Adaptive schemes for milling forces control
Luis Rubio*, 2Manuel De La Sen and 1 László E. Kollár
1
Savaria Institute of Technology, ELTE, Szombathely, Hungary; *lrr@inf.elte.hu
1
2
IIDP, UPV/EHU, Leioa, Bizkaia, Spain
Abstract This paper presents a study of different approaches to implement
adaptive control in milling peak forces. The milling process is modeled as second
order continuous transfer function which varies the axial depth of cut when machining
a work-piece. Conventional adaptive control scheme is applied using a fractional
order holds with different correcting gains to discretize the continuous transfer
function of the milling process, and implement the discrete adaptive control law.
Furthermore, multi-estimation control scheme combine the use of different order hold
gains working in parallel to track the reference signal. Each discretization has an
associate controller, and a supervisory system chooses the adequate controller of the
parallel scheme at each residence time which minimize a defined cost function. In this
mode, every residence time the controller is updated, choosing the one among the
proposed in parallel which minimize the proposed cost function to adjust the control
law. Simulation results display the performance of the different approaches.
Keywords: Adaptive control, Milling Peak Forces Control, Multi-estimation scheme.
1. Introduction
Milling is a cutting process widely used in the manufacturing of mechanical
components. It consists of the relative movement between feeding a workpiece
clamped on a table while rotating multi-tooth cutter. In order to avoid machine tool
malfunctions such as tool wear or breakage and to achieve certain degree of quality in
the finishing of the workpiece, the peak cutting force on the workpiece has to be
maintained below a prescribed safety upper-bound. This fact implies that a control
strategy has to be implemented on the system in order to fulfill such safety and
performance requirements.
2
In practice, batch and high volume milling environments are characterized by
fixing or varying with a known range tool-part combinations. Meanwhile, job-shop
environments require adaptive techniques since tool-part combinations are different at
each operation [1]. Previous paper by the authors dealt with the case of high volume
operations [1]. In this paper, the design of a a discrete time control of milling forces is
presented considering job-shop operations, where the milling system model is uncharacterized and the plant parameters are unknown or varied in a unknown way,
such as even though sudden changes in the tool-part combinations. We consider the
milling machine model to be a linear plant with unknown time varying parameters
which is a typical context in milling processes.
Different kinds of holds are increased control designer attention due to their
enhanced properties respect to the traditional zero order hold (ZOH), which are
commonly used in manufacturing floors. In this work, the strategies for controlling
the milling system are based on the use of fractional order holds (FROH) devices of
the correcting gain, βϵ[-1,1], (β-FROH). Thus, the continuous milling system
described in [2] is discretized under a β-FROH, first in a basic model reference closed
loop and, after that, obtaining a series of discrete-time models of the system. Each
different discretization of the plant has an associated controller [3,4]. Hence, a basic
model reference control is first proposed. The step ahead over the traditional model
reference schemes is the use of FROH to discretize the continuous plant instead of
traditional ZOH. Furthermore, a supervised multi-estimation adaptive control scheme
is presented. The idea behind this is to incorporate a logic based on how the controller
works and modify it in real time. Those kinds of schemes can be useful, for instance,
when the system work in different states governed by different equations or, to
modify the closed-loop structure in one working point achieving best tracking
performance. In this paper, it is presented and applied a multi-estimation adaptive
control scheme which uses different possible discretization models of the continuous
plant to obtain a parallel series of discrete time adaptive control of the system. The
different models of the system are obtained from a set of different discretization of the
continuous transfer function under a fractional order hold of correcting gain βϵ[-1,1],
(β-FROH) .
3
The objective of the study is to design a supervisory scheme which is able to find
the most appropriate value for the gain β in an intelligent design framework. A
tracking performance index is evaluates each possible discretization, and the scheme
chooses which lowest value has, for implementing the FROH device and the control
law.
The use of FROH in both schemes modifies the overall closed-loop of the system
which allows achieving better transient responses in terms of overshoots and settling
times [3,4], improving the properties of the zeros of the discrete time [5,6] and getting
better inter-sample behavior [6] which can lead, for example, to better surface finish
and maintenance of the tool and machine tool components and to minimize wear of
the tool.
