Robust Active Chatter Control in the High-Speed Milling Process

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 4, JULY 2012

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Robust Active Chatter Control in the High-Speed Milling Process Niels J. M. van Dijk, Nathan van de Wouw, Ed J. J. Doppenberg, Han A. J. Oosterling, and Henk Nijmeijer, Fellow, IEEE

Uncertainty model input/output.

Abstract—Chatter is an instability phenomenon in machining processes which limits productivity and results in inferior workpiece quality, noise and rapid tool wear. The increasing demand for productivity in the manufacturing community motivates the development of an active control strategy to shape the chatter stability boundary of manufacturing processes. In this work a control methodology for the high-speed milling process is developed that alters the chatter stability boundary such that the area of chatterfree operating points is increased and a higher productivity can be attained. The methodology developed in this paper is based on a robust control approach using -synthesis. Hereto, the most important process parameters (depth of cut and spindle speed) are treated as uncertainties to guarantee the robust stability (i.e., no chatter) in an a priori specified range of these process parameters. Effectiveness of the proposed methodology is demonstrated by means of illustrative examples.

Generalized plant input/output vector. Actuator/cutter displacements—m. System/controller state vector. Periodic solution. Perturbations about periodic solution. Controller input signal—m. Scalar uncertainty. Uncertainty set. Structured singular value. Delay—s.

Index Terms—Active control, delay systems, high-speed milling, machining chatter, magnetic bearings, robust controller synthesis.

I. INTRODUCTION NOMENCLATURE

C

Axial depth of cut—mm. Constant for selecting controller input. Tooth passing frequency—Hz. Actuator/tooltip force—N. Spindle-actuator transfer function matrix. Averaged cutting force matrix—N/m . Controller output—A. Controller transfer function matrix—A/m. Spindle speed—rpm. Generalized plant. Manuscript received February 02, 2011; revised April 17, 2011; accepted April 18, 2011. Manuscript received in final form May 14, 2011. Date of publication June 20, 2011; date of current version May 22, 2012. Recommended by Associate Editor R. Landers. This work was supported by the Dutch Ministry of Economic affairs within the framework of Innovation Oriented Research Programmes (IOP) Precision Technology. N. J. M. van Dijk was with the Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. He is now with the Philips Innovation Services, 5656 AE Eindhoven, The Netherlands (e-mail: [email protected]). N. van de Wouw and H. Nijmeijer are with the Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]). E. J. J. Doppenberg and J. A. J. Oosterling are with TNO Science and Industry, 2600 AD Delft, The Netherlands (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2157160

HATTER is an instability phenomenon in machining processes which must be avoided at all times. The occurrence of (regenerative) chatter results in an inferior workpiece quality due to heavy vibrations of the cutter. Moreover, a high level of noise is produced and the tool wears out rapidly. The occurrence of chatter can be visualized in so-called stability lobes diagrams (SLD). In a SLD the chatter stability boundary between a stable cut (i.e., without chatter) and an unstable cut (i.e., with chatter) is visualized in terms of spindle speed and depth of cut. In the present day manufacturing industry, an increasing demand for high-precision products at a high productivity level is seen. This motivates the desire for the design of dedicated control strategies, which are able to actively alter the chatter stability boundary and therewith enable high material removal rates. Hereto, this paper presents an active control strategy which alters the stability lobes diagram in a selective spindle speed range and, therewith, ensures a priori chatter-free milling operations for a predefined domain of process parameters (spindle speed and depth-of-cut) such that (chatter-free) operating points of higher material removal rate become feasible. Herein, an important challenge is to transform the model of the high-speed milling process (which in general is described by a set of nonlinear time-variant delay differential equations) into a generalized plant formulation making it suitable for robust control design [1]. Basically three methods exist in literature to control chatter. The first method to avoid chatter is to adjust process parameters (i.e., spindle speed, feed per tooth, or chip load) such that a stable working point is chosen [2]–[4]. Although chatter can

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be eliminated by adaptation of process parameters, the methodology does not enlarge the domain of stable operation points towards those of higher productivity. A second method is to disturb the regenerative effect by continuous spindle speed modulation, see, e.g., [5], [6]. Although the stability boundary is altered by spindle speed modulation [7], the method cannot be used in the case of high-speed milling. Namely, in order to disturb the regenerative effect, the spindle speed variation should be extremely fast, while the speed of variation is limited by the inertia and actuation power of the spindle. The third method is to passively or actively alter the machine dynamics to alter the chatter boundary. There are passive chatter suppression techniques that use dampers [8] or vibration absorbers [9]. Passive dampers are relatively cheap and easy to implement and never destabilize the system. However, the practically achievable amount of damping is rather limited. Moreover, vibration absorbers require accurate tuning of their natural frequencies and, consequently, lack robustness to changing machining conditions. Active chatter control in milling has mainly been focused on active damping of machine dynamics. In [10], active damping of a milling spindle with piezoelectric actuators is demonstrated for low spindle speeds. Kern et al. [11], [12] applied active damping on a milling spindle equipped with an active magnetic bearing (AMB). Minimization of the tooltip compliance using -synthesis, which will result in an increase of stable machining points, is presented in [13]. An approach taking a different perspective is presented in [14]. Herein, it is assumed that chatter originates from workpiece flexibilities. Active damping is applied by using piezoelectric actuators and sensors, which are mounted to the thin-walled workpiece. In general one can say that damping the machine or workpiece dynamics, either passively or actively, results in a uniform increase of the stability boundary for all spindle speeds. All aforementioned active chatter control approaches aim to attenuate chatter vibrations by applying damping to the spindle or the tool in an active way. In general, one can say that damping the machine or workpiece dynamics, either passively or actively, results in a uniform increase of the stability boundary for all spindle speeds. To enable more dedicated shaping of the stability boundary (e.g., lifting the SLD locally around a specific spindle speed), the regenerative effect should be taken into account during chatter controller design. In [15], an optimal state feedback-observer controller with integral control in the case of turning was designed taking the regenerative effect into account. The “infinite-dimensional” regenerative delay term is written as a rational function via Padé approximation. Recently, Chen and Knospe [16] developed three different chatter control strategies for the case of turning: speed-independent control, speed-specified control, and speed-interval control. Moreover, it has been shown that significant improvement in tailoring the stability lobes can be obtained using dedicated controllers, obtained via -synthesis, as compared to proportional-integral-differential (PID)-like controllers. Although the experimental setup discussed in [16] exhibits some aspects encountered in high-speed milling, a comprehensive active chatter control strategy tailored to the

