# Evidence of Spin-Glass state in Molecular Exchange-Bias System

Suman Mundlia  Karthik V. Raman<EMAIL_ADDRESS>Tata institute of
fundamental research, Hyderabad 500107, India

###### Abstract

In conventional exchange-bias system comprising of a bilayer film of
ferromagnet (FM) and antiferromagnet (AFM), investigating the role of spin-
disorder and spin-frustration inside the AFM and at the interface has been
crucial in understanding the fundamental mechanism controlling the exchange-
bias – an effect that leads to a horizontal shift in the magnetization
hysteresis response of the FM. Similarly, in the recently reported monolayer
molecular exchange-bias effect requiring no AFM layer, probing magnetic-
disorder at the FM/molecule interface or inside the FM layer can provide new
insights into the origin of molecular exchange-bias and the associated
physics. In this article, by cooling the Fe/metal-phthalocyanine devices in
oscillating magnetic field, we demonstrate a characteristic temperature
dependent response of exchange-bias shift and ferromagnet coercivity that is
supportive of a spin-glass behavior. Here, the origin of spin-glass is
attributed to the spin-frustration created in the magnetic structure of the Fe
layer, which was absent in our reference-Fe studies. These results highlight
the strong influence of FM/molecule interface $\pi-d$ hybridization on the
magnetic exchange interactions extending deeper into the FM layer.

##  I Introduction

Molecular exchange-bias effect [1, 2, 3, 4, 5] is a particular manifestation
of ferromagnet/molecule spin-interface study [6] in a widely investigated
topic of interface-assisted molecular spintronics [7, 8]. Here, complex charge
transfer and hybridization effects between the molecular orbitals and the
spin-polarised bands of the ferromagnet (FM) surface create new hybridized
interface states that provide novel electronic and magnetic properties [9, 7,
8]. Importantly, studies indicate the possibility to tune the magnetic-
exchange interaction [10] and the magnetic anisotropy at the surface [7] with
the latter receiving wider attention through the experimental reports of
enhancement in film coercivity (Hc) [11, 12, 13]. In recent studies, such a
surface magnetic hardening response is found to be directly responsible for
molecular exchange-spring [14] and exchange-bias effect [1, 2, 3, 4]. In the
case of latter, the effect was recently shown to appear in a monolayer
molecular film on FM [5], suggesting its mechanism to arise from the
magnetization response of a hard-FM (surface)/soft-FM (bulk) bilayer system
(see figure 1a) which is evidently different from the previous studies of
exchange-bias effect in FM/AFM bilayers. However, despite these differences,
existing literature for FM/AFM systems [15, 16] can provide a better
understanding of the possible magnetization dynamics that may be playing a
role in molecular exchange-bias effect.

In conventional FM/AFM systems [17], a number of parameters are known to
influence exchange bias – these include magnetic anisotropy, crystal structure
[18], magnetic domains [19, 20, 21], interface roughness [22, 23, 24], spin-
configuration [25, 26] and exchange coupling [27, 28, 29]. However, in
molecular exchange-bias effect, absence of an AFM layer allow us to reduce the
above parameter space, thereby limiting our study to the properties of
interface and the underneath FM layer. In particular, surface roughness,
magnetic domain state, hybridization induced anisotropy and magnetic exchange
coupling near the surface are expected to play a dominant role in molecular
exchange-bias.

Figure 1: (a) Schematic of molecular exchange bias in a hard-surface/soft-Fe
bilayer showing magnetization switching of bottom Fe (yellow) relative to the
pinned surface Fe (grey) causing a horizontal shift in AMR by Hex. (b) Figure
shows the literature data of molecular exchange bias showing Hex vs FM layer
thickness measured before training (BT) and after training (AT) cycles. (c)
Schematic showing deposition procedure of reference-Fe and Fe/MPc devices in a
single deposition run using shadow mask.

In literature, molecular exchange-bias effect is reported in different
thickness range of FMs and also of the molecule [1, 2, 3, 4, 5]. Here, thicker
molecular spin-chains (acting as the AFM layer) are believed to couple with
the surface hard layer during the field cooling process and temporarily
stabilize the surface magnetism [1, 2, 3, 4]. However, during magnetization
hysteresis studies, the stability of the molecular spin-chain alignment is
weakened (or lost), leading to a strong dependence of exchange-bias shift
(Hex) with training cycle. This behavior is noted in figure 1b for a number of
previous published reports on molecular exchange-bias showing the value of Hex
during 1st cycle with a significant drop after sufficient training period
(AT). In thicker films ($>$ 4 nm), exchange bias is weak with strong training
effect leading to an almost complete loss of Hex. However, recent studies in
thinner FM films ($<$ 4 nm) show sufficient enhancement in Hex and robustness
in their training response [3, 4]. More importantly, as shown in figure 1b
(solid symbols), monolayer molecular study [5] provides considerably weaker
training effect perhaps due to the absence of such molecular spin-chains. In
conventional FM/AFM systems, similar enhancement in Hex at lower temperatures
and in thinner AFM layers ($<$ 2 nm) is observed [19]. Here, they are
attributed to the formation of perpendicular domain walls (perpendicular to
the surface) in the AFM layer that become energetically unfavorable upon
increasing the AFM layer thickness [19, 24]. Lack of an AFM layer in molecular
exchange bias may rule out the above mechanism. However, similar possibility
of perpendicular domain wall formation at the surface of FM that freeze with
cool down and contribute to enhancement in Hex of an FM/molecule system may
not be completely ruled out. Especially in the case of thinner FM films
($<\sim$ 4 nm), we expect these domain walls to extend deeper into the bottom
FM layer. Occurrence of such perpendicular domain walls will in-principle
cause Hex to depend on the strength of in-plane magnetic field cooling since
such fields can modulate the density of perpendicular domain walls. Up till
now, this was however not observed in our previous studies [5] indicating that
perpendicular domain walls, although likely to be present, do not play a
limiting role in magnetization dynamics of our devices. In effect, the
phenomena of surface magnetic hardening naturally favor the formation of
planar domain walls, parallel to the film surface, during the magnetization
switching of the bottom Fe layer [30]. This switching response can be modeled
using the Stoner-Wolfarth (SW) model [31], as was done in Ref. [5], that
provided a qualitative understanding of the role of surface anisotropy in the
origin of Hex. Nevertheless, it is possible that both domain walls coexist in
thinner films [32, 33] which may require sensitive spectroscopy techniques for
careful investigation. The fact that domain wall dynamics do play a
contributory role in molecular exchange-bias is evident from the study of even
thinner films of Fe ($<$ 2 nm) showing no Hex but only an enhancement in Hc
[5]. This response can be explained by the unfavorable energy conditions for
the formation of domain walls.

In this article, we focus our study towards the investigation of magnetization
response and possible domain structure dynamics in Fe/MnPc devices after
cooling them in oscillating magnetic fields. Such a procedure gives us access
to magnetic disorder studies showing spin-glass behavior [34, 35] that was not
possible in earlier studies of zero-field or field-cooled measurements. Our
studies reveal the presence of surface $\pi-d$ hybridization induced strong
modification in magnetic exchange coupling in the FM layer that may provide
additional insights into the recent report of molecular crane-pulley effect
[5].

Figure 2: (a) AMR in % of Fe (3.5 nm)/CoPc (4 ML) devices at 5 K (top), 15 K
(middle), and 25 K (bottom) in ZFC (pink), FC+AFC (green) and AFC (purple)
conditions. In the inset of each AMR plot magnitude of $\Gamma$ (shown in the
middle) is plotted for different field cooling. (b) Hex vs T (top) and Hc vs T
(bottom) of Fe (3.5 nm)/CoPc (4 ML) device extracted from the AMR plots in ZFC
(pink triangle), FC + OFC (green squares) and OFC (purple circles). (c) Hex vs
T (top) and Hc vs T (bottom) of Fe (3.5 nm)/MnPc (1 ML) device in ZFC (pink
triangle) and OFC (purple circles).

##  II Experimental details

For our current study, two kinds of samples, _viz._ reference-Fe and Fe/metal-
phthalocyanine (MPc, M: Co and Mn) [36] bilayers were prepared in an ultra-
high vacuum thin film deposition system. In figure 1c, we show the deposition
procedure of our patterned films using shadow mask technique. First, the Fe
film is grown simultaneously on the two substrates. One of the substrate is
then covered by an in-situ mask before the molecule is evaporated. Finally,
both the films are capped with Al (capping layer) to prevent oxidation of
underneath layers. Analysis of a complete capping layer coverage is performed
using Energy dispersive X-ray elemental mapping. These details can be found in
Ref. [5]. Such a growth procedure allow us to avoid any variation in the Fe
film thickness between the reference-Fe and Fe/MPc devices that are loaded
together in our variable temperature insert cryostat. Here, the samples are
cooled to the base temperature of 2 K under different field cooling conditions
and isothermal anisotropic magnetoresistance (AMR) measurements are recorded
during the warm-up.

