same in practice. There is always some mismatch within a batch of virtually the same plants. Changing environmental and operating conditions: Plants are subject to wear, changes in temperature and humidity, and changing operating conditions. By lumping together several sources of parametric uncertainty. Uncertainty models There are several possibilities to quantify model uncertainty. Three possibilities are listed below. P u represents the true plant, which is subject to uncertainties and P represents the nominal plant. Scaling filters V u and W u are used to normalise the magnitude of the uncertainty block u to one ( u 1): Additive plant uncertainty, see Figure 2.4: P u = P + W u u V u. (2.15) Multiplicative uncertainty at the plant input, see Figure 2.5: P u = P(I + W u u V u ). (2.16) Multiplicative uncertainty at the plant output, see Figure 2.6: P u = (I + W u u V u )P. (2.17)

25 2.6 Robust Stability and Robust Performance 15 More possibilities are possible, for example, the inverse forms of the uncertainty types listed above. The uncertainty loop is then closed in the reverse direction. P u W u u V u K z u P w u Figure 2.4: Additive plant uncertainty. P u W u u V u K z u w u P Figure 2.5: Multiplicative uncertainty at the plant input. P u W u u V u K P z u w u Figure 2.6: Multiplicative uncertainty at the plant output. Sometimes the choice of what uncertainty model to use is quitte straightforward. Uncertainty at the actuators is well modelled with input uncertainty, while uncertainty at the sensors is well modelled with output uncertainty. However, sometimes choosing the right type of uncertainty is not that obvious. Besides choosing a suitable uncertainty description, a nominal plant model has to be chosen. In general, the uncertainty description and nominal plant model that lead to the least conservative controller should be used. 7 In practice a nominal plant 7 The term conservatism is used to denote that the controller is robust for candidate plants that are not likely to arise in practice. Hence, the achieved performance might be unnecessarily limited.

26 16 Robust Control model that leads to satisfactory results is found by averaging several plant FRFs over the operating range [50]. The final step is to choose the frequency dependent weighting functions V u (s) and W u (s). It is possible to use both weighting functions, but in most cases one of them is set to identity and the other is used to bound the estimated size of the uncertainty. Note that if µ-synthesis is used is does not matter whether W u (s) or V u (s) is set to identity, but if H is used it does matter, [50]. The unscaled uncertainty description u can be obtained for each uncertainty type using (2.15) (2.17) and setting both weighting functions to identity. The weighting function(s) has to be chosen such that it encompasses σ( u ). The above discussion holds for both SISO and MIMO plants. In case of a MIMO plant, it is often desired to model uncertainty for each entry of P separately. This leads to a so called structured uncertainty block, which has several advantages. Unstructured uncertainty Imagine the plant P to be square with n inputs and n outputs. For any of the uncertainty representations in Figure , the uncertainty block has the same dimension as the plant P, i.e., n n. Choosing weighting function V u (s) or W u (s) can be simplified by using scalar transfer functions v u (s) or w u (s): V u (s) = v u (s)i n, W u (s) = w u (s)i n. (2.18) The scaler transfer functions v u (s) and w u (s) are preferably low order and are used to encompasses σ( u ). Structured uncertainty A structured uncertainty description can be used to describe the uncertainty in each plant entry separately, leading to a potentially less conservative controller. The uncertainty block is then not an n n block, but a diagonal matrix of size n 2 n 2 with the n 2 entries of u on the main diagonal. Each entry of u is now approximated by a - preferably low order - transfer function v uk (s) or w uk (s), and these are lined up to form diagonal matrices V u (s) and W u (s) of dimension n 2 n 2. To make these weighting matrices compatible with the plant dimension, two permutation matrices are needed as well [50] Robust stability The RS problems boils down to finding a stabilising controller K that is stable for all plant in the set P u. The closed-loop system M is depicted in Figure 2.3, where M is partitioned as in (2.5). Suppose that there are no performance requirement, i.e., z p and w p are absent. The control problem is then reduced to stability problem as depicted in Figure 2.7. Asymptotic stability can be guaranteed by the Small Gain Theorem (Theorom 4.12 in [48]):

27 2.6 Robust Stability and Robust Performance 17 u w u z u M 11 Figure 2.7: The robust stability problem. Theorem 2.1 Small gain theorem. Consider a system with a stable loop transfer function L(s). Then the closed-loop system is stable if L(jω) < 1 ω (2.19) where L denotes any matrix of satisfying the submultiplicativity property AB A B. In the robust stability problem of Figure 2.7 the (stable) loop transfer function is given by M 11 u. Since the H -norm satisfies the submultiplicativity property, robust stability is achieved if M 11 u < 1. Because u 1, RS is achieved if: M 11 < 1 σ(m(jω)) < 1 ω. (2.20) Condition (2.20) is a necessary and sufficient condition for a full complex disturbance matrix u. In the next two sections it is shown that (2.20) is overly conservative when u exhibits structure. Remark 2.2 An important reason for using the H -norm to analyse robust stability is the submultiplicativity property. For example, an H 2 -norm does not satisfy the submultiplicativity property is RS cannot be analysed using (2.20). Remark 2.3 Stability can also be proven by using the generalised nyquist theorem. Theorem 8.1 in [48] states that - assuming M 11 and u stable - the M 11 u system is asymptotically stable if and only if det(i M 11 (jω) u (jω)) 0, ω, u. (2.21) Introducing the spectral radius, which is defined as the maximum eigenvalue of a matrix: ρ(l(jω)) := max λ i (L(jω)) (2.22) i and under the assumption that u 1, (2.21) can be rewritten:

28 18 Robust Control ρ(m 11 (jω) u (jω)) < 1 ω u max u ρ(m 11 (jω) u (jω)) < 1, ω, (2.23) = max σ(m 11 (jω) u (jω)) < 1, ω, (2.24) u = max σ(m 11 (jω)) σ( u (jω)) < 1, ω, (2.25) u = max σ(m 11 (jω)) < 1, ω. (2.26) u The step from (2.23) to (2.24) is only allowed when u is an unstructured (full and complex) matrix, see Lemma 8.3 in [48] Motivation for the structured singular value As stated earlier is this report, u can also be structured, i.e., u is a norm-bounded (block) diagonal matrix. In many practical applications u exhibits some sort of structure, e.g., when the uncertainty at each plant entry is evaluated separately. So, in case of structured uncertainty, u is only allowed to lie in a certain set u that is composed of complex-valued blocks: u = { diag(δ u1 I r1,...,δ k I rk, u1,..., ul ) : δ ui C; ui C s i t i }. (2.27) Here, the ith scalar block has dimension r i, and the ith full uncertainty block has dimension s i t i. If structured uncertainty is treated as an unstructured uncertainty to evaluate RS, condition (2.20) is sufficient since σ 1. However, it is not a necessary condition anymore, since the structure of u has not been taken into account. 8 To take the structure into account the Structured Singular Value (SSV) is introduced ([13, 39, 54]), for which the definition is given by: µ(m) := and for which the interpretation is: 1 min u u ( σ( u ) : det(i M 11 u ) = 0), (2.28) Find the smallest structured u (measured in terms of σ) that makes the matrix I M 11 u singular; then µ(m) = 1/ σ. Clearly, µ(m 11 ) depends not only on M 11 but also on the allowed structure of u. This is sometimes shown explicitly by using a slightly different notation: µ (M 11 ). The reason why the SSV makes use of the structure in u can be made plausible by stating that scaling of the uncertainty matrix u and closed-loop matrix M 11 does not influence the stability, but changes the maximum singular value of u. For meer details, see [48, Section 8.7]. Remark 2.3 In case of unstructured uncertainties: µ(m 11 ) = σ(m 11 ). 8 In case of structured uncertainty max u ρ(m 11(jω) u(jω)) max u σ(m 11(jω) u(jω))

