# Time-delay based output feedback control of fourth-order oscillatory systems

Michael Ruderman<EMAIL_ADDRESS>Department of Engineering Sciences,
University of Agder, P.B. 422, Kristiansand, 4604, Norway

###### Abstract

We consider a robust stabilization of the fourth-order oscillatory systems
with non-collocated output sensing. Worth recalling is that the fourth-order
systems are relatively common in mechatronics as soon as there are two-mass or
more generally two-inertia dynamics with significant elasticities in the link.
A novel yet simple control method is introduced based on the time-delayed
output feedback. The delayed output feedback requires only the oscillation
frequency to be known and allows for a robust control design that leads to
cancelation of the resonance peak. We use the stability margins to justify the
transfer characteristics and robustness of the time-delay control in frequency
domain. The main advantage of the proposed method over the other possible
lead-based loop-shaping strategies is that neither time derivatives of the
noisy output nor the implementation of transfer functions with a numerator
degree greater than zero are required to deploy the controller. This comes in
favor of practical applications. An otherwise inherently instable
proportional-integral (PI) feedback of the non-collocated output is shown to
be stabilized by the proposed method. The control developed and associated
analysis are also confirmed by the experimental results shown for the low
damped two-mass oscillator system with uncertainties.

###### keywords:

time-delay system , feedback control , stabilization by delay , output control
design

††journal: Mechatronics

##  1 Introduction

Time delays in a feedback control loop are usually associated with degradation
of performance and robustness and, in worth case, with destabilization due to
the evoked phase ’deficit’ of the loop transfer function. At the same time,
there are situations where time delays are used as controller parameters, cf.
[1]. Several important classes of dynamic systems, including different type of
oscillators, cannot be stabilized by static output feedback, although they
might be stabilizeable by inserting an artificial time delay into the
feedback, cf. e.g. [2]. While for a well developed (Lyapunov based) stability
analysis of time-delay systems we refer to seminal literature, see e.g. [3],
[4], [5] and references therein, a purposeful use of a time-delayed feedback
for stabilization is a less studied topic in the control applications. A
former work [6] used a positive delayed output feedback for stabilizing the
second-order oscillatory system. Later, the output feedback stabilization
problem of a chain of integrators using multiple delays was addressed in [7].
More specifically, it was shown in [7] that a chain of $n$ integrators can be
stabilized by a proportional plus delay controller including $n-1$ delays, or
by a chain of $n$ delay blocks. As early as two decades ago, it was already
recognized that delay properties can be also useful, since introducing delays
voluntarily can benefit the control, see an overview of some advances and open
problems with time-delay systems in [8]. Still, to the best of the author’s
knowledge, a positive delayed feedback was used only by the Pyragas control
[9] for stabilization of unstable periodic orbits of a chaotic system. Later,
a similar delay-based strategy was pursued in [10] to control the noise-
induced oscillations in nonlinear second-order systems. However, unlike the
approach proposed in the present work, an opposite sign of the difference
between the delayed and current state of the system and the output rate
instead of the output value itself were used in [10] for feedback.

While most of the known works on time delay systems (some of which were
mentioned above) deal with stabilization of the delay-affected plants and
networks, the approach provided in the present work is rather to use a
purposefully injected delay as a control parameter. The main motivating idea
behind the proposed control is the fact of anti-phase between the input $u$
and oscillatory mode of a fourth-order system. Here, it is worth recalling
that the fourth-order systems are relatively common in mechatronics as soon as
there are two-mass or more generally two-inertia dynamics with significant
elasticities in the link. Notable applications can be found, for example, with
payloads driven via the ropes or cables, such as elevators, cranes, winches
and others. It is also worth emphasized that the proposed method uses only the
output information $x$ but not its time derivatives or other internal dynamic
states, cf. with [3]. This comes in favor of different practical applications
like, for example, with non-collocated sensing and actuation, or noisy output
sensing where the time derivatives are not available. One of the main
contributions of this work can be viewed in the introduced control

$u(t)=K_{d}\bigl{(}x(t)-x(t-\tau)\bigr{)},$ (1)

with the dedicated time-delay parameter $\tau$. We have to notice that some
preliminary results were reported in [11], where (1) was shown for the first
time. In view of this, the claimed novelty and results of the present work
are:

  * -

detailed analysis of the closed-loop with (1);

  * -

design of the robust time-delay-based stabilizer for PI-controlled non-
collocated fourth-order plants;

  * -

experimental confirmation of the effectiveness and performance of the
oscillation compensator (1).

