Improving the Closed-Loop Tracking Performance Using the FirstOrder Hold Sensing Technique with Experiments
Chifu Yang1, Shuang Gao2, Zhu Xue3
1
2

Harbin Institute of Technology, Harbin, 150006, China
McGill University, Montreal, Quebec, H3A 0E9, Canada
3
National University of Singapore, 117576, Singapore

Abstract
This 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
1. Introduction
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 [1], 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 [2-5].
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 [6] 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 [7-9]. 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 [10, 11]. 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 [12-14]. The paper [15] 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 [16-19].
2. First-order Hold
The motivation in this paper is on the better estimation of the analog signal based on
the digital signal 𝜃"/$ 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) [20]. As mentioned previously, the FOH signal is a reconstructed piecewise linear
approximation to the original sampled signal (see Figure 1).

Figure 1 [21]. Ideally sampled signal and corresponding piecewise linear FOH signal.

The ideally sampled signal could be represented as,
𝑥' (𝑡) = 𝑇 0
(1)
1230 𝑥(𝑛𝑇)𝛿(𝑡 − 𝑛𝑇)
where 𝑥(𝑡) is the original signal, 𝑥' (𝑡) is the ideal signal, 𝛿(𝑡) is the Dirac impulse
function. Since a sequence of Dirac impulses, representing the discrete samples, is lowpass 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,
931:
𝑥456 (𝑡) = 0
(2)
1230 𝑥(𝑛𝑇)𝑡𝑟𝑖( : )
where 𝑡𝑟𝑖 is the triangular function defined as,
𝑡𝑟𝑖 𝑡 = max 1 − 𝑡 , 0
(3)
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 [21]. Delayed piecewise linear FOH signal

The delayed first-order hold, also known as causal first-order hold, as shown in Figure 2
can be represented as,
93:31:
𝑥456 (𝑡) = 0
)
(4)
1230 𝑥(𝑛𝑇)𝑡𝑟𝑖(
:
The delayed output renders the system a causal system [22-25]. The corresponding
delayed piecewise linear reconstruction is physically realizable with the assistance of a
digital filter[26-28].
2.1 First-order 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 [29-32]. The
mathematical model of the ZOH is written as,
931:
F
𝑥A56 (𝑡) = 0
(5)
1230 𝑥[𝑛]𝑟𝑒𝑐𝑡( : − G)
where x[n] is the discrete samples, 𝑟𝑒𝑐𝑡()is the rectangular function as follows,
0,
𝑖𝑓 𝑡 > 1/2
𝑟𝑒𝑐𝑡 𝑡 =

F
G

,

𝑖𝑓 𝑡 = 1/2

(6)

1,
𝑖𝑓 𝑡 < 1/2
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 𝜋/2𝑇) signals, although the FOH has larger
amplitude distortion than the ZOH does, the FOH has significantly less phase lag than the

ZOH does [20]. 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.
3. 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 under-actuated pendulum)
as shown in Figure 4.

Figure 4. Experimental setup
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,
𝜃M = 0.1 sin 4𝜋𝑡 𝑟𝑎𝑑
(7)
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 T = 0.001s. The desired frequency 2Hz is well
below π/2T 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 -6×103Y and -6×103Z 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 [1].
A 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 angle, θ, from its

equilibrium position [33, 34]. 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 [35, 36]
x = Ax + BVa
(8)
0
1
0
KgKh
θ
K
b+
h
x=
,
A=
B=
mgl
Ra ,
θ
−
−
RaJ
2J
J
where Va 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.
Table 1. Parameters of the experimental setup
Parameter
m (pendulum mass)
l (pendulum length)
g (gravitational constant)
J (rotational inertia)
Ra (armature resistance)
Kb (back-emf constant)
Ki (torque constant)
b (torsional damping coefficient)

Value
0.125
0.33
9.807
0.0042
3.4
0.0592
0.0592
0.0045

Unit
kg
m
m/s2
kgm2
Ω
Vs/rad
Nm/A
Nms/rad

In order for the pendulum to follow the desired trajectory, a control structure shown in
Figure 5 is implemented with the highlighted FOH sensing block

Figure 5. Block Diagram of the System Using the FOH

The feed-forward logic in the control is calculated as follows:
xj = Axj + BVj

(9)

K K
b+ g h
RaJ
mgl
Ra
Vj =
θj +
θj +
θ
Kh
J
2J j
θj
is the desired trajectory and Vj is the feed-forward control voltage. An
θj
important point to mention is that the amplitude of θj should be kept small in order to
maintain the linearity in
(8).
where xj =

Subtracting (8) from (9) gives the error dynamics as
e = Ae + B Vj − Va = Ae + Bu
(10)
θj − θ
where e =
is the state vector describing the error, and u = Vj − Va = −Ke is
θj − θ
the full state feedback control law.

Figure 6. ZOH and FOH output signals
Figure 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 [37-40]. To estimate the analog signal between two quantized
values, a first-order-hold (FOH) equivalent is applied to the ZOH signal θn/o . 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 θn/o
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.

Figure 7. Comparison of various outputs with sinusoidal inputs.
Figure 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
Outputs

Peak to peak error

ZOH

1.9757%

FOH

1.6986%

ZOH+Filter

1.0135%

FOH+Filter

0.8662%

From 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
Outputs

Closed-loop peak to peak error

ZOH

2.0286%

FOH

1.5693%

ZOH+Filter

1.0755%

FOH+Filter

0.9303%

The 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
Outputs

Closed-loop peak to peak error

ZOH

3.3404%

FOH

2.7345%

ZOH+Filter

2.2293%

FOH+Filter

2.1790%

Finally, 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 steadystate error, as depicted in Figure 8.

Figure 8. DFT of the closed-loop error using ZOH and FOH.
The 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 𝐸𝑟𝑟𝑜𝑟A56 (𝜔𝑖) /
𝑚𝑎𝑥 𝐸𝑟𝑟𝑜𝑟A56 (𝜔𝑖) 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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