29 2.6 Robust Stability and Robust Performance 19 A similar condition as (2.20) for robust stability can now be stated for the situation where u exhibits structure. If M 11 and u are stable, and u 1, then RS is guaranteed if and only if: M 11 µ := sup ω µ(m 11 (jω)) < 1. (2.29) Robust performance As stated earlier, Robust Performance (RP) means that the closed-loop system is stable and it achieves the performance specifications in the presence of model uncertainty. Obviously, RP requires NS, NP, and RS. To evaluate RP, a similar approach can be used as for RS, see Figure 2.8, meaning that RP is evaluated using a new -block: p, (P for performance) which is always a full matrix. Figure 2.8 shows that a new p block is pulled out of the closed loop plant, and combined with the uncertainty block u in a new structured uncertainty block : with u u as given by (2.27), and p p, where: = diag( u, p ), (2.30) p = { p C nwp n zp }. (2.31) w u u p z u w p M 11 M 12 z p M 21 M 22 Figure 2.8: The robust performance problem. It is crucial to note that RP implies, but is not implied by, joint NP ( M 22 < 1) and RS ( M 11 µ < 1). The difference is caused by the terms M 21 and M 12, which are generally not zero, but play a role for RP. To evaluate RP, the H -norm ( M < 1) can be used, but since u and hence exhibit structure, this would be an overly conservative approach. Therefore, a good RP condition is given by: M µ := sup ω µ(m(jω)) < 1. (2.32)

30 20 Robust Control Restatement of control problems In this subsection the control problems of Section 2.4 are stated again, but in a more manageable form: Robust Stability (RS): Consider Figure 2.7. Let M 11 and u be stable and let u be structured and bounded by u 1. RS is achieved if and only if: M 11 µ := sup ω µ(m 11 (jω)) < 1. (2.33) Nominal Performance (NP): Consider Figure 2.8. Let M 22 and p be stable and let p be unstructured and bounded by p 1. NP is achieved if and only if: M 22 µ := sup ω µ(m 22 (jω)) = sup ω σ(m 22 (jω)) < 1. (2.34) Robust Performance (RP): Consider Figure 2.8. Let M and be stable and let by bounded by 1. RP is achieved if and only if: M µ := sup ω µ(m(jω)) < 1. (2.35) Note that NS must still hold for all the control problems listed above. 2.7 Solutions to the H optimal control problem In the sections above, it is shown how the control problem is set up in order to design a controller K that minimises a closed-loop system M = F l (G,K), in the presence of uncertainties and performance weights. If a controller is already given, e.g., by manual loop shaping, conditions for NP, RS, and RP can easily be checked by using (2.33) (2.35). This procedure is called µ-analysis. In general, however, the problem is to synthesise a controller K that minimises the closed-loop system M. Several methods to compute H controllers exist. Before 1988, computing H controllers was considered a complex task, see [16]. A general, reliable and computationally effective method is proposed in [13, 19, 54]. This method is often referred to as the DGKF solution or the two-riccati solution and many commercial software tools have implemented this method, see [2, 20] DGKF solution to H control problem In the following, a state-space solution is given to the H control problem. Details about the solution or the derivation can be found in literature, see, e.g., [10, 13, 19, 54].

31 2.7 Solutions to the H optimal control problem 21 Assumptions Some assumptions are generally made in H optimal control, see, e.g., [10, 48]: (A.1) (A,B 2,C 2 ) is stabilisable and detectable. (A.2) D 12 and D 21 have full rank. [ ] A jωi B2 (A.3) has full column rank for all ω. C 1 D [ 12 ] A jωi B1 (A.4) has full row rank for all ω. C 2 D 21 (A.5) D 11 = 0 and D 22 = 0. [ ] 0 (A.6) D 12 = and D I 21 = [ 0 I ]. (A.7) D12 T C 1 = 0 and B 1 D21 T = 0 (A.8) (A,B 1 ) is stabilisable and (A,C 1 ) is detectable The first four assumptions are needed to solve the Algebraic Riccati Equations (AREs) that are introduced later in this section. Assumption (A.1) is required for the existence of stabilising feedback, assumption (A.2) is a sufficient condition to ensure the controllers are proper and hence realisable. Assumptions (A.3) and (A.4) prevent pole/zero cancellations on the jω-axis, which would results in closed-loop instability. Assumption (A.5) simplifies H control and is conventional in H 2 control. D 11 = 0 ensures G 11 is strictly proper (required in H 2 control) and D 22 = 0 simplifies the formulas in the H 2 -algorithm and is made without loss of generality. For H control assumption (A.5) is not required, but simplifies the formulas significantly. Assumption (A.6) can be achieved by scaling of u and y and is often assumed for simplicity. Assumption (A.7) is common in LQG control and means no cross terms in the cost function. If assumption (A.7) holds, then assumptions (A.3) and (A.4) can be replaced by assumption (A.8). None of these assumptions are considered restrictive in practice, since most sensible control problems fulfill them (or can be adjusted to fulfill them), see [10, 37]. H Optimal output feedback control Computing a controller that minimises M is an unsolved problem. Instead, a suboptimal H control problem may be solved: Find a stabilising controller K such that M < γ. By reducing γ the optimal solution is approached. The optimal controller is based on two AREs: A T X + X A X (B 2 B T 2 1 γ 2B 1B T 1 )X + C T 1 C 1 = 0, (2.36) A T Y + Y A Y (C T 2 C 2 1 γ 2CT 1 C 1 )Y + B 1 B T 1 = 0, (2.37)

32 22 Robust Control with associated Hamiltonian matrices: [ H := 1 A γ B 2 1 B1 T B 2B2 T C1 TC 1 A T ] [, J := A T 1 γ C T 2 1 C 1 C2 TC 2 B 1 B1 T A ]. (2.38) A solution of the suboptimal control problem exist is the following conditions are fulfilled: 1. X 0 is a solution of the Controller Algebraic Riccati Equation (HCARE) (2.36). 2. Y 0 is a solution of the Filter Algebraic Riccati Equation (HFARE) (2.37). 3. The coupling condition is fulfilled: ρ(x Y ) < γ 2, (2.39) where ρ is the largest eigenvalue as defined in (2.22), but here X and Y are constant matrices. 4. The Hamiltonian matrices (2.38) do not have eigenvalues on the jω-axis. With the above conditions satisfied, a controller of similar form as in (2.4) that satisfies M < γ is given by: A K = A + 1 γ 2B 1B1 T X + B 2 F + Z L C 2, (2.40) B K = Z L, (2.41) C K = F, (2.42) D K = 0, (2.43) where: F := B T 2 X, (2.44) L := Y C2 T, (2.45) Z := (I 1γ ) 2Y X. (2.46) µ-synthesis As shown in Section 2.6 by (2.33) (2.35), the SSV is used to evaluate RS and RP, whilst the DGKF solution only considers the H -norm. Calculating a controller directly while evaluating RS and RP by using the less conservative µ-norm, is still impossible. Only iterative procedures exist, consisting of a sequence of optimisation steps. This design approach is called µ-synthesis. One example of such an approach is DK-iteration, which is discussed in [39].

