Delft University of Technology
'Constant in gain lead in phase' element-application in precision motion control
Saikumar, Niranjan; Sinha, Rahul Kumar; Hossein Nia Kani, Hassan
DOI
10.1109/TMECH.2019.2909082
Publication date
2019
Document Version
Accepted author manuscript
Published in
IEEE/ASME Transactions on Mechatronics
Citation (APA)
Saikumar, N., Sinha, R. K., & Hossein Nia Kani, H. (2019). 'Constant in gain lead in phase' element-
application in precision motion control.
IEEE/ASME Transactions on Mechatronics
,
24
(3), 1176-1185.
https://doi.org/10.1109/TMECH.2019.2909082
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1
‘Constant in gain Lead in phase’ element -
Application in precision motion control
Niranjan Saikumar, Rahul Kumar Sinha, S. Hassan HosseinNia
Precision and Microsystems Engineering,
Faculty of Mechanical Engineering, TU Delft, The Netherlands
Abstract—This work presents a novel ‘Constant in gain Lead
in phase’ (CgLp) element using nonlinear reset technique. PID is
the industrial workhorse even to this day in high-tech precision
positioning applications. However, Bode’s gain phase relationship
and waterbed effect fundamentally limit performance of PID
and other linear controllers. This paper presents CgLp as a
controlled nonlinear element which can be introduced within the
framework of PID allowing for wide applicability and overcoming
linear control limitations. Design of CgLp with generalized first
order reset element (GFORE) and generalized second order reset
element (GSORE) (introduced in this work) is presented using
describing function analysis. A more detailed analysis of reset
elements in frequency domain compared to existing literature
is first carried out for this purpose. Finally, CgLp is integrated
with PID and tested on one of the DOFs of a planar precision
positioning stage. Performance improvement is shown in terms
of tracking, steady-state precision and bandwidth.
Index Terms—Reset control, Precision control, Motion control,
Mechatronics, Nonlinear control
I. INTRODUCTION
P
ID continues to be popular in the industry due to its wide
applicability, simplicity and ease of design and imple-
mentation. PID is used in high-tech applications from wafer
scanners for production of integrated circuits and solar cells to
atomic force microscopes for high-resolution scanning. With
well-designed mechanisms and feed-forward techniques, high
precision, bandwidth and robustness are being achieved. PID
also lends itself to industry standard loop shaping technique for
designing control using frequency response function obtained
from the plant. However, the constantly growing demands on
precision and bandwidth are pushing PID to its limits. PID
being a linear controller suffers from fundamental limitations
of Bode’s gain phase relationship and waterbed effect [1],
[2]. It is self-evident that these can only be overcome using
nonlinear techniques. However, most nonlinear techniques in
literature presented for precision control [3]–[6] are more
complicated to design and/or implement and do not fit within
techniques like loop shaping which are popular and widely
used in the industry.
Reset control is a nonlinear technique which has gained
popularity over the years and has the advantage of fitting
within the framework of PID for improved performance. Reset
involves the resetting of a subset of controller states when a
reset condition is met. Reset was first introduced by J C Clegg
in [7] for integrators to improve performance. Advantage of
reset is seen in reduced phase lag compared to its linear
counterpart [8]. This work has been extended over the years
with other reset elements from First Order Reset Element
(FORE) [9], Generalized FORE (GFORE) [10] and finally
to Second Order Reset Element (SORE) [11] introduced and
used in control applications. Significant work can be found
in literature showing the advantages of reset control [12]–
[21]. However, in most of these cases, reset control has mainly
been used for it’s phase lag reduction advantage. Some works
exist where reset has been used for phase compensation.
In [22], Ying et al. use the reset element to overcome the
waterbed effect through mid frequency disturbance rejection
by lowering the sensitivity peak. In this case, reset is used
to achieve a narrowband phase compensator, hence improving
phase margin and performance. This compensator was further
modified for improved performance and phase compensation
in [23] allowing for the use of notch for disturbance rejection
without affecting stability margins. Reset control with opti-
mized resetting action for improved performance has also been
presented in [24]. Some preliminary work towards broadband
phase compensation can be found in [25], [26].
In this work, we present a novel reset element termed ‘Con-
stant in gain Lead in phase (CgLp)’ element which extends
the use of reset to be used for broadband phase compensation.
The element is designed using describing function analysis to
work well within existing framework of PID, thus achieving
industry compatibility. Improvement in precision and tracking
is shown on a precision positioning stage. In Section II, basics
of reset systems are provided along with the definitions of reset
elements present in literature. The novel GSORE element is
presented in Section III. Further, while reset elements have
mainly been analysed for their phase lag reduction in literature,
other properties of generalized reset elements in frequency
domain critical to CgLp design are discussed. Design and
analysis of CgLp are presented in Section IV followed by the
inclusion of CgLp within framework of PID for broadband
phase compensation. The application of this modified CgLp-
PID controller on a precision positioning stage is dealt with
in Section V to show improvement in performance. The
conclusions and future work are provided in Section VI.
