As a disturbance develops in an open-loop system, the error signal increases.

6.2.1 Transfer Function of Disturbance Response for Traditional Systems

Disturbance response of a control system both with sensor feedback and with observed-state feedback can be evaluated using transfer functions. The transfer function of the disturbance response of the traditional system shown in Figure 6-9 is easily calculated since there is only one loop.

L1=−GC(s)×G PC(s)×GP(s)×GS(s)

There is a single path from D(S) to C(S):GP(S). The transfer function is then:

(6.8)C(s)D( s)=GP(s)11+GC(s)×GPC(s)×GP(s)×GS(s)⋅

An algebraic manipulation yields

(6.9)C(s)D(s)=GP(s)(1−GC(s)×GPC(s)×GP(s)×GS(s)1+GC(s)×GPC(s)×GP(s)×GS(s)).

Equation 6.9 can be rewritten in terms of the control-law closed-loop transfer function, GCL(S) =Y(S)/R(S):

(6.10)C(s)D(s)=GP(s)(1−GCL(s)).

Understanding that the ideal disturbance response is 0, the closer that GCL(S), the closed-loop response, is to unity, the better the disturbance response. The closed-loop response will be closest to one at low frequency and, correspondingly, the disturbance response will be the best. Raising the control-system bandwidth improves disturbance by keeping GCL(S) approximately unity for a wider range of frequencies; the disturbance response of Equation 6.10 will be lower over a wider frequency range, rejecting more of the disturbance input.

Well above the control-loop bandwidth, the closed-loop response will be near zero and the disturbance response will be GP(S); that is, the disturbances will be limited only by the plant gain. At high frequencies, disturbance response is passive as, for example, when a large capacitor in a power supply prevents high-frequency voltage ripple or when a large inertia prevents high-frequency velocity ripple.

8.1.1 Performance Measures

The main measures of performance in servo systems are noise generation, disturbance response, command response, and stability. Servo systems excel in command and disturbance response. Servomotors are often the technology of choice in discrete operations, where the moves must be rapid, and in continuous operations, where the system must hold constant speed in the presence of disturbances.

Noise susceptibility and stability are often problems for servo systems. The high gains needed to respond to rapidly changing commands or high-frequency disturbances also respond to noise on the sensor and command inputs, often generating considerable noise to the system output. Stability is a problem because designers often push the loop gains as high as possible to maximize response and, in doing so, press the limits of stability margins. As has been discussed in earlier chapters, Luenberger observers are a practical way to deal with stability limitations and so indirectly allow servo gains to be increased. On the other hand, as was the focus of Chapter 7, observers can exacerbate problems with sensor noise.

7.2 Disturbance Response of a Velocity Controller

Consider the velocity controller from Chapter 4, shown in Figure 7.11, as another example of disturbance response. This system is modeled in Visual ModelQ, Experiment 7A; displays from this model are used in the discussion that follows. This model is based on the model in Experiment 4A but with three modifications:

A waveform generator (“Torque Dist”) has been added to inject torque disturbances.

The DSA has been added in line with the disturbance to measure disturbance response (“Disturbance DSA”).

The KT/J multiplier has been separated into two terms, since torque disturbances are injected between KT and J.

This system is similar to the power supply discussed earlier except that the controller is digital, and the feedback is from the position sensor and thus is delayed a half sample interval compared with measuring velocity directly (Section 4.2.3). Finally, because the velocity controller requires the conversion of electrical to mechanical energy, the torque conversion term (KT) must be included. This example uses KT = 1 Nm/amp (as have similar earlier examples) so that the conversion does not add confusion. Finally, motion variables (J, torque, velocity) replace electrical parameters (C, current, voltage). Those differences are all small, and the two controllers perform about the same.

The model of Figure 7.11 will allow more thorough study than was done with the power supply. First, response in either the time or frequency domain can be investigated. Also, the imperfections of phase delay from the digital controller and a realistically limited current loop are included in the simulation.