33 2.8 Feedforward design Feedforward design If there are no disturbances and modelling errors, a well-designed feedforward signal leads to the desired response. However, disturbances and modelling errors are always present and feedback control must be used to guarantee stability and tight performance. In ASML applications, the main task of the feedback controller is to keep the servo errors small during exposure. Reduction of the settling-time is a side-effect of a good feedback controller. If improved settling behaviour is desired, as for a wafer stage, feedforward control should be used as well, see [41, 50]. In [30] trajectory planning and feedforward design for electromechanical motion systems is explained by means of a simple example and experimental results are shown to illustrate the advantages of using a well-tuned feedforward signal. 2.9 Summary and conclusions In this chapter the theory behind H -optimisation and µ-synthesis is described. H -Optimisation and µ-synthesis allow the control engineer to design MIMO controllers by specifying the closed-loop performance, while taking model uncertainties into account. When MIMO systems need to be operated at their physical limit, it is not sufficient to use a set of simple SISO controllers. To be able to deal with interaction between plant entries, MIMO control techniques like H -optimisation and µ-synthesis are required. A useful feature of the general control configuration is the separation of measured and performance variables. If internal dynamics between the measured and performance outputs cause a significant servo error, an explicit distinction between these variables can be beneficial. The next chapter describes how the control problem for an ASML wafer stage can be stated, such that it can be solved using µ-synthesis.

34

35 Chapter 3 Robust control for an ASML wafer scanner 3.1 Introduction This chapter discusses the application of the theory introduced in Chapter 2, on an ASML wafer scanner. In [50], the controller design procedure is described more elaborately. The goal is to control the Short-Stroke Device (SSD) of the T-5 wafer stage. The SSD has 6 DOFs, three translations (x,y,z) and three rotations (R x,r y,r z ), leading to the following partitioning: y = x y R z R x R y z P x x P z x =..... P x z P z z F x F y T Rz T Rx T Ry F z = Pu. (3.1) Only three DOFs (y,r x,z) are subject to MIMO controller design, which simplifies the controller design procedure. The other three DOFs are controlled by SISO controllers. Reasons for choosing these three variables are given in [50], but it is fair to say that every possible 3 3 subsystem has interaction with the other DOFs in the system, which are controlled by the SISO controllers. 3.2 Control goal and control structure The control goal is to design a feedback controller that stabilises the closed-loop system and keeps the servo error and feedback control action within a certain bound, under the influence of model uncertainties and disturbances. A suitable control structure used to tackle this problem is given in Figure 3.1, where the performance weights are set to identity. The servo 25

36 26 Robust control for an ASML wafer scanner error and feedback action are denoted by z p1 and z p2, respectively, and disturbances at the plant input and plant output are denoted by w p2 and w p1, respectively. z p1 z p2 u ff w p2 w p1 W p1 W p2 V p2 V p1 r y z p1 z p2 w p2 w p1 K u P Figure 3.1: Control structure with performance weights W p1,w p2,v p2,v p1. The control structure of Figure 3.1 has three control DOFs: the reference trajectory r that is assumed to be given, the feedforward signal u ff such that nominal reference tracking is achieved, and the controller design K such that robust tracking is achieved along the setpoint r under the influence of disturbances and model uncertainties. This is motivated by looking at the relationship between the variables r,u ff,w p1,and w p2 on the one hand, and the servo error z p1 on the other hand. This relationship is given by: z p1 = S(r Pu ff ) Sw p1 SPw p2, (3.2) }{{}}{{}}{{} (i) (ii) (iii) where S = (I +PK) 1 is the sensitivity. The feedforward signal only turns up in (i), whereas the feedback controller K turns up in all parts of (3.2) via S. Since the reference trajectory r is given (designed off-line), u ff can also be designed off-line such that r = Pu ff. Since both r and u ff are designed in advance, delays, and right half-plane zeros of P are not necessarily limiting factors. Therefore (i) will probably be small. If this is assumed, a regulator problem remains, which only involves designing a feedback controller. Additional assumptions are listed below. 1. P is square, i.e., P has the same number of inputs and outputs. 2. P is approximately rigid body decoupled up to the target BW for each of the separate loops. 3. The diagonal entries of P have a -2 slope at least till the target BW. Assumptions 1 and 2 are imposed to justify multiloop SISO design and to facilitate MIMO design, since useful ideas from SISO design, e.g., bandwidth, phase- and gain margin, can be adopted. For mechanical positioning devices as discussed here (force input, position output), the plant P typically exhibits rigid body behaviour, which appears as a -2 slope for low and midrange frequencies, combined with resonant behaviour and roll-off for the higher frequencies (strictly proper plant). Assumption 2 and 3 imply that there is only little interaction between

37 3.3 Plant modelling 27 different control loops up to the target BW and that the rigid body modes are the dominant modes for frequencies below the target BW. Around the target BW, other modes (flexible modes) become visible and there is interaction between the several DOFs. 3.3 Plant modelling A model for the six DOFs of the SSD of the T-5 wafer stage is obtained experimentally by using white noise (FRF measurement). For the three individual SISO controllers the three corresponding diagonal plant entries are needed, while for the 3 3 MIMO controller a 3 3 model of the corresponding subsystem is needed. In total, twelve FRF models are needed: six for the diagonal components and six for (part of) the off-diagonal components. The twelve FRF models are then individually approximated by SISO transfer functions in the relevant frequency region, which is somewhere between between 20 Hz and 1400 Hz. The approximation can of course be done manually, but special curve fitting algorithms exist that lead to very accurate fits. The 3 3 MIMO plant model is obtained by stacking together the nine SISO fits, which leads to a very high order system. Since, too high model orders give problems during controller design, model reduction is used to decrease the order of the model. Several types of model reduction techniques exist, but in [50] a rather ad hoc approach is used. 3.4 Performance quantification In H -optimisation and µ-synthesis, weighting functions can be used to quantify control goals. Figure 2.2 shows this idea in the general control configuration. The control structure of Figure 3.1 can be written in a similar form, where the exogenous disturbance variables w pi, and regulated variables z pi are stacked in vectors w p and z p, respectively 1 : z p = [ zp1 z p2 ] [ wp1,and w p = w p2 ]. The weighted closed-loop system M of Figure 2.2, that relates the exogenous inputs to the regulated outputs then becomes: [ Wp1 SV M = p1 W p1 SPV p2 W p2 KSV p1 W p2 KSPV p2 ], (3.3) where S is again the sensitivity: S = (I + PK) 1. The nominal performance criterium given by (2.34) can now be used to synthesise a controller for this so called: four-block control problem. Four closed-loop transfer function matrices appear in (3.3): the sensitivity S, the control sensitivity KS, the process sensitivity, SP, and KSP that equals the complementary sensitivity. There are a few reasons that make the four-block control problem a sound problem 1 Note that z p1 and w pi can be vectors themselves.

