Improving Inferential Performance of Flexible Motion Systems. S.L.H. Verhoeven

Transcription

1 Improving Inferential Performance of Flexible Motion Systems S.L.H. Verhoeven DCT Report 2.6 APT Master of Science Thesis Committee: Dr. ir. M.M.J. van de Wal (main supervisor) Dr. ir. J.J.M. van Helvoort (supervisor) Ir. T.A.E. Oomen (supervisor) Prof. ir. O.H. Bosgra (graduate professor) Dr. ir. C.M.M. van Lierop Prof. dr. ir. M. Steinbuch Philips Applied Technologies Mechatronics Program Drives and Control Group Eindhoven University of Technology Department of Electrical Engineering Control Systems Group Eindhoven University of Technology Department of Mechanical Engineering Control Systems Technology Group Eindhoven, February 2

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3 Summary Philips Applied Technologies has a long history of research on advanced control for high precision mechatronic positioning systems. This research has especially been done for ASML, which is a leading company in the market for chip manufacturing machines, i.e., wafer scanners. New trends cause such systems to become more flexible and taking these flexibilities into account will eventually become necessary. Due to this increased flexibility, the actual performance, which is also referred to as the inferential performance, is not necessarily represented well by the measured variables. This situation occurs if internal dynamics between the measured variables and the performance variables becomes relevant. In addition, interaction is inevitable when systems become more flexible and MIMO controller design techniques are crucial to achieve high performance. The goal of this research is to analyze, by using simulation models, whether flexible behavior is important in next generation high performance motion systems whereby the performance variables are not measured during normal operation and, if so, to investigate what the new limitations on performance are and how these limitations can be reached by control design. By performing simulations on a next generation wafer stage FEM model it is shown that conventional controller design, i.e., controller design whereby the system is assumed to behave as a rigid body, is not sufficient; the performance is severely limited while this is not observed by the sensors. Moreover, this research shows that the single-degree-of-freedom control structure, which is used in conventional control, is inadequate for the control of next generation high performance motion systems. An extra control degree-of-freedom therefore needs to be included in the control structure and three alternative structures are introduced. One of these structure, which is referred to as the inferential control structure, is used in the same simulation environment and increases the inferential performance in z-direction by 8%. Furthermore, important interpolation and integral constraints are derived for the standard plant setup that are able to deal with the inferential nature of the problem. Classical performance limitations, including the Poisson integral constraints, are also valid in the standard plant setup. However, it is shown that these constraints do not necessarily limit performance in case the performance variables are not measured. The reverse is also true: there may be strong limitations on the performance variables that do not limit the measured variables. Finally, it is established in this research that actuator/sensor selection is important for flexible motion systems, since it determines the extent to which the flexible dynamics is actuated, sensed and observed in the performance variables. The optimal actuator/sensor configuration strongly depends on the system (and its disturbances) and there is a need for profound tools for actuator/sensor selection in next generation high performance motion systems. iii

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5 Nomenclature Abbreviations ATO CLHP COG CRHP DOF FB FF FEM FRF GB GM GS IC IO I/O LQG LTI MIMO MP MSD MM NMP OLHP ORHP PM POC PSD RHS SISO SS SVD TF TFM Actuator-To-Origin Closed Left Half Plane Center of Gravity Closed Right Half Plane Degree-Of-Freedom Feedback Feedforward Finite Element Method Frequency Response Function Gain Balancing Gain Margin Gain Scheduling Integrated Circuit Input-Output Input/Output Linear Quadratic Gaussian Linear Time Invariant Multi-Input Multi-Output Minimum Phase Mass-Spring-Damper Modulus Margin Nonminimum Phase Open Left Half Plane Open Right Half Plane Phase Margin Point-Of-Control Power Spectral Density Right Hand Side Single-Input Single-Output State-Space Singular Value Decomposition Transfer Function Transfer Function Matrix v

6 vi Nomenclature WS Wafer Stage Notation and symbols R C C and C C + and C + field of real numbers field of complex numbers open and closed left half plane open and closed right half plane belong to end of proof = equal to identically equal to defined as asymptotically greater than asymptotically less than much greater than much less than α α I n diag(a,...,a n ) A T A H A A det(a) λ i (A) σ i (A) σ(a) σ(a) c(a) rank(a) A p F l (G,C) complex conjugate of α C absolute value of α C n n identity matrix an n n diagonal matrix with a j as its i-th diagonal element transpose of A complex conjugate transpose of A inverse of A Moore-Penrose pseudoinverse of A determinant of A i-th eigenvalue of A i-th singular value of A maximum singular value of A minimum singular value of A condition number of A Rank of A p-norm of A, e.g., A angle lower linear fractional transformation

7 Contents Summary Nomenclature Contents iii v vii Introduction. Background Project motivation Problem formulation Outline of the report Elements of linear system theory 7 2. System descriptions State-space description Transfer function description Frequency response function Coprime factorization Controllability and observability Directionality Singular value decomposition Rank Condition number Stability Poles Poles and stability Pole direction Zeros Zero direction Signal and system norms Signal norms System norms Conclusions vii

