Control Design for Relative Stability in a PWM-Controlled Pneumatic System

July 3, 2017 | Autor: Eric Barth | Categoria: Engineering, Control Design, Dynamic systems, Dynamic Systems
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boom sinks slightly due to gravity as the boom weight is compensated by the build up of pressure in the lower part of the cylinder. The boom motion excites the chassis which in turn excites the cab. The relative motion between chassis and cab makes the valve move due to the linkage kinematics. There is a small deadzone where valve motion does not create any flow area. Once this deadzone is exceeded, some valve flow area exists and hydraulic flow can commence. As can be seen in Figs. 6 – 8, with the conventional cab mounts, the system is unstable. Figure 6 shows the valve motion nondimensionalized by the maximum valve opening. The valve motion quickly builds to the maximum and bangs continuously between maximum positive and negative opening. Figure 7 shows the unstable angular motion of the boom. This has the appearance of what was observed on the actual backhoe. Figure 8 shows the corresponding chassis angular motion. It is bouncing around at plus–minus 4°. The mount stiffnesses at the front and rear of the cab were doubled from their nominal values and the simulation was run again. The results are also shown in Figs. 6 – 8. As can be seen, the entire system is stable and well behaved. This also corresponds to the trend observed on the actual backhoe. Although the instability in the backhoe can be resolved by using stiffer cab mounts, this will degrade the ride motion at the idle speed. Another way to stabilize the backhoe is with automatic control. Here the response of the backhoe system with a feedback control is demonstrated. It is assumed that a displacementtype actuator is installed in the valve control rod as shown in Fig. 9. Here the actuator is proposed to be a DC motor with a ballscrewused to convert rotary to linear motion. For controller design purposes, this actuator is represented by the flow source, S f ⫺, shown in the bond graph fragment of Fig. 9. Once the desired boom position is set by the operator, the error between the desired valve position and actual valve position is fed back to the control unit and the flow source is activated to reduce the error. Here it was found the Proportional plus Integral control worked fine. As shown in Figs. 6 – 8, the entire system is stable and the responses are similar to the stable passive system with the stiffer mounts. While it is not possible to publish the measured motion-time histories of the system response, the model presented here is qualitatively accurate and was very useful in the design of a ‘‘fix’’ for the instability of the vehicle. The model allowed testing of many different approaches to solving the stability problem without requiring hardware realizations for each.

Conclusions A complete pitch/plane model of a backhoe was developed that includes the hydraulic dynamics and kinematics of the control linkage. The model was developed in pieces using bond graph fragments, and the overall model was assembled by straightforwardly assembling the bond graph fragments. Equations were derived directly from the bond graph and programmed for simulation using a digital computer. Simulations were run for an initial condition response from near equilibrium. The model predicts the instability observed on the actual backhoe, and is now ready to be used as a design tool for future backhoe development. It was shown that the backhoe can be stabilized passively by using stiffer mounts between chassis and cab. This solution will cause an increase in cab acceleration during engine idle tests. An automatic control solution was also demonstrated. This consisted of an actuator in series with the valve control rod. By measuring and feeding back valve position, the controller was demonstrated to stabilize the system without requiring stiffer cab mounts.

References 关1兴 Vaha, P. K., and Skibniewski, M. J., 1993, ‘‘Dynamic Model of Excavator,’’ J. Aerospace Eng., 6共2兲, pp. 148 –158.