38 28 Robust control for an ASML wafer scanner formulation, see [15]. The most obvious reason is that pole/zero cancellations are excluded, due to the inclusion of SP as a closed-loop Transfer Function Matrix (TFM). The four-block control problem therefore exhibits some robustness against uncertain resonances, which is beneficial for positioning devices with resonant behaviour. An important part of the controller design procedure, is specifying the performance weighting filters of Figure 3.1. Every weighting filter is a series connection of three individual filters: one for loop shaping, one for proper scaling, and an optional one to account for power spectra. In Figure 3.2 this series connection is depicted for V p1. V p1 w p1 V pw p 1 V ls p 1 V sc p 1 w p1 Figure 3.2: Internal structure of performance filter V p Weighting filters for scaling The importance of good scaling for MIMO system can be illustrated best by means of a simple example. Consider the TFM S of a 2 2 MIMO system. This TFM relates the output disturbances (or the reference signal) to the servo error. For good disturbance rejection, the diagonal entries of S should be small, since these entries relate variables of the same unit. However, for the off-diagonal entries this makes no sense, since they relate variables of different units. Norm-based controller design methods like H -optimisation and µ-synthesis would spend a lot of effort trying to reduce a single, though not relevant, entry, at the cost of an increase in all other entries. Several ways to scale variables exist. One possibility is to scale signals by their maximum allowed magnitudes. Servo errors with tight performance requirements are then penalised more severely. Another option is to scale signals, especially disturbances, by the expected magnitudes, but this information is not readily available. A fair assumption that can be used is that output disturbances are comparable in magnitude to the servo errors. In the controller design procedure for the SSD the maximum allowed servo error is used to scale both the servo error and output disturbances: Vp sc 1 = (Wp sc 1 ) 1. The plant input disturbances and controller actions could be scaled as well, but for the controller design discussed here, they drop out of the control problem formulation. So, these scaling filters can be simply set to identity. Remark 3.1 Scaling filters are diagonal matrices, since the off-diagonal components make no sense Weighting filters for loop shaping The loop shaping filters are in place to specify the desired closed-loop response, and are therefore very import. In principle any kind of filter can be used for loop shaping, but

39 3.4 Performance quantification 29 choosing unrealistic filters leads to an unsolvable control problem. Hence, it is wise to set up the filter design such that it has a strong link with manual loop shaping for SISO controllers. In order to accomplish this link with manual loop shaping some remarks can be made: 1. The target bandwidth f BW. Based on the assumptions that the MIMO plant is approximately decoupled up to the target BW, the target BW for the ith control loop can be defined in a similar way as for SISO systems: the ith open-loop transfer function crosses the 0 db line from above for the first time. Here, the ith open-loop is defined as L ii = P ii K ii, with {.} ii the ith diagonal entry of a TFM. 2. The frequency f I. Below this frequency the controller must have integral action to suppress low-frequency and constant disturbances. 3. The frequency f R. Above this frequency the controller must roll-off to suppress noise and achieve robustness against model uncertainties. From here on the following diagonal TFMs are defined ( X contains the diagonal elements of X): P = diag(p ii ), K = diag(kii ), L = diag(lii ), S = diag(sii ), (3.4) with i = 1,...,n and n the number of control loops. The first loop shaping filter that is specified is V ls p 1, which is set to be the identity matrix: Next, V ls p 2 at the plant input is specified as follows: V ls p 1 = I n. (3.5) Vp ls 2 = (Vp sc 2 ) 1 P 11 (j2πf BW1 )... Pnn(j2πfBWn) 1 V sc p 1, (3.6) }{{} P fbw with P fbw a diagonal matrix containing the gains of the diagonal plant entries around the target BWs for each control loop. When the weighting filters for Vp ls 1 and Vp ls 2 are filled in into (3.3), the closed-loop matrix M becomes: M = [ Wp ls 1 Wp sc 1 SVp sc 1 Wp ls 2 Wp sc 2 KSVp sc 1 Wp ls 1 Wp sc 1 SP P fbw 1 Vp sc 1 Wp ls 2 Wp sc 2 KSP P fbw 1 Vp sc 1 ]. (3.7) When assuming P P and S S at the target BWs, the left and right columns of M in (3.7) have about the same size. Below the target BWs, the right column dominates the left one, and above the target BWs the left column dominates the right one. These properties

40 30 Robust control for an ASML wafer scanner are exploited for choosing the other loop shaping filters W ls p 1 and W ls p 2, which are depicted in Figure 3.3. Note that the weighting filters to account for power spectra are set to identity to facilitate the discussion of loop shaping filter. At low frequencies, where P has a -2 slope, the following reasoning holds: L = PK P K. If K must have a -1 slope, L must have a -3 slope. Moreover, S = (I + L) 1 L 1 L 1 and SP L 1 P. Since the SP-part of the right column dominates the S-part at low frequencies, W ls p 1 should enforce SP to have a +1 slope: ( ) Wp ls s + 2πf Ii 1 = diag k Ii, (3.8) s with i = 1,...,n, and n the number of control loops. Typically, f Ii is at least four times smaller that f BWi. The parameter k I can be used to set a maximum allowed magnitude for the diagonal sensitivity entries. Sensitivity peaks typically need to be lower than 6 db, so k I = 1/2 is a suitable value. At high frequencies a similar reasoning is valid. The plant magnitude P approaches zero and S I. In the second row of M in (3.7), the KS-part dominates the KSP-part and requiring K to have a roll-off with a -2 slope, implies that KS should have a -2 slope. This can be achieved by choosing the following weighting filter (with a +2 slope) for each entry of W ls p 2 : ( Wp ls 2 = diag k Ri α 2 s 2 + 4πβ Ri f Ri s + (2πf Ri ) 2 ) i s 2 + 4πν Ri f Ri s + (α i 2πf Ri ) 2. (3.9) Typically, f Ri is at least four times larger than f BWi. In general, all weighting filters need to be proper to have a state space realisation. So, Wp ls 2 need to be cut-off at a fairly high frequency f = αf R, e.g., α = 10. Parameters β R and ν R are damping parameters and are set to 0.7. Parameters k Ri are chosen such that the first and second row of M in (3.7) are the same at the target BWs. With K = P 1 L,P P, and L I n, this leads to: diag(k Ri ) = W sc p 1 P fb W (W sc p 2 ) 1. (3.10) The controller output scaling W sc p 2 in (3.7) then drops out Weighting filters to account for power spectra If knowledge is available on the frequency contents of the exogenous and regulated variables, this information can be used in the controller synthesis. Suppose that a controller has been synthesised that leads to servo errors that are dominated by one or two frequencies. The freedom of an additional weighting filter can be used to design a new controller that explicitly accounts for disturbances at these frequencies.