##  III Results and Discussion

In our previous studies [5], we observed that zero-field cooling (ZFC) or
field-cooling (FC) the Fe/MPc devices in different magnetic fields (ranging
from 0 T to 0.6 T) do not lead to any visible change in Hex suggesting the
dominant role of planar domain walls in magnetization switching dynamics. In
comparison, reference-Fe films do not show any Hex even under FC conditions
with a significantly lower Hc. In the present work, we have extended the AMR
studies to oscillating field cooling (OFC) conditions –wherein the devices are
cooled in an in-plane oscillating field with an amplitude of 50 mT at
$\sim$0.1 Hertz. Figure 2a shows the temperature dependence of AMR in Fe (3.5
nm)/CoPc (4ML) devices acquired during warm-up in ZFC, OFC and FC + OFC (i.e.
the field was made to oscillate between 70 mT and -30 mT) cooling conditions.
Clearly, these cooling conditions lead to a change in the value of Hex
dropping down to nearly zero for OFC measurements (see figure 2b).
Additionally, the magnitude of Hc and the width ($\Gamma$) of the AMR drops
(inset of figure 2) at Hc are found to decrease in moving from ZFC to OFC
plots. In contrast, the AMR plots of our reference-Fe films after OFC show
similar value of Hc with a marginal increase in $\Gamma$ compared to the ZFC
measurements (not shown). The above behavior in AMR response of Fe/MPc devices
suggest that OFC conditions lead to changes in the magnetic structure that
could arise in the top hard-surface layer and possibly extend into the bottom
Fe layer.

We now examine the possible effects of OFC on the domain dynamics– this may
include (a) random orientation of the hard-surface layer domains mostly
parallel or anti-parallel to the magnetic field axis leading to an overall
drop in Hex and/or (b) drop in average domain size in the hard-surface layer
and/or (c) changes in the magnetic domain structure of the bottom soft-Fe
layer. If we assume that OFC only leads to case (a) with no change in domain
size dynamics, we would expect a similar value of Hc and an increase in the
width $\Gamma$ of the AMR peaks compared to ZFC measurements (as noted in
reference-Fe films). This is also suggested by our simulation studies using
the SW model (see Appendix B). However, our experiments show a drop in Hc and
$\Gamma$ suggesting domain size dynamics to possibly contribute. In order to
investigate the above in greater detail, in figure 2b & 2c, we plot the
temperature dependence of the extracted values of Hex and Hc for the Fe/CoPc
and Fe/MnPc device cooled under different conditions. Here, the enhancement in
Hc, above the exchange-bias onset temperature (TEB), relative to reference-Fe
is attributed to surface magnetic hardening effects giving rise to a molecular
crane-pulley response below TEB [5]. In figure 2, in the case of ZFC
measurements, Hc monotonically increases with decreasing temperature with an
increase in Hex below TEB of $\sim$50 K in CoPc devices and 30 K in the case
of MnPc devices. Here, based on our understanding of FM/AFM [37, 19, 38] or
FM/spin-glass exchange-bias system [29], the enhancement in Hc relative to the
reference-Fe can be understood to arise from two additive but competing
effects – surface exchange induced magnetic anisotropy (K${}^{i}_{ex}$) in the
bottom Fe layer and softer domains or frustrated spin centers in the hard
surface layer matrix that couple with the bottom Fe layer and partially rotate
upon magnetization reversal. Generally, the latter effect depends on
interface-disorder and spin-frustration and its strong presence leads to a
drop in Hc at lower temperatures caused by surface domain freezing, whereby
they stop contributing to the Hc enhancement [29, 19, 39]. While on the other
hand, K${}^{i}_{ex}$ enhances at lower temperature contributing to an increase
in Hc, as noted in previous studies [25] and in our ZFC data.

In the case of FC + OFC procedure (see figure 2b), Hex shows a decrease during
warm-up with a plateau-like response at around 30 K. Additionally, at 2K, its
value remain lower than in the case of ZFC. Interestingly, in this temperature
range, Hc shows an opposite trend with warm-up compared to the ZFC data. Here,
Hc at 2K is at $\sim$ 8 mT that is considerably lower than the ZFC value (16
mT), and show an increase with warm-up reaching $\sim$ 14 mT. In the 30K - 50K
temperature range, FC + OFC show a larger value of Hc than in ZFC hinting
towards the possible presence of softer domains frustrated in the magnetically
hard surface matrix that couple to the bottom Fe layer during magnetization
reversal thereby enhancing its Hc. At lower temperatures, these domains and
the associated domain walls may remain frozen and do not (or weakly)
contribute to Hex. Such a trend in Hc and Hex may suggest a stronger
interfacial spin-glass contribution to the magnetization switching dynamics as
typically observed in conventional FM/spin-glass exchange-bias systems [29].
Ideally, if this was the only scenario, we would expect OFC measurements (see
figure 2a and 2b) to give a stronger enhancement in Hc, just below TEB, over
the values in ZFC and FC + OFC measurements. However, our results in figure 2b
do not support such a response suggesting that the origin of non-monotonic
temperature dependence of Hc in OFC may not be completely driven by a
conventional interfacial spin-glass behavior caused by the formation of softer
(or superparamagnetic) smaller surface domains or isolated spin centers.

On the contrary, in OFC, lack of Hex in the complete temperature range below
TEB suggest that the spin-frustration caused by OFC is strong and must be
creating a strongly pinned frustrated domain matrix at the surface, forming
enhanced density of perpendicular domain walls that extend deeper into the
bottom Fe layer. Formation of these frozen domains walls will, in-principle,
also make domain wall motion harder during magnetization switching of the
bottom Fe layer suggesting an enhancement in Hc and $\Gamma$. This is however
contrary to our experimental observations implying that the above domain
dynamics also do not explain the temperature dependent AMR response in the
devices. Ideally, we expect that OFC does not affect the local strength of
surface hybridization induced KUD (in per unit interface area) since the
energy scales involved here are very different with the hybridization effects
being much stronger than the magnetic energy. We believe a better measure of
average response from the unidirectional exchange anisotropy over the
interface is not by extracting Hex which in this case is zero, but by probing
the presence of such anisotropy using in-plane angle-dependent AMR studies.
These studies allow us to analyse the asymmetric magnetization switching
response with in-plane field rotation caused by the presence of interfacial
unidirectional anisotropy (see appendix C). Interestingly, our experimental
measurements show a difference in response compared to the conventional FM/AFM
system i.e. Fe/FeO bilayer device. In our Fe/FeO system, the asymmetry in the
AMR response disappear (appear) when the exchange-bias is absent (present)
(see figure 5d). On the contrary, the Fe/MPc devices continue to show
asymmetric AMR response even when the extracted H${}_{ex}=0$ implying the
presence of interfacial unidirectional anisotropy of similar strength as in
ZFC (see figure 5b). These observations, different from conventional FM/AFM
systems, seem to suggest a different magnetic structure in Fe/MPc devices
which may require further exploration in the future.

The suggested existence of a strong interfacial exchange anisotropy but with a
drop in Hc and $\Gamma$ of the bottom Fe layer in OFC studies may indicate a
relative loss in the magnetic anisotropy of the bottom Fe layer. The fact that
Hc remain higher than the value in reference-Fe, even at 2 K, suggest that the
effects of K${}_{ex}^{i}$ has reduced but is not completely absent. Such a
response is often observed in reentrant spin-glass phases in weak ferromagnets
[40, 35, 41, 42]. Here, a low-temperature mixed state is formed in which the
spin-glass behavior coexists along with the long-range ferromagnetic order.
These phases form due to weakening of magnetic exchange strength leading to
cluster freezing at lower temperatures contributing to the spin-glass
response. In our Fe/MPc devices, formation of such a mixed state with a spin-
glass nature that is decoupled from the exchange coupling of the top surface
(i.e. KUD) corroborate well with the experimental observation of a reduction
in Hc and $\Gamma$ in OFC measurements with no Hex. With an increase in
temperature, cluster freezing weakens leading to a dominant ferromagnetic
phase with an increase in magnetic anisotropy. This leads to an increase in Hc
with an overall no change in Hex since the domains in the frustrated surface
matrix grow equally for the parallel and the anti-parallel magnetization
alignments. Hence, these above results highlight the formation of an
unexpected spin-glass magnetic layer due to spin-frustration introduced at the
molecule-absorbed surface offering new interesting physics at the fundamental
level. Importantly, the work demonstrates the possibility to achieve strong
modification in the strength of magnetic exchange coupling in an FM layer that
is tunable by field cooling procedures. We believe, in the future, more
sensitive techniques such as depth-resolved neutron reflectometry, ac-
susceptibility, and Mössbauer spectroscopy [42] studies may provide direct
insights into the rich magnetization dynamics in such FM/molecule hybrid
bilayers.