full complexity of the HSM process is missing to this date. Moreover, except for the work in [11] and [12], all research on active chatter control is limited to low spindle speeds (i.e., below 5000 rpm). In this paper, an active chatter controller methodology for the high-speed milling process is presented, which can guarantee chatter-free cutting operations in an a priori defined range of process parameters such as spindle speed and depth of cut by employing an active magnetic bearing as an actuator. Current chatter control strategies for the milling process cannot provide such a strong guarantee of a priori stability for a predefined range of working points. In general, the existing techniques require a posteriori calculation of the set of stable working points. The methodology developed in this paper is based on a robust control approach using -synthesis. Hereto, the most important process parameters (depth of cut and spindle speed) are treated as uncertainties. The proposed methodology will allow the machinist to define a desired working range (in spindle speed and depth of cut) and lift the SLD locally in a dedicated fashion. In practice the maximum actuator force is limited. Hence, we propose a methodology for the robust stabilization of high-speed milling operations while minimizing the control effort. Effectiveness of the proposed control methodology is shown by means of an illustrative example. The paper is organized as follows. Section II presents a comprehensive model of the milling process. Moreover, stability properties of the model will be discussed. Section III presents the problem statement of the active chatter control problem. Then, in Section IV, a comprehensive analysis is performed to select an appropriate feedback signal for the active chatter controller input, such that the actuator forces needed for stabilization are significantly reduced. The model of the milling process as presented in Section II cannot be directly used in the robust controller design procedure. Therefore, in Section V, some model simplifications will be discussed in order to construct a model suitable for controller design. Section VI present the robust control design procedure, based on a -synthesis approach. Results of the proposed strategy, when applied to an illustrative example, are presented in Section VII. Finally, conclusions are drawn in Section VIII. II. MILLING PROCESS This section presents a comprehensive model of the milling process which can be used to predict the occurrence of regenerative chatter. Moreover, stability properties of the model will be discussed. The model of the milling process as discussed in this section is originally presented in [17]–[20]. In Fig. 1, a schematic representation of the milling process is given. A block diagram of the milling process, with controller, is given in Fig. 2. Each of the blocks in this figure will be explained in more detail as follows. As can be seen from the block diagram in Fig. 2, the milling process is a closed-loop position-driven process. The setpoint of the milling process is the predefined motion of the tool with respect to the workpiece, given in terms of the static chip thickness , where is the feed per tooth and the rotation angle of the th tooth of the tool with respect to the (normal) axis (see Fig. 1). However, the total

VAN DIJK et al.: ROBUST ACTIVE CHATTER CONTROL IN THE HIGH-SPEED MILLING PROCESS

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tangential and radial direction for a single tooth are described by the following exponential cutting force model:

(1) where and are cutting parameters which depend on the workpiece material and is the axial depth of cut. The function describes whether a tooth is in or out of cut else Fig. 1. Schematic representation of the milling process.

(2)

and are the entry and exit angle of the cut, rewhere spectively. Via trigonometric functions, the cutting force can easily be converted to (feed)- and (normal)-direction (see Fig. 1). Hence, cutting forces in - and -direction, and , respectively, can be obtained by summing over all teeth, as shown in (3) at the bottom of the page, where and

B. Spindle Dynamics and Actuator Dynamics Fig. 2. Block diagram of the milling process.

chip thickness also depends on the interaction between the cutter and the workpiece. Since, in general, the machine tool is not infinitely stiff, the interaction between the cutter and the workpiece leads to cutter vibrations resulting in a dynamic displacement of the tool which is superimposed on the predefined tool motion. This results in a wavy surface on the workpiece. The next tooth encounters the wavy surface left behind by the previous tooth and generates its own waviness. This is called the regenerative effect and results in the block Delay in Fig. 2. The difference between the current and previous wavy surface is denoted as the dynamic chip thickness with the delay. Hence, the total chip thickness encountered by tooth , , is the sum of the static and dynamic chip thickness: . In the next sections, the components of the milling model will be described in more detail. A. Cutting Force Model The cutting force model (indicated by the Cutting block in Fig. 2) relates the cutting forces acting at the tool tip of the machine spindle to the total chip thickness. The cutting forces in

The cutting force interacts with the spindle rotor and tool dynamics (block Spindle in Fig. 2). For the purpose of active chatter control, an actuator is implemented in the spindle rotor. The controller output is dictated to the actuator which, in turn, generates a force on the spindle. In general the spindle rotor and tool dynamics (jointly called the spindle dynamics) can be modelled by a linear multi-input-multi-output (MIMO) model. The model has four inputs and four outputs. The inputs consist of the cutting forces acting at the tool-tip in -/ -direction and the actuator forces in -/ -direction induced at some point in the spindle, which generally differs from the location at which the cutting forces are acting (the tooltip). This leads to an inherent flexibility between the actuator/sensor system and the cutting forces. The outputs of the spindle rotor dynamics model are the displacements of the tooltip and displacements measured at some position on the spindle, which are used for feedback. In this paper, the machine spindle-toolholder-tool dynamics is modelled by two decoupled subsystems (representing the dynamics in two orthogonal directions perpendicular to the spindle axis) consisting of two mass-spring-damper systems to mimic the inherent compliance between actuator and tooltip, see Fig. 3,

(3)

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dynamic systems and in [11] feasibility of using such actuator in the scope of high-speed milling has been shown. This motivates to pay special attention to this kind of actuator (model). The nonlinear model of an AMB driven in differential mode is given as follows [22]:

Fig. 3. Schematic overview of spindle dynamics model, . : forces displacements at tooltip. forces displacements at actuator,

TABLE I MILLING MODEL PARAMETERS

with masses

(5)

:

, eigenfrequencies and dimension-

less damping ratios . This is done in order to capture the inherent dynamics between the actuator/sensor system (denoted by subscript ) and the cutting tool (denoted by subscript ). The parameters of the machine spindle model and cutting force coefficients are listed in Table I. Herein, the cutting model parameters ( and ) are taken from [21] and spindle parameters are chosen such that these represent realistic machine spindle dynamics for high-speed milling machines. The state-space equations describing the rotor dynamic model are given as follows:

(4) where is the state vector (the order of this model primarily depends on the order of the spindle-tool dynamics model, which in this case equals ), , , and . The state-space matrices of the spindle rotor dynamic model are given in the appendix. Next to the spindle rotor dynamics, an actuator model is included. The active chatter control design procedure will be developed for a model incorporating a nonlinear actuator model for an active magnetic bearing (AMB). As described in the introduction an AMB is a common type of actuator applied to rotor

are the specific AMB coefficients, is the where so-called premagnetizing current (to compensate for gravity, etc.), the corresponding nominal gap displacement and is the controller output (i.e., the input currents to the actuator) and the bearing displacements. In general, the displacements in the actuator journal are significantly smaller than the gap width . In addition, the controller output will be limited by the controller design methodology. Then, for control design the nonlinear model of the AMB may be linearized about , which has already been successfully performed for many applications as is described in [22]. This results in the following linear model of the AMB: (6) where (7) (8) C. Total Milling Model In the previous sections, the submodels for the cutting force and spindle rotor, toolholder, tool, and actuator dynamics representing the different blocks in the milling model as given in Fig. 2 are introduced. Substitution of the cutting force model and actuator model, given in (3) and (6), respectively, into the model of the spindle rotor, toolholder, and tool dynamics, given in (4), yields the total milling model, shown in (9) at the bottom of the page. It can be seen that the model describing the milling process is set of nonlinear, time-dependent delay differential equations (DDE). In the next section, the stability properties of the model will be analyzed. D. Stability of the Milling Process In this section, we will briefly address the stability analysis exploited to determine chatter boundaries in the stability lobes

(9)


diagram. In the milling process the static chip thickness is periodic with period time . Here is the spindle speed in revolutions per minute (rpm). In general, the uncontrolled (i.e., ) milling model (9) has a periodic solution with period time [23]. To validate this fact let us adopt the following decomposition of : (10) where is a -periodic motion that can be considered as the ideal motion when no chatter occurs and the perturbation term. When no chatter occurs, and the tool motion is described by the following ordinary differential equation (ODE):

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which results in an uncertainty in the delay .1 Moreover, chatter is defined as the loss of stability of this periodic solution and stability of the milling process is based on the stability of the model describing the perturbations of the milling process around the periodic solution. The aim of this paper is to design a finite-dimensional linear controller , which guarantees the following: • robust stability of the milling process (12) for the given uncertainties in depth of cut , time delay ; • performance by minimizing the total amount of actuator energy needed to stabilize the uncertain milling process. Hereby, it is assumed that the controller , with controller input and output current , has the following state-space description: (14)