The rest of the paper is organized as follows. In Section 2, the problem
formulation is given. The main results of the introduced feedback control (1),
its stability analysis, and parameterizations are presented in section 3. The
stabilization problem of a PI-feedback controlled fourth-order plant with non-
collocated sensing and actuation is addressed in section 4. This is followed
by the associated experimental example demonstrated in section 5. Brief
summary and discussion are given in section 6.

Notation. Throughout the text, the uppercase italic Latin letters denote the
vectors and matrices of the appropriate dimension, while their lowercase
counterparts denote the variables. The dynamic variables in time domain are
with argument $t$, and in Laplace domain with argument $s$. The italic Latin
letters with time argument are used for denoting the measured signals. The
Latin and Greek letters denote the constants and parameters, while the control
gains are denoted by $K$ with a subindex. $I$ is an identity matrix of an
appropriate dimension. Unless other specified, $\omega$ is the angular
frequency, $j$ is the imaginary unit of a complex number, and $|\Omega|$ and
$\angle\Omega$ are the magnitude and phase of a complex function $\Omega$.

##  2 Problem formulation

We consider a class of single-input-single-output (SISO) oscillatory systems
of the fourth-order with one free integrator. Note that the integrative
behavior of the system output is purposefully required for elucidating the
issue of a feedback destabilization due to the lack of phase characteristics.
Otherwise, instead of the pole in origin, the systems considered can also have
an additional negative real pole. For the measurable system input and output
$u(t)$ and $x(t)$, respectively, the transfer characteristics in Laplace
domain are given by

$G(s)=\frac{x(s)}{u(s)}=F(sI-A)^{-1}B.$ (2)

The exemplified system matrix and coupling vectors are

$A=\left(\begin{array}[]{cccc}-a_{1}&-a_{2}&a_{3}&-a_{4}\\\ 1&0&0&0\\\
a_{5}&a_{6}&-a_{7}&-a_{8}\\\ 0&0&1&0\end{array}\right),\quad
B=\left(\begin{array}[]{c}b\\\ 0\\\ 0\\\ 0\end{array}\right),$

$F=(0,0,0,1)$, respectively, with the constant parameters
$a_{1},\ldots,a_{8},b>0$ which are satisfying $a_{3},a_{5},a_{7}\ll
a_{1},a_{2},a_{4},a_{6},a_{8}$. A typical configuration of the poles of the
system (2), (2) is exemplary shown in Figure 1, without numerical values.

Figure 1: Typical poles configuration of oscillatory system (2), (2).

Then, the associated fourth-order input-output transfer function (2) can be
rewritten as

$G(s)=\frac{N(s)}{D(s)}=\frac{k(s+z_{1})}{s(s+p_{1})(s^{2}+2\zeta\omega_{0}s+\omega_{0}^{2})}.$
(3)

Here the system parameters are accommodated in the gain factor $k$, natural
frequency $\omega_{0}$, damping ratio $\zeta$, and the roots’ coefficients of
one real zero $z_{1}$ and one real pole $p_{1}$, while $z_{1}>\omega_{0}$; all
coefficients are strictly positive. Worth emphasizing that $0<\zeta\ll 1$,
while for $\zeta=1$ the conjugate complex pole pair collapses into the double
real pole, and the system (3) becomes critically damped without having to
compensate for oscillatory behavior.

Applying any type of the output feedback controller $R(s)$, yet without pole-
zero cancelation within $R(s)G(s)$, results in the closed-loop transfer
function

$H(s)=\frac{N(s)}{D(s)+R(s)N(s)}=\frac{N(s)D(s)^{-1}}{1+R(s)N(s)D(s)^{-1}}.$
(4)

Recall that the loop transfer function $R(s)N(s)D(s)^{-1}$ contains all
information about the closed-loop behavior and is (usually) used for analysis
and design of the feedback control $R(s)$, which should render a desired
$H(s)$-behavior, cf. e.g. [12, 13]. Due to the phase lag of the fourth-order
dynamics (3), the output feedback capacities are relatively limited, featured
by the stability margins, and more specifically – gain and phase margins. An
increase of the loop gaining factor, i.e. either of $k$ in (3) or $R(0)$ of
the control, leads unavoidably to destabilization of $H(s)$. This is due to
the lack of the gain margin (GM) of the loop transfer function
$R(s)N(s)D(s)^{-1}$ once a proportional feedback is included in $R(s)$. Worth
emphasizing is that without compensating explicitly for the resonance peak at
$\omega_{0}$ and, therefore, improving the GM criteria of the loop transfer
characteristics, any type of $R(s)$ with proportional feedback action will
lead to an unstable, or at least largely oscillating, behavior of $H(s)$. This
becomes best way visible when comparing the loop transfer functions
$N(s)D(s)^{-1}$ for two largely differing damping ratios, e.g.
$\zeta=[0.01,0.7]$. This results in a frequency response with and without
resonance peak as exemplary shown in Figure 2.