II. PRELIMINARIES
A. Definition of Reset control
A general reset controller can be defined using the following
differential inclusions:
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2
Σ
R
=
˙x
r
(t) = A
r
x
r
(t) + B
r
e(t) if e(t) 6= 0
x
r
(t
+
) = A
ρ
x
r
(t) if e(t) = 0
u(t) = C
r
x
r
(t) + D
r
e(t)
(1)
where A
r
, B
r
, C
r
, D
r
are state-space matrices of the base
linear system, A
ρ
is reset matrix determining the state after
reset values. e(t) is the error signal fed to the controller and
u(t) is the output of controller which is used as control input
for plant. While other forms of reset like reset band and fixed
instant reset exist in literature, the form provided above is the
most popular, widely applied and tested. The reset controller
of Eqn. 1 generally consists of both linear and nonlinear reset
part. The A
ρ
matrix is defined to reset only the appropriate
states of controller.
B. Describing function
The nonlinearity of reset elements creates the problem of
designing controllers in frequency domain especially using
industry popular loop shaping technique which uses Bode,
Nyquist and Nichols plots. In literature, sinusoidal input
describing function analysis has been used to analyse reset
elements in frequency domain. In fact, the phase lag reduction
advantage was seen by Clegg in 1958 using this technique.
Although describing function does not accurately capture all
the frequency domain aspects of reset, it is useful in providing
necessary information for design and analysis.
The describing function of generic reset systems as defined
by Eqn. 1 is provided in [10] and this is used to obtain under-
standing of the system in frequency domain. The sinusoidal
input describing function is obtained as
G(jω) = C
T
r
(jωI − A
r
)
−1
(I + jΘ
ρ
(ω))B
r
+ D
r
(2)
where
Θ
ρ
=
2
π
(I + e
πA
r
ω
)
I − A
ρ
I + A
ρ
e
πA
r
ω
A
r
ω
2
+ I
−1
C. Stability of reset elements and systems
Stability conditions given in [27] can be used to check
closed-loop stability of reset control systems for SISO plants.
The following condition has to be satisfied for ensuring
quadratic stability:
Theorem 2.1: There exists a constant β ∈ <
n
r
×1
and
positive definite matrix P
ρ
∈ <
n
r
×n
r
, such that the restricted
Lyapunov equation
P > 0, A
T
cl
P + P A
cl
< 0 (3)
B
T
0
P = C
0
(4)
has a solution for P , where C
0
and B
0
are defined by
C
0
=
βC
p
0
n
r
×n
nr
P
ρ
, B
0
=
0
n
p
×n
r
0
n
nr
×n
r
I
n
r
(5)
A
cl
is the closed loop matrix A-matrix
A
cl
=
A
p
B
p
C
r
−B
r
C
p
A
r
(6)
in which (A
r
, B
r
, C
r
, D
r
) are the state space matrices of the
controller defined by Eqn. 1 with n
r
being the number of states
being reset and n
nr
being the number of non-resetting states.
(A
p
, B
p
, C
p
, D
p
) are the state space matrices of the plant.
D. Reset elements
The reset part of controllers defined by Eqn. 1 have been
presented as different reset elements in literature.
1) Clegg Integrator (CI): Clegg or Reset integrator is the
first introduction of reset technique in literature [7]. The action
of resetting integrator output to zero when input crosses zero
results in favoured behaviour of reducing phase lag from
90
◦
to 38.1
◦
. CI is the most extensively studied and applied
reset element in literature due to advantages seen in reduced
overshoot and increased phase margins.
The matrices of CI for Eqn. 1 are
A
r
= 0, B
r
= 1, C
r
= 1, D
r
= 0, A
ρ
= 0
2) First Order Reset Element - FORE and its general-
ization: CI was extended to a first order element as FORE
by Horowitz et al. in [9]. FORE provides the advantage
of filter frequency placement unlike CI and has been used
for narrowband phase compensation in [22]. The matrices
of FORE for Eqn. 1 where the base linear filter has corner
frequency ω
r
are
A
r
= −ω
r
, B
r
= ω
r
, C
r
= 1, D
r
= 0, A
ρ
= 0
FORE was generalized in [10] to obtain Generalized FORE
(GFORE) which provides the additional freedom of having
a non-zero resetting parameter A
ρ
and hence controlling the
level of reset. This is achieved by using an additional reset
parameter γ such that A
ρ
= γ, where γ = 1 results in a linear
filter. γ is used to influence the amount of nonlinearity and
hence phase lag. The influence of γ on phase lag and other
properties is studied in the next section.