(11) which follows from (9) by exploiting the fact that . The ODE in (11) is a linear system with a periodic excitation with period time . Hence, when has no eigenvalues at , for and all , the solution exists, is unique and is -periodic [24]. Therefore, the periodic solution is (at least locally) asymptotically stable when no chatter occurs and when chatter occurs it is unstable. Therefore, the chatter stability boundary can be found by studying the (local) stability of the periodic solution . To this end, the uncontrolled milling model is linearized about the periodic solution which yields the following linearized dynamics in terms of the perturbations :

(12) where

(13) As can be seen from (12) and (13), the linearized model is a delayed, periodically time-varying system. Stability of these kind of systems can be assessed using, e.g., the semi-discretization method of [25]. The main point of semi-discretization is that only the delay term is discretized, instead of the actual time-domain terms. Unless stated differently, all stability lobes diagrams, presented throughout this paper, are determined using the semi-discretization method. III. PROBLEM STATEMENT Recall from the previous section that the SLD is determined using the model which describes the perturbation vibrations about the (chatter-free) periodic solution of the milling process. Therefore, the controller design, as presented in this paper, will be based on the model which is linearized about the periodic solution with uncertainties in depth of cut and spindle speed

, , , , Herein, with the order of the controller. The and choice of the controller input signal will be discussed in Section IV. The linearized uncertain model of the milling process, given by (12), (13), cannot be directly used in the standard robust controller design procedure. Therefore, after discussing the selection of the controller input signal , two model simplifications will be presented in Section V such that the infinite-dimensional time-varying model (12), (13), is transformed into a finite-dimensional linear time-invariant (LTI) model. In this way, the model can be used in a robust control design procedure, which will be presented in Section VI. IV. CONTROLLER INPUT SIGNAL An important part of any control system is the choice of the feedback signal used for control. From the discussion in Section II-D it becomes clear that the nominal (chatter-free) solution of the milling model is periodic with period time . Moreover, chatter is defined as the loss of stability of this periodic solution and stability of the milling process is based on the stability of the model describing the perturbations of the milling process around the periodic solution. Then, two possibilities arise in selecting the feedback signal which serves as an input to the controller in (14), namely as follows: 1) full output feedback, i.e., the total (measured) displacements are used for feedback: in (14); 2) perturbation feedback, i.e., the perturbation (chatter) vibrations are used as feedback signal, where denotes the periodic solution of the nominal model given by (9): in (14). In Section VI, the design of a linear dynamic output feedback control law characterized by the transfer function and with a state-space description as defined in (14), is pursued. Next, the controller input signal will be denoted as (15) 1Note that and are not uncertain in practice, but since we aim to stabilize , we treat the parameters the milling process in a range of working points as uncertainties in the control design.

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where is the periodic solution at the measured output. Moreover, is a constant indicating whether full output or perturbation feedback is applied. The implication of the choice for either one of the two controller input signals will be demonstrated next. It can easily be shown that the stability properties of the closed loop, in case of the linear actuator model are the same for both choices of the feedback signal [either , or in (15)]. First, it will be shown that the chatter-free -periodic solution of the closed-loop system (9), (14); will be different for both choices of the controller input signal. To show this, consider the following decompositions of the state vectors and :

(i.e., ) the periodic solution will be equal to that of the uncontrolled system (and , i.e., the nominal control action is zero). On the other hand, for full output feedback (i.e., ) the periodic solution will be different from that of the uncontrolled system (and will in general be non-zero, i.e., the steady-state control action does not vanish). Second, it can be shown that for both choices of the feedback signal the linearization of (9) with (14) about is given by

(16) (18) Then, let us combine (9) with (14) and substitute and . Using the fact that is -periodic (i.e., ), this results in the following closed-loop dynamics:

(17) It can be seen that the closed-loop dynamics in (17) governing the periodic solution is an LTI system with a -periodic disturbance since both and are periodic with . Then it can be concluded that when full output feedback is applied (i.e., ), with the assumption that has no eigenvalues at , for and all , the solution exists, is unique and is -periodic [24]. Note that this solution differs from the periodic solution of the open-loop dynamics given in (11). The conditions imposed on the eigenvalues of will in general be satisfied, since the controller has to render the closed-loop system stable also for (i.e., the eigenvalues of will lie in the open left half plane). However, when and consequently perturbation feedback is applied, the eigenvalues of are given by the eigenvalues of and . Then, there exists a unique, -periodic, solution when has no eigenvalues at , for and has no eigenvalues with real part equal to zero. The conditions imposed on will typically be satisfied since, in general, the AMB actuator is designed such that the decrease in stiffness, due to the negative stiffness effect of an electromagnetic actuator, see [22], is significantly smaller than the stiffness of the spindle rotor. Consequently, under such conditions, in case of perturbation feedback, is the only solution of (17) satisfying . Therewith, the periodic solution of (17) becomes equal to the solution of the open-loop periodic solution of (11). Hence, in case of perturbation feedback

with as defined in (13). Clearly, for both choices of the control input signal, the resulting SLD will be the same (since the perturbation dynamics (18) does not depend on the constant ). Moreover, in the case of perturbation feedback the nominal control action vanishes in steady-state, which is not the case for full output feedback . As a result, the choice for perturbation feedback is favorable from the point of view of bounding the control action. From the analysis, presented above, it can be concluded that the SLD with active chatter controller (14) does not depend on the chosen controller input signal. This is due to the fact that the variable , indicating whether full output feedback or perturbation feedback is applied, does not appear in the linearized equations of motion [see (18)]. Moreover it is shown that the actuator forces, needed to stabilize the milling process, will be zero in steady state in case of perturbation feedback whereas the actuator forces will be non-zero in steady state in case of full output feedback. Based on the previous discussion and the fact that for an AMB it is important to limit the input current in order not to exceed the maximum amount of carrying force, in the remainder of this paper, perturbation feedback will be considered. In practice, the perturbation displacements can be obtained by using a chatter detection algorithm based on a parametric model of the milling process, as, e.g., described in [4]. V. MODELLING FOR ROBUST CONTROL DESIGN The model of the milling process, discussed in Section II, can readily be employed for stability analysis (i.e., determination of the SLD). However, the presence of time-delay and the explicit time-dependency of the right-hand side of the DDE (9) complicate the development of robust control synthesis techniques. Therefore, we apply two model simplifications to construct a finite-dimensional, time-invariant model, which will be more suitable for controller synthesis. Moreover, the effect of these model simplifications on the SLD is demonstrated. This section will discuss two model simplifications to construct a finite-dimensional time-invariant model of the milling process, which will be more suitable for controller synthesis.