##  IV Conclusion

In conclusion, OFC studies in FM/molecule devices has supported the formation
of an unexpected spin-glass magnetic structure in the FM layer that seem to
dominate the temperature dependent response of magnetization dynamics. These
observations provide new insights into the strong effect of surface $\pi-d$
hybridization on the strength (weakening) of magnetic exchange interactions
extending deeper into the FM layer and bringing them closer to the critical
threshold to induce a long-range magnetic order. We believe our present work
provides new insights into the important role of magnetic exchange
interactions at the surface that may help to understand the origin of
molecular exchange bias in tailoring them for potential device applications in
2D spintronics.

##  V Appendix

###  V.1 Modeling magnetization switching of bottom Fe layer with exchange-
bias

We model the magnetization switching response of a hard-FM/soft-FM bilayer
system by considering the top hard-FM layer to be pinned along its internal
magnetization axis which is responsible for the exchange-bias coupling on the
bottom soft-FM layer. Stoner-Wolfarth (SW) model [31] is used to model the
magnetization switching response of a domain in the bottom soft-FM layer.

Figure 3: Schematic shows the representation of the SW model for simulating
in-plane angular-dependent AMR.

Here, we assume that the contribution of domain wall motion towards the value
of Hc is much lesser compared to the Hc enhancement (relative to the value in
reference-Fe) resulting from the surface hybridization induced magnetic
anisotropy, K${}^{i}_{ex}$. The interfacial exchange-bias coupling is
introduced in the model by assuming an unidirectional exchange anisotropy KUD
(see figure 3). K${}_{ex}^{i}$ was previously shown to be biaxial in nature
[5]. In the following analysis, the results are independent of the nature of
K${}_{ex}^{i}$ but for convenience we consider a biaxial anisotropy term in
the SW model. Here, the free energy per unit area (E) of bottom Fe layer can
be estimated by [15]:

$E=-K_{UD}.cos(\theta-\phi)-K^{i}_{ex}.cos(4(\theta-\phi-x))\\\
-\mu_{o}H_{a}.M_{Fe}.t_{Fe}.cos(\theta)$

where $\theta$ is the angle between M and Ha, $x$ is the angle between KUD and
K${}_{ex}^{i}$ anisotropy axes (see figure 3). Here, the contribution of the
internal anisotropy of the bottom Fe film is neglected and a random
orientational absorption of the MPc molecules on Fe surface is considered to
simulate the average AMR response, given by $<$AMR$>$ = $<cos^{2}\theta>$ \-
1, where $<>$ represents an average over the in-plane azimuthal rotation of
angle $x$.

###  V.2 Effect of randomly oriented hard surface domains on the AMR response
of bottom Fe layer

Figure 4: Simulated AMR plots for bottom Fe film using KUD= 70 $\pm$ 10 mT
showing exchange bias (bottom black) in uni-directional response of KUD in all
the domains and a domain-averaged response over parallel and anti-parallel
direction of KUD in different domains (top red) i.e. average over KUD = 70 and
-70 mT domains.

In order to model the effect of OFC on the AMR response, we first consider the
scenario that OFC only leads to random orientation of the surface domains
primarily along the parallel or anti-parallel direction to the magnetic field
axis. Here, we consider that the magnitude of KUD in each domain is unaffected
by OFC and its strength is determined primarily by the $\pi-d$ hybridization.
For such a scenario, we plot the simulated AMR response in figure 4 which
shows similar value of Hc and an enhancement in $\Gamma$ with no Hex in
comparison to the case when the domains are aligned only in the parallel
direction.

###  V.3 In-plane angle-dependent AMR studies

Figure 5:  (a) Angle ($\phi$ values in inset) dependent AMR plots of Fe (3.5
nm)/MnPc (1 ML) device in ZFC (pink) and OFC (green) conditions. Top-panel
inset shows the device measurement geometry. (b) Magnitude of $\lvert\Delta
R\rvert$ (shown in (a)) vs $\phi$ in ZFC (pink) and OFC (green) showing
similar strength. (c) $\lvert\Delta R\rvert$ vs Kud extracted using the SW
model is plotted for different values of $\phi$: 0o (purple circles), 30o
(green squares), and 60o (pink triangles). (d) Angle ($\phi$ values in inset)
dependent AMR plots of Fe/FeO device show $\lvert\Delta R\rvert$ only in the
presence of exchange bias.

Angle-dependent AMR studies of FM/AFM devices in presence of exchange-bias (or
KUD) often given rise to asymmetric features due to the difference in
magnetization rotation pathways in the ascending and descending branches of
field sweep [43, 44]. When imposed over the internal anisotropy of the FM
layer, the asymmetric features can take different trend with rotation of
applied magnetic field. Figure 5a shows the angle-dependent AMR measurements
performed on Fe/MnPc devices after cooling them in ZFC and OFC conditions.
Here, in the case of OFC measurements, no Hex is observed. In our analysis,
the asymmetric response with $\phi$ is captured by $\lvert\Delta R\rvert$
(shown in figure 5b) whose origin can be established directly with KUD. This
is shown in figure 5c where we plot the value of $\lvert\Delta R\rvert$
extracted from the AMR plots simulated at different $\phi$s by varying KUD.
Clearly, for all values of $\phi$, $\lvert\Delta R\rvert$ approaches zero with
a reduction in KUD suggesting its origin to the unidirectional interface
exchange-anisotropy. In addition, we also experimentally confirm the
dependency of $\lvert\Delta R\rvert$ on KUD by performing angle dependent AMR
studies in conventional FM/AFM (Fe/FeO) system (shown in figure 5d). Here, Fe
(10nm)/FeO devices are prepared by in-situ oxygen plasma oxidation after Fe
growth. In these devices, $\lvert\Delta R\rvert$ at $\phi\neq 0$ is observed
only when Hex is non-zero.

Surprisingly, in our Fe/MnPc devices, we observe $\lvert\Delta R\rvert$ to
exist even after OFC conditions (showing negligible Hex) with its magnitude
similar to the values observed after ZFC. These above observations seem to
suggest that (i) AMR response in Fe/MPc devices is different from conventional
FM/AFM bilayers and (ii) OFC procedure do not modify the local strength of
interfacial unidirectional exchange anisotropy i.e. KUD. We believe further
investigations using neutron reflectometry techniques may be needed to examine
these differences in greater details.

##  VI Acknowledgements

We acknowledge extra mural funding from the Department of Atomic Energy, TIFR
Hyderabad and Science and Engineering Research Board of the Government of
India (SERB sanction no. ECR/2015/000199).