Figure 2: Exemplary Bode diagrams of the loop transfer functions
$N(s)D(s)^{-1}$ with different damping ratios $\zeta=[0.01,0.7]$.

While the phase margin (PM) appears sufficient and nearly unsensitive to the
damping ratio, for the assigned loop gaining factor, the GM is drastically
reduced in case of the resonance peak. The GM of the loop transfer function
with $\zeta=0.7$, marked by the double-arrow in Figure 2, will only gradually
decrease with an increasing gain factor. Quite opposite, for the loop transfer
function with $\zeta=0.01$, an infinitesimally small gain enhancement will
already lead to losing completely GM and, therefore, to destabilization of
$H(s)$.

Against the background provided above, the objective is to design the time-
delay-based output feedback compensator $R(s)$, which will largely attenuate
the resonance peak of the system (3) without significantly affecting the
residual shape of the loop transfer function. This way, a time-delay-based
compensation should allow for using other outer feedback control loops which
can guarantee for $x(t)-r(t)\rightarrow 0$, where $r=\mathrm{const}\neq 0$ is
a reference set value, and that after possibly oscillation-free transient
response of the control system.

##  3 Time-delay feedback control

The proposed control (1) is parameterized by the gain factor $K_{d}>0$ and the
time-delay

$\tau=-\bigl{[}\arg G(j\omega)|_{\omega=\omega_{0}}\bigr{]}\omega_{0}^{-1}.$
(5)

The latter corresponds to the phase angle at the resonance frequency, where
the output value is in anti-phase (i.e. $-\pi$ rad/s) to the input value.
Though an explicit use of the time delay in output feedback was proposed
previously for the second-order systems in [6], the control (1), (5) is
principally differing to that one provided in [6]. The difference between the
current and anti-phase-delayed output values in (1) provides a nearly zero or
some low-constant control action for all angular frequencies other than
$\omega_{0}$. On the contrary, in vicinity to $\omega_{0}$ the proposed
control aims to suppress the resonance peak, cf. Figure 2, through the
positive feedback of the control law (1).

Despite the dynamic behavior of systems with time delay are usually analyzed
in time domain, cf. [8], [7], [2], we purposefully use the frequency domain
consideration of the system transfer characteristics, this way allowing for
stability margin analysis and corresponding loop shaping via time delay in
feedback. Here it is worth mentioning that the stability of time-delay systems
by considering the induced-gain of a time-delay operator in the loop and the
associated Bode plots were successfully analyzed in [14].

Taking the ratio between the uncompensated loop transfer function (3) and
feedback-compensated (4) results in

$\frac{G(s)}{H(s)}=1+\frac{R(s)N(s)}{D(s)}.$ (6)

For achieving the above stated compensation goals, cf. with Figure 2, one
needs to ensure that (6) has the magnitude response satisfying
$|G(j\omega)||H(j\omega)|^{-1}=\mathrm{const}$ for $\omega<\omega_{0}$ and
$|G(j\omega)||H(j\omega)|^{-1}=\mathrm{const}\approx 1$ for
$\omega>\omega_{0}$. This is in the sense of loop shaping where the resonance
peak associated with $\omega_{0}$ needs to be suppressed. Recall that here,
$H(j\omega)$ constitutes a resonance compensated loop transfer function, cf.
with exemplary frequency characteristics shown in Figure 2 for the well damped
case of $\zeta=0.7$. Note that at lower frequencies $\omega<\omega_{0}$, the
loop gaining factor of (6) can be adjusted afterwards, this way settling the
required crossover frequency once the resonance peak is compensated. For
angular frequencies around $\omega_{0}$, the (6) ratio must be of the same
magnitude as the resonance peak of $G(j\omega)$, meaning its maximal possible
compensation.

