Type: Critical Essays
Sample donated: Roberta Stone
Last updated: July 11, 2019
AbstractThispaper proposes a new perspective in the enhancement of the closed-loop trackingperformance by using the first-order hold (FOH) sensing technique. Firstly, theliterature review and fundamentals of the FOH are outlined. Secondly, theperformance of the most commonly used zero-order hold (ZOH) and that of the FOHare compared.
Lastly, the detailed implementation of the FOH on a pendulumtracking setup is presented to verify the superiority of the FOH over the ZOHin terms of the steady state tracking error. The results of the simulation andthe 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 amathematical model to reconstruct the sampled signals that could be done by aconventional digital-to-analog converter (DAC) and an analog circuit which iscalled an integrator.
The FOH signal is reconfigured as a piecewise linearapproximation of the original sampled signal. In (Darby, Blakeborough,and Williams 2001), the FOH signal is used for real-time substructuretesting, which is a novel method of testing structures under dynamic loading.An extrapolation of a first-order-hold discretization is used which increasesthe accuracy of the numerical model over more direct explicit methods. Theimprovements are demonstrated using a series of substructure tests on a simpleportal frame.
Some deep mathematical relationship between ZOH and FOH isrevealed 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 fromrapid sampling of continuous-time systems preceded by a ZOH are the roots ofthe Euler-Frobenius polynomials. The simplicity, negative realness, andinterlacing properties of the sampling zeros of ZOH and FOH sampled systems areproven for the first time in literature. The paper (Lozano, Rosell, andPallas-Areny 1992) deals with the quality requirements of thesynthesized sine waves reconstructed through a ZOH and FOH for testingpurposes, especially when a switching demodulator is used. Results show that aFOH implies a decrease of total harmonic amplitude distortion, but the measuredspurious harmonics are kept lower or equal when using a ZOH in the 15 closestcomponents. It is concluded in the paper that in testing applications a ZOHyields better results, thus the benefits of using a FOH need furtherinvestigation.
The effects of various sensing techniques on the performance ofthe 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 atarget, while an operator can obtain tactile sensation of the remote target (Gao, Cepeda-Gomez, andOlgac 2014; Gao, Zalluhoglu, and Olgac 2014).
Results show that a FOH yields better performancecompared 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 neededin order to eliminate the adverse effects of the FOH. The mathematicalstructure of new discretization schemes are proposed and characterized asuseful methods of establishing concrete connections between numerical andsystem 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 forthe reachability of the sampled-data system obtained by the discretization of alinear time-invariant continuous-time system with a FOH. The equivalence of thereachability and controllability of the system is shown and similar results aregiven for observability and re-constructability.(Gao, et al.
2015a;Gao, et al. 2015b; Gao and Olgac 2015a; Gao and Olgac 2015b) 2. First-orderHold The motivation in this paper is on thebetter estimation of the analog signal based on thedigital signal read from the analog-to-digitalconverter (ADC). As a result, theclosed-loop tracking performance could be improved.
For this, thefirst-order-hold (FOH) is a better method to approximate the continuous analogsignal than the zero-order hold (ZOH) (Franklin 1994) pg.154.As mentioned previously, the FOH signal is a reconstructed piecewise linearapproximation to the original sampled signal (see Figure 1). Figure 1. Ideally sampled signal and correspondingpiecewise linear FOH signal The ideally sampledsignal could be represented as, (1)where is the original signal, is the ideal signal, is the Dirac impulse function. Since asequence of Dirac impulses, representing the discrete samples, is low-passfiltered, the mathematical model for FOH is necessary. The impracticality ofoutputting a sequence of Dirac impulses foster the development of devices thatuse a conventional DAC and some linear analog circuitry, to reconstruct thepiecewise linear output for the FOH signals.
The commonly used analyticalpiecewise linear approximation is written as, (2)where is the triangular function defined as, (3) However, the system represented in (2) isnot achievable in realty. In fact, the typical FOH model used in practice isthe delayed first-order hold, which is identical to the FOH except for the factthat its output is delayed by one sample period, resulting in a delayedpiecewise linear output signal (see Figure 2). Figure 2. Delayed piecewise linear FOH signal The delayedfirst-order hold, also known as causal first-order hold, as shown in Figure 2can be represented as, (4) The delayed output renders the system acausal system (Gao 2015; Gao andOlgac 2016a; Gao and Olgac 2016b; Gao and Olgac 2016c).
The corresponding delayed piecewise linear reconstruction isphysically realizable with the assistance of a digital filter(Gao, Zhang, and Yang2017; Gao and Olgac 2017; Schmid, Gao, and Olgac 2015). 2.1 First-orderHold VS Zero-order Hold The zero-order hold (ZOH) is a mathematicalrepresentation of the practical signal reconstruction done be a conventionaldigital-to-analog converted (DAC). It works in a way that the signal is held ateach sample value for each and every sample interval while converting adiscrete-time signal to a continuous-time signal.