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关2兴 Koivo, A. J., Ramos, M. C., and Thoma, M., 1994, ‘‘Dynamic Model for Excavators 共and Backhoe兲,’’ IFAC Symposium on Robot Control, pp. 763– 768. 关3兴 Koivo, A. J., 1994, ‘‘Kinematics of Excavators 共Backhoes兲 for Transferring Surface Material,’’ J. Aerospace Eng., 7共1兲, pp. 17–32. 关4兴 Koivo, A. J., Thoma, M. C., Kocaoglan, E., and Andrade-Cetto, J., 1996, ‘‘Modeling and Control of Excavator Dynamics During Digging Operation,’’ J. Aerospace Eng., 9共1兲, pp. 10–18. 关5兴 Vaha, P. K., and Skibniewski, M. J., 1993, ‘‘Cognitive Force Control of Excavators,’’ J. Aerospace Eng., 6共2兲, pp. 159–166. 关6兴 Koivo, A. J., and Song, B., 1998, ‘‘Neural Adaptive Control of Excavators,’’ IEEE Intl. Conf. Of Robotics and Automation, 1, 8120817. 关7兴 Plonecki, L., Trampczynski, W., and Cendrowicz, J., 1998, ‘‘A Concept of Digital Control System to Assist the Operator of Hydraulic Excavators,’’ Autom.Constr., 7, pp. 401– 411. 关8兴 Nguyen, Q. H., Ha, Q. P., Rye, D. C., and Durrant-Whyte, H. F., 2000, ‘‘Force/ Position Tracking for Electrohydraulic Systems of a Robotic Excavator,’’ IEEE Conf. On Decision and Control, Sydney, Australia, pp. 5224 –5229. 关9兴 Haga, M., Hiroshi, W., and Fujishima, K., 2001, ‘‘Digging Control System for Hydraulic Excavator,’’ Mechatronics, 11, pp. 665– 676. 关10兴 Chang, P. H., and Lee, S., 2002, ‘‘A Straight-Line Motion Tracking Control of Hydraulic Excavator System,’’ Mechatronics, 12, pp. 119–138. 关11兴 Arai, F., Tateishi, J., and Fukuda, T., 2000, ‘‘Dynamical Analysis and Suppression of Human Hunting in the Excavator Operation,’’ Proc. of the 2000 IEEE Intl. Workshop on Robot and Human interactive Commu. Osaka, Japan, pp. 394 –399. 关12兴 Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., System Dynamics: Modeling and Simulation of Mechatronic Systems, 3rd ed., John Wiley & Sons, NY, 2000. 关13兴 Margolis, D., and Hennings, C., 1994, ‘‘Stability of Hydraulic Motion Control Systems,’’ ASME FPST, 1, pp. 65–74.

Control Design for Relative Stability in a PWM-Controlled Pneumatic System Eric J. Barth, Jianlong Zhang, and Michael Goldfarb Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235

This paper presents a control design methodology that provides a prescribed degree of stability robustness for plants characterized by discontinuous (i.e., switching) dynamics. The proposed control methodology transforms a discontinuous switching model into a linear continuous equivalent model, so that loop-shaping methods may be utilized to provide a prescribed degree of stability robustness. The approach is specifically targeted at pneumatically actuated servo systems that are controlled by solenoid valves and do not incorporate pressure sensors. Experimental demonstration of the approach validates model equivalence and demonstrates good tracking performance. 关DOI: 10.1115/1.1591810兴

Introduction Many papers that treat the control of pneumatic systems offer some introductory claim about the low cost of pneumatic actuation. Such a claim is in part justified, since a typical pneumatic actuator costs on the order of tens of U.S. dollars. The claim of low cost, however, is also somewhat misleading, since the components required for the servo control of a pneumatic actuator, specifically the proportional valve and pressure sensors, have a combined cost typically an order of magnitude greater than the actuator itself. Specifically, servo control of pneumatic systems Contributed by the Dynamic Systems, Measurement, and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division June 17, 2002; final revision, December 27, 2002. Associate Editor: N. Manring.