The transfer function of disturbance response for the model of Figure 7.11 can be approximated, as was the power supply in Equation 7.5. In order to make Equation 7.9 clearer, the current loop dynamics and sampling delays have been ignored; however, those effects are included in all simulations that follow.

(7.9)TDIST(s)=VM(s)TD(s)≈sJs2+KVPKTs+KVIKVPKT

As with the power supply, the motion controller disturbance response can be approximated over ranges of frequency, as shown in Equations 7.10 through 7.12.

(7.10)TDIST−HIGHFREQ(s)≈sJs2=1Js

(7.11)TDIST−MEDFREQ(s)≈sKV PKTs=1KVPKT

(7.12)T DIST−LOWFREQ(s)≈sKVIKVPKT

7.2.1 Time Domain

A time-domain disturbance response is shown in Figure 7.12. The disturbance waveform, ±1 Nm at 10 Hz, is shown above and the velocity response below. At the vertical edges of the disturbance, the motor moves rapidly. The peak velocity of the excursion is bounded by the proportional gain (KVP); the integrator term (KVI) requires more time to provide aid, although given long enough, KVI will eliminate all DC error. In Figure 7.12, after about 3 msec, the integral has returned the system to near ideal (i.e., zero speed) response.

Figure 7.12. Dynamic stiffness: velocity in reaction to a disturbance torque.

The most immediate way to improve disturbance response is to increase controller gains. Experiment 4A showed that KVP can be increased to about 1.2 while remaining stable. Increasing KVP from 0.72 (the default) to 1.2 cuts the peak response almost proportionally: from about 21 RPM (Figure 7.12) to 14 RPM (Figure 7.13a). Restoring the KVP to 0.72 and increasing KVI to 150 improves the response time of the integral, again approximately in proportion, from about 1.5 time divisions in Figure 7.12 to 1 division in Figure 7.13b. However, changing the integral gain does not improve the peak of the disturbance response, which is a higher-frequency effect, out of reach of the integrator; both Figures 7.12 and 7.13b show excursions of about 21 RPM.

Figure 7.13. Disturbance response improves compared with Figure 7.12. (a) KVP is raised to 1.2. (b) KVP is restored to 0.72 and KVI is raised to 150.

Note that for motion systems, the area under the velocity response is often more important than the peak excursion. Area under the velocity curve (i.e., the integral of the velocity) represents position error. You could just as easily monitor the position deviation from a torque pulse. In that case, the disturbance response would be PM/TD rather than VM/TD.

7.2.1.1 Proportional Controller

The controller from Experiment 7A can be modified to become a proportional controller by setting KVI = 0. Removing the integral term demonstrates its benefit. The time-domain response of the proportional controller is shown in Figure 7.14. Notice that the lack of an integral term implies that the controller does not provide ideal DC disturbance response. Unlike that in Figures 7.12 and 7.13, the velocity response does not decay to the ideal value of zero.

Figure 7.14. Disturbance response of proportional controller (KVP = 0.72).

7.2.2 Frequency Domain

Experiment 7A can be used to generate Bode plots of frequency. The concern here will be with gain; the phase of the disturbance response matters less. The system is configured to display disturbance response. The result of the default system is shown in Figure 7.15, with the approximations for the disturbance response (Equations 7.10 through 7.12) drawn in the appropriate zones.

Figure 7.15. Frequency-domain plot of velocity controller.

The effect of higher KVP and higher KVI can also be seen in the frequency domain. The effects demonstrated in the time domain in Figure 7.13 are shown in the Bode plot of Figure 7.16. In Figure 7.16a, KVP has been raised from 0.72 to 1.2; the result, with the plot for higher KVP being a bit lower, shows that higher KVP improves (lowers) the disturbance response in both the medium and low frequencies. In Figure 7.16b, KVP is restored to 0.72 and KVI is raised from 100 to 150; note that only the low-frequency disturbance response is improved. In both cases, the high-frequency response, which is governed by J, is unchanged.

Figure 7.16. Disturbance response with (a) KVP =  1.2 and KVI = 100 and (b) KVP =  0.72 and KVI = 150.