41 3.5 Uncertainty quantification 31 W ls p 1 W ls p k I k R f I f BW f f BW f R αf R f Figure 3.3: Asymptotes of the loop shaping filters W ls p 1 and W ls p Uncertainty quantification As described in Chapter 2, various sources for uncertainty exist. In [50], two types of experiments are performed: controller design for a single operating point (machine stand still), and controller design for a grid of operation points on a line (machine scan). For both experiments the additive uncertainty description of Figure 2.4 is used. However, the amount of uncertainty is different for both experiments. Including uncertainty weights in the control problems leads to a new closed-loop system M: M = W u KSV u W u KSV p1 W u S I V p2 W p1 SV u W p1 SV p1 W p1 SPV p2 W p2 KSV u W p2 KSV p1 W p2 KSPV p2 with S I the input uncertainty, which is defined as: S I := (I + KP) 1., (3.11) For the machine standstill scenario, it is assumed that there is only uncertainty due to the performed model reduction. The amount of uncertainty for each plant entry, i.e., the uncertainty weight, is determined by subtracting the reduced order model from the measured FRF. For the machine scan scenario, multiple FRFs are determined for five positions along the y-axis. For the diagonal plant entries the difference between the FRFs are small, especially for the three translation directions (x, y, z). For the off-diagonal entries, the differences are relatively large and for some entries even greater than 100%. For both scenarios not all the entries of the uncertainty matrix are sufficiently large compared to the nominal plant and are therefore neglected. 3.6 Results Without describing all the experiment done in [50], some important design steps, findings, and conclusions drawn in [50] are given. For a stand still experiment, the MIMO controller performs better than the multiloop

42 32 Robust control for an ASML wafer scanner SISO controller, when only looking at the sensitivity TFM. Performance is then judged irrespective of the type of time domain verification experiment. The off-diagonal sensitivity peaks are better suppressed with the MIMO controller, at the cost of a lower achieved BW and an increase in other - less relevant - off-diagonal entries. 2. This is not surprising, since plant interaction has not been accounted for in the SISO controller design; it can only be checked after a controller has been designed. However, when looking at the power spectra of the error signals for a 100 [s] stand still experiment, it appears that the MIMO controller is only slightly better in a small region between 140 [Hz] and 180 [Hz] for the z-direction, where plant interaction occurs. In other regions the performance is worse. For the y- and z-direction it is unclear which controller performs best and for the R x -direction, the multiloop SISO controller is much better than the MIMO controller for almost all frequencies. However, the MIMO controller still performs within the performance requirements. Additional weighting filters to account for power spectra are useful to get some extra reduction in servo error at local frequencies. However, for the system studied, disturbances with time-varying frequencies and amplitudes seem to occur. These disturbances cannot be handled well by applying some additional weighting. In addition, a better servo error - especially at high frequencies - is no guarantee that the actual performance at the exposure spot has improved. For a scan experiment along the y-axis, position dependent plant dynamics need to be taken into account, leading to a bigger sized uncertainty matrix. When the setpoints for the R x - and z-direction are set to zero, the scan performance in y-direction is better than with the MIMO controller, but is worse in the R x - and z-direction. Nevertheless, the R x and z errors are still within the performance requirements. The reason for this is that the feedback controller K is designed for a pure disturbance rejection problem, which is only justified for an ideal feedforward. Since plain acceleration feedforward is used, there is a mismatch in the feedforward signal and the term S(r Pu ff ) in (3.2) is non-zero. Because the sensitivity components S y Rx and S y z are much greater for the MIMO controller, the effect of the mismatch in the feedforward signal is greater. A similar scan experiment along the y-axis is performed, but with wafer setpoints for the R x - and z-direction 3. The MIMO controller now performs considerably better than the multiloop SISO controllers, since the MIMO controller explicitly accounts for plant interaction. 3.7 Summary and conclusions In this chapter the controller design procedure for an ASML wafer stage is described according to [51] and some comments are made regarding experiments on the SIRE T-5 short stroke test rig. The design process as presented in this chapter seems like a fixed design scheme, where the steps need to be taken in a pre-defined order. This is, however, not true, since the 2 In order to achieve enough robustness, the uncertainty weights had to be multiplied by a factor two, which is a rather ad hoc approach. 3 At ASML this is called levelling. The goal is to keep the exposed field better into focus.

43 3.7 Summary and conclusions 33 procedure is an iterative process of adjusting filters, judging closed-loop performance, and re-adjusting the filters again. In general, shaping closed-loop TFMs gives the control engineer a lot of design freedom. Choices for the weighting filters for performance are, however, not that straightforward as for SISO loop shaping. In principle, any kind of filter can be used for loop shaping, but choosing unrealistic filters leads to an unsolvable control problem. Hence, it is wise to set up the filter design such that it has a strong link with manual loop shaping for SISO controllers. For the control of the wafer scanner, this is done by choosing appropriate scaling- and loop shaping filters to specify the closed-loop performance, see Section 3.4, which allows for a performance specification in terms of BW, integral action and controller roll-off. However, the controllers designed in [50] do not all satisfy the robust performance criterium of (2.35), meaning that performance cannot be guaranteed for all possible plants in the uncertainty set. In addition, when flexible modes are present below the target BW, the weighting filter selection procedure as presented in [50] has to be adjusted. The models used for controller design are created by stacking together separate SISO plants, leading to - unnecessarily - high order models that make the DK-iteration procedure infeasible. Model reduction leads to lower order models, but because of common dynamics between separate loops, the order of the model is still too high. Ideally, a good direct MIMO model should be identified, leading to a lower order plant model. A similar reasoning can be applied to the modelling of uncertainties. It is expected that better models of the plant and uncertainties can improve the achieved performance significantly. For more information on this topic, see [38]. Because the wafer stage has six rigid body modes, six actuators and six sensors are used to control the six rigid body modes. Control effort is not used to actively control the flexible modes, e.g., by adding damping to them, but the target BWs are set low enough such that the flexible modes do not cause instability. However, not satisfying the robust performance criterium means that the flexible modes can cause problems, since the performance criterium is not satisfied for all possible plants. A possible way to control the flexible modes is to add extra actuators and sensors, leading to an n m plant, where n,m N = 6 and possible n m. In the next chapters different ways to explicitly control flexible modes are discussed. In Chapter 4 the focus is on the control of flexible modes in structures, i.e., systems without rigid body modes, and in Chapter 5 vibration control for flexible motion systems is discussed.