## References

  * Gruber  _et al._ [2015] M. Gruber, F. Ibrahim, S. Boukari, H. Isshiki, L. Joly, M. Peter, M. Studniarek, V. Da Costa, H. Jabbar, V. Davesne, _et al._ , Exchange bias and room-temperature magnetic order in molecular layers, Nature materials 14, 981 (2015).
  * Boukari  _et al._ [2018] S. Boukari, H. Jabbar, F. Schleicher, M. Gruber, G. Avedissian, J. Arabski, V. Da Costa, G. Schmerber, P. Rengasamy, B. Vileno, _et al._ , Disentangling magnetic hardening and molecular spin chain contributions to exchange bias in ferromagnet/molecule bilayers, Nano Letters 18, 4659 (2018).
  * Jo  _et al._ [2018] J. Jo, J. Byun, I. Oh, J. Park, M.-J. Jin, B.-C. Min, J. Lee, and J.-W. Yoo, Molecular tunability of magnetic exchange bias and asymmetrical magnetotransport in metalloporphyrin/ Co hybrid bilayers, ACS nano 13, 894 (2018).
  * Jo  _et al._ [2020] J. Jo, J. Byun, J. Lee, D. Choe, I. Oh, J. Park, M.-J. Jin, J. Lee, and J.-W. Yoo, Emergence of multispinterface and antiferromagnetic molecular exchange bias via molecular stacking on a ferromagnetic film, Advanced Functional Materials 30, 1908499 (2020).
  * Mundlia  _et al._ [2020] S. Mundlia, S. Chaudhary, L. Peri, A. Bhardwaj, J. J. Panda, S. Sasmal, and K. V. Raman, Robust monolayer exchange-bias effect in molecular crane-pulley response at magnetic surface, Phys. Rev. Applied 14, 024095 (2020).
  * Sanvito [2010] S. Sanvito, Molecular spintronics: The rise of spinterface science, Nature Physics 6, 562 (2010).
  * Raman [2014] K. V. Raman, Interface-assisted molecular spintronics, Applied Physics Reviews 1, 031101 (2014).
  * Atodiresei and Raman [2014] N. Atodiresei and K. V. Raman, Interface-assisted spintronics: Tailoring at the molecular scale, MRS Bull 39, 596 (2014).
  * Atodiresei  _et al._ [2013] N. Atodiresei, V. Caciuc, P. Lazic, and S. Blügel, Chemical and van der waals interactions at hybrid organic-metal interfaces (2013).
  * Callsen  _et al._ [2013] M. Callsen, V. Caciuc, N. Kiselev, N. Atodiresei, and S. Blügel, Magnetic hardening induced by nonmagnetic organic molecules, Phys. Rev. Lett. 111, 106805 (2013).
  * Raman  _et al._ [2013] K. V. Raman, A. M. Kamerbeek, A. Mukherjee, N. Atodiresei, T. K. Sen, P. Lazić, V. Caciuc, R. Michel, D. Stalke, S. K. Mandal, _et al._ , Interface-engineered templates for molecular spin memory devices, Nature 493, 509 (2013).
  * Decker  _et al._ [2013] R. Decker, J. Brede, N. Atodiresei, V. Caciuc, S. Blügel, and R. Wiesendanger, Atomic-scale magnetism of cobalt-intercalated graphene, Phys. Rev. B 87, 041403 (2013).
  * Bairagi  _et al._ [2015] K. Bairagi, A. Bellec, V. Repain, C. Chacon, Y. Girard, Y. Garreau, J. Lagoute, S. Rousset, R. Breitwieser, Y.-C. Hu, Y. C. Chao, W. W. Pai, D. Li, A. Smogunov, and C. Barreteau, Tuning the magnetic anisotropy at a molecule-metal interface, Phys. Rev. Lett. 114, 247203 (2015).
  * Moorsom  _et al._ [2020] T. Moorsom, S. Alghamdi, S. Stansill, E. Poli, G. Teobaldi, M. Beg, H. Fangohr, M. Rogers, Z. Aslam, M. Ali, B. J. Hickey, and O. Cespedes, $\pi$-anisotropy: A nanocarbon route to hard magnetism, Phys. Rev. B 101, 060408 (2020).
  * Nogués and Schuller [1999] J. Nogués and I. K. Schuller, Exchange bias, Journal of Magnetism and Magnetic Materials 192, 203 (1999).
  * Zhang and Krishnan [2016] W. Zhang and K. M. Krishnan, Epitaxial exchange-bias systems: From fundamentals to future spin-orbitronics, Materials Science and Engineering: R: Reports 105, 1 (2016).
  * Meiklejohn and Bean [1957] W. H. Meiklejohn and C. P. Bean, New magnetic anisotropy, Phys. Rev. 105, 904 (1957).
  * Yuan  _et al._ [2016] W. Yuan, T. Su, Q. Song, W. Xing, Y. Chen, T. Wang, Z. Zhang, X. Ma, P. Gao, J. Shi, _et al._ , Crystal structure manipulation of the exchange bias in an antiferromagnetic film, Scientific reports 6, 28397 (2016).
  * Ali  _et al._ [2003] M. Ali, C. H. Marrows, M. Al-Jawad, B. J. Hickey, A. Misra, U. Nowak, and K. D. Usadel, Antiferromagnetic layer thickness dependence of the IrMn/Co exchange-bias system, Phys. Rev. B 68, 214420 (2003).
  * Nowak  _et al._ [2001] U. Nowak, A. Misra, and K. D. Usadel, Domain state model for exchange bias, Journal of Applied Physics 89, 7269 (2001).
  * Stiles and McMichael [1999] M. D. Stiles and R. D. McMichael, Model for exchange bias in polycrystalline ferromagnet-antiferromagnet bilayers, Phys. Rev. B 59, 3722 (1999).
  * Moritz  _et al._ [2016] J. Moritz, P. Bacher, and B. Dieny, Numerical study of the influence of interfacial roughness on the exchange bias properties of ferromagnetic/antiferromagnetic bilayers, Phys. Rev. B 94, 104425 (2016).
  * MAI [2012] Interfacial roughness and temperature effects on exchange bias properties in coupled ferromagnetic/antiferromagnetic bilayers, Journal of Magnetism and Magnetic Materials 324, 403 (2012).
  * Malozemoff [1987] A. P. Malozemoff, Random-field model of exchange anisotropy at rough ferromagnetic-antiferromagnetic interfaces, Phys. Rev. B 35, 3679 (1987).
  * Gökemeijer  _et al._ [2001] N. J. Gökemeijer, R. L. Penn, D. R. Veblen, and C. L. Chien, Exchange coupling in epitaxial CoO/NiFe bilayers with compensated and uncompensated interfacial spin structures, Phys. Rev. B 63, 174422 (2001).
  * Takano  _et al._ [1997] K. Takano, R. H. Kodama, A. E. Berkowitz, W. Cao, and G. Thomas, Interfacial uncompensated antiferromagnetic spins: Role in unidirectional anisotropy in polycrystalline Ni81Fe19/CoO bilayers, Phys. Rev. Lett. 79, 1130 (1997).
  * Fan  _et al._ [2013] Y. Fan, K. Smith, G. Lüpke, A. Hanbicki, R. Goswami, C. Li, H. Zhao, and B. Jonker, Exchange bias of the interface spin system at the Fe/MgO interface, Nature nanotechnology 8, 438 (2013).
  * Zhou  _et al._ [2017] G. Zhou, C. Song, Y. Bai, Z. Quan, F. Jiang, W. Liu, Y. Xu, S. S. Dhesi, and X. Xu, Robust interfacial exchange bias and metal–insulator transition influenced by the LaNiO3 layer thickness in La0.7Sr0.3MnO3/LaNiO3 superlattices, ACS applied materials & interfaces 9, 3156 (2017).
  * Ali  _et al._ [2007] M. Ali, P. Adie, C. H. Marrows, D. Greig, B. J. Hickey, and R. L. Stamps, Exchange bias using a spin glass, Nature Materials 6, 70 (2007).
  * Mauri  _et al._ [1987] D. Mauri, H. Siegmann, P. Bagus, and E. Kay, Simple model for thin ferromagnetic films exchange coupled to an antiferromagnetic substrate, Journal of Applied Physics 62, 3047 (1987).
  * Stoner and Wohlfarth [1948] E. C. Stoner and E. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 240, 599 (1948).
  * Wee  _et al._ [2001] L. Wee, R. L. Stamps, and R. E. Camley, Temperature dependence of domain-wall bias and coercivity, Journal of Applied Physics 89, 6913 (2001).
  * Nowak  _et al._ [2002] U. Nowak, K. D. Usadel, J. Keller, P. Miltényi, B. Beschoten, and G. Güntherodt, Domain state model for exchange bias. i. theory, Phys. Rev. B 66, 014430 (2002).
  * Schlenker  _et al._ [1986] C. Schlenker, S. Parkin, J. Scott, and K. Howard, Magnetic disorder in the exchange bias bilayered FeNi-FeMn system, Journal of magnetism and magnetic materials 54, 801 (1986).
  * Mydosh [1993] J. A. Mydosh,  _Spin glasses: an experimental introduction_ (CRC Press, 1993).
  * Lu and Kobayashi [2016] H. Lu and N. Kobayashi, Optically active porphyrin and phthalocyanine systems, Chemical Reviews 116, 6184 (2016).
  * Nayak  _et al._ [2020] S. Nayak, P. K. Manna, T. Vijayabaskaran, B. B. Singh, J. A. Chelvane, and S. Bedanta, Exchange bias in Fe/Ir20Mn80 bilayers: Role of spin-glass like interface and ‘bulk’antiferromagnet spins, Journal of Magnetism and Magnetic Materials 499, 166267 (2020).
  * Ding  _et al._ [2013] J. Ding, O. Lebedev, S. Turner, Y. Tian, W. Hu, J. Seo, C. Panagopoulos, W. Prellier, G. Van Tendeloo, and T. Wu, Interfacial spin glass state and exchange bias in manganite bilayers with competing magnetic orders, Physical Review B 87, 054428 (2013).
  * Xi and White [2000] H. Xi and R. White, Antiferromagnetic thickness dependence of exchange biasing, Physical Review B 61, 80 (2000).
  * Binder and Young [1986] K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
  * Pradheesh  _et al._ [2012] R. Pradheesh, H. S. Nair, C. M. N. Kumar, J. Lamsal, R. Nirmala, P. N. Santhosh, W. B. Yelon, S. K. Malik, V. Sankaranarayanan, and K. Sethupathi, Observation of spin glass state in weakly ferromagnetic Sr2FeCoO6 double perovskite, Journal of Applied Physics 111, 053905 (2012).
  * Sato  _et al._ [2001] T. Sato, T. Ando, T. Ogawa, S. Morimoto, and A. Ito, Spin freezing and the ferromagnetic and reentrant spin-glass phases in a reentrant ferromagnet, Phys. Rev. B 64, 184432 (2001).
  * Fitzsimmons  _et al._ [2000] M. R. Fitzsimmons, P. Yashar, C. Leighton, I. K. Schuller, J. Nogués, C. F. Majkrzak, and J. A. Dura, Asymmetric magnetization reversal in exchange-biased hysteresis loops, Phys. Rev. Lett. 84, 3986 (2000).
  * Krivorotov  _et al._ [2002] I. Krivorotov, C. Leighton, J. Nogués, I. K. Schuller, and E. D. Dahlberg, Relation between exchange anisotropy and magnetization reversal asymmetry in Fe/MnF2 bilayers, Phys. Rev. B 66, 100402R (2002).