The effect of removing the integral controller shows up in the low frequencies. Figure 7.17 shows the disturbance response with KVI set to 100 (PI control) and to 0 (P control). The response of the two controllers is similar in the medium and high frequencies; in the low frequencies, the difference is clear. The response of the PI controller is superior; it improves (declines) as the frequency drops. The P controller does not improve with lower frequency.

Figure 7.17. Comparing the disturbance response of a PI controller and a P controller.

7.4.2 Reducing Noise in Disturbance-Decoupled Systems

Lowering the observer bandwidth will produce a similar benefit for systems with disturbance decoupling, as is predicted by Equation 7.26. However, lowering the observer bandwidth will degrade the disturbance response in the case where the observer bandwidth is lower than the power converter bandwidth, as shown in Equation 6.15. The primary limits on the disturbance response are the observer bandwidth, which delays the disturbance signal, and the power converter, which delays the decoupling signal entering the control system. The slower of the two will be the primary limit. Lowering the primary limit will degrade the disturbance response.

An alternative means for reducing noise sensitivity when using disturbance decoupling is to add a low-pass filter in line with KDD, as in Experiment 7D as shown in Figure 7-16. The low-pass filter bandwidth is set with the Live Constant DD LPF. The improvement of noise susceptibility from the low-pass filter is demonstrated in Figure 7-17. When the filter is reduced from 250 to 50 Hz, the noise is attenuated accordingly.

The degradation of disturbance response from the low-pass filter is shown in Figure 7-18. The response to disturbances worsens (grows) with lower bandwidth filtering on the observed disturbance. However, as long as the filter bandwidth is more than about four times the power converter bandwidth, lowering the filter bandwidth will have a negligible effect on disturbance response.

Adding a low-pass filter in line with the observed disturbance provides an observed disturbance signal similar to that which would have been produced by lowering the observer bandwidth. The primary benefit to the additional filter compared to reducing observer bandwidth is that the filter can be used to lower the bandwidth for decoupling, where noise susceptibility is so great, without reducing the observer bandwidth. If observed-state feedback is used for other purposes, the benefits ofhigh-observer bandwidth for that signal can be maintained. Since the noise sensitivity of the observed-state feedback is so much less than that of the observed disturbance, having the extra degree of design freedom brought by the decoupling filter will be of benefit in some of the applications where both signals from the observer are used simultaneously.

18.3.2 Experiment 18E: Using Observed Acceleration Feedback

Experiment 18E models the acceleration-feedback system of Figure 18.21 (see Figure 18.22). The velocity loop uses the observed velocity feedback to close the loop. A single term, KTEst/JTEst, is formed at the bottom center of the model to convert the current-loop output to acceleration units; to convert the observed acceleration to current units, the term 1 + KAFB is formed via a summing block (at bottom center) and used to scale the control law output, consistent with Figure 18.20. An explicit clamp is used to ensure that the maximum commanded current is always within the ability of the power stage. The Live Scope shows the system response to a torque disturbance (the command generator defaults to zero). The observer from Experiment 18A is used without modification.

The improvement of acceleration feedback is evident in the step-disturbance response as shown in Figure 18.23. Recall that without acceleration feedback, the control law gains were set as high as was practical; the non-acceleration-feedback system took full advantage of the reduced phase lag of the observed velocity signal. Even with these high control law gains, acceleration feedback produces a substantial benefit as can be seen by comparing Figures 18.23a and 18.23c.

The disturbance response Bode plot of the system with and without acceleration feedback is shown in Figure 18.24. The acceleration feedback system provides substantial benefits at all frequencies below about 500 Hz. Figure 18.24 shows two levels of acceleration feedback: KAFB = 1.0 and KAFB = 5. This demonstrates that as the KAFB increases, disturbance response improves, especially in the lower frequencies, where the idealized model of Figure 18.19 is accurate. Note that acceleration feedback tests the stability limits of the observer. Using the observer as configured in Experiment 18E, KAFB cannot be much above 1.0 without generating instability in the observer. The problem is cured by reducing the sample time of the observer through changing the Live Constant “TSAMPLE” to 0.0001. This allows KAFB to be raised to about 15 without generating instability. It should be pointed out that changing the sample time of a model is easy but may be quite difficult in a working system. See Question 18.5 for more discussion of this topic.

