paper proposes a new perspective in the enhancement of the closed-loop tracking
performance by using the first-order hold (FOH) sensing technique. Firstly, the
literature review and fundamentals of the FOH are outlined. Secondly, the
performance of the most commonly used zero-order hold (ZOH) and that of the FOH
are compared. Lastly, the detailed implementation of the FOH on a pendulum
tracking setup is presented to verify the superiority of the FOH over the ZOH
in terms of the steady state tracking error. The results of the simulation and
the experiment are in agreement.
Keywords: First-order hold, Zero-order hold, Tracking performance,
Sensing technique, Pendulum experiment
The first-order-hold (FOH) method is a
mathematical model to reconstruct the sampled signals that could be done by a
conventional digital-to-analog converter (DAC) and an analog circuit which is
called an integrator. The FOH signal is reconfigured as a piecewise linear
approximation of the original sampled signal. In (Darby, Blakeborough,
and Williams 2001), the FOH signal is used for real-time substructure
testing, which is a novel method of testing structures under dynamic loading.
An extrapolation of a first-order-hold discretization is used which increases
the accuracy of the numerical model over more direct explicit methods. The
improvements are demonstrated using a series of substructure tests on a simple
portal frame. Some deep mathematical relationship between ZOH and FOH is
revealed in (Weller, et al. 2001;
Fulai Yao 2017; Gao, Dong, and Jia 2017; Gao, et al. 2017). It is shown that the zeros of sampled-data systems resulting from
rapid sampling of continuous-time systems preceded by a ZOH are the roots of
the Euler-Frobenius polynomials. The simplicity, negative realness, and
interlacing properties of the sampling zeros of ZOH and FOH sampled systems are
proven for the first time in literature. The paper (Lozano, Rosell, and
Pallas-Areny 1992) deals with the quality requirements of the
synthesized sine waves reconstructed through a ZOH and FOH for testing
purposes, especially when a switching demodulator is used. Results show that a
FOH implies a decrease of total harmonic amplitude distortion, but the measured
spurious harmonics are kept lower or equal when using a ZOH in the 15 closest
components. It is concluded in the paper that in testing applications a ZOH
yields better results, thus the benefits of using a FOH need further
investigation. The effects of various sensing techniques on the performance of
the motion canceling bilateral control (MCBC) are studied (Nakajima, et al. 2012;
Gao, Cepeda-Gomez, and Olgac 2012; Gao, Zalluhoglu, and Olgac 2013). MCBC is a method to synchronize motion of a teleoperation robot and a
target, while an operator can obtain tactile sensation of the remote target (Gao, Cepeda-Gomez, and
Olgac 2014; Gao, Zalluhoglu, and Olgac 2014). Results show that a FOH yields better performance
compared with a ZOH, but it has a peak gain near the Nyquist frequency.
Therefore, in order to make full use of FOH, additional techniques are needed
in order to eliminate the adverse effects of the FOH. The mathematical
structure of new discretization schemes are proposed and characterized as
useful methods of establishing concrete connections between numerical and
system theoretical properties (Zheng and Kil To 2007;
Gao, et al. 2014a; Gao, et al. 2014b). The paper (Hagiwara 1995) deals with the necessary and sufficient condition for
the reachability of the sampled-data system obtained by the discretization of a
linear time-invariant continuous-time system with a FOH. The equivalence of the
reachability and controllability of the system is shown and similar results are
given for observability and re-constructability.(Gao, et al. 2015a;
Gao, et al. 2015b; Gao and Olgac 2015a; Gao and Olgac 2015b)
The motivation in this paper is on the
better estimation of the analog signal based on the
digital signal read from the analog-to-digital
converter (ADC). As a result, the
closed-loop tracking performance could be improved. For this, the
first-order-hold (FOH) is a better method to approximate the continuous analog
signal than the zero-order hold (ZOH) (Franklin 1994) pg.154.
As mentioned previously, the FOH signal is a reconstructed piecewise linear
approximation to the original sampled signal (see Figure 1).