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First, the discussion will focus on an autonomous approximation of the linearized nonautonomous DDE describing the linearized perturbation milling dynamics, obtained by linearising (9) about , given by (12)

(19) A characteristic feature of a milling process is that the direction . As a of the cutting forces is a function of the rotation angle result, time-dependent functions appear in the describing model equations. In [17], a method is described which approximates by means of a Fourier series expansion. The number of harmonics to be considered for an accurate reconstruction of depends on the immersion conditions (which indicates the percentage of the tool diameter used during cutting) and the number of teeth in cut. In this paper, we will consider full immersion cuts (i.e., the entire tool diameter is used for cutting). Then, as described in [17], it is sufficient to take the average (zero-order) component of the Fourier series expansion over one tooth passing, i.e. (20) is valid only between the entry and exit angles Since of the cutter (i.e., when ), it becomes equal to the average value of at cutter pitch angle (21) , and can be where the integrated functions determined analytically in case of a linear cutting model ( , see [17]) and have to be computed numerically in case of an exponential cutting model . At this point we have obtained a time-invariant milling model in which the dependency on the rotation angle is eliminated. Secondly, a finite-dimensional approximation of the time delay, using a Padé approximation, is applied (see also [15] and [16] where Padé approximation is used for controller design in case of turning). Hereto, the delayed tool vibrations are approximated using a Padé approximation and the resulting approximation is denoted by , such that . The choice for a suitable order of the Padé approximation depends on the

Fig. 4. Stability lobes diagram for the milling process with Padé approximation of order and the milling model with exact time delay.

eigenfrequencies of the spindle-toolholder-tool dynamics. In order to accurately approximate the regenerative effect, given, in the frequency domain by , an approximation function should be chosen which has magnitude 1 at the frequencies , and accurately approximate the phase of . Consequently, the order of the Padé approximation should be chosen such that it approximates the phase of up to at least the highest frequency of the spindle-toolholder and tool modes which are relevant for chatter. The milling model in (19) with cutting force averaging, defined in (21) and Padé approximation is given as shown in (22) at the bottom of the page, where , and denote matrices of the state-space description of the Padé approximation. The size of these matrices depends on the chosen order for the Padé approximation. Since the delayed output vector has two elements ( - and -direction), the state-space description of the Padé approximation has two times the number of states of the Padé approximation order . The order of the Padé approximation will be based on a desired level of accuracy regarding the predicted chatter stability boundary using the model with Padé approximation. In Fig. 4 the chatter stability boundary is given for the autonomous model with time-delay and for different orders of the Padé approximant with the parameters of the model listed in Table I. From Fig. 4 it can be observed that, for increasing order of the Padé approximant, the error between the stability lobes determined using the exact delay term and the approximated delay term becomes smaller. Moreover, since the delay is inversely proportional to the spindle speed, the approximation becomes more accurate as the spindle speed increases.

(22)

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Fig. 5. Generalized plant interconnection.

In this work we focus on high-spindle speeds (i.e., above 20 krpm). Hence, it is sufficient to choose the order of the Padé approximant for equal to be , which will be used throughout this paper. VI. ROBUST CONTROLLER DESIGN In the previous section, a model has been derived which is suitable for robust controller design. Therefore, in this section, the actual controller design for an active chatter control methodology, which will alter the chatter stability boundary, is presented, such that stable operating points of higher productivity can be attained while avoiding chatter. The control goal, as presented in Section III, can be cast into the generalized plant framework and solved using -synthesis techniques [26]. Fig. 5 shows the configuration of this framework. The generalized plant is a given system with three sets of inputs and three sets of outputs. The signal pair denote the inputs/outputs of the uncertainty channel. The signal represents an external input in which possible disturbances, measurement noise and reference inputs are stacked. The signal is the control input. The output can be considered as a performance variable while denotes the measured outputs used for feedback. The remainder of this section will be devoted to uncertainty modelling, in depth of cut and spindle speed and the specification of the performance requirement for the active chatter control problem as stated in Section III. A. Nominal Model Equation (22) gives the nominal plant model used during -synthesis. Note that, in contrast to most active chatter control methods discussed in the introduction, in this work we do not only consider the spindle dynamics during the control design, but also take the interaction between the spindle dynamics and the cutting forces (and therewith the regenerative effect responsible for chatter) into account. It is expected that this is a more profound and promising method to make dedicated modifications to the chatter stability boundary by means of feedback control. B. Uncertainty Modelling This section describes the modelling of the uncertainties in the process parameters, which can be considered as a key step in achieving the control objective defined above: robust stability

(i.e., chatter avoidance) in a predefined range of process parameters. The control design will be based on the milling model (22) presented in the previous section. 1) Uncertainty in Process Parameter—Depth of Cut : First, the uncertainty in depth of cut is considered which is modelled as a parametric uncertainty. An important (practical) aspect is that robust control design should provide stability for small as well as (relatively) large values of the depth of cuts, see also [16]. Hereto, the uncertain depth of cut is modelled such that it specifies a range from zero up to a maximum value , i.e., . Let us define a real scalar uncertainty set . The uncertainty for the depth of cut is then defined by

(23) where is the maximal depth of cut for which stable cutting is desired. 2) Uncertainty in Process Parameter—Spindle Speed : Next, the uncertainty model for the spindle speed is considered. As described before, the delay is inversely proportional to the spindle speed. Hence, uncertainty in spindle speed is modelled as an uncertainty in the delay , where . Since a Padé approximation is a rational function of two polynomials in the Laplace operator and delay , modelling the interval delay via a parametric uncertainty would result in an overall uncertainty of very large dimensions (due to the relatively large order of the Padé approximations). Here an alternative approach will be used to model the delay uncertainty. Hereto, note that for arbitrary frequency , the value set of the frequency-domain delay operator for all can be represented in the complex plane as a circular arc extending along the unit circle. This time-delay interval can be approximated by choosing any pair of stable transfer functions and such that , with and , covers the uncertainty set with . Several alternatives exist to determine transfer function and satisfying these conditions. Chen and Knospe [16] propose to choose and such that at each frequency: 1) the arc length covered by the disk is nearly that of the delay element , for and 2) the area of the disk lying outside the unit circle is minimized. Doing so results in a transfer function which has twice the order of the chosen Padé approximation. Since the Padé approximation needed to accurately describe the delay term is already of a relatively high order, the generalized plant will be of an even higher order which is not desired due to possible computational and implementation issues for the resulting controller. Moreover, the size of the circle covering the circular arc of the delay uncertainty is rather large which is due to the fact that the area of the disk lying outside the unit circle is chosen to be minimized. This approach may therefore give conservative results as illustrated for the milling process in [1]. Hence, here a different approach is presented to model the delay uncertainty. In contrast to the approach discussed above,


we model the delay uncertainty based on a Padé approximation of the nominal model. The total delay uncertainty interval is then overapproximated using a low-order transfer function which covers the circular arc of the delay uncertainty interval along the unit disk about the nominal delay. Hereto, consider the linearized autonomous milling model with a delay uncertainty only. Basically, this model can be represented by the following state-space model:

(24) , , where and uncertainty set . It is easy to show that (24) can be written as a feedback interconnection between the dynamics

(25) and uncertainty term (26) where

, the delay operator is defined as and . The representation of the time-domain operator in (26) can be given in the Laplace domain as

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Hereto, in [27] rational transfer functions for several orders are derived such that . Since, in this work, the high-speed milling process is considered, the delay intervals will be relatively small (typically of ) s. for typical spindle speed ranges of rpm for spindle speeds rpm). Using this fact, together with the fact that the dominant spindle dynamics resonances lie in general between Hz, implies that typically and, consequently, an accurate approximation of the frequency-dependent upper bound is required. Moreover, from a numerical point of view, proper transfer functions are desired. Then, based on the results in [28], is chosen as (30) which ensures a tight over bound of (by ) especially in the frequency region which is relevant in the case of high-speed milling, as is described above. Hence, by using the results presented above, the delay uncertainty is approximated by two rational transfer functions and , where is the Padé approximation of and , with as in (30), such that

(31)

(27) where , the Laplace transforms of and , respectively. Let be the gain bound of the uncertainty operator (27) in the frequency domain, given as (28) is analytic and Since the transfer function bounded in the open right half of the complex plane, the -norm of can be determined by evaluating the transfer function on the imaginary axis, i.e., for . Consequently, in order to determine a bound on , it should be determined for . It can be shown, see [27], that the upper bound on the delay uncertainty is represented as follows: (29) . The frequency-dependent upper bound where on the delay uncertainty is not a rational function and can therefore not readily be used during controller synthesis.