8 viii Contents 3 Controller design for flexible motion systems 9 3. Controller design procedure Control objective Control architecture design I/O selection Control structure design Controller configuration selection Norm-based controller design Selecting a norm Incorporating dynamic weighting filters Selecting exogenous outputs and inputs Internal stability and well-posedness Control problem formulation Performance evaluation Time domain performance Frequency domain performance Control structures for inferential servo problems Limitations of the single-dof control structure Alternative control structures for inferential servo problems Formulating a nine-block control problem Conclusions H loop-shaping Motivation for loop-shaping The design procedure Shaping filter selection Robust stabilization Relation with H optimization Using γ min as performance parameter Stability Guarantees on the achieved open-loop shape Design examples: H loop-shaping applied to a two MSD system SISO controller design MIMO shaping filters Guidelines for shaping filter selection Conclusions Fundamental performance limitations Fundamental limitations in SISO systems S plus T is one Interpolation constraints Bode integrals for S and T Poisson integrals for S and T Bode gain-phase relation LHP zeros Design examples: limitations due to a NMP zero Fundamental limitations in MIMO systems

9 Contents ix 5.2. S plus T is identity Interpolation constraints Bode integrals for S Poisson integral for S Design example: Bode sensitivity constraint in a MIMO system Performance limitations for non-square plants Performance limitations in the standard plant setup Preliminaries Systems reducible to feedback loop Interpolation constraints due to CRHP zeros in G zu or G yw A generalized Bode integral Design example: limitations in the standard plant setup Conclusions Control of a two MSD system Conventional control Rigid body assumption Collocated control Non-collocated control Limitations in the standard plant setup Inferential control State observer design Collocated measurement, non-collocated performance Non-collocated measurement, collocated performance Motivation for a general two-dof control structure Conclusions Control of a wafer stage FEM model 7. Notation Collocated and non-collocated control I/O selection for flexible stages Controller design and performance evaluation for P Control objective Model reduction Controller design for I/O set Performance evaluation for I/O set Error-based shaping filter selection Conclusions Conclusions and recommendations 3 8. Conclusions Recommendations

10 x Contents A More on transmission zeros 35 A. Blocking effect of a zero A.2 Physical interpretation of transmission zeros in mechanical systems A.2. Collocated actuators and sensors A.2.2 Non-collocated actuators and sensors A.2.3 Distributed parameter systems A.3 Interpretation using wave propagation theory A.4 Initial undershoot A.5 Creating transmission zeros by manipulating the plant inputs and outputs.. 4 A.6 Zero assignment problem B Case study: a flexible cart system 45 B. Introduction B.2 Conventional feedback control B.3 Explicit distinction between y and z (option ) B.4 Explicit distinction between y and z (option 2) B.5 Effect of changing mass and inertia B.6 Discussion C Simulation parameters 55 C. Flexible cart system C.2 Mass-spring-damper systems D FEM model 57 D. FEM Model D.2 Input and output selection D.3 Constructing the physical plant D.4 Rigid body decoupling D.4. Static sensor transformation D.4.2 Static actuator transformation D.4.3 Effect of decoupling E Frequency domain results for the WS FEM model 69 Bibliography 77

11 Chapter Introduction. Background Philips Applied Technologies has a long history of research on advanced control for high precision motion systems. This research has especially been done for ASML, which is a leading company in the market for chip manufacturing machines, i.e., wafer scanners. Chips, or Integrated Circuits (ICs), are miniaturized electronic circuits that are produced on silicon wafers by a photolithographic process. This process is schematically depicted in Figure.. Single wavelet light bundle Reticle stage containing the reticle Lens that projects the reticle pattern on the wafer Wafer stage containing the wafer Figure.: Schematic representation of the photolithographic process in a wafer scanner. A wafer is first covered with a photo resistant layer and then placed on the so-called wafer stage. To etch the IC pattern in the wafer, a monochromatic light bundle is sent through a reticle containing an enlarged version of the IC pattern. An advanced lens and mirror system