Figure 1. Ideally sampled signal and corresponding
piecewise linear FOH signal
The ideally sampled
signal could be represented as,
where is the original signal, is the ideal signal, is the Dirac impulse function. Since a
sequence of Dirac impulses, representing the discrete samples, is low-pass
filtered, the mathematical model for FOH is necessary. The impracticality of
outputting a sequence of Dirac impulses foster the development of devices that
use a conventional DAC and some linear analog circuitry, to reconstruct the
piecewise linear output for the FOH signals. The commonly used analytical
piecewise linear approximation is written as,
where is the triangular function defined as,
However, the system represented in (2) is
not achievable in realty. In fact, the typical FOH model used in practice is
the delayed first-order hold, which is identical to the FOH except for the fact
that its output is delayed by one sample period, resulting in a delayed
piecewise linear output signal (see Figure 2).
Figure 2. Delayed piecewise linear FOH signal
first-order hold, also known as causal first-order hold, as shown in Figure 2
can be represented as,
The delayed output renders the system a
causal system (Gao 2015; Gao and
Olgac 2016a; Gao and Olgac 2016b; Gao and Olgac 2016c). The corresponding delayed piecewise linear reconstruction is
physically realizable with the assistance of a digital filter(Gao, Zhang, and Yang
2017; Gao and Olgac 2017; Schmid, Gao, and Olgac 2015).
Hold VS Zero-order Hold
The zero-order hold (ZOH) is a mathematical
representation of the practical signal reconstruction done be a conventional
digital-to-analog converted (DAC). It works in a way that the signal is held at
each sample value for each and every sample interval while converting a
discrete-time signal to a continuous-time signal. The most commonly used
sensing feature in practice is the ZOH due to its ease of implementation(F Yao and Q Gao 2017;
Fulai Yao and Qingbin Gao 2017; Zhang, Olgac, and Gao 2017; Zhang and Gao 2017). The mathematical model of the ZOH is written as,
where xn is the
discrete samples, is
the rectangular function as follows,
Next, the properties
of the FOH and the ZOH are compared as shown in Figure 3,
Figure 3. Magnitude and phase of ZOH and FOH filters
From Figure 3, for
low frequency (below )
signals, although the FOH has larger amplitude distortion than the ZOH does,
the FOH has significantly less phase lag than the ZOH does (Franklin 1994) pg.120. This property is utilized in this paper to reduce
the level of steady state tracking error based on the fact that a more precise
sensing signal is being utilized for the feedback. The details of the
implementation are illustrated in the following section.
Implementation of the FOH on an Experimental Setup
The enhanced tracking
performance while using the FOH instead of the ZOH is demonstrated over a
single axis manipulator test platform (an actuated pendulum) as shown in Figure
Figure 4. Experimental
The trajectory tracking
performance of the pendulum is investigated. The desired trajectory is selected
as a sinusoidal motion, for simplicity, at approximately twice the pendulum’s
natural frequency, that is,
which is at 2 Hz frequency (about twice the natural
frequency of the uncontrolled system). The feedback control loop is performed
at 1000 Hz sampling speed corresponding to a sampling period .
The desired frequency 2Hz is well below and thus is a low frequency signal compared
with sampling frequency. From the inset of figure 1, the magnitudes of ZOH and
FOH at the desired frequency are 0.999993 and 1.000065 respectively, both of
which are very close to 1. Thus the magnitude distortions are negligible. On
the other hand, the phases of ZOH and FOH at the desired frequency are -and
– respectively. The FOH has significantly less
phase lag (about 10 to the 4th power less) than the ZOH does. Because of the
excellent phase responses, the FOH discretization has been shown to increase
the accuracy of the numerical model over more direct explicit methods in the
real-time substructure testing (Darby, Blakeborough,
and Williams 2001).
combination of feed-forward and feedback control is implemented on the
pendulum. A DC servo-motor (Minertia Motor, FB5L20E) serves as the actuator
while an optical encoder (with 0.09 resolution) is used to measure the pendulum
from its equilibrium position(Zhang and Olgac 2013a;
Zhang and Olgac 2013b). The control action is performed at 1000 Hz sampling
rate on the pendulum that has a natural frequency of 1.1 Hz.