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typically requires some type of proportionally controllable valve, which enables control of cylinder pressures via proportional constriction of the pneumatic flow 共i.e., throttling兲. Most frequently, these valves are a proportional servovalve type, which is a 4-way spool valve with a position-controllable spool. Control approaches for pneumatic servo systems that utilize proportional servovalves are described in papers by Liu and Bobrow 关1兴, Bobrow and Jabbari 关2兴, Bobrow and McDonell 关3兴, and Drakunov et al. 关4兴, and Kimura et al. 关5兴, among others. Other works treating the servo control of pneumatic actuators incorporate 共noncommerciallyavailable兲 alternative configurations of proportional valves, such as jet pipes valves 共Jacobsen et al. 关6兴, Henri et al. 关7兴兲 or flapper valves 共Ben-Dov and Salcudean 关8兴兲. Regardless of the valve type, the proportional valve required to implement a proportional control approach is one of the most expensive components of a pneumatic servo system, typically costing on the order of several hundred U.S. dollars. In addition to a proportional servovalve, pressure sensors are typically incorporated in pneumatic servo controllers to measure the pressure states for feedback control. Due to the compressibility of air, the pressures on each side of a pneumatic cylinder constitute two states of the basic dynamic model of a pneumatic servo system. Though pressure feedback is utilized for various purposes, the most fundamental reason requiring its use in servo control is to distinguish between the choked and unchoked flow conditions that result from compressible flow through the control valve. Such conditions are unobservable from measurement of motion and/or force at the output of the actuator. This paper presents a control methodology that enables servo control of a pneumatic actuator without a proportional valve or pressure sensors. Specifically, the methodology provides flow control via binary-position solenoid valves rather than proportional valves, and concomitantly enables a prescribed degree of stability robustness in order to adequately compensate for the loss of information that results from the absence of pressure sensors. Since the cost of solenoid valves is on the order of ten U.S. dollars, the use of solenoid instead of proportional valves, along with the elimination of pressure sensors, can reduce the cost of pneumatic servo control implementation by an order of magnitude and can therefore provide high-power actuation at a significantly lower cost relative to a power-comparable DC motor-actuated system. Several prior works have demonstrated the viability of servocontrol of pneumatic actuators via solenoid on–off valves, including the work of Ye et al. 关9兴, Kunt and Singh 关10兴, Lai et al. 关11兴, Royston and Singh 关12兴, Paul et al. 关13兴, Norigitsu 关14 –16兴, Shih and Hwang 关17兴, and van Varseveld and Bone 关18兴. None of these prior works, however, enable a prescribed degree of stability robustness. As previously mentioned, such stability robustness is particularly important in the absence of pressure sensors, since the controller cannot otherwise distinguish between choked and unchoked flow through the valve, and thus the robustness of the control approach must adequately and reliably compensate for the loss of pressure information. This paper provides a method to transform the nonanalytic, nonlinear description of pulse-widthmodulated 共PWM兲 based control of a pneumatic system into an analytic, linear model, which in turn enables the application of frequency domain loop shaping to address issues of performance and stability robustness. The switching aspect enables use of solenoid 共rather than proportional兲 valves, and the robustness enables elimination of pressure sensors, both of which effectively reduce the cost of implementation by an order of magnitude. This approach is illustrated by way of example on a single degree-offreedom pneumatic servo system.

Modeling a Pneumatic Servo System Controlled by Switching Valves A schematic of a single degree-of-freedom pneumatic actuation system is shown in Fig. 1. The general objective of this section is Journal of Dynamic Systems, Measurement, and Control

to cast the dynamics of the PWM-controlled pneumatic actuator into a linear continuous form, in order to apply control design techniques that incorporate frequency domain concepts of stability robustness, and more specifically that enable the implementation of prescribed gain and phase margins. A lumped-parameter model that treats all unmodeled forces, such as Coulomb and viscous friction and external disturbance forces, as a single disturbance term is given as M x¨ ⫽ P B A B ⫺ P A A A ⫹F disturbance


where P A and P B are the 共gage兲 pressures inside chambers A and B of the pneumatic actuator respectively and A A and A B are the areas of the piston seen by each chamber. The pressure in each chamber will be controlled by a two-position three-way solenoid valve that serves to connect the chamber to either a high-pressure supply or to atmospheric pressure. The pressure response in each chamber should be first order in character, assuming that the primary energetic behavior will result from flow resistance of the valve and flow capacitance of the cylinder volume. The pressure dynamic in each cylinder can therefore be reasonably modeled by

␶ A,B P˙ A,B ⫹ P A,B ⫽ v A,B 共 t⫺T D 兲


where the discrete control input v A,B 苸 兵 0,P s 其 is the pressure boundary condition imposed by the valve state 共either atmospheric or supply pressure兲; the time constant ␶ A,B is a typically nonlinear function of the upstream and downstream pressures, the cylinder displacement (x), and various other geometric and thermodynamic quantities; and T D is the time delay exhibited between the control command and the pressure dynamic as a byproduct of the spool/sleeve overlap required to inhibit leakage flow in the spool valves. In order to provide linear system dynamics and preclude dependence on pressure sensors, the pressure dynamic time constant was assumed for this treatment to be invariant 共i.e., constant so that ␶ A,B ⫽ ␶ ). Such an assumption captures the fundamental dynamics of the system, but assumes that the robustness of the control approach can adequately and reliably compensate for the loss of information. The controller can command one of four switched modes to the pneumatic actuator, corresponding to one of the four valve state permutations ( v A , v B ) given by 兵 0,P s 其 ⫻ 兵 0,P s 其 : Mode 1: v A ⫽ P s and v B ⫽0 Mode 2: v A ⫽0 and v B ⫽ P s Mode 3: v A ⫽ P s and v B ⫽ P s Mode 4: v A ⫽0 and v B ⫽0 Each mode results, respectively, in the following system dynamics 共assuming the disturbance force to be zero兲: Mode 1: M 共 ␶ ត x 共 t 兲 ⫹x¨ 共 t 兲兲 ⫽ P s eˆ 共 t⫺T D 兲 A A