Before deriving the corresponding optimal $K_{d}$-gain, for which the
resonance peak is attenuated as possible at $\omega_{0}$, let us first examine
the principal ability of

$R(s)=K_{d}\bigl{(}\exp(-s\tau)-1\bigr{)}$ (7)

to meet the above requirements at lower and higher frequencies, i.e. for
$\omega<\omega_{0}$ and $\omega>\omega_{0}$. Recall that the time-delay-based
compensator (7), as infinite-dimensional operator, is upper bounded by a lead
transfer element with the gaining factor $2K_{d}$, cf. Figure 3 and e.g. [13,
ch. 4].

Figure 3: Exemplary magnitude plot of (7) with $K_{d}=1$, $\tau=0.3142$.

Since for $\omega<\omega_{0}$ the $|D(j\omega)|$ has also an incremental slope
of one decade per decade and $|N(j\omega)|\rightarrow\mathrm{const}=kz_{1}$,
it is apparent that $|R(j\omega)N(j\omega)D^{-1}(j\omega)|=\mathrm{const}$ at
lower frequencies, cf. (6). Note that this constant value depends on both, the
system parameter $kz_{1}$ and the control parameter $\tau$. At higher
frequencies, $|D(j\omega)|$ has an incremental slope of four decades per
decade, owing to four integrators in a chain, see (3). At the same time,
$|N(j\omega)|$ has an incremental slope of one decade per decade and
$|R(j\omega)|\leq\mathrm{const}$ for $\omega>\omega_{0}$, cf. Figure 3.
Therefore, $|R(j\omega)N(j\omega)D^{-1}(j\omega)|\rightarrow 0$ as $\omega$
increases within the range larger than $\omega_{0}$. An exemplary
$G(s)H(s)^{-1}$ ratio is demonstrated by the magnitude plot in Figure 4, here
for the sake of visualization.

Figure 4: Exemplary plot of $|GH^{-1}|$ ratio according to (6), (7).

In order to determine the optimal $K_{d}$-gain, consider first the resonance
peak of $G(j\omega)$ comparing to the transfer characteristics of $G$ without
resonance peak, i.e. for $\zeta=0.7$. For the natural frequency, one obtains

$\frac{|G(j\omega_{0})|}{|G_{\zeta=0.7}(j\omega_{0})|}=\frac{0.7}{\zeta},$ (8)

which indicates the desired ratio of (6) at $\omega=\omega_{0}$ when applying
a suitable $R(s)$. Replacing in (6) the left-hand-side by (8) and $R(s)$ by
(7), substituting $j\omega$ instead of $s$, and solving the obtained equation
for $\omega=\omega_{0}$ results in

$K_{d}=\frac{\omega_{0}^{3}(0.7-\zeta)}{k}\,\Bigl{|}\frac{p_{1}+j\omega_{0}}{z_{1}+j\omega_{0}}\Bigr{|}.$
(9)

##  4 Stabilization of non-collocated PI-control

For non-collocated output of the system (3) to be controlled, an inherent
stability problem lies in the fact of a phase ’deficit’ due to the system
relative degree $>2$. The $\angle G$ is always crossing the $-180$ deg phase
limit, cf. Figure 2, thus making the GM to a sensitive stability criteria.
Needless to say is that additional unmodeled lag properties in the loop with
system $G(s)$, such as due to even minor dynamics of sensing and actuating
elements and signal transmission delays, will further increase the phase
’deficit’ and, thus, impair stability margins in general.

In order to guarantee the controlled output $x(t)$ can follow the reference
value $r(t)$, i.e. to solve not only a set value stabilization problem
$r=\mathrm{const}$, an integral control term is usually required. When
applying a standard PI (proportional-integral) controller, with the design
parameters $K_{p},K_{i}>0$, the loop transfer function is written

$L(s)=C(s)G(s)=K_{p}\,\frac{s+K_{i}K_{p}^{-1}}{s}\,\frac{N(s)}{D(s)}.$ (10)

It is remarkable that independent of the assigned $K_{p},K_{i}$ a conjugate-
complex pole pair of $L(s)$ does not vanish, cf. Figure 1, and is migrating to
the right towards the unstable right-hand-side half plane when increasing the
loop gain factor $K_{p}$. Using, for example, the root locus analysis or other
conventional tools of the linear control theory, cf. [12], one can find the
critical $\max K_{p}$ beyond which the closed loop of (10) becomes unstable.
Even if $K_{p}<\max K_{p}$ is guaranteed, the uncertainties in $k$ or its
temporal variations, cf. (3), can destabilize the closed-loop system.
Furthermore, a lower loop gain cannot improve the transient oscillating
behavior since the damping ratio of the conjugate-complex pole pair is not
directly affected by the design of $K_{p},K_{i}$.