C. Performance Requirement This section discusses the specification of a performance requirement for the active chatter control design. In essence, the chatter control problem at hand is a robust stabilization problem rather than a performance problem. As outlined in the problem statement in Section III, the robust stability requirement has to be achieved with limited control effort, since actuator forces have to satisfy practical saturation limits (of, e.g., AMB). Therefore, the control gain should be bounded during -synthesis, which reflects the most relevant performance requirement for chatter control. Limiting the control gain is done by applying an upper bound on the control sensitivity transfer function , where we have the equation at the bottom of the page, which gives the transfer function representation from to of the nominal plant given by (22). Here, the control sensitivity is defined as the transfer function from a input signal (which can, e.g., be interpreted as measurement noise on the measured perturbation displacements entering the feedback loop) to the control input . The bound on the control sensitivity is

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As often in robust control, fashion.

is chosen in an iterative

D. Generalized Plant Formulation

Fig. 6. Block diagram of linearized approximated autonomous milling model with performance weighting.

enforced by defining a weighting function , which will be described below, such that the performance output of the generalized plant is the weighted control sensitivity . A schematic overview of the closed-loop approximated autonomous milling model with performance weighting is given in Fig. 6 where the transfer function of the spindle-actuator dynamics is denoted by and given by

Based on the discussion on uncertainty modelling and the specification of a performance requirement in the previous sections and the milling model for control, presented in Section V, the control problem can now be transformed into the generalized plant framework [26]. In order to derive the generalized plant, first, consider the following state-space descriptions of the systems , , where and , respectively (34) (35) (36)

(32) Then, the problem in which it is aimed to find a -optimal controller which stabilizes the milling process in the face of modelled uncertainties (in and ) while minimizing the peak magnitude of the weighted control sensitivity; that is a controller which achieves . Of course, minimizing the weighted control sensitivity actually enforces a frequency-dependent upper bound on the magnitude of control gain rather than on the magnitudes of the actual control input . Hence, by estimating the magnitude of the inputs to the controller, i.e., of the chatter-related tool displacements in - and -direction, an appropriate bound on the control gain can in practice be chosen such that the actuator forces are satisfying given saturation limits. In this work, the weighting transfer function matrix is chosen to be diagonal, because of the 2-D nature of the control input , i.e., . Moreover, its structure is chosen such that is a lead-lag filter with high-pass characteristics. This means that, for frequencies below the roll-off frequency , the control gain is limited by a certain value and that, for frequencies larger than , the inputs to the controller are attenuated in order to reduce (undesired) influences on the control action, due to sensor noise and aliasing effects due to discretization of the continuous-time controller. These (undesired) influences need to be attenuated for any practical (digital) implementation of the controller. Moreover, in general, an actuator has a limited bandwidth in which it can operate. Consequently, the roll-off frequency should be chosen such that it accounts for the actuator bandwidth limitation, high-frequent measurement noise sensor noise and respects sampling frequency influences. Based on the discussion above, weighting function is written as

where the size of depends on the order of the Padé approximation which should be chosen such that it approximates the phase of up to at least the highest resonance frequency of the spindle-toolholder and tool modes which are relevant for chatter. Next, consider the linearized autonomous milling model as described by (22)

(37) Then, by adding the uncertainty and performance channel and , respectively, input/output, denoted by to the system and rearranging terms, the generalized plant is given as follows:

(38) with the state vector input vector and output

, , output vector . The uncertainty channel input are defined as

where the subscripts and denote the input/output of the delay and depth of cut uncertainty, respectively. The definition of the state-space matrices of the generalized plant can be found in the appendix of the paper. Combining all the sources of uncertainty as described in Section VI-B, the total uncertainty block is given as

(33) denotes the gain of the weighting function. A pole, where at frequency (such that ), is added to obtain a proper weighting function, necessary for implementation.

(39) and correFrom the definition of the generalized plant sponding uncertainty set it becomes clear that the control


problem at hand is a robust performance problem which contains structured uncertainties, i.e., the uncertainty is not a full complex matrix but has specific elements which contain uncertainties. Hence, it is recommended to solve the problem using -synthesis, which will be briefly discussed in the following section. E. Controller Synthesis In this section, the problem of finding controllers, which satisfy the requirements as defined in Section III, will be discussed. As discussed in the previous section, the control problem at hand is a robust performance problem. It is well known that there is no direct method available yet to synthesis a -optimal controller, see [26]. From the generalized plant model, presented in the previous section, it can be concluded that we are dealing with a so-called mixed -synthesis problem, i.e., both complex and real uncertainties are present. Although mixed -synthesis can be employed via D,G-K-iteration, it will in general result in high-order controllers due to high-order fits required for the G-scales. Moreover, as demonstrated in [13], D,G-K-iteration does not guarantee an increase in performance. As the general plant in this work is of relatively high order (since a relatively high-order Padé approximation is needed to accurately approximate the time delay), the uncertainty in depth-of-cut is considered as a complex uncertainty and controller design is employed using D-K-iteration. We accept the possible conservatism introduced by considering only complex uncertainties during the controller design in order to avoid the design of a controller of even higher order. The complex uncertainty set, denoted by , is given as (40) By using an upper bound on the structured singular value , the controller synthesis problem is transformed into an optimization problem which tries to minimize the peak value over frequency for this upper bound, namely (41) denotes a (frequency-dependent) scaling matrix to Herein, represent an upper bound on and denotes the set of functions that are analytic and bounded in the open right half plane. The optimization problem (41) is iteratively solved for and . For a fixed scaling transfer matrix , the problem reduces to a standard synthesis problem, which can be turned into a convex optimization problem. The optimization problem for a fixed controller matrix, i.e., the problem of determining the optimal scaling matrix for a given frequency can also be recast in to a convex optimization problem. Both the as well as the step in the D-K-iteration can be solved using algorithms from the Robust Control Toolbox of MATLAB [29]. F. Controller Order Reduction Due to the relatively high order of the Padé approximation needed to accurately describe the delay term, the resulting con, see [1]. As trollers will be of relatively high order discussed above, the spindle dynamics typically has resonances

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TABLE II PARAMETERS OF THE AMB MODEL

lying between Hz, which will generally result in relatively fast controller poles which in turn require relatively large sample frequencies in a digital implementation. Hence, for the purpose of the feasibility of the implementation of the proposed active chatter control methodology in practice, controller order reduction should be applied. Balanced truncation is an order reduction procedure which is often applied to tackle such model reduction problems. However, balanced truncation can only be applied in case the system to be reduced is stable. The control synthesis procedure discussed in the previous section does, however, not guarantee the design of stable controllers. To deal with this fact, closed-loop balanced truncation can be applied, see [30]. The controller states which do not contribute significantly to the closed-loop input/output of the generalized plant will be removed from the controller using closed-loop balanced truncation. After that, robust performance for the closed-loop system with the reduced-order controller is evaluated by determining -values. The acceptable amount of reduction is defined as the smallest controller order for which . As already outlined above, the robust control problem under consideration has structured uncertainties, which will be solved via D-K-iteration. Hence, during closed-loop balanced truncation, the D-scaling matrices obtained during controller synthesis are absorbed into the generalized plant. VII. RESULTS In this section, the results for actual controller synthesis for a realistic model of a high-speed milling machine is addressed. In order to demonstrate the feasibility of the -synthesis approach proposed in the previous section, control design is performed for an illustrative example. Hereto, consider the parameters of the milling process as given in Table I. The spindle dynamics is modelled, as before, by two decoupled subsystems consisting of a two mass-spring-damper model in order to capture the inherent compliance between the actuator/sensor system (with mass ) and the cutting tool (with mass ). The parameters of the AMB model are listed in Table II. Moreover, a four-fluted tool is considered. Consequently, as already discussed in Section V, a 10-th order Padé approximation is used to approximate the time delay in the milling model. We aim to design controllers that stabilize milling operations (i.e., guarantee the avoidance of chatter) for two different spindle speed ranges, for a range of depth-of-cut which should be as large as possible for a given performance requirement (i.e., for a given limitation of the control gain). Hereto, D-K-iteration is employed within a bi-section scheme. The performance requirement, presented in the previous section, is used to limit the control forces. For an AMB it is important to limit the input current in order not to exceed the maximum amount of carrying force. Here, we choose to limit