Finally, the tracking performance of the continuous-time signal is intended to be
studied by means of a cost function. It will be used to compare the basic control
scheme when just a ZOH and fixed gain FROH devices are used, and when the multiestimation adaptive scheme is applied [1].
In manufacturing literature, adaptive control has been classified into two classes,
the adaptive control with optimization (ACO) and the adaptive control with
constraints (ACC). Extensive research has been carried out in both fields [7]. The
most successful realizations are with ACC. In those systems, the feed rate is
manipulated to maintain a constant cutting force torque or power independently of the
variations in machining. These approaches indirectly measure the cutting forces for
adaptive control the milling systems, such as using the current drawn from different
ac/dc feed drive servos [7-12]. The earliest ACC systems applied the fixed gain
integral control algorithms [7,8]. The mayor problem existing in these ACC systems
is that the use of a fixed control method might cause instability with a large transient
overshoot when machining conditions are changed significantly. To solve this
problem, the parameter adaptive control approaches have been proposed. In those,
some parameters of the controller can be adjusted to match the change of the
dynamics of machining processes. There are three typical approaches to the parameter
adaptive control that have been applied for machining process control, including the
variable gain PI control [9], the model reference adaptive control (MRAC) [10,11]
and the self-tunning regulator [2,12].
4
However, very few adaptive control systems have been accepted by machine tool
manufacturers so far, all the research works have common practical drawbacks:
(1) Cutting force sensors, such as dynamo-meters, which are presently available,
set many limits with respect to cost, wiring, reliability, and machine strokes.
(2) The controller structure should be more robust so that it can be applied
reliably to various combinations of machining processes, work-piece
materials and tools, without major changes of the control algorithm and
parameters.
Despite these limitations the control of forces in milling processes is still an open
research topic [13-20].
The work is organized as follows. In the next section, the continuous model and its
discretization under a β-FROH is presented, and the model reference transfer function
is displayed. In the third section, basic multi-estimation control scheme with fixed
gain of the β-FROH and, a parameterization of model reference multi-estimation
control scheme based on multiple discretization of the continuous plant using a βFROH is addressed. Section four presents some simulation results, ending the paper
with conclusions.
2. System description
2.1 Continuous Model
The milling system can be modeled as the series decomposition of a Computerized
Numerical Control (CNC), which includes all the circuitry involving in the table
movement (amplifiers, motor drives), and the tool-work-piece interaction model
itself. A feed rate command f c (which plays the role of the control signal) is sent to
the CNC unit. This feed rate represents the desired velocity for the table movement.
Then, the CNC unit manages to make the table move at an actual feed velocity of ��
according to the CNC dynamics. Even though the machine tool drive servos are
typically modeled as high order transfer functions, they can usually be approximated
as a second order transfer function within the range of working frequencies. Besides,
5
they are tuned to be over-damped without overshoot, so that they can be modeled as
the first order system [2]:
�� � =
�� �
�� �
=
1
(1)
�� �+1
Where �� and �� are the actual and command velocity values of the table in
�� � respectively and �� is an average time constant, which depends on the type
of the machine tool.
In addition, the chatter vibration and resonant free cutting process can be
approximated as the first order system [2]:
�� � =
�� �
�� �
=
�� ��� � ∅��,∅�� ,��
� � ��
1
(2)
�� �+1
Where �� � ��2 is the cutting pressure constant, ��� �� is the axial depth of
cut, � ∅�� , ∅�� , �� is a non-dimensional immersion function, ranging between 0 and
~�� depending on the immersion angle and the number of teeth in cut, �� is the
number of teeth on the milling cutter and �� ��� � is the spindle speed. The axial
deep of cut function � ∅�� , ∅�� , �� in (2) may be time-varying leading to a potential
time-varying system. In particular, the cutting process is assumed to be in this work
piece-wise constant, admitting sudden changes in the cutting parameters at certain
time instants while remaining invariant between changes. In the first part of the work,
potential changes in the axial depth of cut are assumed to be completely known.
While in the second part, the changes in the axial depth of cut are dealt as unknown.
This assumption allows us to consider the cutting process to be described by the
transfer function of equation (2) with time intervals between changes.