In order to see the stabilizing properties of the time-delay-based compensator
(7), the $G(s)$ and $D(s)$ must be substituted in (10) by $H(s)$ and
$D(s)+R(s)N(s)$, respectively. Then, the phase characteristics and, thereupon
based, stability margins are visible when comparing
$\angle\,\bigl{(}D(j\omega)\bigr{)}^{-1}$ and
$\angle\,\bigl{(}D(j\omega)+R(j\omega)N(j\omega)\bigr{)}^{-1}$, see Figure 5.

Figure 5: Exemplary phase characteristics of $D^{-1}$ and $(D+RN)^{-1}$.

One can recognize that the lifted and reshaped phase response of the loop
transfer function with $H(j\omega)$ does not cross $\pm 180$ deg phase limits
over the whole frequency range. This allows for larger variations of the loop
gains $K_{p}$ and $k$, in addition to the resonance peak cancelation and,
thus, improvement of GM criterion, cf. section 3.

Combining the outer PI control $C(s)$ and the time-delay-based compensation
$R(s)$ results in the overall two-degrees-of-freedom feedback controller

$u(s)=C(s)r(s)+\bigl{(}R(s)-C(s)\bigr{)}x(s).$ (11)

Figure 6: Block diagram of the control loop with feedback controller $C(s)$
and stabilizing compensator $R(s)$

This is purposefully denoted to have two degrees-of-freedom since the
compensator $R(s)$ is designed fully independently of $C(s)$. This is
equivalent to the outer control of the resonance compensated system
$H(s)=x(s)/v(s)$, where the new (virtual) input is $v(s)=u(s)-R(s)y(s)$. The
structure of the overall two-degrees-of-freedom control loop with (11) is
visualized in the block diagram in Figure 6, for convenience of the reader.

##  5 Experimental example

In the following, a series of control experiments performed on the laboratory
system [11], [15] is provided for evaluating the time-delay-based compensator
in accord with sections 3 and 4. The fourth-order system plant is the two-mass
oscillator with non-collocated contactless sensing of the load position and
actuation by the voice-coil-motor, see Figure 7.

Figure 7: Two-mass oscillator system with non-collocated contactless sensing
and voice-coil-motor actuation (laboratory view).

The available actuator displacement is bounded by $0\leq y\leq 0.02$ m. The
well-balanced free hanging load (with one vertical degree of freedom) is
subject to the oscillations with an extremely low structural damping of the
connecting spring, cf. [11] and Figure 9 (a). The input control voltage $U(t)$
of the voice-coil-motor and the relative position $X(t)=x(t)+x_{0}$ of the
hanging load are real-time available with the set sampling rate of 10 kHz. The
steady-state elongation offset $x_{0}$ is due to the gravity force when
$U(t)=0$. The nominal values of the system parameters, partially identified
and partially taken over from the technical data sheets, are listed in Table
1, while for more details on the system model we refer to [16]. Since the
gravity force of both moving masses $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ is
known and does not change over the operation range, it is pre-compensated,
thus resulting in

$U(t)=u(t)+u_{g}=u(t)+\frac{\mathrm{R}\mathrm{g}}{\Psi}\Bigl{(}\mathrm{m}_{1}+\mathrm{m}_{2}\Bigr{)}.$
(12)

Here $u(t)$ is the applied feedback control law (11). Further we note that the
electromagnetic dynamics of the voice-coil-motor is reasonably neglected, so
that the coupling factor between the input (terminal) voltage and the produced
electro-magnetic force (EMF) is $\Psi\mathrm{R}^{-1}$.

Table 1: Nominal values of the system parameters. parameter |  unit |  value |  meaning
---|---|---|---
$\mathrm{m}_{1}$ |  kg |  0.6 |  actuator mass
$\mathrm{m}_{2}$ |  kg |  0.75 |  load mass
$\mathrm{k}$ |  N/m |  200 |  spring constant
$\sigma$ |  kg/s |  200 |  actuator damping
$\delta$ |  kg/s |  0.01 |  spring damping
$\mathrm{R}$ |  V/A |  5.23 |  coil resistance
$\Psi$ |  Vs/m |  17.16 |  EMF constant
$\mathrm{g}$ |  m/s2 |  9.81 |  gravity constant

For the vector of the state variables $z\equiv(\dot{y},y,\dot{x},x)^{\top}$,
the corresponding state-space model is