12 2 Introduction then focusses the light bundle on a small part of the wafer and only those parts of the wafer that need to be removed are exposed. Because an IC contains multiple layers with different patterns, the exposure process needs to be repeated several times with different reticles. In addition, only parts of the wafer can be exposed at the same time due to limitations of the lens system. These two conditions require that the reticle and wafer need to be positioned with high accuracy with respect to each other and the lens system. High-accuracy stages are nowadays controlled in six Degrees-Of-Freedom (DOFs): three rigid body translations and rotations. Controller design is mainly based on the plant Input- Output (IO) behavior and six independent actuators and sensors are used to control the six rigid body DOFs, while the flexible dynamics is regarded as parasitic. In addition, to keep FeedBack (FB) controller design simple, PID-like Single-Input Single-Output (SISO) controllers are used, despite the Multi-Input Multi-Output (MIMO) nature of the problem (Van de Wal et al., 22). However, the actual performance of a motion system is not necessarily represented well by the plant IO behavior. This situation occurs if internal dynamics between the measured variables and the performance variables becomes relevant. For a wafer stage, the truly relevant performance is in terms of positioning that part of the silicon wafer that is subject to light exposure. The measured variables can only be used to estimate the performance variables by using geometric relations, since it is based on laser interferometer data measured at the edges of the wafer stage. A similar reasoning applies to the actuator side, where internal deformations refute the validity of the traditional actuator transformation..2 Project motivation Because of fierce competition in the IC market, it is desirable to put more and smaller electronic components on a single IC and to increase machine throughput. ASML is therefore faced with the industrial challenge to build bigger and lighter stages, while at the same time the performance requirements become ever demanding, see, e.g., Jansen (28). Similar trends emerge for general high precision motion systems. Demanding a higher accuracy motivates contactless operation and hence weight minimization. In addition, by virtue of Newton s law F = m a, decreasing the mass of the system allows higher accelerations without more energy consumption, which saves money. Unfortunately, these trends imply that the next generation motion systems become more flexible, see, e.g., Balas (99), and hence it will eventually become necessary to explicitly address the internal plant dynamics. Such systems are referred to as flexible motion systems in the remainder of this research. In 25, Philips Applied Technologies recognized the importance of taking flexible dynamics into account and this resulted in the Ph.D assignment of Tom Oomen, see Oomen (2). The main focus herein is on model identification for advanced control design, i.e., obtaining accurate plant and uncertainty models, but more motion control challenges exist. A complete overview of these challenges can be found in Van de Wal (29) and Verhoeven (29). The challenges that are elaborated on in this research are introduced in the next section. Input as used here refers to the physical plant inputs (actuator forces) and output to the measured plant outputs (displacements).

13 .3 Problem formulation 3.3 Problem formulation Due to the increased flexibility of next generation motion systems, conventional controller design, i.e., controller design whereby the system is assumed to behave as a rigid body, eventually does not lead to the desired performance anymore. To illustrate this, consider the control structure depicted in Figure.2. Herein z p denotes the unmeasured performance variables, y p are the measured variables, u p are the plant inputs, u are the controller outputs, and r is the reference. The structure depicted by the solid lines is common practice in the field of motion control and is therefore also referred to as the conventional control structure. Since the system is assumed to behave as a rigid body, the assumption that y p = z p can be made without loss of generality. Relatively simple PID-like SISO controllers are then designed to keep r y p small (reference tracking) under the influence of disturbances d u and measurement noise η. However, due to the presence of internal dynamics, the actual system performance should be evaluated at z p and cannot be measured directly. A model-based controller design approach is therefore essential. Furthermore, notice that z p and y p do not necessarily have the same dimensions. Hence, the dashed lines in Figure.2. The goal of this research project is then formulated as: Analyze, by using simulation models, whether flexible behavior is important in next generation high performance motion systems whereby the performance variables are not measured during normal operation. And, if so, investigate what the limitations on the achievable performance are and how these limitations can be reached by control design. u ff C ff d u z p e z P z r e y C u u p P y p η Figure.2: Conventional control structure (solid line). The dashed structure points out the actual control goal of making e z small. Notice that e z cannot be determined unless y p and z p have the same dimensions. It is well-known that a well-designed FeedForward (FF) signal (u ff in Figure.2) greatly improves the performance of conventional servo systems. Using FF control, however, is basically nothing more than using setpoint information and plant knowledge 2 to steer the system without evaluating the effect of the plant inputs during normal operation. Since model uncertainties and unknown disturbances are always present in real systems, a FB loop is essential and FF controller design 3 is regarded as a tool to further boost the performance after a FB controller has been designed. Hence, this research mainly focusses on FB controller design. 2 Although not explicitly visible in Figure.2, known disturbances can also be accounted for in u ff. 3 FF controller design for flexible motion systems is considered in different research project, see, e.g., Lunenburg (29).

14 4 Introduction In the context of this research project, the following three research issues are addressed:. Explicit distinction between the measured and performance variables. The conventional control structure, which is depicted by the solid lines in Figure.2, does not represent the true control goal of keeping e z small. It is merely an approximation and is only justified when y p z p. In this research, shortcomings of the conventional control structure are investigated and alternative two-dof control structures are proposed that are better suited for the control of next generation high performance motion systems. 2. Norm-based controller design. FB controllers are often designed using SISO controller design techniques, despite the MIMO nature of the problem. When systems become more flexible, interaction is inevitable and MIMO controller design techniques are essential to achieve high performance. The standard plant setup, which is depicted in Figure.3, offers the possibility to formulate a wide variety of control problems that are able to deal with the MIMO nature of the problem. In addition, it allows for an explicit distinction between sensed outputs y and (weighted) exogenous outputs z on one side, and (weighted) exogenous inputs w and control signals u on the other side, which makes this setup extremely useful to distinguish between measured and performance variables. The goal of control is then to minimize a norm of the (closed-loop) mapping between w and z. Hence, the term norm-based control. In literature, several examples can be found that demonstrate the usefulness of this setup, see, e.g., McFarlane and Glover (99) and Steinbuch and Norg (998). Within Philips Applied Technologies, norm-based controller design has already been used to design MIMO controllers for ASML wafer stages, see Van de Wal et al. (22). (Weighted) exogenous inputs w G z (Weighted) exogenous outputs Control signals u y Sensed outputs C Figure.3: Standard plant setup 3. Fundamental limitations on achievable performance. Fundamental limitations such as the Bode sensitivity integral are well understood for the conventional control structure, see, e.g., Freudenberg and Looze (985, 986) and Skogestad and Postlethwaite (25). However, for the standard plant setup these limitations are less well understood. Obtaining insights in these limitations is considered relevant, since limitations in the standard plant setup may severely limit the performance (at z p ), even when there are no limitations on the measured variables (y p ). Possible limitations in the stan-