The combined transfer function of the system, obtained from (1) and (2) is
�� � =
�� �
�� �
=
��
�� �+1 �� +1
(3)
where the process gain is �� ��∗� �� = ����� � � � .
� �
Figure 1 shows the sample workpiece depicting basic cutting geometry features
with changes in the axial depth of cut used in the simulations.
The spindle speed remains constant, 715 ��� ; the workpiece is made of
Aluminium 6067 whose specific cutting pressure is assumed to be �� =
6
1200 �
��2
.
A 4-fluted carbide mill tool, half-immersed and rouging milling
operation will be taken into consideration in the present paper.
tool
feed
3mm
2 mm
5mm
3mm
workpiece
5.87 mm
5.87 mm
5.87 mm
7.55mm
Figure 1: Work-piece profile to test control algorithms.
Also, note that the desired final geometry of the piece to be milled involves
changes in the axial deep of cut which implies suddenly changes in its value,
according to the sudden changes assumption presented before. On the other hand, it
has been considered that the control law computes new feed-rate command value at
each sampling interval. Furthermore, it is worth to be mentioned that the CNC unit
has its own digital position law executed at small time intervals in comparison with
the sampled time of the control law, even though if high speed milling tool drives are
used [2].
2.2 Discrete model under � − ����
In this paper, the problem of controlling a continuous plant is addressed by using a
discrete controller. The discrete controller is obtained applying a model-reference
pole-placement based control design to a discrete model of the plant (3) obtained by
means of a ���� with a certain correcting gain � . The additional “degree of
freedom” � provided by the ���� can be used with a broad variety of objectives
such as to improve the transient response behaviour, to avoid the existence of
oscillations in the continuous time output of the system or to improve the stability
properties of the zeros of the discretized system [5, 6]. In this way, this work is
especially focused on the use of these devices to improve the transient response of the
7
closed-loop system by selecting an adequate value of the fractional order hold. Thus,
in the following sections a comparative study of the behaviour of the closed-loop
system under different values of the correcting gain is developed. The influence of the
value of b will be extended to the adaptive case in the subsequent paper. Hence, the
discretization of (3) under a FROH is calculated as [21,22]:
�� = � ℎ� � ∗ �� �
(4)
Where ℎ� � = 1 − ��−�� +
� 1−�−��
��
1−�−��
��
is the transfer function of a � −
����, where � is the argument of the � − ���������, being formally equivalent to
the one step ahead operators, �, used in the time domain representation of difference
equations. This allows us to keep a simple unambiguous notation for the whole paper
content. The sampling time � has been chosen to be the spindle speed, �� , as it is
usual for this kind of systems [1,4, 8-10]. Note that when � = 1 , the ���� hold
becomes a first order hold ��� and when � = 0 , the zero-order hold ��� is
obtained, being both cases of � ∈ −1,1 . Furthermore, �� �
using just ��� devices in the following way:
�� � =
(5)
�� �
�
� �� �
Where ℎ� � =
=
�−�
�
1−�−��
�
� ℎ� � � � � +
� �−1
��
� ℎ� �
�� �
�
=
may be calculated
�� �
�
�
� � �−� ��
�
�−� ��
is the transfer function of a ��� and �� = 1 if � ≠ 0 and
�� = 0 if � = 0, which means that a fractional order hold with � ≠ 0 adds a pole at
the origin (z=0).
2.3 Desired response: model reference
A second order system �� � =
�2�
�2+2��� s+�2�
(6) is selected to represent the
system model reference. This system is characterized by the desired damping ratio, �
and its natural frequency, �� . It is known that small � leads to a large overshoot and a
large setting time. A general accepted range value for � to attain satisfactory
performance is between 0.5 and 1, which corresponds to the so-called under-damped
systems. In this way, a damping ratio of � = 0.7 and a rise time, �� , equal to three
8
spindle periods is usually selected for practical applications [1,2]. Furthermore, the
natural frequency is then usually suggested to be �� = 2.5 � ��� � . This continuous�
time reference model is then discretized with the same ���� as the real system was
to obtain the corresponding discrete-time reference model for the controller. Thus, a
few different discrete models obtained from a unique continuous reference model are
considered depending on the value of � used to obtain the discretization.