# Tuning of a Class of Reset Elements Using Pseudo-Sensitivities*

Ali Ahmadi Dastjerdi1, Niranjan Saikumar1 and S.H. HosseinNia1 *This work was
supported by NWO, through OTP TTW project #16335.1A. Ahmadi Dastjerdi,
N.saikumar and S.H HosseinNia are with the Faculty of Department of Precision
and Microsystems Engineering, Delft University of Technology, Delft, The
Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

Currently, the demand for a better alternative to linear PID controllers is
increasing due to the rising expectations of the high-tech industry. In
literature, it has been shown that Constant in gain Lead in phase (CgLp)
compensators, which are a type of reset element, have high potential to
improve the performance of systems. Although there are few works which
investigate tuning of these compensators, the high order harmonics and steady-
state performances have not yet been considered in these methods. Recently, a
frequency-domain framework has been developed to analyze closed-loop
performances of reset control systems which includes high order harmonics. In
this paper, this frequency-domain framework is combined with loop-shaping
constraints to provide a reliable frequency-domain tuning method for CgLp
compensators. Finally, different performance metrics of a CgLp compensator are
compared with those of a PID controller on a precision positioning stage. The
results show that the presented tuning method is effective, and the system
with the CgLp compensator achieves superior dynamic performance to that of the
PID controller.

##  I INTRODUCTION

The fast rising high-tech industry leads researchers to find a better
alternative for linear controllers [1]. One of the appropriate alternatives is
reset element which has gained a lot of attention due to its simple
configuration [2, 3, 4, 5, 6, 7]. In 1958, the first reset element was
introduced by Clegg [4]. Clegg Integrator (CI) is an integrator which resets
its state to zero when its input crosses zero. Then, First Order Reset Element
(FORE) [2, 8] and Second Order Reset Element (SORE) [8, 6] have been developed
to provide more design freedom and applicability. Other reset conditions such
as reset band [9, 10] and fixed reset instants [11] have also been studied. In
order to soften non-linearities of reset elements, several techniques like
partial reset and PI+CI approaches have been proposed [12].

Based on Describing Function (DF) analysis, it can be seen that reset
controllers provide less lag phase in comparison with their base linear
structures. This phase advantage is utilized to introduce new compensators
[13, 14, 8]. One of these reset compensators is ’Constant in gain Lead in
phase’ (CgLp) whose gain is constant while providing a phase lead [8, 15]. In
these works, CgLp has been used as an alternative for the derivative to
compensate part of the required phase lead. This is advantageous because the
open-loop will have higher gains at low frequencies and lower gains at high
frequencies which results in higher precision performances.

There are few studies which investigate tuning of CgLp compensators [16, 8,
15]. In those works, CgLp is tuned to get a specific amount of phase lead at
the cross-over frequency. However, as a result of the design flexibility of
reset controllers, various combinations of tuning parameters could be used to
provide the same open-loop phase compensation at the cross-over frequency
based on the DF analysis. However, not all sets of tuning parameters result in
performance improvement. In addition, stability has not been assessed in
tuning method and has to be checked with non-linear stability methods,
separately. Furthermore, the existence of the steady-state performance of the
closed-loop has not been assured in those works. Thus, there is a lack of
reliable tuning method for CgLp compensators.

Recently, a new frequency-domain framework is developed which analyzes the
closed-loop steady-state performances of reset control systems considering
high order harmonics [17]. Moreover, a frequency-domain method for assessing
the stability of reset elements has been proposed [18]. In this paper, we
combined the frequency-domain framework [17], the frequency-domain stability
method [18], the DF method, and loop-shaping constraints to provide a reliable
tuning method for CgLp compensators. Finally, to show the effectiveness of the
proposed tuning method, a CgLp compensator is tuned and implemented on a
precision positioning stage.

In the remainder of this paper, the tuning method is elaborated in Section II.
In Section III, a tuned CgLp compensator is applied to a precision positioning
stage, and its performance is compared with a PID controller. Conclusions and
remarks for further study are provided in Section IV.

##  II Tuning Method

In this section, first, frequency-domain descriptions for reset elements, CgLp
compensators, the stability condition, and pseudo-sensitivities (sensitivity
functions defined for nonlinear controllers) are briefly recalled. Then the
structure of the controller is introduced, and the tuning method is proposed.

###  II-A Frequency Analysis of Reset Elements

The state-space representation of reset elements is

$\left\\{\begin{aligned} \dot{x}_{r}(t)&=A_{r}x_{r}(t)+B_{r}e(t),&e(t)\neq
0,\\\ x_{r}(t^{+})&=A_{\rho}x(t),&e(t)=0,\\\
u(t)&=C_{r}x(t)+D_{r}e(t),\end{aligned}\right.$ (1)

in which $A_{r}$, $B_{r}$, $C_{r}$ and $D_{r}$ are the state matrices of the
base linear system, $e(t)$ and $u(t)$ are the error input and control input,
respectively. The resetting matrix $A_{\rho}$ determines states’ values after
reset action. Since reset elements are non-linear, the DF analysis is
popularly used in literature to study their frequency behaviour. To have a
well-defined steady-state response, it is required that $A_{r}$ has all
eigenvalues with negative real part and $A_{\rho}e^{\frac{A_{r}\pi}{\omega}}$
has all eigenvalues with magnitude smaller than one [3]. The sinusoidal input
DF of reset elements (1) is given in [3] as

${\mathcal{N}(j\omega)=C_{r}\left(j\omega
I-A_{r}\right)^{-1}B_{r}\left(I+j\Theta(\omega)\right)+D_{r}},$ (2)

where $\Theta$ is

$\displaystyle\Theta(\omega)$ $\displaystyle{=}$
$\displaystyle\frac{-2\omega^{2}}{\pi}(I+e^{\frac{\pi
A_{r}}{\omega}})\Bigg{(}(I+A_{\rho}e^{\frac{\pi
A_{r}}{\omega}})^{-1}A_{\rho}(I+e^{\frac{\pi A_{r}}{\omega}})$ (3)
$\displaystyle{-}\>I\Bigg{)}(\omega^{2}I+A_{r}^{2})^{-1}.$

###  II-B CgLp Compensator

A CgLp compensator (4) is constructed using a FORE or a SORE with the series
combination of a corresponding order of a lead filter. Considering the DF
analysis, this compensator has a constant gain with a lead phase (Fig. 1) ([8,
19]). In this paper, we only consider the first order CgLp which is

$C_{CgLp}(s)=\left(\cancelto{\gamma}{\frac{1}{\frac{s}{\omega_{r}}+1}}\right)\left(\frac{\frac{s}{\omega_{d}}+1}{\frac{s}{\omega_{t}}+1}\right),$
(4)

where $\omega_{r}$ is the corner frequency of the reset element,
$A_{\rho}=\gamma$ is the reset matrix, and $\omega_{d}$ and $\omega_{f}$ are
the corner frequencies of the lead filter. To have a constant gain, corner
frequencies $\omega_{d}$ and $\omega_{r}$ are almost equal (there is a small
correction factor which is provided in [8]) and $\omega_{f}\gg\omega_{r}$.