Mode 2: M 共 ␶ ត x 共 t 兲 ⫹x¨ 共 t 兲兲 ⫽⫺ P s eˆ 共 t⫺T D 兲 A B


Fig. 1 Schematic diagram of a pneumatic inertial positioning system actuated with a double-acting pneumatic cylinder and controlled with two binary „2-position… 3-way pilot-assisted solenoid valves.

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Mode 3: M 共 ␶ ត x 共 t 兲 ⫹x¨ 共 t 兲兲 ⫽ P s eˆ 共 t⫺T D 兲 A A ⫺ P s eˆ 共 t⫺T D 兲 A B (3c) Mode 4: ␶ ត x 共 t 兲 ⫹x¨ 共 t 兲 ⫽0


where eˆ (t) is the unit step.

Average Model State-space averaging can be utilized to convert the switching model given in Eqs. 共3a-d兲 to a continuous average model 关19兴, which can be utilized for implementing a robust control design. Denoting as d 1 , d 2 , d 3 , and d 4 the fractions of a normalized switching period that each mode is active 共i.e., the duty cycle of each mode兲, a state-space average representation of the system results in the following average model: M ␶ត x 共 t 兲 ⫽⫺M x¨ 共 t 兲 ⫹ P s eˆ 共 t⫺T D 兲关 A A d 1 ⫺A B d 2 ⫹ 共 A A ⫺A B 兲 d 3 兴 (4) where d 1 ⫹d 2 ⫹d 3 ⫹d 4 ⫽1.


Note that the switching fraction d 4 drops out of Eq. 共4兲, due to the constraint of Eq. 共5兲 and a common term present in all modes of Eqs. 共3a-d兲. Denoting the input term as uˆ 共 t 兲 ⫽A A d 1 ⫺A B d 2 ⫹ 共 A A ⫺A B 兲 d 3


the average model is given in the s-domain as ˆ 共 s 兲⫽ G

X共 s 兲 ˆ 共s兲 U

e ⫺T D s P s M ␶ s 3 ⫹M s 2


The average model given by Eq. 共7兲 assumes that the input uˆ (t) can vary continuously in time, which is not the case. Specifically, once a given duty cycle is commanded, the control command cannot be changed until the next PWM period. The control command is therefore subjected to a sample-and-hold operation. Given the standard frequency domain approximation of a sampleand-hold, the transfer function of the average model from the continuous control command u(t) to the motion of the output x(t) can be given as: G共 s 兲⫽

X 共 s 兲 1⫺e ⫺Ts e ⫺T D s P s ⫽ U共 s 兲 Ts M ␶ s 3 ⫹M s 2


It should be emphasized that the zero-order sample-and-hold effect is not the result of digital implementation but instead due to the PWM nature of the control signal. As an aside, electrical PWM amplifiers typically switch on the order of 10 kHz. In such applications, it may be assumed that the sample-and-hold effect is negligible. In pneumatic applications the sample-and-hold can have a non-negligible effect on the dynamics of the servopositioning system due to its longer switching period, and so must be included.

d 1 ⫽sat共 u/A A 兲 , d 1 ⫽0,

d 2 ⫽0; for u⭓0

d 2 ⫽sat共 ⫺u/A B 兲 ; for u⬍0

(9b) (9c)