44

45 Chapter 4 Vibration control of flexible structures 4.1 Introduction Methods for vibration control can be divided into two groups: passive vibration control and active vibration control. Passive vibration control is aimed at modifying the plant properties, like stiffness, mass, and damping, and is not discussed further in this report. Active vibration control uses actuators to actively compensate for flexible modes or to add damping to the flexible modes. The field of science that is interesting for motion control is often called structural vibration control. Structural vibration control is frequently discussed in literature, see, e.g., in [18, 40], but often only focusses on flexible structures, i.e., systems without rigid body modes. Although these rigid body modes are the key element in motion system, control methods that focus only controlling the flexible modes can still give valuable insights. Structures can also have rigid body modes. For example, a deep space network antenna that can rotate around its azimuth (vertical) and elevation (horizontal) axis. Control forces can then be divided in tracking forces to follow a reference and damping forces to suppress vibrations. 1 Typically, the controller design is split up in two parts and both parts are designed independently. The first part, sometimes referred to as the high-authority controller, focusses on tracking the reference signal and the second part, sometimes referred to as the low-authority controller, focusses on suppressing the flexible modes. The separation is made because of the large difference in control effort for the high- and low-authority controller, see [18, Chapter 10]. A lot of work has been done by G.J. Balas in the field of vibration attenuation for large space structures, see [1, 3 7]. In this work the flexible modes of a structure, called the Caltech experimental flexible structure, are suppressed by using µ-synthesis. Although such a structure is very different from a wafer stage, similarities occur in the form of closely spaced flexible modes and uncertainty in the high-frequency modes. Other applications of active vibration control can, for example, be found in [26] and [21]. In [26], flexible modes of 1 Also in systems with no rigid body modes it is possible to let part of the system follow a reference signal. For example, the swing arm in an optical drive [12]. 35

46 36 Vibration control of flexible structures a circular plate are suppressed using piezo actuators and sensors. Robustness to parameter uncertainty is acquired by using Lyapunov stability theory. In [21], vibration of a flexible beam is suppressed with two piezoelectric actuators and a piezoelectric sensor, using robust model reference control. The outline of this chapter is as follows. In the next section, the modal system description is introduced, because this leads to valuable insights in the system dynamics. Putting a system in modal coordinates does not change the input/output behaviour, but merely adds structure to the states of the system. After that, various types of modal control techniques are discussed, as well as the effect of neglecting modes in the controller design (spillover). Finally, the work done by G.J. Balas is briefly described. 4.2 System description Distributed parameter system Since mechanical systems are distributed-parameter systems, they are infinite dimensional 2 and can be represented by the generalised wave equation [33, 35, 45]: M(p) δ2 w(p,t) δt 2 + Lw(p,t) = u(p,t), (4.1) which must be satisfied at every point p. The displacement of a point p along the body is denoted as w(p,t), M(p) denotes the distributed mass of the system, L is a linear self-adjoint positive definite differential operator 3, and u(p,t) is the distributed control vector, e.g., an external force distribution. In theory, the system described by (4.1) consists of infinitely many modes, corresponding to a set of natural frequencies, ω i, and mode shapes, φ i (i 1,..., ). Note that w(p,t) (and u(p,t)) are scalar functions of time and position. Since it is not possible (and necessary) to model infinitely many modes, the system is modelled by a finite number of modes, n (discretised), resulting in a lumped parameter model. The justification for this assumption is that the bandwidth of the actuators and sensors is too low to respond to the highest frequency modes and that the highest modes in a physical structure are harder to excite. Therefore, they are not really part of the control problem [8]. A distinction can therefore be made between the modelled modes, corresponding to mode shapes φ i (i 1,...,n) and the unmodelled modes, corresponding to φ i (i n + 1,..., ). From here on, it is assumed that the number of modelled modes is chosen sufficiently large to accurately describe the system. 2 This is not entirely true because at molecular level there is no material continuum. This is further discussed in the end of this section. 3 A self-adjoint matrix (or Hermitian matrix) is a square matrix with complex entries that is equal to its own conjugate transpose, that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j

47 4.2 System description Nodal models A system description that is commonly used to analyse structural dynamics is the second order structural model. For a system with n DOFs, m inputs, and l outputs, the system is represented by second order linear differential equations: M q + D q + Kq = B o u, y = C oq q + C ov q. (4.2) In this equation, q is the n 1 nodal displacement vector, q the n 1 nodal velocity vector, q the n 1 nodal acceleration vector, u the m 1 input vector, y the l 1 output vector, B o the n m input matrix, C oq and C ov the l n output matrices 4, M the n n positive definite mass matrix, and K and D the n n positive semidefinite stiffness and damping matrices, respectively. It is also possible to represent the second order system as a set of linear first order differential equations, i.e., a state space representation (without direct feedthrough term). This state space description is characterised by the matrix triple (A,B,C): ẋ = Ax + Bu, y = Cx. (4.3) In order to obtain a state representation, the nodal model of (4.2) is rewritten: q + M 1 D q + M 1 Kq = M 1 B o u, y = C oq q + C ov q, (4.4) where it is assumed that the mass matrix is nonsingular. Introducing the state vector x as a combination of the structural displacements q and structural velocities q: x = [ x1 x 2 ] = [ q q ]. (4.5) The state vector x contains the minimal number of physical variables that allow for a unique calculation of the output variables y from the applied inputs u. The state vector x is not unique and the states are linearly independent, i.e., a state is not allowed to be a combination of other states. Combining (4.4) and (4.5) leads to the following state-space matrices: [ A = 0 I M 1 K M 1 D ] [, B = 0 M 1 B o ], C = [ C oq C ov ], (4.6) where A has dimensions 2n 2n, B is of size 2n m, and C has size l 2n. 4 Note that in this representation only position and rate sensors are taken into account.

48 38 Vibration control of flexible structures Modal models Modal coordinates are often used in the analysis and control of flexible structures. The power of the modal system description is that all the independent modes of the system are directly visible, in either the second order description, or in the state space description. Transforming the second order system of (4.2) into modal coordinates can be accomplished by using a modal transformation: q = Φq m. (4.7) The transformation matrix Φ can be derived by solving (4.2) for the undamped situation (D = 0) and without external excitation (u = 0): M q + Kq = 0. (4.8) The solution of (4.8) is q = φe jωt, so the second derivative of the solution is q = ω 2 φe jωt. Introducing these expressions for q and q in (4.8) gives: (K ω 2 M)φe jωt = 0. (4.9) This is a set of homogeneous equation, for which a nontrivial solution exists if the determinant of (K ω 2 M) is zero: det(k ω 2 M) = 0. (4.10) Condition (4.10) is satisfied for n values of frequency ω. These frequencies are called natural frequencies and are denoted as ω 1,...,ω n. Substituting ω i into (4.9) gives the corresponding set of vectors φ i,...,φ n that satisfy this equation. The ith vector φ i, corresponding to the ith natural frequency, is called the ith natural mode, or mode shape, of the system. Mode shapes are not unique since they can be arbitrarily scaled. For notational conveniences, the matrix Ω is defined as an n n matrix with the natural frequencies on its main diagonal: ω 1 0 Ω =....., (4.11) 0 ω n and the matrix of mode shapes, or modal matrix Φ, is defined as: Φ = [ φ 11 φ n1 ] φ 1 φ n =....., (4.12) φ 1n φ nn