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Fig. 7. Closed-loop -values for reduced controllers using closed-loop balanced truncation. Black bars indicate an unstable closed-loop: (a) rpm; (b) rpm.

the input current to 2 A. Typical bearing displacements , for the modelled spindle under consideration, related to onset of chatter, are of the order of mm. Then, an upper bound on the control gain due to physical constraints of the actuator is set to N/mm and consequently in (33) is set equal to mm/A. The high-frequent roll-off frequency is chosen to account for actuator bandwidth limitation (which is typically a few kHz in case of an AMB), high-frequent sensor noise and aliasing effects due to discretisation of the continuous-time controller (sampling frequencies will be typically set to a value larger than 15 kHz). Based on this is set to 7500 Hz. The remaining parameter of the weighting filter is set to Hz, the additional pole is added such that, first, is well-posed and, second, the generalized plant fulfills the rank conditions typically made in the problem, see [26, p. 354]. Controllers are designed for two different ranges of spindle speeds, namely a relatively small interval given as rpm and a relatively large interval given as rpm. Controller synthesis using D-K-iteration yields a 40-th order controller for a maximal depth of cut of mm for rpm and a 36-th order controller for a maximal depth of cut of mm for rpm. The difference between the controller orders is due to a difference in the D-scales. In Fig. 7, the -value for different controller orders (after reduction) are depicted for the two ranges of spindle speeds. In general, closed-loop stability cannot be guaranteed after controller order reduction. Therefore, before determining the -value for a specific reduced-order controller, first stability of the nominal closed-loop system is checked. When the closed-loop system is stable the corresponding -value is determined. From these results, the lowest controller order is selected for which robust performance can be guaranteed, i.e., the lowest controller order for which . This yields a 24-th order controller for and a 16-th order controller for rpm. So, it can be concluded that a significant reduction of the controller order can be achieved while still guaranteeing robust stability

and performance which warrant the feasibility of the practical implementation of such controllers in practice. Frequency response functions (FRF) of the full- and reduced-order controllers together with the inverse of the frequency bound imposed on the control sensitivity (i.e., ) are given in Fig. 8. It can be seen that the resulting controllers exhibit highly dynamical characteristics indicated by the inverse notches in the FRF. Although the magnitude of the controllers do not exactly fulfill the imposed bound (since the bound is imposed on the control sensitivity and not the controller itself), it can be seen that the magnitude is bounded. Moreover, it can be seen that the full- and reduced-order controller have similar FRF magnitudes. Hence, it is expected that robust performance is maintained under controller-order reduction. To verify whether robust performance is maintained, stability lobes diagrams (SLDs) are determined using the (closed-loop) linearized non-autonomous milling model (18), as outlined in Section II-D, for the case with the reduced-order controller and without control. Note that for the case without control, (12) and (13) are used for the stability analysis. The resulting SLDs can be found in Fig. 9. It can be seen that the SLD of the controlled milling are shaped such that it contains a lobe in the desired spindle speed range. Stability is ensured up to a depth of cut 3.055 mm (an increase of approximately 760% compared to the case without control) and 2.686 mm (an increase of approximately 241% compared to the case without control) where controllers are designed for rpm and rpm, respectively. Herein, denotes the maximal achievable depth of cut in the SLD in the desired spindle speed range. Fig. 9 clearly illustrates the power of the proposed approach, as the SLD is shaped locally to be able to increase at a specific spindle speed (while avoiding chatter and satisfying a specified bound on the control gain). This is contrary to the application of active damping which lifts the SLD over the entire spindle speed range at the cost of high required levels of actuation energy. Whereas stability is increased at the desired spindle speeds, it decreases significantly at other spindle speeds. The characteristics of the controller design and its ability to shape the SLD in a dedicated fashion can be explained by fur-


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Fig. 8. Magnitude of FRF of the full-order (black) and reduced-order (grey) controllers obtained by D-K-iteration for two different range of spindle speeds and rpm.

Fig. 9. Stability lobes diagrams, determined using the linearized nonautonomous milling model (18), for reduced-order controllers designed and for two different range of spindle speeds, rpm.

cated at 1667 Hz. A better damped resonance around 2000 Hz is created in case of the larger spindle speed range rpm , which lies near the edge of the range of desired tooth passing frequencies. As a matter of fact, the location of the closed-loop resonances, which are dominant for chatter occurrence, is closely related to the tooth excitation frequencies for milling operations within the defined spindle speed ranges. Hence, it can be concluded that, in order to create a stability lobe at a certain spindle speed, the natural frequency of the spindle dynamics should be set equal the corresponding tooth passing excitation frequency, see also [18]. The fact that a closed-loop spindle resonance situated at a tooth-passing excitation frequency is beneficial for avoiding chatter can be explained as follows. In the milling process the highest depth of cut can be obtained (corresponding to a peak in the SLD) when the dynamic chip thickness is equal to zero. This relation can be transformed to the frequency domain as follows: (42) and are the Fourier transforms of and , respectively. Hence, the difference between the tooltip displacements of the present and previous cut is actually characterized by a filter, denoted by , with zeros at . Moreover, for the milling process, the dominant (chatter) frequency of the perturbation vibrations lies in general close to the eigenfrequency of the spindle dynamics. Then, by designing the controller such that the closed-loop resonance is close to a tooth-passing frequency and due to the filter properties of the (in particular the location of the zeros of at -related frequencies), the dynamic chip thickness is enforced to be zero at the desired spindle speed. This, in turn, results in a large depth of cut within the desired spindle speed range and a peak in the SLD at that spindle speed. So, by applying robust control design techniques, a controller is obtained which tailors the tooltip spindle dynamics, such that a resonance is created near a tooth passing harmonic which in turn results in a peak in the SLD. where

ther examining the controlled spindle dynamics. The FRF of the closed-loop tool-tip spindle dynamics (i.e., the FRF to ) is given, together with the original (unconfrom trolled) spindle dynamics, in Fig. 10. While the original (uncontrolled) spindle dynamics only has - and -components (due to decoupled spindle dynamics), the controlled machine dynamics also has off-diagonal components. This can be explained by the fact that controller design is performed using the complete milling model where coupling between - and -direction is introduced by the cutting force model (resulting in a full matrix in (21) and consequently in a full 2 2 controller ). A striking characteristic displayed in Fig. 10 is the fact that the controller has tailored the spindle dynamics such that the resonances are shifted. A similar conclusion was drawn in [16] for active chatter control in case of the turning process. For the small spindle speed range rpm , a dominant weakly damped resonance can be seen which is lo-

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Fig. 10. Controlled (black) and uncontrolled (open-loop) and different range of spindle speeds, spindle speed range is indicated by the grey area.