3) Second Order Reset Element - SORE: SORE has been
recently developed by Hazelgar et. al. [11] opening new pos-
sibilities for reset controllers in the shape of notch and second
order low pass filters. SORE has the advantage of an additional
parameter, damping coefficient β
r
as seen in the base matrix
definitions below. This provides an extra degree of freedom in
the design of nonlinear resetting element. 2 identical FOREs in
series is a special case of SORE with β
r
= 1. The additional
parameters β
r
allows for achieving properties not possible by
combination of FOREs. The matrices of SORE as applicable
to Eqn. 1 are given as
A
r
=
0 1
−ω
2
r
−2β
r
ω
r
, B
r
=
0
ω
2
r
C
r
=
1 0
, D
r
=
0
where, ω
r
is the corner frequency of the filter; β
r
is the
damping coefficient
3
III. FREQUENCY DOMAIN BEHAVIOUR OF RESET
ELEMENTS
A. Generalized SORE (GSORE) and generalization of reset
controller
Hazelgar et. al. introduced SORE in [11] where both states
are reset to zero when the reset condition is met. This can
be considered as traditional reset control system. Such a
system similar to FORE provides less flexibility of design and
overall design becomes dependent on the base linear system.
Hence, we present generalized SORE (GSORE) where A
ρ
∈
R
2×2
can be an arbitrary resetting matrix. While such a system
provides greater freedom in design, 4 additional parameters
also add to the complexity of analysis during design. Hence
we limit this freedom to one parameter by defining A
ρ
similar
to the manner in GFORE as
A
ρ
= γI
2×2
Broadly, reset controllers can be generalized such that
resetting matrix A
ρ
in Eqn. 1 is no longer a zero matrix as
originally conceived, but is of form
A
ρ
=
γ
1
0 . . . . 0 0
0 γ
2
. . . . 0 0
0 0 . . . . γ
nr
0
0 0 . . . . 0 I
n
nr
×n
nr
where n
r
and n
nr
are number of resetting and non-resetting
states of overall controller respectively. Each resetting state has
its own factor γ determining its after reset value as a fraction of
its pre-reset value. It must be noted that while this generalized
form provides a large degree of freedom in design, this might
not be useful or convenient in all cases. This is specially true
with loop shaping technique which is generally carried out by
experienced engineers and not algorithms; and hence having
too many variables for tuning might impede design rather than
aid it.
B. Analysis of reset elements using describing function
Frequency domain behaviour analysis of reset elements
in literature has mainly focussed on phase lag reduction.
However, loop shaping requires a more comprehensive un-
derstanding of the behaviour. This knowledge is also essential
for design of CgLp presented in the next section. This analysis
is carried out using describing function method explained in
Sec II-B for the reset elements. Describing function based
frequency behaviour is obtained for GSORE for different
values of γ ∈ [−1, 1] and is shown in Fig. 1. There are three
important characteristics, two in gain and one in phase, which
needs to be noted. While the change in phase behaviour has
been studied greatly in literature, the effect of reset on gain is
not found in literature to the best of authors’ knowledge.
• Shift in corner frequency: From the figure, it can be seen
that for values of γ ∈ [0, 1], gain behaviour of reset
element is similar to that of its linear counterpart (γ = 1).
However, for values of γ < 0, while the slope of gain is
still −40 db/decade at high frequencies, there is a shift
in the corner frequency of the filter. While this is shown
10
0
10
1
10
2
10
3
10
4
-100
-50
0
Magnitude (dB)
= 1
= 0.5
= 0
= -0.5
= -0.99
10
0
10
1
10
2
10
3
10
4
Frequency (Hz)
-200
-100
0
Phase (deg)
Fig. 1: Describing function based frequency response of
GSORE for different values of γ with ω
r
= 2π100 and β
r
= 1
here for GSORE, this is also true for filters of other orders
[28]. This shift is parametrized as fraction α where
α =
Corner frequency of reset element
Corner frequency of base linear element
The value of α as a function of γ is plotted in Fig. 2
for GFORE and GSORE (β
r
= 1). While the value of α
can be used to modify the base linear system to ensure
that the corner frequency of reset filter is at the desired
value, these α values are close to 1 for values of γ ≥ 0.
However, they increase in an almost exponential manner
as value of γ is reduced further. This limits the values of
γ for which GFORE and GSORE can be effectively used
in practice.