15 .4 Outline of the report 5 dard plant setup should thus be taken into account when altering the actuator/sensor configuration. In this research, an overview is given of the fundamental performance limitations in the conventional control structure and the standard plant setup that are considered relevant (at present) for the control of next generation high performance motion systems. The word fundamental is used here to indicate that only limitations are considered that are caused by poles and zeros in the closed right half plane. Causality and practical limitations, like, e.g., a finite sampling frequency, are not yet considered..4 Outline of the report This thesis is organized as follows. Chapter 2 introduces basic elements of linear system theory that are considered relevant for the control of flexible motion systems and are needed in the remainder of this thesis. Chapter 3 discusses (part of) the controller design procedure for flexible motion systems. The focus is limited to steps related to FB controller design. Although the procedure to design FB controllers seems trivial when a plant model is available, the presence of internal dynamics creates new aspects that need to be considered. The main focus of Chapter 3, however, is on formulating sensible norm-based control problems for conventional (stiff) and flexible motion systems (research issues and 2). For the latter, it is shown that an extra control DOF needs to be included in the control structure and alternative two-dof control structures are proposed that are better suited for the control of flexible motion systems. Chapter 4 proposes H loop-shaping as an alternative for conventional norm-based controller design in which the desired closed-loop transfer functions between w and z need to be specified. One of the benefits of H loop-shaping, which was originally introduced by McFarlane and Glover (99), is that the open-loop singular values are shaped, such that knowledge from manual loop-shaping can be used (research issue 2). In addition, design examples are given for a simple flexible motion system and these examples are used to derive guidelines for tuning controllers. In Chapter 5, an overview is given of fundamental performance limitations in the conventional control structure and the standard plant setup that are caused by poles and zeros in the closed right half plane and are considered relevant (at present) for the control of next generation motion systems. Illustrative examples are included as well to show the effect of these limitations. Subsequently, FB controllers are designed in Chapters 6 and 7 for two flexible motion systems. In Chapter 6, for a relatively simple two Mass-Spring-Damper (MSD) system and in Chapter 7 for a Finite Element Method (FEM) model of a Wafer Stage (WS) short stroke device. For both systems, the effect of applying current state-of-the-art control and adding an extra control DOF is compared in the time and frequency domain. Finally, in Chapter 8 the main conclusions and recommendations are stated.

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17 Chapter 2 Elements of linear system theory The goal of this chapter is to summarize important results from linear system theory and to introduce definitions that are needed in the remainder of this thesis. For a more thorough treatment, see, e.g., Skogestad and Postlethwaite (25) and Zhou et al. (996). 2. System descriptions 2.. State-space description A linear system with n u inputs (vector u), n y outputs (vector y), and an internal description of n x state variables (vector x), can be represented by a linear State-Space (SS) model of the form ẋ(t) = Ax(t) + Bu(t), x() = x, (2.a) y(t) = Cx(t) + Du(t), (2.b) where ẋ(t) dx/dt. Matrix A R nx nx is called the state matrix, B R nx nu the input matrix, C R ny nx the output matrix, and D R ny nu the feedthrough matrix. If, just as in (2.), these matrices are independent of time, the system is called a Linear Time Invariant (LTI) system. Remark 2.. To avoid notational confusion, variables y(t) and u(t) are sometimes replaced by y p (t) and u p (t), respectively. At the sensor side, the subscript p is added to distinguish between the measured variables y p (t) and the sensed variables y(t) in the standard plant setup. At the actuator side, the subscript is used to differentiate between controller outputs u(t) and plant inputs u p (t) in the presence of input disturbances. When there is no danger of confusion, the subscript p is omitted. The SS representation is not a unique system description, because multiple realizations (A,B,C,D) can give the same IO behavior. Firstly, because there can be additional unobservable and/or uncontrollable states, which never show up in the IO behavior. Secondly, even if there are no unobservable and uncontrollable states, there are still infinitely many 7