3. Control schemes
3.1 Basic model following adaptive controller
The aim of the model-following control strategy is to force the closed-loop system
to behave as a prescribed reference model. For this purpose, the control scheme
displayed in figure 2 is applied to the practical milling transfer function represented in
� �,�
the first part of the paper. In the figure, ��� �, � = � �,� represents the feed-forward
� �,�
filter from the reference signal, ��� �, � = � �,�
the feedback controller, � �, � is
the discrete plant, �� �, � the model reference and ��� is the reference force. The
recursive least square (RLS) estimation algorithm is used for updated parameter
estimation.
9
Figure 2: Adaptive basic model following control scheme.
The transfer function of the reference model can be extracted from the control
closed loop scheme as,
�� � =
�− � �'� � �� �
�� � �� �
=
�� � � � �
�� � �� �
(7)
where �'� � contains the free-design reference model zeros, �− � is formed by
the unstable (assumed known) plant zeros and �� � is a polynomial including the
eventual closed-loop stable pole-zero cancellations which are introduced when
necessary to guarantee that the relative degree of the reference model is no less than
that of the closed-loop system so that the synthesized controller is casual. Then, it will
�
�
be considered for each ��ℎ controller the polynomials �� , �� and � (� depends only
on the reference model zeros polynomial which is of constant coefficients) where
�
�
= �'� �� , �� (monic) and �� are unique solutions with degrees fulfilling
�
��� ��
�
= �, ��� ��
= � − 1, ��� �� �� = 2� − 1 (8)
of the polynomial Diophantine equation
�
�
�
�
� +
�
�
�
�� �� + �� �� = �� �� �� ↔ �� �1,� + �− �� = �� �� (9)
Which have unique solutions for the above polynomial degree constraints provided
�
�
that �� , ��
�
� +
�
are all co-prime for � = 1,2, . . , �� with �� = �� �1,� and for all
10
at every sampling instant, being ��
1 ≤ � ≤ ��
the number of estimation
schemes/adaptive controllers parameterizations. � = 2 if a ��� is used in view of the
second order milling plant and reference model continuous transfer function. For
���� not being ���, � = 3 since a new pole at the origin is automatically added to
the plant.
From (7) - (8), perfect matching is achieved through the control signal:
��,� =
� �
� �
��,� −
� �
� �
��,� (10)
3.2 Multi-estimation scheme
The parallel multi-estimation scheme is composed by different estimators. Each
estimator is used to identify a different discretization of the continuous plant under a
� − ���� . The main idea for the scheme implementation is that all the
estimator/controller pairs are running in parallel at the same time while calculating
each control law, but only one of them actually generates the control law. Each
controller parameterization is updated for all time although only one is generating the
control signal. The strategy is to use the controller obtained from the best estimation
model at each time interval. The closed loop is guaranteed if the time interval between
consecutive switching is larger than an appropriate residence time. The estimated
output for each ��ℎ identifier can be calculated as,
�
� �
�
�� = �� ��
�
�
for 1 ≤ � ≤ �� and � ≥ 0 where �� and ��
are the parameter
�
estimation vector and associate regressor, respectively. For each estimator, ��
possesses the discrete estimated parameters of each discretization of the continuous
transfer function and the regressor is composed by the plant input and output signals
at different sampling times.
Figure 3 displays a typical multi-estimation scheme, where the estimated outputs
are compared with the real plant output. � � �
�
continuous transfer function, while �� �
function at ��ℎ sampling instant.
for 1 ≤ � ≤ �� denotes the set of the
is used to indicate their estimated transfer
11
Figure 3: Multi-estimation scheme.
The following estimation method is proposed. In this method, each estimation
vector is updated at each sampling time by the estimation algorithm,
�
�
�
�
��+1 = �� +
�
��+1 = �� −
�
�
�
�
�� �� ��
1+
�
�
��
�
� �
�� �� ��
�
�
��
�
�
��
� �
��
� �
1+�� �� ��
�
�
; �� ���������
� �
; �� = ��
> 0 (11)
Where �� = ��,� − ��,� is the identification error for the ��ℎ sample ∀� ∈ �� =
1,2, . . , �� .