49 4.2 System description 39 where φ ij is the jth displacement of the ith mode shape. The model transformation matrix Φ has an interesting property: it diagonalises the mass and stiffness matrices: M m = Φ T MΦ, (4.13) K m = Φ T KΦ, (4.14) where M m is called the modal mass matrix and K m the modal stiffness matrix. 5 If the same transformation is applied to the damping matrix D, the modal damping matrix is obtained: D m = Φ T DΦ. (4.15) Unlike the modal stiffness and modal mass matrix, the modal damping matrix is not always a diagonal matrix. If the modal damping matrix is a diagonal matrix the system is said to be proportionally damped. The proportionality of damping is often assumed, since off-diagonal terms often have a negligible impact on the structural dynamics, see [18]. Proportional damping is often achieved by assuming that the damping matrix is a linear combination of the mass and stiffness matrices. Transformation from the variable q to q m can be accomplished by using (4.7). Introducing this transformation to (4.2) and left-multiplying by Φ T leads to: Φ T MΦ q m + Φ T DΦ q m + Φ T KΦq m = Φ T B o u, y = C oq Φq m + C ov Φ q m. (4.16) Assuming proportional damping, and using the expressions for the modal matrices, M m, D m, and K m, (4.16) can be rewritten as: M m q m + D m q m + K m q m = Φ T B o u, y = C oq Φq m + C ov Φ q m. (4.17) Left multiplying (4.17) by M 1 m, and introducing a new notation, leads to a more insightful system description: q m + 2ZΩ q m + Ω 2 q m = B m u, }{{} u m y = C mq q m + C mv q m, (4.18) 5 Since the modal transformation diagonalises the mass matrix, it is often said that the mode shapes are orthogonal with respect to the mass matrix of the system. If a system has a mass matrix that is a scalar multiple of the unity matrix the mode shapes Φ i are orthogonal as well. The modal matrix is often normalised such that the modal mass matrix or modal stiffness matrix is a unity matrix, but other type of normalisation are possible.

50 40 Vibration control of flexible structures where Ω given by (4.11) and Z is a diagonal matrix with the modal damping factors 6 ζ i on the main diagonal. Analogous to (4.2), B m is the modal input matrix and C mq and C mv are the modal output matrices. The column vector u m is the modal control vector containing modal control forces, i.e., some abstract forces corresponding to modal coordinates. Note that the upper part (4.18) is a set of n uncoupled equations from input u m to q m, due to the diagonality of Ω and Z. Each modal input force works on a separate mode. This decoupling is referred to as internal decoupling. The set of decoupled equations can also be written for i = 1,...,n as: q mi + 2ζ i ω i q mi + ωi 2q mi = b mi u, }{{} u mi y i = c mqi q mi + c mvi q mi, (4.19) y = n i=n y i, where b mi is the ith row of the input matrix B m, and c mqi and c mvi are the ith columns of the input matrices C mq and C mv, respectively. An important property of linear systems is clearly visible in (4.19), i.e., the total response y of a linear system is a sum of the individual modal contributions y i. Similar as for the second order system in nodal coordinates, the second order system in modal coordinates can also be written in state-space form. Several forms exist, see [18], but the most straightforward form is the one with state vector: x = and each component x i consisting of two states: x 1. x n, (4.20) x i = [ xi1 x i2 ] = [ qmi q mi ]. (4.21) The modal state space form is then a triple (A,B,C) characterised by a 2n 2n block diagonal system matrix A m, consisting of 2 2 blocks A mi. In a similar fashion, the modal input and modal output matrices are divided, correspondingly: B m = B m1. B mn, C m = [ C m1 C mn ], (4.22) where B mi and C mi are 2 m and l 2 blocks, respectively. Considering the modal description with a state vector as in (4.21), leads to the following blocks: 6 Modal damping factors are also called (dimensionless) damping ratios.

51 4.2 System description 41 [ A mi = 0 1 2ζ i ω i ω 2 i ] [ 0, B mi = b mi ], C m = [ C mqi C mvi ]. (4.23) Relevance of modal analysis Although the models discussed in this section tend to describe the real system, they never exactly describe the real system. This also holds for the Partial Differential Equation (PDE) given by (4.1). A PDE is just a mathematical model of the real structure and to come up with the model a set of idealisations is made. All these assumptions are reasonable, but collectively they bring forth a more significant difference between model and real structure. The modal analysis focusses on solving the PDE by seeking solutions in the form: w(p,t) = φ i (p)q mi (t). (4.24) i=1 Here φ i (p) is the analytical mode shape and q mi (t) the modal coordinate. 7 Both are also defined in the previous subsection, but here the ith mode shape is a function of position p, instead of an n 1 column vector. Although a PDE can be solved in this way, the question at hand is whether the real system also possesses these modes. In [28] an attempt is made to answer this question. Some relevant conclusions of this work hare listed below: 1. Neither a PDE model nor any other mathematical model of a system is exact, even for a very simple system such as a cantilever beam. If a structure or system is more complex, it is in general harder to formulate a (PDE) model. Even when a PDE model exist, it is still very difficult to extract numerical information from it. 2. A Finite Element Model (FEM) is a very powerful numerical method to solve a PDE model. A nice feature of a FEM is that it converges to the exact solution, which is the solution of the PDE model only. However, it should be kept in mind that the PDE model is only an approximation of the real physical structure. 3. No real structure has an infinite number of modes. A system can never have more modes than molecules and natural frequencies that require particles to move faster than the speed of light also have no practical relevance. 4. The idea of a mode is a pure mathematical one and it is highly unlikely that any real structure can vibrate exactly in a mode shape. In other words, it is highly unlikely that a real structure has any modes. However, as an approximation the idea of a mode is an excellent one. Especially for the lower modes. 5. Model reduction can be done by truncating modes with high frequencies, resulting in a model containing the lower modes. A better approach is to use mode selection, based on an appropriate error criterion. 7 This type of solution is based on the more general mathematical idea of separation of variables.

52 42 Vibration control of flexible structures 4.3 Modal control In this section, two modal control techniques are discussed. The essence of modal control is to control the individual modes of a system. If a system is written in modal coordinates, a set of decoupled second order equations, similar to (4.18), can be used to describe the system. The system is said to be internally decoupled, i.e., decoupled from modal input u m to modal output q m. A distinction can then be made regarding the controllers that have to be designed. If feedback forces for a certain mode are not allowed to depend on states of other modes than the mode to be controlled, then the set (4.18) is also externally decoupled. This is the case in Independent Modal Space Control (IMSC), see, e.g., [34, 47]. If the controller forces are allowed to depend on all modes of the system model, there is no external decoupling and this is called coupled control. In the following, it is assumed that the modelled system is fully observable and controllable, regardless of the number of actuators and sensors. The reason for this assumption is given in the next subsection Independent modal space control A relatively simple way to control flexible modes is to use IMSC, as discussed in [34, 47]. Linear second order systems can be written in modal form as is discussed in Section 4.2. The resulting set of independent second order systems can then be controlled by designing controllers for every separate sub-system. Hence, the sub-systems are internally and externally decoupled, which is the essence of IMSC. The obtained control gains then lead to modal control forces u m, i.e., some abstract forces corresponding to modal coordinates. The benefit of this approach is that controllers can be designed easily, regardless of the size of the model. In [18] a distinction is made between high- and low-authority controllers. This distinction allows the problem to be divided in a servo problem, i.e., following a command precisely (which is done by the high-authority controller) and a vibration suppression problem (which is done by the low-authority controller). In [18, Chapter 10] rate sensors are used in combination with force actuators in a collocated setting (actuators are placed at the same location as the sensors) for vibration suppression, which is called dissipative control. In classical loop shaping terms, using rate sensors is the same as using a pure derivative action (D-action) when the position is measured. In this way, damping can be added to the flexible modes. In [18, 34] a method is presented to determine the required gains of the controller (D-action) to place the closed loop poles of the system. However, manual loop shaping or other types of controller design methods are possible, e.g., pole placement, LQG, H -optimisation or Efficient Modal Control (EMC) [47]. The obtained modal control forces u m, need to be transformed back into real actuator forces u. For a system with the same number of actuators as modes (m = n), this can be accomplished by: u(t) = B 1 m u m. (4.25) However, for systems where the number of actuators is smaller that the number of modes (m n), the inverse in (4.27) can be replaced by the pseudo-inverse: B m. Since the pseudo-