8.2.2 Increasing the Power Converter Bandwidth vs. Feed-Forward Compensation

The effects on command response of raising the power converter bandwidth are similar to using the bi-quad filter for cancellation. Given a choice between raising the power converter bandwidth or compensating for it in the feed-forward path, designs can benefit more by raising the bandwidth. First, a higher bandwidth will replace higher loop gains and thus provide superior disturbance response. The feed-forward path is outside the loop and thus provides no response to disturbances. Also, the actual power converter is known only to a limited accuracy. For example, the dynamic performance of a power converter may vary over its operating range. Variation between the power converter and the compensation function in the feed-forward path will degrade performance, especially when high feed-forward gains are used. This topic will be covered in a later section. So, the preference is to raise the power converter bandwidth; when that alternative is exhausted, the power converter can be compensated for in the feed-forward path.

Increased Loop Gain

The term loop gain is used to describe the negative feedback loop that controls ventilation to regulate blood gas levels within narrow limits (Younes et al., 2007). This feedback loop incorporates pulmonary, circulatory and central responses, and works to counteract any change in blood gas tension (increase or decrease) with a corresponding change in ventilation. Loop gain has three key gain components. Controller gain, equivalent to central plus peripheral chemosensitivity, governs hypercapnic and hypoxic ventilatory responses to changes in blood gases (controller gain = Δventilation/ΔPCO2) (Wellman et al., 2011). Plant gain is the “pulmonary filter” that processes blood gas variations in the lungs to a sudden change in ventilation. Thus, plant gain is an estimation of the effectiveness of the lungs to alter blood gases (plant gain = ΔPCO2/ventilation) and is influenced by factors such as changes in lung volume. Finally, mixing gain, a secondary “smoothing filter,” further regulates blood gas tensions when the blood passes from the pulmonary capillary to the larger chest vessels. Circulation delay times (i.e. lung to chemoreceptors) are also crucial components of loop gain and the resultant time course of unstable breathing patterns.

In unstable systems, an elevated dynamic loop gain (Hudgel et al., 1998) reacts to breathing disturbances such as hypopneas or apneas with an exaggerated response (such that the response/disturbance ratio is ≥ 1). This heightened response itself then becomes a new disturbance and propagates ventilatory instability at a rate determined by the characteristics of the plant (i.e. components of the lung). The amplitude of the response is determined by the controller (i.e. chemoreceptors). Thus, loop gain = plant × controller gain. While high loop gain underpins unstable breathing, low loop gain acts to progressively dampen the original disturbance. Thus, the subsequent, smaller ventilatory response is lower. This then forms the next cycle such that the ventilatory response/disturbance ratio is < 1 (Khoo et al., 1982).

High loop gain occurs when there is an alteration in one or more gain components. The common denominator is the increase in ventilatory drive, which can ultimately contribute to oscillations in PCO2 to below the apnea threshold. Given that PCO2 is the main driver of breathing during sleep, if PCO2 falls below a critical threshold during sleep, breathing ceases (Dempsey, 2005). These factors promote OSA through several mechanisms. First, continuous oscillations in ventilatory drive to the respiratory pump muscles lead to breathing instability with concomitant intermittent periods of reduced “mechanical drive” to the upper airway muscles. This can result in a mismatch between drive to the pump versus pharyngeal dilator muscles and further contribute to cycles of upper airway closure-reopening (Eckert and Wellman, 2015). Another contributor to unstable breathing is a heightened ventilatory response to arousal (Jordan et al., 2003). This can perpetuate transient blood gas disturbances and unstable breathing during sleep (Eckert and Younes, 2014). Third, large negative inspiratory pressure swings to small increases in CO2 can cause “negative effort dependence” (Eckert and Wellman, 2015; Genta et al., 2014b). This is where the upper airway is effectively “sucked closed” during inspiration.