Switching rule for multi-estimation scheme. The switching rule for the model
(�)
reference control re-parameterization is obtained from the performance index , �� ,
presented
in
next
subsection,
as
follows:
1. Let the switching sampling time sequence be denoted by �� = � 1 , �(2) , . . , �(��)
where nl is the number of switchings, and �(�+1) − �(�) ≥ �� = �� �, �� ��, where T is
the sampling time, �� a known minimum residence time (or if it is unknown, then, it is
replaced by some available upper-bound), for all �(�) , �(�+1) ��� . The minimum
12
residence time, �� , is given by the necessary time to ensure the stability of the closedloop system. During this period of time, it is not allowed to switch to other parallel
controller in order to guaranty the stability of the closed loop system. Then, the
chosen controller to generate the control signal to the plant is fixed during Nr times
the
sampling
time.
2. Thus, the ck-estimation scheme with 1 ≤ ck ≤ nl , which parameterize for all k > 0
the basic controller at any switching time, , is updated as follows. Assume that the last
switching time for the controller re-parameterization was t(i). Thus, for each current kth
�
�
sampling time, define the auxiliary integer variable: �� = ���[�: �� = ��� �� ; �, � ∈
(12),
��
1.
2.
if
if
�� ≠ ��+1
for
all
then
kT ≥ t i + τR
then
�
�+1
integer
← ��
and
modify
k≥1
�� ← �� , �
,
.
�+1
.
Note that it does not necessary means that the update of the β-value has to be at each
switching instant. The value of the minimum residence time which ensures the closed
loop stability could be obtained from the parameters of the system or from ‘a priori’
knowledge through an algorithm [22]. However, it is know that there exists a
minimum value of the residence time which guaranties the stability of the closed-loop
system
[22].
This
1. For each k > k1 , if min
can
be
k
Ji
j=k−k1 j
mathematically
expressed
as:
≥ JM , 1 ≤ i, j ≤ nl then �� ← �� + ∆��1 until
closed loop stability is guaranteed by an admissible range �� ∈ ���(�� ), ∞ .
2. But, max
�
�=�−�1
�� � − ��
��
≤ ���(�� ) and �� ∈ ���(�� ), ∞ , 1 ≤ � ≤ �� ,
then �� ← �� − ∆��2 for each k ≥ k2 being �� and ���(�� ) prefixed real positive
constants, sufficiently large and small respectively and the initial value �� arbitrary.
The parameters �� , ���(�� ) , ∆��1 and ∆��2 are supposed to be programmed by the
designer according to previous knowledge and insight of the problem at hand. k1 and
k2 are integers. This part of the algorithm has to be run on-line or off-line during a
short period of time being significantly less than the relevant transient response
duration.
Identification performance index. A general performance index is now proposed
�
�
which is able to evaluate the performance of the closed loop ��� − ��
, the jump
13
�
�−1
− ���,� , the cost of
in the output associates to a switching between controls, ��,�
�
(�)
�
�−1
− ��,� and the output signal error �� − �� . Then, the following
the control ��,�
cost
�
�� = ��
��
�−�
�
�
�=�−� 1
�−�
�
�
�=�−� 2
function
�
is
2
�
��� − ��
�
�� − ��
2
�
�−1
− ���,�
+ �� ��,�
suggested:
2
�
(�)
�−1
− ��,�
+ �� ��,�
2
+
(13)
for 1 ≤ l ≤ nl , where λ ∈ 0,1 and M > 0 are designed as real parameters, being M
big enough to evaluate the performance of the system. λ1 and λ2 are forgetting factors
which allow to give more importance to the last time interval values, �� + �� + �� +
�
�� = 1; �� , �� , �� , �� ≥ 0 , and ��� represents the force output of the reference model
�
in the l-parallel control, ��
�
�−1
the estimated output of the plant, ��,�
the estimated
output associated to the ck-1 model in the kth-sample, ���,� the estimated output
�
�−1
associated to the l-model at the kth-sample, ��,�
the control signal associated to the ck-
(�)
1 model in the kth-sample, ��,� the control signal associated to the l-model at the kthsample, and Fp the output of the system.