• Phase lag reduction: While the difference in gain is only
seen for lower values of γ, reduction in phase lag is seen
to be sensitive and is seen for all values of γ < 1. The
phase lag of GSORE for different values of γ is shown in
Fig. 3. Phase lag achieved with GFORE is also shown in
the same figure for comparison. It can be seen that large
phase lag reductions are seen for smaller values of γ with
phase lag being zero at γ = −1. However, due to the
corresponding change in corner frequency of GSORE as
seen in Fig. 1, use of these generalized elements becomes
limited.
• Change in damping factor: Another interesting charac-
teristic of GSORE is seen in change in the damping
factor of the designed linear filter and achieved GSORE
filter. This is shown in Fig. 4 for different values of β
r
,
where it can be seen that even for β
r
= 0 (resulting in a
Q factor = ∞ in the case of linear filter), the resonance
peak is less than 10 dB. Although change in gain plot
vs β
r
is negligible, there is change in phase plot with
changing β
r
values and this can be used advantageously
to obtain a sharp change in phase without the cost of a
resonance peak. This additional advantage is only seen
with GSORE due to presence of damping factor β
r
and
not in GFORE.
In the above frequency domain analysis of reset elements,
sinusoidal input describing function analysis has been used
to obtain the frequency response. While this pseudo-linear
technique is useful, it is only an approximation technique. To
4
10
0
10
1
10
2
-1
-0.5
0
0.5
1
GFORE
GSORE
Fig. 2: Fraction α denoting extent of change in corner fre-
quency as a function of γ for GFORE and GSORE (β
r
= 1)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-200
-150
-100
-50
0
Phase lag (deg)
GSORE
GFORE
Fig. 3: Reduction in phase lag with reset for both GSORE and
GFORE
verify the accuracy of this method, we obtained the frequency
response of GSORE directly by applying chirp and step
inputs and using the tfestimate function of MATLAB and
comparing the response to the one obtained from describing
function. The coherence C
xy
which gives a measure of the
accuracy of obtained frequency response using tfestimate
is also obtained and plotted in Fig. 5. The plots show good
match between the describing function based results and those
obtained through estimation in MATLAB.
IV. CONSTANT-GAIN LEAD-PHASE (CGLP)
Reset is used in controls for its phase lag reduction. SORE
helps in this regard, with generalization further providing the
10
0
10
1
10
2
10
3
10
4
-100
-50
0
50
Magnitude (dB)
10
0
10
1
10
2
10
3
10
4
Frequency (Hz)
-60
-40
-20
0
Phase (deg)
reset
= 1
reset
= 0.1
reset
= 0.01
reset
= 0
Fig. 4: Change in damping value of GSORE with γ = 0
10
1
10
2
10
3
10
-5
10
0
10
5
Magnitude[Abs]
=1
=0.5
=0
10
1
10
2
10
3
-200
0
200
Phase [Degrees]
10
1
10
2
10
3
Frequency[Hz]
0
0.5
1
Coherence
Fig. 5: Gain, Phase and Coherence relation for different values
of γ. Dashed line represents the values from the describing
function. ω
r
= 2π200
freedom to choose the level of reset and hence the level of
nonlinearity introduced. However, low pass filters are generally
used in controls at high frequency for noise attenuation. While
these can be replaced by reset low pass filters in the form of
GFORE or GSORE advantageously, this use of reset results
only in phase lag reduction. Reset for phase lead which can
be used advantageously in region of bandwidth has not been
explored sufficiently in literature. The main works in this
regard as noted earlier are [22] and [23]. Here, we introduce
a new reset element termed Constant in gain Lead in phase
(CgLp) which uses GFORE (or GSORE) to provide broadband
phase compensation in the required range of frequencies.
A. Definition
Broadband phase compensation is achieved in CgLp by
using a reset lag filter R (GFORE or GSORE) in series with
a corresponding order linear lead filter L as given below.
R(s) =
1
:
γ
(s/ω
rα
)
2
+ (2sβ
r
/ω
rα
) + 1
or
1
:
γ
s/ω
rα
+ 1
(7)
and
L(s) =
(s/ω
r
)
2
+ (2sβ
r
/ω
r
) + 1
(s/ω
f
)
2
+ (2s/ω
f
) + 1
or
s/ω
r
+ 1
s/ω
f
+ 1
(8)
correspondingly with ω
f
>> ω
r
, ω
rα
. The arrow indicates the
resetting nature of R. ω
rα
= ω
r
/α accounting for the shift in
corner frequency with reset as noted in Sec. II-B and can be
obtained from Fig. 2 for the chosen value of γ.
The reset state matrices of CgLp using GFORE are given
below as
A
r
=
−ω
rα
0
ω
f
−ω
f
, B
r
=
ω
rα
0