The linearized state space
representation of the test setup is given as in (Franklin, Powell, and
Emami-Naeini 2006; Zhang, Diaz, and Olgac 2013)
where is the control voltage (motor armature
voltage) and the other parameters are electro-mechanical properties of the
motor-pendulum assembly as listed in Table 1.
In order for the pendulum to follow the desired
trajectory, a control structure shown in Figure 5 is implemented.
The feed-forward logic in the control is calculated as
where is the desired trajectory and is the feed-forward control voltage. An
important point to mention is that the amplitude of should be kept small in order to maintain the
linearity in (1).
Subtracting (8) from (9) gives
the error dynamics as
where is the state vector describing the error, and is
the full state feedback control law.
ZOH and FOH output signals
6 illustrates one actual signal with ZOH sampled signal and delayed FOH signal
and it can be seen that the latter one yields a better approximation of the
actual signal. As mentioned previously, the optical encoder with 4000 pulses
per revolution has a sensor resolution of 0.09 deg (Zhang 2012; Huang, et
al. 2016; Tang, et al. 2016; Babinski, et al. 2016). To estimate the analog signal between two quantized values, a
first-order-hold (FOH) equivalent is applied to the ZOH
signal. The extrapolated signal is
a piecewise linear approximation to the original analog signal that was sampled
as shown in Figure 6. The slope of the previous step of the ZOH signal is used to estimate the output of the current
step and the estimated value is obtained at the beginning of each sampling
period. Since the FOH output is still
not smooth enough (but yields much smaller errors in amplitude which is shown
later), a second order low pass filter could be added to the FOH output.
Comparison of various outputs with sinusoidal inputs.
7 shows various ways of sensing, i.e., ZOH, FOH, ZOH with filter and FOH with
filter. In reality, a low-pass filter is usually used to eliminate high-frequency
noise. The position of the FOH in the whole system is shown in Figure 5 and it
is shown that the FOH is implemented for the signal obtained from the encoder. In
order to compare the performance of the FOH output and ZOH output with/without
the filter, a simulation is made to analyze the performance of the above
methods on the sensing side as shown in Figure 7. The peak to peak errors
between the various outputs and the sinusoidal input signal are obtained as
listed in Table 2.
Table 2. Peak to peak error between the
output of different sensing schemes and the sinusoidal input
Peak to peak
Table 2, the FOH output yields smaller error than the ZOH output does. Also,
adding a filter to the output yields apparent smaller errors than the
corresponding original output. Out of all the listed methods, the filtered FOH
output produces the best approximation to the continuous sinusoidal input.
Based on this analysis, the closed-loop peak to peak errors with respect to
different outputs on the sensing side are obtained on the simulation model
(Figure 5). The highlighted FOH block is modified according to Figure 7 to get
various outputs. The performance of different outputs from the peak to peak
tracking error perspective is shown in the Table 3.
Table 3. Simulation result for the
closed-loop peak to peak tracking error
peak to peak error
agreement between Table 3 and Table 2 shows that better sensing and
reconstruction scheme yield smaller peak to peak tracking error. Finally,
experimental results were done to verify the finding and show that the filtered
FOH equivalence produces the best approximation to the continuous system out of
all the methods examined (Table 4).
Table 4. Experimental result for the
closed-loop peak to peak tracking error
Closed-loop peak to peak
the degree of the reduction of the closed-loop error for a simple trajectory
tracking example is visualized in the discrete Fourier transformation (DFT) of
the steady-state error, as depicted in Figure 8.
Figure 8. DFT of the closed-loop error using ZOH and
scale of the vertical axis is normalized with respect to the maximum magnitude
of the closed-loop error using ZOH sensing scheme, i.e. the ratio of expressed in percent is shown in the figure.
The light line represents the DFT of the steady-state error using the ZOH
sensing scheme. The bold line depicts the DFT of the steady-state error using
the FOH sensing scheme. It is observed that the dominant frequency component of
2 Hz (which is the desired frequency) is suppressed by about 40%, while the
rest of the frequency spectrum remains practically unchanged.
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