(grey) tooltip spindle dynamics for reduced-order controllers designed for the two rpm. The interval of tooth passing excitation frequencies corresponding to the

Fig. 11. Displacements at the tooltip for a time domain simulation performed for nonlinear AMB model. The controller is designed for a spindle speed range of

In the final part of this section some results from time-domain simulations (TDS) will be discussed. Hereto, the nonlinear nonautonomous delay differential equations describing the total milling model, given by (9), along with the nonlinear bearing model, given by (5), have been implemented in MATLAB/SIMULINK. The purpose of the TDS is to further illustrate the effectiveness of the controller design, the benefit of perturbation feedback and investigate whether it is justified to apply the linear AMB model for controller design. As described before, for an AMB it is important to limit the controller input in order to avoid actuator saturation. In order to apply perturbation feedback, the periodic solution with period time has to be known. Existence of the periodic solution in case of the nonlinear bearing model is difficult to prove. Here, the periodic solution is approximated using the finite difference method, as illustrated for the milling model in [23]. In practice, the perturbation displacements can be obtained by using a chatter detection algorithm based on a parametric model of the milling process, as, e.g., described in [4]. For the sake of brevity, we do not describe these algorithms here. In order to compare the performance of the milling process with and without chatter control the simulation is performed for an operating point which is originally unstable (i.e., exhibits chatter), but is stabilized by means of control. Here, the opera-

31 000 rpm and 2 mm with and without control using the rpm. (a) Control off. (b) Control on, .

tion point under consideration has the process parameters 31 000 rpm and 2 mm, which is originally an unstable working point (see Fig. 9). The results are gathered in Figs. 11 and 12. Fig. 11 presents the displacements at the tooltip with control off and control on with perturbation feedback . Furthermore, the -sampled tool displacements is depicted by dots. Fig. 12 presents the actuator currents generated by the controller for the case with control on. From the figures, it can be seen that without control, the amplitude of the displacements becomes relatively large (approximately 35% of the tool radius which is chosen as 5 mm), which will result in the tool jumping in and out of cut resulting in an inferior workpiece quality. When the controller is switched on, the motion for the initially unstable working point is stabilized, which can be seen from the -sampled displacements. Moreover, it can be seen that the amplitude of the displacements is considerably smaller for the case with active chatter control as compared to the uncontrolled case. The actuator input currents are given in Fig. 12. Due to the fact that perturbation feedback is applied, the (steady-state) actuator current, after some transients at the start of the simulation, are (almost) zero. From the results of the time-domain simulations, it can be seen that the assumptions, for which the linear AMB model is a good approximation of the nonlinear AMB model, as discussed in Section II-B, remain valid. Moreover, the results


Fig. 12. Actuator input currents for a time domain simulation per31 000 rpm and 2 mm where perturbation feedback formed for is considered. The controller is designed for a spindle speed range of rpm.

from time-domain simulations clearly demonstrate the benefit of applying perturbation feedback.

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In addition, it is shown that the actuator forces, needed to stabilize the milling process, will be zero in steady state in case of perturbation feedback (i.e., only chatter vibrations are used as a feedback signal) whereas the actuator forces will be non-zero in steady state in case of full output feedback. This result is exploited for a milling model incorporating a nonlinear Active Magnetic Bearing model, where it is important to limit the actuator input current in order to avoid actuator saturation. Results, for illustrative examples, clearly illustrate the power of the proposed control methodology. The chatter stability boundary is locally shaped to stabilize the desired range of working points. This is contrary to the application of active damping which lifts the SLD over the entire spindle speed range at the cost of high required levels of actuation energy. By means of illustrative examples it is shown that this control strategy can render working points of significantly higher productivity stable. APPENDIX Spindle Rotor Dynamics: The state-space matrices of the spindle rotor dynamic model, given in (4), are defined as follows:

VIII. CONCLUSION In this paper, an active chatter control design methodology for the suppression of regenerative chatter in the high-speed milling process has been developed. The main purpose achieved is the suppression of chatter (i.e., stabilization of the milling process) in an a priori specified range of process parameters (spindle speed and depth of cut), such that working points of significantly higher productivity become feasible while avoiding undesirable chatter vibrations. Herein, the requirement for a priori stability guarantee for a predefined range of process parameters is cast into a robust stability requirement. Moreover, a performance requirement is imposed on the control sensitivity in order to limit the actuator forces. Current chatter control strategies for the milling process cannot provide such a strong guarantee of a priori stability for a predefined range of working points. The control problem is solved via -synthesis using D-K-iteration.

where

for Generalized Plant: The state-space matrices of the generalized plant, given in (38), are defined as shown in the equation at the bottom of the page.

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REFERENCES [1] N. J. M. van Dijk, N. van de Wouw, E. J. J. Doppenberg, J. A. J. Oosterling, and H. Nijmeijer, “Chatter control in the high-speed milling process using -synthesis,” in Proc. Amer. Control Conf., 2010, pp. 6121–6126. [2] S. Smith and T. Delio, “Sensor-based chatter detection and avoidance by spindle speed,” J. Dyn. Syst., Meas. Control, vol. 114, no. 3, pp. 486–492, 1992. [3] R. E. Haber, C. R. Peres, A. Alique, S. Ros, C. Gonzalez, and J. R. Alique, “Toward intelligent machining: Hierarchical fuzzy control for the end milling process,” IEEE Trans. Control Syst. Technol., vol. 6, no. 2, pp. 188–199, Mar. 1998. [4] N. J. M. van Dijk, E. J. J. Doppenberg, R. P. H. Faassen, N. van de Wouw, J. A. J. Oosterling, and H. Nijmeijer, “Automatic in-process chatter avoidance in the high-speed milling process,” J. Dyn. Syst., Meas. Control, vol. 132, no. 3, p. 031006(14), 2010. [5] A. Yilmaz, E. AL-Regib, and J. Ni, “Machine tool chatter suppression by multi-level random spindle speed variation,” J. Manuf. Sci. Eng., vol. 124, no. 2, pp. 208–216, 2002. [6] E. Soliman and F. Ismail, “Chatter suppression by adaptive speed modulation,” Int. J. Mach. Tools Manuf., vol. 37, no. 3, pp. 355–369, 1997. [7] T. Insperger and G. Stépán, “Stability analysis of turning with periodic spindle speed modulation via semi-discretisation,” J. Vibr. Control, vol. 10, pp. 1835–1855, 2004. [8] K. J. Liu and K. E. Rouch, “Optimal passive vibration control of cutting process stability in milling,” J. Mater. Process. Technol., vol. 28, no. 1–2, pp. 285–294, 1991. [9] Y. S. Tarng, J. Y. Kao, and E. C. Lee, “Chatter suppression in turning operations with a tuned vibration absorber,” J. Mater. Process. Technol., vol. 105, no. 1, pp. 55–60, 2000. [10] J. L. Dohner, J. P. Lauffer, T. D. Hinnerichs, N. Shankar, M. E. Regelbrugge, C. M. Kwan, R. Xu, B. Winterbauer, and K. Bridger, “Mitigation of chatter instabilities in milling by active structural control,” J. Sound Vibr., vol. 269, no. 1–2, pp. 197–211, 2004. [11] S. Kern, C. Ehmann, R. Nordmann, M. Roth, A. Schiffler, and E. Abele, “Active damping of chatter vibrations with an active magnetic bearing in a motor spindle using -synthesis and an adaptive filter,” presented at the 8th Int. Conf. Motion Vibr. Control, Daejeon, Korea, 2006. [12] S. Kern, A. Schiffler, R. Nordmann, and E. Abele, “Modelling and active damping of a motor spindle with speed-dependent dynamics,” presented at the 9th Int. Conf. Vibr. Rotat. Mach., Exeter, Great Britain, 2008. [13] R. L. Fittro, C. R. Knospe, and L. S. Stephens, “ -synthesis applied to the compliance minimization of an active magnetic bearing HSM spindle’s thrust axis,” Machin. Sci. Technol., vol. 7, no. 1, pp. 19–51, 2003. [14] Y. Zhang and N. D. Sims, “Milling workpiece chatter avoidance using piezoelectric active damping: A feasibility study,” Smart Mater. Structures, vol. 14, no. 6, pp. N65–N70, 2005. [15] M. Shiraishi, K. Yamanaka, and H. Fujita, “Optimal control of chatter in turning,” Int. J. Mach. Tools Manuf., vol. 31, no. 1, pp. 31–43, 1991. [16] M. Chen and C. R. Knospe, “Control approaches to the suppression of machining chatter using active magnetic bearings,” IEEE Trans. Control Syst. Technol., vol. 15, no. 2, pp. 220–232, Mar. 2007. [17] Y. Altintas, Manufacturing Automation. Cambridge, U.K.: Cambridge Univ. Press, 2000. [18] Y. Altintas and M. Weck, “Chatter stability of metal cutting and grinding,” CIRP Annals—Manuf. Technol., vol. 53, no. 2, pp. 619–642, 2004. [19] R. Faassen, “Chatter prediction and control for high-speed milling: Modelling and experiments,” Ph.D. dissertation, Dept. Mech. Eng., Eindhoven Univ. Technol., Eindhoven, The Netherlands, 2007. [20] G. Stépán, “Modelling nonlinear regenerative effects in metal cutting,” Philosophical Trans. Royal Soc. A: Math., Phys., Eng. Sci., vol. 359, no. 1781, pp. 739–757, 2001. [21] R. P. H. Faassen, N. van de Wouw, J. A. J. Oosterling, and H. Nijmeijer, “Prediction of regenerative chatter by modelling and analysis of high-speed milling,” Int. J. Mach. Tools Manuf., vol. 43, no. 14, pp. 1437–1446, 2003. [22] G. Schweitzer, H. Bleuler, and A. Traxler, Active Magnetic Bearings. Zürich: Verlag der Fachvereine, 1994.