The most commonly usedsensing 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 thediscrete samples, isthe rectangular function as follows, (6)Next, the propertiesof 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, forlow 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 reducethe level of steady state tracking error based on the fact that a more precisesensing signal is being utilized for the feedback. The details of theimplementation are illustrated in the following section. 3.Implementation of the FOH on an Experimental SetupThe enhanced trackingperformance while using the FOH instead of the ZOH is demonstrated over asingle axis manipulator test platform (an actuated pendulum) as shown in Figure4. Figure 4.
ExperimentalsetupThe trajectory trackingperformance of the pendulum is investigated. The desired trajectory is selectedas a sinusoidal motion, for simplicity, at approximately twice the pendulum’snatural frequency, that is, (7)which is at 2 Hz frequency (about twice the naturalfrequency of the uncontrolled system). The feedback control loop is performedat 1000 Hz sampling speed corresponding to a sampling period .The desired frequency 2Hz is well below and thus is a low frequency signal comparedwith sampling frequency. From the inset of figure 1, the magnitudes of ZOH andFOH at the desired frequency are 0.999993 and 1.000065 respectively, both ofwhich are very close to 1. Thus the magnitude distortions are negligible.
Onthe other hand, the phases of ZOH and FOH at the desired frequency are -and- respectively. The FOH has significantly lessphase lag (about 10 to the 4th power less) than the ZOH does. Because of theexcellent phase responses, the FOH discretization has been shown to increasethe accuracy of the numerical model over more direct explicit methods in thereal-time substructure testing (Darby, Blakeborough,and Williams 2001). Acombination of feed-forward and feedback control is implemented on thependulum. A DC servo-motor (Minertia Motor, FB5L20E) serves as the actuatorwhile an optical encoder (with 0.09 resolution) is used to measure the pendulumangle, ,from its equilibrium position(Zhang and Olgac 2013a;Zhang and Olgac 2013b). The control action is performed at 1000 Hz samplingrate on the pendulum that has a natural frequency of 1.
1 Hz. The linearized state spacerepresentation of the test setup is given as in (Franklin, Powell, andEmami-Naeini 2006; Zhang, Diaz, and Olgac 2013) (8)where is the control voltage (motor armaturevoltage) and the other parameters are electro-mechanical properties of themotor-pendulum assembly as listed in Table 1.In order for the pendulum to follow the desiredtrajectory, a control structure shown in Figure 5 is implemented. The feed-forward logic in the control is calculated asfollows: (9)where is the desired trajectory and is the feed-forward control voltage. Animportant point to mention is that the amplitude of should be kept small in order to maintain thelinearity in (1). Subtracting (8) from (9) givesthe error dynamics as (10)where is the state vector describing the error, and isthe full state feedback control law.
Figure 6.ZOH and FOH output signals Figure6 illustrates one actual signal with ZOH sampled signal and delayed FOH signaland it can be seen that the latter one yields a better approximation of theactual signal. As mentioned previously, the optical encoder with 4000 pulsesper revolution has a sensor resolution of 0.09 deg (Zhang 2012; Huang, etal.
2016; Tang, et al. 2016; Babinski, et al. 2016). To estimate the analog signal between two quantized values, afirst-order-hold (FOH) equivalent is applied to the ZOHsignal. The extrapolated signal isa piecewise linear approximation to the original analog signal that was sampledas shown in Figure 6. The slope of the previous step of the ZOH signal is used to estimate the output of the currentstep and the estimated value is obtained at the beginning of each samplingperiod.
Since the FOH output is stillnot smooth enough (but yields much smaller errors in amplitude which is shownlater), a second order low pass filter could be added to the FOH output. Figure7. Comparison of various outputs with sinusoidal inputs.
Figure7 shows various ways of sensing, i.e., ZOH, FOH, ZOH with filter and FOH withfilter. In reality, a low-pass filter is usually used to eliminate high-frequencynoise.
The position of the FOH in the whole system is shown in Figure 5 and itis shown that the FOH is implemented for the signal obtained from the encoder. Inorder to compare the performance of the FOH output and ZOH output with/withoutthe filter, a simulation is made to analyze the performance of the abovemethods on the sensing side as shown in Figure 7. The peak to peak errorsbetween the various outputs and the sinusoidal input signal are obtained aslisted in Table 2. Table 2. Peak to peak error between theoutput 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% FromTable 2, the FOH output yields smaller error than the ZOH output does. Also,adding a filter to the output yields apparent smaller errors than thecorresponding original output. Out of all the listed methods, the filtered FOHoutput produces the best approximation to the continuous sinusoidal input.
Based on this analysis, the closed-loop peak to peak errors with respect todifferent outputs on the sensing side are obtained on the simulation model(Figure 5). The highlighted FOH block is modified according to Figure 7 to getvarious outputs. The performance of different outputs from the peak to peaktracking error perspective is shown in the Table 3. Table 3. Simulation result for theclosed-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% Theagreement between Table 3 and Table 2 shows that better sensing andreconstruction scheme yield smaller peak to peak tracking error. Finally,experimental results were done to verify the finding and show that the filteredFOH equivalence produces the best approximation to the continuous system out ofall the methods examined (Table 4). Table 4. Experimental result for theclosed-loop peak to peak tracking error Outputs Closed-loop peak to peak error ZOH 3.3404% FOH 2.
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