$\bm{\omega_{r}=\omega_{d}}$$\bm{\theta}$ Figure 1: The DF of a CgLp
compensator

###  II-C $H_{\beta}$ Condition

There are several theories to determine the stability of reset control systems
[12, 7, 5, 20]. Among those, the $H_{\beta}$ condition presented in [5, 7, 18]
gets a lot of attention. In [18], a method is developed to examine the
$H_{\beta}$ conditions utilizing the frequency response of the plant. To this
end, consider $L(j\omega)$ and $C_{R}(j\omega)$ as the base linear frequency
responses of the open-loop and of the reset element, respectively. Then, the
Nyquist Stability Vector (NSV=$\vv{\mathcal{N}}(\omega)\in\mathbb{R}^{2}$),
for all $\omega\in\mathbb{R}^{+}$, is
$\vv{\mathcal{N}}(\omega)=[\mathcal{N}_{\chi}\quad\mathcal{N}_{\Upsilon}]^{T}$
in which

$\displaystyle\mathcal{N}_{\chi}$ $\displaystyle{=}$ $\displaystyle\left\lvert
L(j\omega)+\frac{1}{2}\right\rvert^{2}-\frac{1}{4},$
$\displaystyle\mathcal{N}_{\Upsilon}$ $\displaystyle{=}$
$\displaystyle\Re(L(j\omega)\cdot C_{R}(j\omega))+\Re(C_{R}(j\omega)).$

###### Theorem 1

Considering
$\theta_{1}=\underset{\omega\in\mathbb{R}^{+}}{\min}\phase{\vv{\mathcal{N}}(\omega)}$
and
$\theta_{2}=\underset{\omega\in\mathbb{R}^{+}}{\max}\phase{\vv{\mathcal{N}}(\omega)}$.
Suppose $-1<\gamma\leq 1$, then, the $H_{\beta}$ condition for a reset control
system is satisfied and its response is uniformly bounded-input bounded-state
(UBIBS) for any bounded input if [18]

$\left(-\dfrac{\pi}{2}<\theta_{1}<\pi\right)\ \land\
\left(-\dfrac{\pi}{2}<\theta_{2}<\pi\right)\ \land\
(\theta_{2}-\theta_{1}<\pi).$ (5)

###  II-D Pseudo-Sensitivities for Reset Control Systems

In linear systems, the relation between reference signal $r(t)$ to error
$e(t)$ can be calculated by sensitivity transfer functions [21]. Although it
is possible to use the DF of the reset elements in those sensitivity transfer
functions to analyze the tracking performance of CgLp compensators, it is not
a reliable approach because high order harmonics are neglected. In order to
analyze reset control systems more accurately, a pseudo-sensitivity function
$S_{\infty}(j\omega)$ for a sinusoidal reference $r=r_{0}\sin(\omega t)$ is
defined in [17].

###### Theorem 2

A closed-loop reset control system has a well-defined steady-state solution
for any Bohl function input if the $H_{\beta}$ condition is satisfied and
reset instants have the well-posedness property [17].

In addition, if Theorem 2 holds, the tracking error of the reset control
system is a periodic function with the period $\dfrac{2\pi}{\omega}$. Thus,
the pseudo-sensitivity for a reset control system is defined as the ratio of
the maximum tracking error of the system to the magnitude of the reference at
each frequency.

###### Definition 1

Pseudo-sensitivity $S_{\infty}$

$\begin{array}[]{*{35}{c}}\forall\omega\in\mathbb{R}^{+}:\
S_{\infty}(j\omega)=e_{\max}(\omega)e^{j\varphi_{max}},\end{array}$

where

$e_{\max}(\omega)=\left(\dfrac{\underset{t_{ss_{0}}\leq t\leq
t_{ss_{m}}}{\max}(r(t)-y(t))}{|r|}\right)=\sin(\omega
t_{\max})-\dfrac{y(t_{\max})}{r_{0}},$

$\varphi_{max}=\frac{\pi}{2}-\omega t_{max}$, $y(t)$ is the response of the
closed-loop reset control system, and $t_{ss_{0}}$ and
$t_{ss_{m}}=t_{ss_{0}}+\frac{2\pi}{\omega}$ are the steady-state reset
instants of the closed-loop reset control system
$(e(t_{ss_{0}})=e(t_{ss_{m}})=0)$. In a similar way, the pseudo-control
sensitivity $CS_{\infty}(\omega)$, the pseudo-complementary sensitivity
$T_{\infty}(\omega)$, and the pseudo-process sensitivity $PS_{\infty}(\omega)$
are defined in [17]. These calculations are embedded in a user-friendly
toolbox [22].

###  II-E Problem Formulation

In this section, the tuning procedure is explained. For this purpose, a CgLp
compensator along with a PID controller is considered for tuning as

$C_{\text{CgLp}}(s)=K_{p}\underbrace{\left(\cancelto{\gamma}{\dfrac{1}{\frac{\alpha
s}{\omega_{r}}+1}}\right)\left(\dfrac{\frac{s}{\omega_{r}}+1}{\frac{s}{\omega_{f}}+1}\right)}_{\mathrm{CgLp}}\underbrace{\overbrace{\left(1+\frac{\omega_{i}}{s}\right)}^{\mathrm{PI}}\overbrace{\left(\dfrac{\frac{s}{\omega_{d}}+1}{\frac{s}{\omega_{t}}+1}\right)}^{\mathrm{Lead}}}_{\mathrm{PID}},$
(6)

in which $\gamma=A_{\rho}$ determines the value of the reset state after the
reset action and
$(K_{p},\omega_{i},\omega_{r},\omega_{t},\omega_{d},\omega_{f},\gamma)$ is the
tuning parameter set. It has been shown that the sequence of controller
filters has effects on the performance of reset control systems [23]. In this
research, the sequence of control filters is the traditional approach in which
the tracking error is the input of the reset element and other linear parts
following in series. In this tuning method, the controller is tuned given the
following specifications: cross-over frequency $\omega_{c}$, phase margin
$\varphi_{m}$, and modulus margin $M_{m}$. Note that these specifications are
based on the DF analysis or defined pseudo-sensitivities. Furthermore, since
the scope of this paper is tuning of the CgLp part, $\omega_{i}$ and
$\omega_{f}$ are tuned as $\dfrac{\omega_{c}}{10}$ and $8\omega_{c}$,
respectively, to have acceptable tracking and noise rejection performances
[21, 24, 8]. To assure stability and use of pseudo-sensitivities, the
$H_{\beta}$ condition (Theorem 1) has to be satisfied. In addition, a
robustness requirement in the form of iso-damping behaviour [1, 25] requires
that the phase behaviour of the system must follow

$\dfrac{d(\phase{\mathcal{N}_{\text{CgLp}}(j\omega)\text{PID}(j\omega)G(j\omega)})}{d\omega}\Big{|}_{\omega=\omega_{c}}=0.$
(7)

All constraints are summarized as

  * •

Cross-over frequency constraint:

$|\mathcal{N}_{\text{CgLp}}(j\omega_{c})\text{PID}(j\omega_{c})G(j\omega_{c})|=1$

  * •

Phase margin constraint:

$\phase{\mathrm{\mathcal{N}_{CgLp}}(\omega_{c})}+\phase{\mathrm{PID}(\omega_{c})}+\phase{G(\omega_{c})}=\varphi_{m}$

  * •

Modulus margin constraint: $\text{max }|S_{\infty}(j\omega)|<M_{m}$

  * •

Iso-damping Behaviour: Equation (7)

  * •

The $H_{\beta}$ condition: Equation (5) and $-1<\gamma\leq 1$

Eventually, we define a suitable cost function for the tuning of the control
structure (6). According to [26], to have an appropriate tracking performance
in the interested region of frequencies, the following cost function is
obtained as

$J=\underset{\omega\leq\omega_{l}}{\max}\Big{|}\dfrac{S_{\infty}(j\omega)}{\omega}\Big{|}_{\text{dB}},$
(8)

in which $\omega_{l}$ determines the interested region of frequencies over
which the reset control system is expected to track references and reject
disturbances. There are several methods such as grid search, gradient methods,
Genetic Algorithm, etc., for solving this problem. Here, since the performance
of the controller is not so sensitive to a small change of the tuning set
parameter, we suggest to use a grid search method for completing the tuning
procedure. The parameter $K_{p}$ is determined by the cross-over frequency
definition. In addition, it is possible to find vectors $l_{B}$ and $u_{B}$ to
set lower and higher limits for $(\omega_{r}\,\omega_{d}\,\omega_{t})$ by the
phase margin definition and considering stability of the base linear of the
system (i.e. $\omega_{c}l_{B}<[\omega_{r}\ \omega_{d}\
\omega_{t}]^{T}<\omega_{c}u_{B}$). Then, with a small resolution, we grid the
parameters and provide a parameter space. Now, we calculate constraints
(3)-(6) for every point in this space, and eliminate the points which do not
satisfy the constraints. Finally, suppose there are N tuning parameter sets
$(K_{p},\omega_{r},\omega_{t},\omega_{d},\gamma)$ which satisfy the
aforementioned constraints, then the parameter set which has the minimum $J$
value is selected for designing the control structure (6).