4. Simulation results
There is an extensive literature which carefully explains the algorithms here
developed, for example, and prove the stability robustness of the adaptive law [2326]. The novelty of the control relies on the use of fractional order holds instead of the
usual ��� appearing in the manufacturing literature. In this paper, the correcting gain
of � − ���� is handled to show that the system transient response may be enhanced
respect to the use of ���. This can lead to avoid overloading of the insert, because
the maximum removed chip-thickness would not increase the principal tensile stress
in the cutting wedge beyond the ultimate tensile strength of the tool material, this can
also lead to prevent fracture of the shank, and fulfil the machine tool requirements,
such as power and torque availability [27]. Moreover, if the reference force is selected
near the tool breakage limit, the large overshot lead to tool breakage [2, 12,27]. Then,
14
if the overshoot of the system response is reduced, the reference force can be
increased, improving time production requirements.
An adaptive model following controllers have been developed using different
correcting gains of the fractional order hold. The milling system and the model
reference are discretized via fractional order hold, and arbitrary values of the initial
parameter vector, θi, where i is the number of estimated parameters, are considered.
As example, some representative cases are plotted in figures from 4 to 6. The
figures show four sub-plots; the discrete-time milling system response over the
sampling time compared to the reference (dashed red), the reference error, the control
law, and the continuous-time milling system response. The continuous time response
is obtained by introducing the continuous-time feed rate (control law) as input into the
continuous transfer function of the milling system. The figures present the resultant
force keeping at the reference force, which is set to 1200 N. The system registers
large overshoots in the transient responses, depending on the b -value and the initial
values of the parameter vector. The initial parameter vector has the ability that if it is
near to the real values of the plant, the transient response of the system will be smooth
and feasible. In contrast, if the initial value of the parameter vector has been selected
in arbitrary manner the transient is normally oscillated with a great maximum
overshoot and large setting time. In any case, fractional order holds can help to reduce
large overshoots.
On the other hand, there are abruptly overshoots in the output when the axial depth
of cut changes suddenly. It is due to the intrinsic structure of the closed-loop output. It
is not the main purpose of this study at this stage to reducing or avoiding these jumps.
But, in that case, some ‘a priori’ information about the work-piece geometry is
required to design a successful control, as in [12], where a CAD model of the workpiece is used to modify the control command when the axial depth of cut changes to
minimize the overshoots due to abrupt changes in the transfer function
Simulations have been developed for the milling system. Figure 4 outputs the
response of the closed loop system using a ZOH, Figure 5 represents the the closed
loop system using b - FROH with β=0.5, and Figure 6 is obtained by applying the
explained multi-estimation scheme building from a set of b - FROH devices with
15
correcting gains,β=[-0.3,0,0.3,1], taking the residence time equal to one sampling
time, and �� = 1, �� , �� , �� = 0.
Figure 4: Outputs of the adaptive closed loop controller using ZOH.
Figure 5: Outputs of the adaptive closed loop controller using β-FROH with β=0.5.
16
Figure. 6: Outputs of the adaptive closed loop controller using multi-estimation
scheme..
In the figures, it cam be observed that the transient response of the forces are larger
in case of using the multi-estimation scheme over the basic scheme using βϵ[-1,1], (βFROH). The selection of adequate value of β to initialize, in case of figure 6 we use
β=0, the multi-estimation scheme plays a key role to achieve good transient response.
Moreover, the selection of adequate residence time helps to enhance the output.
Finally, figure 7 displays the switching of β-value in the multi-estimation scheme.
17
Figure 7: Switching of β in the multi-estimation scheme.
Conclusion
Conventional and multi-estimation control schemes have been developed to study
the control of peak forces in milling processes. The aim of the research resides on
achieving better transient responses in terms of overshoots and settling times,
improving the properties of the zeros of the discrete time and getting better intersample behavior which can lead, for example, to better surface finish and
maintenance of the tool and machine tool components and to minimize wear of the
tool. For this purpose, a switching rule and identification performance index have
been defined to manage the multi-estimation adaptive milling force control scheme.
First approaches shows just by using a conventional scheme with a β-FROH, better
transient responses may be achieved respect to ZOH, which is a milestone to get
better behaviour with the multi-estimation scheme. Simulation results have been
presented to show up the basic working of the algorithms and the potentiality of this
approach to enhance machining performance indexes.
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