where the saturation functions are implemented to ensure the condition given by Eq. 共5兲. Note that this control specification also addresses the asymmetric influence of unequal piston areas on opposing sides of the pneumatic actuator 共due to the single rod configuration兲. The combination of Eqs. 共8兲 and 共9a-c兲 enable treatment of the PWM-controlled pneumatic actuator as a linear, analytical, continuous-time system, which in turn enables the use of relative stability notions. Though several approaches to the design of such systems are available, the use of loop shaping addresses directly two significant issues in the control design of PWM-controlled pneumatic actuators. The first is stability robustness, which must be present to compensate for the elimination of pressure sensors 共and the resulting loss of information兲. The second is saturation, which is a fundamental aspect of PWM-based controllers. Specifically, as described by Eqs. 共9a-c兲, the duty cycles cannot exceed 100%, and so the control command u(t) is bounded by 关 ⫺A B ,A A 兴 . The former issue, stability robustness, can be addressed directly in a loop-shaping approach by simply shaping the open loop frequency response so that it exhibits desired gain and phase margins 共e.g., with a lead-lag form of compensator兲. The latter issue, saturation, can also be addressed fairly directly in a loop-shaping context. Rather than shape the open-loop transfer function, one can constrain the maximum gain on the loopshaping compensator to avoid saturation, given bounds on the input to the compensator 共i.e., bounds on the tracking error兲. For the case of a pneumatic cylinder, the maximum amplitude of the error signal will be the stroke length of the cylinder, L. In order to avoid the saturation limits 关 ⫺A B ,A A 兴 of the control command u(t), the frequency response of the compensator K(s) must be such that 兩 K 共 s 兲 兩 max⫽

min共 兩 A A 兩 , 兩 A B 兩 兲 L


Control of a Pneumatic Servo System Controlled by Switching Valves The proposed control methodology was implemented on a pneumatic system as depicted in Fig. 1 with a 1.9 cm 共3/4 in兲 inner diameter, 10 cm 共4 in兲 stroke single-rod double-acting pneumatic cylinder 共Bimba 044-DXP兲 equipped with two pilot assisted

Control Design Though the switching system described by Eqs. 共3a-d兲 has been described by the continuous-time frequency domain model of Eq. 共8兲, the PWM-controlled system is still not of the form necessary to apply standard frequency domain techniques. Specifically, the control input, nominally given by Eq. 共6兲, requires an additional constraint to uniquely specify the control. Though one could formulate various constraint equations, for purposes of this paper the third mode was simply eliminated, since it is not required for motion control of the actuator. Accordingly, positive control input values are specified by Mode 1 and negative values specified by Mode 2 as follows: u 共 t 兲 ⫽A A d 1 ⫺A B d 2 where 506 Õ Vol. 125, SEPTEMBER 2003


Fig. 2 Open-loop frequency response plots of the equivalent model G „ s …. Parameters used are those for the particular application of interest with M Ä10 kg, ␶ Ä9 ms, T D Ä19 ms, P Ä586 kPa gage „85 psig… and T Ä38.5 ms „26 Hz PWM switching frequency….

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3-way solenoid-activated valves 共SMC VQ2200H-5B兲. The cylinder rod is rigidly connected to a 10 kg brass block on a track with linear bearings 共Thompson 1CBO8FAOL10兲. The setup is instrumented with a linear potentiometer 共Midori LP-100F兲 for position feedback. Model parameters were estimated or measured to be: M ⫽10 kg, A A ⫽2.8 cm2 , and A B ⫽2.5 cm2 . Additionally, the pressure response described by Eq. 共2兲 was measured at the midstroke position and the pressure time constant and valve time delay were determined to be ␶ ⫽9 ms and T D ⫽19 ms, respectively. The supply pressure was P s ⫽586 kPa gage 共85 psig兲 and the PWM switching period was T⫽38.5 ms 共26 Hz switching frequency兲. The open loop frequency response of this system, as described by Eq. 共8兲, is shown in Fig. 2. Based on the frequency response shown in Fig. 2, a compensator was designed to provide an open-loop frequency response that would provide a desired degree of stability robustness and additionally avoid control saturation, while also maintaining a high low-frequency gain for purposes of command following and disturbance rejection. The resulting compensator was given by: K 共 s 兲 ⫽k