##  III Practical Example

To show the effectiveness of the proposed tuning method, a precision
positioning stage (Fig 2) is used as a benchmark in this paper. In this stage,
which is termed “Spider”, three actuators are angularly spaced to actuate 3
masses (indicated by B1, B2, and B3) which are constrained by parallel
flexures and connected to the central mass D through leaf flexures. Only one
of the actuators (A1) is considered and used for controlling the position of
mass B1 attached to the same actuator which results in a SISO system. A linear
power amplifier is utilized to drive the Lorentz actuator, and Mercury M2000
linear encoder is used to obtain position feedback with the resolution of
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The identified frequency response
data of the system is shown in Fig 3. As shown in Fig. 3, although the plant
is a collocated double mass-spring system, the identified frequency response
data is well approximated by a mass-spring-damper system with the transfer
function

$G(s)\approx\frac{Ke^{-\tau s}}{\frac{s^{2}}{\omega_{r}^{2}}+\frac{2\zeta
s}{\omega_{r}}+1}=\frac{1.14e^{-0.00014s}}{\frac{s^{2}}{7627}+\frac{0.05s}{87.3}+1}.$
(9)

Figure 2: The whole setup including computer, CompactRio, power supply, sensor
power, amplifier, isolator, sensor and, stage Figure 3: Identification of the
stage

Note that to use relations provided in [17], the time delay $(e^{-0.00014s})$
is approximated by the first order Pade method [27] as
$\dfrac{-s+14400}{s+14400}$. The design requirements for this system are:

  * •

the cross-over frequency: $\omega_{c}=100\text{ Hz}$

  * •

the phase margin: $\varphi_{m}=30^{\circ}$

  * •

the modulus margin: $M_{m}\leq 6.5\text{ dB}$

Now, the control structure (6) is tuned based on the described method in
Section II-E. To satisfy constraint (7), $\omega_{c}\leq\omega_{t}$. Also, in
order to not eliminate the low-pass filter effects,
$\omega_{t}\leq\omega_{f}$. Considering $30^{\circ}$ phase margin and
stability of the base linear system, it is obtained that
$\dfrac{\omega_{c}}{5}<\omega_{d}<\omega_{c}$. Furthermore, selecting a very
small value for $\omega_{r}$ leads to increase the amplitude of high order
harmonics at low frequencies which are not desired [15]. Hence, we consider
this parameter range $\omega_{c}[0.05\ 0.2\ 1]^{T}<[\omega_{r}\ \omega_{d}\
\omega_{t}]^{T}<[1\ 1\ 8]^{T}\omega_{c}$ in the tuning procedure. In addition,
we take $\omega_{l}=\dfrac{\omega_{c}}{10}$ as the maximum limit of the
interest region for tracking. The controller is obtained through the proposed
tuning method as

$C_{\text{CgLp}}{=}25.5\left(\cancelto{0.3}{\dfrac{1}{\dfrac{s}{111\pi}+1}}\right)\left(\dfrac{\dfrac{s}{105.2\pi}+1}{\dfrac{s}{1600\pi}+1}\right)\left(1+\dfrac{20\pi}{s}\right)\left(\dfrac{\dfrac{s}{105.2\pi}+1}{\dfrac{s}{260\pi}+1}\right).$

(10)

To compare the performance of the tuned controller with a linear controller, a
PID structure is also tuned with the same method proposed in Section II-E. To
have a fair comparison, the structure of the PID controller is similar to the
control structure (6). Finally, the $C_{\text{PID}}$ is

$C_{\text{PID}}=18.46\left(\dfrac{1}{\dfrac{s}{1600\pi}+1}\right)\left(\dfrac{\dfrac{s}{77\pi}+1}{\dfrac{s}{520\pi}+1}\right)\left(1+\frac{20\pi}{s}\right).$
(11)

Figure 4 shows the open-loop frequency response of the system with controllers
$C_{\text{PID}}$ and the DF of the open-loop of the system with the controller
$C_{\text{CgLp}}$. Two systems have the same phase margin and are robust
against the gain variation (iso-damping behaviour) as shown in Fig. 4.

Figure 4: Open-loop frequency responses of the system with controllers
$C_{{\text{PID}}}$ and $C_{\text{CgLp}}$

(a) Complementary sensitivity

(b) Sensitivity

(c) Control sensitivity

(d) Process sensitivity

Figure 5: The DFs $(.\\_$ DF) and pseudo-sensitivities (.$\\_\ \infty$) of the
closed-loop of the system with the controllers $C_{CgLp}$, and the closed-loop
sensitivities of the system with the controller $C_{\text{PID}}$

The closed-loop frequency responses of the systems with the controller
$C_{\text{CgLp}}$ including the pseudo-sensitivities and the DF methods, and
the closed-loop sensitivities of the system with the controller
$C_{{\text{PID}}}$ are shown in Fig. 5. These frequency responses are obtained
utilizing the toolbox in [22]. By $T_{\infty}$ (Fig. 5(a)), the noise
rejection capability of the system with the controller $C_{\text{CgLp}}$ must
be better than that of the controller $C_{\text{PID}}$. Furthermore, as shown
in Fig. 5(b), the system with the controller $C_{\text{CgLp}}$ has better
tracking performance than that one with the controller $C_{{\text{PID}}}$ at
frequencies less than $10$Hz while the modulus margin of the system with the
controller $C_{\text{CgLp}}$ is less than that of with the controller
$C_{\text{PID}}$. Also, there are discrepancies between the sensitivity DF and
pseudo-sensitivity in the frequency range (30 - 70 Hz) which are due to the
existence of high order harmonics.
Based on $PS_{\infty}$ (Fig 5(d)), the disturbance rejection capability of the
system with the controller $C_{\text{CgLp}}$ is better than that of the
controller $C_{{\text{PID}}}$. As shown in Fig.5(c), there is a significant
difference between the control input of the system with the controller
$C_{\text{CgLp}}$ and what is predicted by the DF method. In addition, the
control input of the system with the controller $C_{\text{CgLp}}$ is more than
one with the controller $C_{{\text{PID}}}$. This is explained by the fact that
reset elements produce jumps in their output and differentiation of jumps
produces a large control input.

###  III-A Time Domain Results

In this part, the time domain results of the designed controllers are compared
with each other. To implement controllers (Fig. 6), each controller is
discretized with sample time $T_{s}=$$100\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ using the Tustin method [21, 26, 24]. Furthermore, to provide the well-
posedness property [12, 7], there are no reset instants in tandem.

$\bm{C(z)}$$\bm{-}$$\bm{r}$$\bm{e}$$\dfrac{\bm{1}}{\bm{ms^{2}+cs+k}}$$\bm{+}$$\dfrac{\bm{1}}{\bm{s}}$$\bm{\gamma}$Linear
Part
ofControllerPlant$\bm{+}$$\bm{n}$Reset$\bm{A_{r}}$$\bm{C_{r}}$$\bm{B_{r}}$$\bm{D_{r}}$$\bm{+}$$\bm{+}$$\bm{w}$$\bm{y}$

Figure 6: The block diagram of the whole system for implementing the designed
controllers (reset matrices are discretized)

The step responses (step of $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$) of the
system with these controllers are illustrated in Fig. 7. To assess iso-damping
behaviour of the system, the gains of the controllers are varied between
$80\%$ to $120\%$ of their nominal values. The step responses have the same
rise time while the overshoot of the system with the controller
$C_{\text{CgLp}}$ are less than that of the controller $C_{\text{PID}}$
because the modulus margin of the system with the CgLp compensator is less
than one with the PID controller. Furthermore, system with the controller
$C_{\text{CgLp}}$ has less settling time in comparison with that of with the
controller $C_{\text{PID}}$. Besides, step responses of the system with these
controllers show iso-damping behaviour indicating. However, $C_{\text{CgLp}}$
provides more robustness against gain variation for the system.

(a) Step responses of controllers

(b) Step responses of $C_{\text{CgLp}}$

(c) Step responses of $C_{\text{PID}}$

Figure 7: The step responses of controllers with gain variation between $80\%$
to $120\%$ of their nominal values

In order to compare tracking performances of the systems with both
controllers, one triangular reference with the amplitude of
$400\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (Fig. 8(a)) and one sinusoidal
reference $r(t)=111\sin(10\pi t)$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
(Fig. 8(c)) are applied to the system. As was predicted by $S_{\infty}$ (Fig.
5(b)), the system with the controller $C_{\text{CgLp}}$ has a better
performance at $5$Hz (Fig. 8(d)). In addition, $S_{\infty}$ (Fig. 5(b))
precisely predicts the maximum error of the system with the controller
$C_{\text{CgLp}}$ for the sinusoidal reference. Note that, for the sake of
brevity, we only show the result at $5$Hz while the tracking performance of
the system with the controller $C_{\text{CgLp}}$ is better than one with the
controller $C_{\text{PID}}$ for the sinusoidal reference for all frequencies
less than $10Hz$. As shown in Fig. 8(b), the system with the controller
$C_{\text{CgLp}}$ also has the better tracking performance than that of the
controller $C_{\text{PID}}$ for the triangular reference (Fig. 8(a)) which is
a combination of several frequencies. For these trajectories, the tracking
performance of the system is improved by 30% using the controller
$C_{\text{CgLp}}$.