[23] R. P. H. Faassen, N. van de Wouw, H. Nijmeijer, and J. A. J. Oosterling, “An improved tool path model including periodic delay for chatter prediction in milling,” J. Comput. Nonlinear Dyn., vol. 2, no. 2, pp. 167–179, 2007. [24] M. Farkas, Periodic Motions. Berlin: Springer-Verlag, 1994. [25] T. Insperger and G. Stépán, “Updated semi-discretization method for periodic delay-differential equations with discrete delay,” Int. J. for Numer. Methods Eng., vol. 61, no. 1, pp. 117–141, 2004. [26] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, 2nd ed. Chichester, U.K.: Wiley, 2005. [27] Y.-P. Huang and K. Zhou, “Robust stability of uncertain time-delay systems,” IEEE Trans. Autom. Control, vol. 45, no. 11, pp. 2169–2173, Nov. 2000. [28] Z.-Q. Wang, P. Lundström, and S. Skogestad, “Representation of uncertain time delays in the H1 framework,” Int. J. Control, vol. 59, no. 3, pp. 627–638, 1994. [29] The MathWorks, Inc., Natick, MA, “Documentation,” 2010. [Online]. Available: http://www.mathworks.com [30] C. Ceton, P. M. R. Wortelboer, and O. H. Bosgra, “Frequency weighted closed-loop balanced reduction,” in Proc. 2nd Euro. Control Conf., 1993, pp. 697–701.

Niels van Dijk received the M.Sc. and Ph.D. degrees in mechanical engineering from the Eindhoven University of Technology, Eindhoven, The Netherlands, in 2006 and 2011, respectively. Currently he is with the Mechatronics Technologies Group, Philips Innovation Services, Eindhoven, The Netherlands. His interests include modelling and control of manufacturing and high-precision systems.

Nathan van de Wouw received the M.Sc. (with honors) and Ph.D. degrees in mechanical engineering from the Eindhoven University of Technology, Eindhoven, The Netherlands, in 1994 and 1999, respectively. From 1999 to 2010, he has been an Associate Professor with the Department of Mechanical Engineering, Eindhoven University of Technology. He was with Philips Applied Technologies, Eindhoven, The Netherlands, in 2000 and at the Netherlands Organization for Applied Scientific Research, The Netherlands, in 2001. He was a visiting Professor with the University of California Santa Barbara, in 2006/2007 and the University of Melbourne, Australia, in 2009/2010. He has published a large number of journal and conference papers and the books Uniform Output Regulation of Nonlinear Systems: A convergent Dynamics Approach with A.V. Pavlov and H. Nijmeijer (Birkhauser, 2005) and Stability and Convergence of Mechanical Systems with Unilateral Constraints with R.I. Leine (Springer-Verlag, 2008).

Ed J. J. Doppenberg received the B.S. degree in control engineering from the Rotterdam University, the Netherlands, in 1974 and the M.Sc. Ing-degree in control engineering from the Delft University of Technology, Delft, the Netherlands, in 1989. He has been working from 1987 until 2002 in the development of active noise and vibration adaptive control system for the transportation sector (aerospace and automotive applications) at TNO Science and Industry, Delft, the Netherlands. He was involved from 2004 to 2010 with the manufacturing technology development at TNO Science and Technology. In 2011 he joined the Department of Precision Motion Systems, TNO. He has published several papers on adaptive (MIMO) control systems and on machining technology, chatter control and micro milling.


Han Oosterling received the M.Sc. degree in mechanical engineering from the Eindhoven University of Technology, Eindhoven, The Netherlands, in 1981. From 1981 to 1992, he has been with Manufacturing Technology Development, Dutch Navy, Philips Tool and Die Shop, and Philips Research. In 1991, he started with TNO Science and Technology, Delft, The Netherlands, as a Research Leader of the Machining Technology Research Group. In 2011, he joined the Department Space and Science, TNO. He has published several papers on coating technology, machining technology, chatter control, and micro milling.

Henk Nijmeijer (F’99) received the M.Sc. and Ph.D. degrees in mathematics from the University of Groningen, Groningen, The Netherlands, in 1979 and 1983, respectively. From 1983 to 2000, he was with the Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. Since 1997, he was also part-time affiliated with the Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands, where since 2000, he has been a Full Professor with and

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chairs the Dynamics and Control section. He has published a large number of journal and conference papers and several books, including the “classical” Nonlinear Dynamical Control Systems (Springer, 1990) coauthored with A. J. van der Schaft, Synchronization of Mechanical Systems (World Scientific, 2003) coauthored with A. Rodriguez, Dynamics and Bifurcations of Non-Smooth Mechanical Systems (Springer-Verlag, 2004) coauthored with R. I. Leine, and Uniform Output Regulation of Nonlinear Systems (Birkhauser 2005) coauthored with A.Pavlov and N. van de Wouw. Dr. Nijmeijer is Editor-In-Chief of the Journal of Applied Mathematics, corresponding editor of the SIAM Journal on Control and Optimization, and board member of the International Journal of Control, Automatica, Journal of Dynamical Control Systems, International Journal of Bifurcation and Chaos, International Journal of Robust and Nonlinear Control, Journal of Nonlinear Dynamics, and the Journal of Applied Mathematics and Computer Science. He was a recipient of the IEE Heaviside premium in 1990. In the 2008 research evaluation of the Dutch Mechanical Engineering Departments the Dynamics and Control Group was evaluated as excellent regarding all aspects (quality, productivity, relevance, and viability).