$\dot{z}=\underset{A}{\underbrace{\left(\begin{array}[]{cccc}-333.4&-333.3&0.033&333.3\\\
1&0&0&0\\\ 0.027&266.7&-0.027&-266.7\\\ 0&0&1&0\end{array}\right)}}z+B\,u,$
(13)

with the coupling vectors $B=(5.47,0,0,0)^{\top}$ and $F=(0,0,0,1)$ of the
input and output, respectively. The natural frequency corresponding to the
oscillations of the hanging load is at $\omega_{0}=16.3$ rad/sec. The
parameters of the feedback control $u(t)$, given by (11), are $K_{p}=100$,
$K_{i}=150$, $K_{d}=100$, $\tau=0.1923$. The corresponding Bode diagrams of
the plant $G(j\omega)$ and the plant extended by the time-delayed feedback
$H(j\omega)$, both connected in series with the PI-feedback controller
$C(j\omega)$, are shown in Figure 8.

Figure 8: Bode diagrams of the loop transfer functions $CG$ and $CH$.

While the phase margin of the $C(s)G(s)$ loop transfer function is relatively
moderate, being $\mathrm{PM}_{CG}=52.3$ deg, the gain margin is not available,
i.e. being already negative $\mathrm{GM}_{CG}=-4$ dB. That means an unstable
closed-loop behavior of the $C(s)G(s)$ loop system. On the contrary, the
$C(s)H(s)$ loop transfer function reveals a sufficiently large phase margin,
as $\mathrm{PM}_{CH}=112$ deg, and a theoretically infinite gain margin since
$\angle CH$ is not crossing the $\pm 180$ deg phase limits, see Figure 8.

The open-loop response of the experimental system to a short rectangular pulse
is shown in Figure 9 (a) and visualizes the low damping properties of the two-
mass oscillator.

Figure 9: Measured load position: oscillatory open-loop response to a short
square-shape pulse excitation in (a), controlled step response with PI
feedback only in (b), controlled step response with PI feedback extended by
the time-delay compensator (11) in (c).

The unstable step response of the PI-feedback controlled system (i.e. when
setting $K_{d}=0$) is shown in Figure 9 (b). On the contrary, the step
response of the time-delay-stabilized PI-feedback control system (i.e. when
allowing for $K_{d}=100$) is shown in Figure 9 (c). Note that here, an
additional external manual disturbance was injected at the time about
$t\approx 16.5$ sec, for additional assessment of robustness of the proposed
time-delay-based compensator.

##  6 Summary and Discussion

In this paper, we presented a new time-delay-based control method which allows
for a robust compensation of resonance oscillations in non-collocated fourth-
order dynamic systems. The compensator has only two parameters, the natural
frequency and an adjustable gain factor, the value of which is determined
based on the knowledge of resonance peak magnitude of the system transfer
characteristics. The analysis of the time-delay-based approach and associated
loop shaping are made in frequency domain, also using the stability margins as
classical criteria for robustness and performance of an output feedback loop.

It is fair to notice that the proposed control (1), (5) has the induced gain
properties which are similar (in magnitude response) to those of the lead
transfer element

$W(s)=\frac{2K_{d}s}{\tau s+1}.$

The main advantage of the proposed method, however, is that neither time
derivatives of the noisy output nor implementation of any transfer functions,
like e.g. $W(s)$, are required for applying (1), (5). Also the analytic form
(1) of the control law in time domain can allow for adaptation and on-line
adjustment of the natural frequency parameter, which speaks for different real
applications.

If a non-collocated output feedback system (3) uses a PI control to guarantee
the steady-state performance, i.e. $x(t)-r(t)\rightarrow 0$, the feedback
control loop yields inherently unstable if not compensating for the resonance
peak. The proposed time-delay-based loop shaping suppresses robustly the
resonance peak without much affecting the loop transfer characteristics at
other frequencies. It is worth noting that the proposed compensator relies on
the knowledge of $\omega_{0}$, cf. (5), (9). A robust estimation of
$\omega_{0}$, as shown in [15], is however available for tuning the required
parameters. It can also be noted that despite uncertainties of $\omega_{0}$,
which are due to effective stiffness in the operational point of the
oscillator, the compensator performs robustly, as confirmed by the
experimental evaluation. A detailed sensitivity analysis of parameter tuning
is beyond the scope of this work and is the subject of future research.

The proposed control method was evaluated experimentally on two-mass
oscillator system with non-collocated contactless sensing and voice-coil-motor
actuation. It was shown, cf. Figure 9, that an extremely low-damped
oscillatory behavior is effectively compensated by the proposed time-delay-
based method, thus allowing also for a standard PI outer feedback control
loop.

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