18 8 Elements of linear system theory state realizations. To see this, consider a state transformation x(t) = Tx(t), where T is an invertible constant matrix. The new SS realization has the same IO behavior, but is of the form à = TAT, B = TB, C = CT, D = D. (2.2) 2..2 Transfer function description An alternative system representation is the Transfer Function (TF) description. A TF may be obtained directly from the SS model by assuming a zero initial state, i.e., x =, and taking the Laplace transform of (2.). This leads to sx(s) x = Ax(s) + Bu(s) x(s) = (si A) Bu(s), x = (2.3a) y(s) = Cx(s) + Du(s) y(s) = (C(sI A) B + D) u(s), }{{} P(s) (2.3b) where P(s) is the TF, or Transfer Function Matrix (TFM) in case of a MIMO system. Because a TF only describes the IO behavior, unobservable and uncontrollable states are not included. Since only physical systems are considered, all SISO TFs (or elements of TFMs) are rational functions of the form P(s) = β n z s nz + + β s + β α np s np + + α s + α. (2.4) For SISO systems, n p is also defined as the order of the system and δ(p) = n p n z as the pole excess or relative degree of the system. Additionally, it is assumed that a TF P(s) is always a minimum realization, meaning that common terms in the denominator and numerator polynomial are omitted from (2.4). Definition 2.2. Consider a TFM P(s) with elements p ij. Then: System P(s) is strictly proper if n p > n z for all its elements p ij (s), i.e., p ij (s) as s. System P(s) is semi-proper or bi-proper if n p = n z for at least one element p ij (s), i.e., p ij (s) c,c R,c > as s, and the other elements are strictly proper. A system P(s) which is strictly proper or semi-proper is proper. A system P(s) is improper if n p < n z for at least on element p ij (s), i.e., P ij (s) as s. Notice that it is always possible to go from a SS representation to a TF description. However, in order to go back from a TF description to a SS representation, the system should at least be proper Frequency response function By evaluating s over jω, the Frequency Response Function (FRF) is obtained. This system representation has the advantage of having a clear physical interpretation. At each frequency ω, the complex number P(jω) (or complex matrix for a MIMO system) relates a

19 2. System descriptions 9 sinusoidal input signal to a sinusoidal output signal. If this input signal is persistent, i.e., applied at t =, and of the form u(t) = u sin(ωt + φ). (2.5) Then, the output signal is also a persistent sinusoid of the same frequency, i.e., y(t) = y sin(ωt + ψ). (2.6) The magnitudes u and y and phase shift ψ φ can be directly obtained from P(jω) by y u = P(jω) and ψ φ = P(jω) = arctan ( ) Im(P(jω)). (2.7) Re(P(jω)) 2..4 Coprime factorization Another useful way to describe systems is by means of a coprime factorization, see, e.g., Damen and Weiland (22, p. 72). Definition 2.3 (Left coprime factorization). Any rational transfer function matrix P can be factored as P = M N, (2.8) in such a way that: Both M and N are stable TFMs. M is square and N has the same dimensions as P. There exist stable transfer function matrices X and Y such that which is called the Bézout identity. NX + MY = I, (2.9) Such a factorization is called a left coprime factorization of P. A right coprime factorization can be defined is a similar way. Left and right coprime factorizations are not unique and therefore often normalized. The left coprime factorization is called a normalized left coprime factorization if M and N are chosen such that NN H + MM H = I, (2.) where {.} H denotes the complex conjugate transpose or Hermitian transpose. Remark 2.4. The terminology comes from number theory where two integers n and m are called coprime if ± is their greatest common divisor. It follows that n and m are coprime if and only if there exist integers x and y such that xn + ym =.

20 Elements of linear system theory 2.2 Controllability and observability Definitions in literature about controllability and observability are often about the states of a system. In this thesis, the definitions from Skogestad and Postlethwaite (25) are used. Definition 2.5 (State controllability). The dynamical system ẋ(t) = Ax(t) + Bu(t) or the pair (A,B) is said to be state controllable if, for any initial state x() = x, any time t >, and any final state x, there exists an input u(t), such that x(t ) = x. Otherwise, the system is said to be state uncontrollable. Definition 2.6 (State observability). The dynamical system ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) or the pair (A,C) is said to be state observable if, for any time t >, the initial state x() = x can be determined from the time history of the input u(t) and the output y(t) in the interval [,t ]. Otherwise, the system is said to be state unobservable. A SS realization (A,B,C,D) of a system is said to be a minimal realization of a system P(s) if all states, or equivalently, all modes of the system are controllable and observable. The state matrix A then has the smallest possible dimension. 2.3 Directionality The main difference between SISO systems and MIMO systems is the presence of directions in the latter. For a SISO system, the gain y / u at a given frequency is given by (2.7). The gain clearly depends on the frequency ω and is independent of the input signal. For MIMO systems, the input and output signals are both vectors and to get a similar interpretation of gain, the elements of these vectors need to be summed up by use of a norm. In Section 2.7, signal and system norms are further discussed. If the vector 2-norm is used for this, the gain of an n m system P at a given frequency is given by y(ω) 2 u(ω) 2 = j u j(ω) 2 j y j(ω) 2 = y yn 2. (2.) u u 2 m Clearly, for MIMO systems the gain still does not dependent on the input magnitude, but it does depends on the direction of u. The maximum gain of P as the direction of the input is varied, is given by the maximum singular value of P, i.e., max u = Pu 2 u 2 = max u 2 = Pu 2 σ(p). (2.2) Similarly, the minimum gain equals the minimum singular value of P, i.e., Pu 2 min u = u 2 = max u 2 = Pu 2 σ(p). (2.3) Besides changing the magnitude of the input signal, P also changes the directionality. The direction of y, i.e., the output direction, is therefore generally different than the input direction. The vector 2-norm (or Euclidian norm) is a spatial norm defined as: a 2 i ai 2.