(a) Triangular reference

(b) Tracking error of the triangular reference

(c) Sinusoidal reference

(d) Tracking error of the sinusoidal reference

Figure 8: Tracking performance of the designed controllers for a triangular
and a sinusoidal references

(a) Disturbance rejection

(b) Noise rejection

Figure 9: Disturbance and noise rejection capabilities of the system with the
designed controllers

Figure 9 compares the noise and disturbance rejection capabilities of the
system with the designed controllers. To study the noise rejection
capabilities of the system with the controllers, a white noise with a maximum
amplitude of $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is applied to the
system. As was expected from $T_{\infty}$ (Fig. 5(a)), the noise rejection
capability of the system with the controller $C_{\text{CgLp}}$ is better than
one with the controller $C_{\text{PID}}$ (Fig. 9(b)). It can be said that
using $C_{\text{CgLp}}$ enhance the noise rejection capability of the system
by 40%. In order to evaluate the abilities of the system with the designed
controllers for attenuating disturbances, a sinusoidal disturbance
$w(t)=190\sin(14\pi t)$ $\text{\,}\mathrm{\SIUnitSymbolMicro A}$ is applied to
the system. Similar to the $PS_{\infty}$ prediction (Fig. 5(d)), the system
with the controller $C_{\text{CgLp}}$ has the optimal disturbance rejection
performance (Fig. 9(a)). The disturbance rejection capability of the system is
improved by 30% using the controller $C_{\text{CgLp}}$.

To wrap up, the system with the tuned CgLp compensator has less overshoot, the
same rise time, better tracking performance for frequencies less than 10 Hz,
less modulus margin, better noise and disturbance rejection capabilities than
those of the system with the controller PID.

##  IV Conclusion

This paper has proposed a frequency-domain tuning method for CgLp compensators
based on the defined pseudo-sensitivities for reset control systems. In this
method, a PID+CgLp structure is considered, and its parameters are tuned such
that the pseudo-sensitivity is minimized under several constraints. Also, the
tuned CgLp compensator with this method, makes the system robust against gain
variations. To show the effectiveness of the proposed approach, the
performance of this tuned CgLp is compared with a linear PID. The results show
that the frequency framework is reliable for tuning CgLp compensators.
Furthermore, the tuned CgLp can achieve more favourable dynamic performance
than the PID controller for the precision motion stage. The tracking
performance, the disturbance rejection capability, and the noise rejection
capability of the system are improved by 30% using the CgLp compensator.
Indeed, this method, which allows for tuning in the frequency-domain, opens
doors for the implementation of reset controllers in industrial applications.

## References

  * [1] A. A. Dastjerdi, B. M. Vinagre, Y. Chen, and S. H. HosseinNia, “Linear fractional order controllers; a survey in the frequency domain,” _Annual Reviews in Control_ , 2019.
  * [2] I. Horowitz and P. Rosenbaum, “Non-linear design for cost of feedback reduction in systems with large parameter uncertainty,” _International Journal of Control_ , vol. 21, no. 6, pp. 977–1001, 1975.
  * [3] Y. Guo, Y. Wang, and L. Xie, “Frequency-domain properties of reset systems with application in hard-disk-drive systems,” _IEEE Transactions on Control Systems Technology_ , vol. 17, no. 6, pp. 1446–1453, 2009.
  * [4] J. C. Clegg, “A nonlinear integrator for servomechanisms,” _Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry_ , vol. 77, no. 1, pp. 41–42, 1958.
  * [5] O. Beker, C. Hollot, Y. Chait, and H. Han, “Fundamental properties of reset control systems,” _Automatica_ , vol. 40, no. 6, pp. 905 – 915, 2004.
  * [6] L. Hazeleger, M. Heertjes, and H. Nijmeijer, “Second-order reset elements for stage control design,” in _American Control Conference (ACC)_ , 2016, pp. 2643–2648.
  * [7] Y. Guo, L. Xie, and Y. Wang, _Analysis and Design of Reset Control Systems_. Institution of Engineering and Technology, 2015.
  * [8] N. Saikumar, R. K. Sinha, and S. H. HosseinNia, “‘Constant in gain Lead in phase’ element-application in precision motion control,” _IEEE/ASME Transactions on Mechatronics_ , vol. 24, no. 3, pp. 1176–1185, 2019.
  * [9] A. Barreiro, A. Baños, S. Dormido, and J. A. González-Prieto, “Reset control systems with reset band: Well-posedness, limit cycles and stability analysis,” _Systems & Control Letters_, vol. 63, pp. 1–11, 2014.
  * [10] A. Baños and M. A. Davó, “Tuning of reset proportional integral compensators with a variable reset ratio and reset band,” _IET Control Theory & Applications_, vol. 8, no. 17, pp. 1949–1962, 2014.
  * [11] J. Zheng, Y. Guo, M. Fu, Y. Wang, and L. Xie, “Improved reset control design for a pzt positioning stage,” _IEEE International Conference on Control Applications_. IEEE, 2007, pp. 1272–1277.
  * [12] A. Baños and A. Barreiro, _Reset control systems_. Springer Science $\&$ Business Media, 2011.
  * [13] S. J. A. M. Van den Eijnden, Y. Knops, and M. F. Heertjes, “A hybrid integrator-gain based low-pass filter for nonlinear motion control,” _IEEE Conference on Control Technology and Applications (CCTA)_ , 2018, pp. 1108–1113.
  * [14] D. Valério, N. Saikumar, A. A. Dastjerdi, N. Karbasizadeh, and S. H. HosseinNia, “Reset control approximates complex order transfer functions,” _Nonlinear Dynamics_ , pp. 1–15, 2019.
  * [15] H. Xiaojun, A. Ahmadi Dastjerdi, N. Saikumar, and S. HosseinNia, “Tuning of ‘Constant in gain Lead in phase (CgLp)’ reset controller using Higher-Order Sinusoidal Input Describing Function (HOSIDF),” _Australian and New Zealand Control Conference (ANZCC)_ , 2020.
  * [16] M. Shirdast Bahnamiri, N. Karbasizadeh, A. Ahmadi Dastjerdi, N. Saikumar, and S. HosseinNia, “Tuning of CgLp based reset controllers: Application in precision positioning systems,” _IFAC World Congress_ , 2020.
  * [17] A. Ahmadi Dastjerdi, N. Saikumar, D. Valerio, and S. Hassan HosseinNia, “Closed-loop frequency analyses of reset systems,” _arXiv e-prints_ , p. arXiv:2001.10487, Jan 2020.
  * [18] A. A. Dastjerdi, A. Astolfi, and S. H. HosseinNia, “A frequency-domain stability method for reset systems,” _IEEE 59th Conference on Decision and Control_ , 2020.
  * [19] “Complex order control for improved loop-shaping in precision positioning,” in _IEEE 58th Conference on Decision and Control_ , 2019, pp. 7956–7962.
  * [20] S. Polenkova, J. W. Polderman, and R. Langerak, “Stability of reset systems,” _Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems_ , 2012, pp. 9–13.
  * [21] R. M. Schmidt, G. Schitter, and A. Rankers, _The Design of High Performance Mechatronics High-Tech Functionality by Multidisciplinary System Integration_. IOS Press, 2014.
  * [22] A. A. Dastjerdi. Toolbox for frequency analysis of reset control systems. [Online]. Available:
https://www.tudelft.nl/en/3me/about/departments/precision-and-microsystems-
engineering-pme/research/mechatronic-system-design-msd/msd-research/motion-
control/toolbox-frequency-analysis-of-reset-control-systems/

  * [23] C. Cai, A. A. Dastjerdi, N. Saikumar, and S. HosseinNia, “The optimal sequence for reset controllers,” _$18^{th}$ European Control Conference (ECC_, 2020.
  * [24] A. A. Dastjerdi, N. Saikumar, and S. H. HosseinNia, “Tuning guidelines for fractional order PID controllers: Rules of thumb,” _Mechatronics_ , vol. 56, pp. 26 – 36, 2018.
  * [25] R. De Keyser, C. I. Muresan, and C. M. Ionescu, “A novel auto-tuning method for fractional order PI/PD controllers,” _ISA transactions_ , vol. 62, pp. 268–275, 2016.
  * [26] J. Sabatier, P. Lanusse, P. Melchior, and A. Oustaloup, _Fractional order differentiation and robust control design_. Springer, 2015, vol. 77.
  * [27] S. Al-Amer and F. Al-Sunni, “Approximation of time-delay systems,” _Proceedings of the American Control Conference. ACC (IEEE Cat. No. 00CH36334)_ , vol. 4. IEEE, 2000, pp. 2491–2495.