冉 冊冉 s⫹ ␣ 3 s

␣ 1 s⫹1 ␣ 1 ␤ 1 s⫹1


␣ 2 s⫹1 ␣ 2 ␤ 2 s⫹1


where k⫽3.98⫻10⫺5 (⫺88 dB), ␣ 3 ⫽7.54, ␣ 1 ⫽3.01, ␤ 1 ⫽3.11 ⫻10⫺2 , ␣ 2 ⫽1.80⫻10⫺1 , and ␤ 2 ⫽4.91⫻10⫺2 . This is a double lead, single lag compensator where the first lead network is selected to add 70° of phase at 0.3 Hz, the second lead network is selected to add 65° of phase at 4.0 Hz, and the lag network is selected to add integral action with a break-point of 1.2 Hz. A bound of 兩 K(s) 兩 ⬍⫺52 dB ensures that the control output should not saturate for errors in the frequency band of interest of approximately 0.02–2 Hz. The frequency response of this compensator is shown in Fig. 3. With this compensator, the frequency response of the open-loop transfer function K(s)G(s) is shown in Fig. 4 along with the uncompensated plant and the compensator. Stability robustness measures are 6.7 dB of gain margin and 33° of phase margin with a crossover frequency of 2.0 Hz.

Fig. 4 Frequency response plots of the uncompensated openloop system, the compensator, and the compensated openloop system. The compensated open-loop response shows a phase margin of 33° and a gain margin of 6.7 dB at a cross-over frequency of 2.0 Hz.

Experimental Results The compensator described by Eq. 共11兲 and Fig. 3 was experimentally implemented on the previously described experimental setup. In order to ensure a reasonable approximation of the sample-and-hold used in Eq. 共8兲, a prefilter F(s) was placed on the command to the control loop. The filter F(s) has the form of a critically damped second-order low-pass filter

Fig. 3 Frequency response plots of the compensator K „ s … obeying the saturation gain limit of À52 dB imposed by PWM control near the targeted cross-over frequency of 2 Hz.

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Fig. 5 Step response. The filtered commanded step is shown as dashed and the measured system response is shown as solid.

Fig. 6 Sinusoidal response at 0.5 Hz

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共Coulomb兲 friction effects. Apart from the well known fact that frictional effects typically influence lower frequencies, further evidence of this attributed unmodeled effect is taken from the fact that the open-loop frequency response of Fig. 2 shows little influence of the equivalent model at frequencies of 0.1 and 0.5 Hz. At higher frequencies, the effects of PWM switching as incorporated into the equivalent model dominate over frictional effects.

Conclusions Solenoid-valve controlled pneumatic actuators without pressure sensors can provide low-cost, high-power servo actuation. Such a system requires a control methodology that can accommodate the switching nature of solenoid valve control, and concomitantly provide for a prescribed degree of stability robustness to compensate for the loss of pressure measurements. This paper presents an approach that transforms a discontinuous switching model into a linear continuous equivalent model, so that loop-shaping methods may be utilized to incorporate a prescribed degree of stability robustness into the system. The method is illustrated by way of example on a single degree-of-freedom pneumatic system. Experimental demonstration of the approach indicates good model equivalence and good tracking performance. The method is generalizable to any linearizable PWM-controlled process.

Fig. 7 Sinusoidal response at 1.0 Hz.


Fig. 8 Frequency response plots of the closed-loop system. The figure shows both the control design model prediction as well as experimentally measured points overlayed.

F共 s 兲⫽

1 ␶ s⫹1 兲2 共 f


where ␶ f ⫽0.0159 seconds was chosen for a filter cut-off frequency of 10 Hz, well below the PWM switching frequency of 26 Hz. It should be noted that this filter does not decrease the stability robustness of the system 共since it is outside the closed loop兲, and also does not adversely affect the closed-loop performance. Specifically, the presence of a sample-and-hold in the closed-loop makes it impossible to achieve a significant open-loop gain at frequencies near the PWM switching frequency, and as a result, the closed-loop response will always have a tracking bandwidth less than the switching frequency 共and in practice, well below it兲. Since the loop cannot track frequencies near or above the switching frequency, the input-shaping filter does not adversely affect closed-loop performance, but simply precludes frequencies that the loop cannot track. Figure 5 shows the measured step response of the system. Figures 6 and 7 show sinusoidal responses at 0.5 and 1.0 Hz, respectively. Figure 8 shows a comparison of the predicted gain and phase characteristics of the closed-loop system to several measured points. Good agreement is exhibited as the phase falls off, indicating validation of the equivalent model formulation. Disagreement between predicted and measured phase at lower frequencies 共0.1 and 0.5 Hz兲 is attributed primarily to unmodeled 508 Õ Vol. 125, SEPTEMBER 2003

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