21 2.3 Directionality 2.3. Singular value decomposition Definition 2.7 (Singular Value Decomposition). Any complex n m matrix A may be factorized using the Singular Value Decomposition (SVD) A = UΣV H, (2.4) where U C n n and V C m m are unitary 2 and matrix Σ R n m contains a diagonal matrix Σ of real, non-negative singular values σ i, arranged in descending order as in [ ] Σ Σ = if n m, (2.5) or where and Σ = [ Σ ] if n m, (2.6) Σ = diag {σ,...,σ k }, with k = min(n,m) (2.7) σ σ σ 2 σ k σ. (2.8) The column vectors of U, denoted by u i, are orthogonal and represent the output directions of a plant. Similarly, the orthogonal column vectors of V, denoted by v i, represent the input directions. If there is an input in direction v i, the output is in direction u i with an amplification of a factor σ i. The maximum gain of any system is thus given by the maximum singular value σ and the minimum gain by the minimum singular value σ. The eigenvalues λ of a matrix also give an indication of the system gain and are related to the singular values by σ i (P) = λ i (P H P). (2.9) Eigenvalues, however, can be a very misleading measure for the system gain as illustrated in Example 2.8 and can only be calculated for square systems. Example 2.8. Consider the system y = Pu, with [ P = ]. (2.2) Input vector u = [ ] T gives output vector y = [ ] T. Although both eigenvalues are zero, the maximum system gain is σ = Rank Definition 2.9 (Rank). The rank of an n m matrix A is equal to the number of non-zero singular values of the matrix. If rank(a) = r, then the matrix A is called rank deficient if r < k = min(n,m), and singular values σ i = for i = r+,...,k. In case of a square matrix, rank-deficiency also implies that the matrix is singular. 2 A complex matrix is unitary if U H = U.

22 2 Elements of linear system theory If a matrix has rank r, it has r independent rows (or columns) and the other k r rows (or columns) are zero or are a linear combinations of the first r rows (or columns). In case of a TFM, the term normal rank is often used, because the rank depends on the Laplace variable s. Definition 2. (Normal rank). The normal rank of an n m TFM P(s) is the maximum rank of the matrix P(s) evaluated over all values of s. In most situations, the normal rank of an n m system is equal to k = min(n,m). However, if two or more sensors (or actuators) are located at exactly the same point, two or more rows (or columns) are equal and the normal rank can be less than k Condition number If the gain of a system varies heavily with the input and output direction, the system has strong directionality and is called ill-conditioned. The condition number of a matrix is a measure for this and is defined as the ratio between the maximum and minimum singular value (see Definition 2.). Definition 2. (Condition number). The condition number of a matrix is defined as the ratio between the maximum and minimum singular value, i.e., c(p) σ(p) σ(p). (2.2) 2.4 Stability Several definitions of stability exist in literature. In this thesis, the notion of asymptotic stability is used. Definition 2.2 (Stability). An LTI system is (asymptotically) stable if the injection of a bounded external signal or a non-zero initial condition does not result in an unbounded output signal. Moreover, if the system is given a fixed, finite input, i.e., a step, then any resulting oscillations in the output will decay at an exponential rate and the output will tend asymptotically to a new final, steady-state value. 2.5 Poles The poles p of a system may be loosely defined as the values s = p, where P(p) equals infinity. In case of a rational TF as (2.4), the poles are equal to the roots of the denominator polynomial. Practically, poles are related to the modes of a system and are therefore independent of the inputs and outputs. Hence, they only depend on the state matrix A. Appendix A discusses the physical interpretation of poles and zeros in more detail. Definition 2.3 (Poles). The poles p i of a system with a SS realization (A,B,C,D) are the eigenvalues λ i (A), i =,...,n x.

23 2.6 Zeros Poles and stability For linear systems, the pole locations determine stability. Denote the open and closed left and right halves of the complex plane by OLHP, CLHP, ORHP, and CRHP, respectively. The imaginary axis, which is exactly in between the two halves, is included in the domain in case of closed half and not included in case of an open half. Theorem 2.4 (Stability). A continuous time system ẋ(t) = Ax(t) + Bu(t) is stable if all poles p i are in the OLHP, i.e., Re(p i ) = Re(λ i (A)) < i. A state matrix with such a property is called Hurwitz. A system is unstable if it has any poles in the CRHP. Proof. The proof follows straightforwardly by considering the time response of x(t), i.e., where x(t) = e A(t t ) x(t ) + with t i and q i the left and right eigenvectors of A, respectively. t t e A(t τ) Bu(τ) dτ, (2.22) n x e At = t i e λit qi H, (2.23) i= Remark 2.5. The term neutrally stable is sometimes used for systems with non-repeated poles (Franklin et al., 22) on the imaginary axis. For example, a single integrator system is neutrally stable, since a zero initial condition results in a constant output. However, a non-zero constant input results in an unbounded response. Hence, the system is unstable by Definition Pole direction As mentioned earlier, directionality is important in multivariable systems and hence it is not sufficient to define multivariable poles in terms where the denominator polynomials of P(s) have a singularity. Multivariable poles have directions associated with them and to quantify these directions the input and output normalized pole vectors are defined as y pi = Ct i Ct i 2 and u pi = BH q i B H q i 2. (2.24) The matrix gain is infinite in the direction of the pole, which can be written as (Skogestad and Postlethwaite, 25, p. 37) P(p i )u pi = y pi (2.25) 2.6 Zeros Zeros of a system arise when competing internal effects are such that the output is zero, even when the inputs (and the states) are not identically zero. Whether or not a system contains