53 4.3 Modal control 43 inverse is not an actual inverse, errors in the actual control force can be expected. The actual control forces u may therefore not lead to the desired modal control forces u m. This is especially true when some modes are uncontrollable or when the number of actuators is much smaller than the number of modes. For the former situation, the required modal control forces can then never be created out of the real actuator forces. It is expected that a small mismatch in modal control forces u m probably leads to a too large performance degradation. In literature, however, it is often assumed that a large number of sensors and actuators is available, or that the number of actuators and sensors is equal to the number of modes. Although good vibration suppression can be achieved by using modal control with many actuators and sensors, it is believed that modal control will probably not work satisfactorily in its current form for vibration suppression purposes in high-performance applications with only a limited number of actuators and sensors. The main reason for this is the error introduced by the pseudo-inverse. However, insight in the dynamics of the system using the modal system description is valuable. For example, for actuator/sensor placement. Other ways than the standard pseudo inverse could be used to derive the real actuator forces. For example, a different algorithm for the pseudo-inverse or the control engineer could use his own experience to manually adjust the real actuator forces obtained by using the pseudo inverse Coupled control The essence of coupled control is to allow for external coupling between the decoupled set of equations of (4.19). The main advantage of coupled control is that the actual control forces are designed, so there is no need for a pseudo-inverse if the number of actuators is smaller than the number of modes (m n), which is true for all applications. It is therefore better suited for situations where the number of actuators is limited (m n). A comparison between coupled control and IMSC for different methods of control (pole placement, linear optimal control, and nonlinear on-off control) is made in [34]. Although the authors draw the conclusion that IMSC is superior to coupled control for various reasons, they hardly pay any attention to the scenario where the number of actuators is limited. Controllers for a limited number of actuators are calculated for coupled control, but the performance is compared to an IMSC based controller with more actuators (m = n). The main conclusions that can be drawn from this work are: It is computationally more expensive to calculate a state feedback controller for the complete system with n modes, than to design n independent state feedback controllers for every mode. However, roughly 25 years later this is not an issue anymore. Especially for high-performance applications. For linear optimal control the same performance is obtained for IMSC and coupled control, if the weighting filters for state error and controller effort are chosen equal for every mode. Although this result is not surprising, it is important to notice that the linear optimal controller can easily cope with the situation where m n, without leading to closed loop instability and still have good performance. It can therefore be concluded that it is not necessary to use one actuator/sensor pair to control (damp) an individual mode. Although this may not be the ideal situation, using a single actuator/sensor pair

54 44 Vibration control of flexible structures to control several flexible modes is possible. In [18, Chapter 12] H -optimisation is used to add damping to flexible modes of several systems. To accomplish this, the procedure listed below is followed: 1. Put the structural model in modal coordinates. 2. Define the performance criteria, either in time (overshoot, settling time) and/or frequency domain (bandwidth) of the whole system. 3. Assign initial values of the disturbance input matrix (B 1 ) and performance output matrix (C 1 ), which are defined in (2.2). Assigning initial values is done on the basis of knowledge of the disturbances working on the system (at a certain node) and the required vibration suppression at a number of nodes. 4. Solve the optimisation problem by solving the AREs (2.36) and (2.37) as discussed in Chapter 2. Simulate the closed-loop performance and check whether the performance criteria are satisfied. If not, continue. 5. Check which modes do not satisfy the performance criteria and scale the corresponding components of B 1 and/or C 1. Scaling of individual components of the input and output matrices in modal coordinates is basically the same as specifying which modes need more suppression (damping). If the corresponding component in B 1 or C 1 are increased, the H -norm is higher and the mode that degrades performance gets more controller attention. 6. Perform controller reduction. This procedure only works well as long as the individual entries of the disturbance matrix or performance matrix only influence a limited number of modes. Using this procedure, damping can be added to the flexible modes in simple structures (2D truss), but also in larger structures like a deep space network antenna, see Figure 4.1. Uncertainties are, however, not taken into account in the controller design process, nor are they put in the model. Robust stability and robust performance as introduced in Chapter 2 are therefore not considered. Although dealing with uncertainties is part of the power of the generalised control configuration, it has not been taken into account in this work. The benefit of the inclusion of explicit performance variables (y z) and exogenous inputs, e.g., references and disturbances, is shown by means of an example. In this example H -optimisation is compared to LQG control. In the latter, there is no distinction between performance and sensed variables and the only plant inputs are the actuator signals. It turns out that the H -based controller outperforms the LQG controller because of the inclusion of disturbance information in the control problem. In addition, the performance weighting in [18] is done by scaling the B 1 and/or C 1 components of the system, while this can also be done by using filters, as discussed in Chapter 2. A reason to directly scale components of the input and output matrices is that it gives direct insight on which modes are suppressed.

55 4.3 Modal control 45 Figure 4.1: Deep space network antenna Spillover Often the number of modes in the mathematical model (n) is quite large. In those cases it may not be feasible to control all n modes. Therefore, n c controlled modes are selected, such that n c n. The set of n modelled system modes can therefore be divided in two subsets, i.e., controlled modes and uncontrolled modes, leading to three sets of modes for the complete system: 1. Modelled and controlled modes, φ i, with i = 1,...,n c 2. Modelled and uncontrolled modes, φ i, with i = n c + 1,...,n 3. Unmodelled (and uncontrolled) modes, φ i, with i = n + 1,..., The modes in set 2 are called residual modes from here on. In literature the set of unmodelled modes (set 3) is often included in the set of residual modes. For the analysis conducted here this does not matter, since it is assumed that the number of modes in the system model is chosen sufficiently large to accurately describe the system behaviour. A modal controller can then be designed based on the controlled modes only and the presence of the residual modes is neglected. Either IMSC or coupled control can be used to design the controller. However, the variable n, denoting the number of modes in the system, needs to be replaced by n c, denoting the number of modes to be controlled. A drawback of this approach is called spillover. Control spillover refers to the phenomenon in which energy intended to go into the controlled modes goes into the residual modes. Note that this is determined by the placement of the actuators, and is therefore considered a plant property. Feedforward signals therefore also suffer from this problem. In a similar fashion, observation spillover can be defined as the contamination on the controlled modes by the residual modes. These effects are illustrated in Figure 4.2. A state feedback controller K is