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 (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)

2. First-order

Hold

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,

(1)

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,

(2)

where is the triangular function defined as,

(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. 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,

(4)

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).

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(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,

(5)

where xn is the

discrete samples, is

the rectangular function as follows,

(6)

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.

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 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,

(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 .

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).

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, ,

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)

(8)

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

follows:

(9)

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

(10)

where is the state vector describing the error, and is

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 (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.

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 steady-state 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 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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