24 4 Elements of linear system theory zeros is therefore determined by the placement of the actuators and sensors relative to the underlying dynamics. For SISO systems, the zeros q i are simply the solutions to P(q i ) =, i.e., the roots of the numerator polynomial in (2.4). However, to capture this blocking nature for MIMO systems is not that straightforward and Definition 2.6 should be used. Definition 2.6 (Transmission zeros). The (transmission) zeros q i of a system P(s) are the locations in the complex plane where the rank of P(q i ) is less than the normal rank of P(s). Definition 2.6 is based on the TFM description, which corresponds to a minimum realization of the system. If no minimum realization is available, numerical computations may yield additional invariant zeros. These invariant zeros plus the transmission zeros then form the so-called system zeros (Skogestad and Postlethwaite, 25, p. 4). Due to the multivariable nature, transmission zeros are also called multivariable zeros to distinguish them from zeros in the elements of the TFM. For simplicity, however, transmission zeros are often simply referred to as zeros Zero direction Similar to pole directions in the previous section, there are also directions associated with the transmission zeros of a system. If P(s) loses rank at s = q i, then there exist non-zero vectors u qi and y qi such that P(q i )u qi = y qi, (2.26) where u qi and y qi are defined as the zero input and output direction, respectively. Remark 2.7 (Pinned zeros). A zero is pinned to a subset of outputs if the zero output direction y qi has one or more elements equal to zero. Pinned zeros are common in practice and their effect cannot be moved freely to any output. For example, the effect of measurement delay at one output cannot be moved to another output. Example 5.2 shows how the effect of a zero can be moved to a specific output. If the i th element of the zero output direction is zero, the effect of the corresponding zero cannot be moved towards this output. Similarly, a zero is pinned to certain inputs if the zero input direction u qi has one or more elements equal to zero. Remark 2.8. For square systems the poles and zeros of P(s) are essentially not the same as the poles and zeros of det(p(s)). Using det(p(s)) to determine the system poles and zeros fails when pole/zero cancellations occur between elements of P(s) when calculating the determinant. 2.7 Signal and system norms The basic idea behind a norm is to have a single number that gives an overall measure of the size of a vector, matrix, signal, or system. Norms are therefore particulary useful to recast the control problem as a mathematical optimization problem (see Section 3.4). The material

25 2.7 Signal and system norms 5 presented here is presented without going into the mathematical details and is mainly based on Oomen (24, Section 2.4). A more comprehensive treatment of norms is given by Damen and Weiland (22, Chapter 5) and Zhou et al. (996, Chapters 2 and 4). Regardless whether e represents a vector, matrix, signal, or a system, the norm of e should satisfy the following definition. Definition 2.9. A norm of e is a real-valued number, denoted by e, that satisfies the following properties:. Non-negative: e. 2. Positive: e = e =. 3. Homogeneous: αs = α s,α C. 4. Triangle inequality: e + e 2 e + e 2 (2.27) For matrices (and systems) the following definition should also hold for the norm to qualify as a matrix norm. Definition 2.2. A norm of matrix E, denoted by E, is a matrix norm if, in addition to the properties in Definition 2.9, it also satisfies the multiplicative property, i.e., AB A + B. (2.28) Property (2.28) is very important when combining systems and forms the basis of the smallgain theorem (Zhou et al., 996, Theorem 9.). Notice that there exist norms on matrices (thus satisfying the properties in Definition 2.9) that do not qualify as a matrix norm Signal norms A signal e is a function that quantifies how a certain variable evolves in time and is given by the mapping e : T W, (2.29) where T denotes the time set and W is the signal space. In this research, only continuous time signals are discussed, i.e., T R. Furthermore, only real-valued multi-dimensional signals are considered. At each time instant t T, e is thus a vector with l entries, representing l real valued quantities, i.e., W = R l. To deal with the multi-dimensional nature of signal, the vector norm is defined first. Definition 2.2 (Vector norm). Consider a vector x C l. The p-norm of x, denoted by x p, is defined as x p ( l i= x i p ) p, max i x i, for p <, for p =. (2.3)

26 6 Elements of linear system theory After the channels of a vector-valued signal are summed up at a given time instant using a vector norm, a second summation is made over the time values using a temporal norm. It is common practice, see, e.g., Skogestad and Postlethwaite (25, p. 537), to use the same p-norm to sum up the vector and the time signal. Definition 2.22 (Signal norm). The temporal norm, denoted as the L p norm 3, of a signal e : T R l is given by ( t T e Lp ( e(t) p) p dt ) p, sup t T e(t), for p <, for p =, (2.3) where t denotes the time instant (either for a finite time set T = [a,b] or an infinite time set T = R). Signal norms can also be determined in the frequency domain as in done in Section 2.3. The first summation is then done by applying a vector norm at a given frequency and the second summation is done over the frequencies. For T = R, several signals norms are of special interest. The -norm (p = ) of a signal equals the integrated absolute value, the squared 2-norm equals the energy of a signal, and the -norm equals the maximum absolute value of a signal. By using Parseval s Theorem, see, e.g., Ambardar (999), it can be shown that the time domain 2-norm as defined by (2.3) is equivalent to frequency domain 2-norm as used in Section 2.3. Since not all signals have a finite energy, like, e.g., periodic signals, an additional norm is introduced: T e pow lim e(t) T 2π 2 dt. (2.32) T The norm e pow is called the power-norm and defined for all signals with a finite power, i.e., e pow <. Note that the power-norm does not satisfy the second property in Definition 2.9. The power-norm is therefore only a semi-norm System norms The norm of a system is used to quantify how large (in terms of a signal norm) the output signals can be if a certain sized (in terms of a norm) input are applied. Two norms are discussed that are popular in the field of norm-based controller design: the H 2 and the H norm. H norm Consider a stable MIMO system H(s) with inputs w and outputs z. 3 L stands for the fact that the signals should be Lebesgue integrable. Sometimes this notations is shortened to the p-norm. The context should then avoid confusion with the vector 2-norm as defined by (2.3).

27 2.7 Signal and system norms 7 Definition 2.23 (H norm). Let H(s) be a stable TFM with FRF H(jω). The H norm of H(s), denoted by H(s), is then defined as H(s) sup ω R σ (H(jω)). (2.33) For SISO systems, the H norm thus equals the peak value in the Bode magnitude diagram and for MIMO systems this peak value is generalized to the maximum singular value. The H norm is a so-called induced norm, see Damen and Weiland (22, Section 5.3), because sup ω R σ (H(jω)) = sup w L 2,w Hw 2 w 2 = sup w z(t) L2 w(t) L2 = sup w z(t) pow w(t) pow. (2.34) The proof involves using Parseval s Theorem and can be found in Damen and Weiland (22, Section 5.3). The H norm can thus be interpreted in the time domain as the largest amplification of energy (or power) for an input signal in a particular direction and at a particular frequency. Hence, the H -norm is also called an L 2 - (or power-) induced norm. In case a system is unstable, i.e., has poles in the CRHP, the output energy is unbounded. Hence, the H norm is not defined for unstable systems. Remark In robust control literature, see, e.g., Zhou et al. (996), the H norm of H(s) is often denoted by H rather than by H H. The context should avoid confusion with the matrix -norm. A similar reasoning holds for the H 2 norm. H 2 norm Consider the stable SISO system H(s) with input w and output z. Definition 2.25 (H 2 norm for SISO systems). Let H(s) be a stable TF with FRF H(jω). The H 2 norm of H(s), denoted by H(s) 2, is then defined as H(s) 2 ( ( H(jω) 2 ) ) 2 dω. (2.35) 2π Although the H 2 norm is not an induced norm, several interpretations exist. By using Parseval s theorem, it can be shown that the H 2 norm equals the 2-norm of the impulse response of H. This interpretation is often referred to as the deterministic interpretation and is not discussed further. A stochastic interpretation also exist. Only the general idea is presented here; the complete derivation is given in Damen and Weiland (22, p. 5 53). Suppose input w has a Power Spectral Density (PSD) S w (ω), it can then be shown that the PSD of z is given by S z (ω) = S w (ω) H(jω) 2. (2.36) Next, it can be shown that (Skogestad and Postlethwaite, 25, Section 9.3.2) z pow = ( ( H(jω) 2 ) ) 2 dω S w (ω), (2.37) 2

28 8 Elements of linear system theory where z pow is defined in (2.32). Now consider a white noise input u with S w (ω) = ω R. Then, (2.37) equals (2.35) and the H 2 norm can be interpreted as the system response to a white noise input signal. In other words: the H 2 norm is the squared area under the Bode magnitude plot of H. 4 To conclude this section, a generalization of the H 2 norm is given for MIMO systems. Definition 2.26 (H 2 norm for MIMO systems). Let H(s) be a stable TFM with FRF H(jω). The H 2 norm of H(s), denoted by H(s) 2, is then defined as ( H(s) 2 2π ( = 2π Trace{H H ( jω)h(jω)} dω k i= ) 2, (2.38) σ i (H(jω)) 2 ) 2, (2.39) where the Trace of a square matrix is the sum of the entries at its diagonal, σ i is the i th singular value, and k is the number of singular values. It follows from (2.39) that the H 2 norm is a measure for the area under all the singular values of H(s) in a singular value diagram. Remark The symbol H in H and H 2 stands for Hardy space. The symbol H stands for the set of TFMs with bounded -norm, which is the set of proper and stable TFMs. Similarly, the symbol H 2 stands for the set of TFMs with bounded 2-norm, which is the set of strictly proper and stable TFMs. 2.8 Conclusions In this chapter, basic elements of linear system theory are introduced and definitions are included that are needed in the remainder of this research. The topics discussed are various types of system descriptions, directionality in multivariable systems, controllability and observability, system stability, (multivariable) poles and zeros, and signal and system norms. The next chapter discusses (part of) the controller design procedure for (flexible) motion systems, whereby the main focus is on norm-based controller design. 4 Note that the magnitude is taken to the second power and the frequency is evaluated over a linear axis.