Elevated loop gain contributes to OSA (Wellman et al., 2011; Eckert et al., 2013; Deacon et al., 2018; Salloum et al., 2010) in at least one third of patients (Eckert et al., 2013).

Targeted therapy for an elevated loop gain includes oxygen (Sands et al., 2018a; Wellman et al., 2008), CO2 (Xie et al., 2013) and carbonic anhydrase inhibitors (Edwards et al., 2012, 2013) (Fig. 3). Strategies to increase lung volume reduce plant gain and can therefore reduce loop gain. These include: CPAP (Messineo et al., 2018; Edwards et al., 2009), positional therapy (Joosten et al., 2015), weight loss (Parameswaran et al., 2006) and lung inflation in restrictive (Kolilekas et al., 2013) or obstructive (Krachman et al., 2016) lung diseases.

4.2 Dynamic simulations

To test the dynamic performance of our approach on the heat exchanger network, we consider three scenarios. In the first two scenarios we use a decentralized control structure, and in the third scenario we use model predictive control to control the Jäschke Temperatures to equal values. The temperature sensor dynamics are modelled as first order dynamics with a time constant of 5s and a delay of Is.

In all three cases we use the difference between the Jäschke Temperature of branches A — E and branch F, i.e. ci = TJ,i — TJ, F for i = A,B,C,D, E, as controlled variables. The reason all controlled variables are taken relative to TJF is that branch F has the largest heat capacity, and thus we expect this to mitigate interactions when controlling the system.

4.2.3 Results

All three approaches keep the controlled variables at their setpoints, and the performance looks quite similar. In the top of Figure 2 we have plotted the end temperature (objective function to maximize). The steady state value is very close (<0.5 K) to the optimal value for the disturbances:

Figure 2. Comparison of control strategies. Top: Final outlet temperature Tend, Bottom: Flow rate through branches F and E

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At 5,000s step in feed temp (+10 °C)

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At 6,000s step in feed flowrate (-10 %)

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At 7,000s step in hot stream inlet temperature branch C (-10 °C)

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At 8,000s step in hot stream flow rate on branch E (-10 %)

All three control structures give excellent performance, as can be seen from the temperature profiles of the end temperature in Figure 2. For comparing the input usage, we show the flow rates to line D and F. Here we see that the unfiltered decentralized approach and the MPC give a more aggressive control action, while the filtered decentralized approach gives a smother, slower performance. The difference is, however, not reflected in the end temperature. From this point of view, it seems that the filtered decentralized control structure works best, because it does not require as much input usage as the other approaches. However, de-tuning the MPC controller may also result in smoother control action.

2.1.4.1 Errors in Feedback Sensors

Feedback sensors measure signals imperfectly. The three most common imperfections, as shown in Figure 2-5, are intrinsic filtering, noise, and cyclical error.

The intrinsic filtering of a sensor limits how quickly the feedback signal can follow the signal being measured. The most common effect of this type is low-pass filtering. For all sensors there is some frequency above which the sensor cannot fully respond. This may be caused by the physical structure of the sensor. For example, many thermal sensors have thermal mass; time is required for the object under measurementsto warm and cool the sensor's thermal mass. Filtering may also be explicit as in the case of electrical sensors where passive filters are connected to the sensor output to attenuate noise.

Whatever the source of the filtering, its primary effect on the control system is to add phase lag to the control loop. Phase lag reduces the stability margin of the control loop and makes the loop more difficult to stabilize. The result is often that system gains must be reduced to maintain stability in order to accommodate slow sensors. Reducing gains is usually undesirable because both command and disturbance response degrade.

Cyclical error is the repeatable error that is induced by sensor imperfections. For example, a strain gauge measures strain by monitoring the change in electrical parameters of the gauge material that is seen when the material is deformed. The behavior of these parameters for ideal materials is well known. However, there are slight differences between an ideal strain gauge and any sample. Those differences result in small, repeatable errors in measuring strain. Since cyclical errors are deterministic, they can be compensated out in a process where individual samples of sensors are characterized against a highly accurate sensor. However, in any practical sensor some cyclical error will remain. Because control systems are designed to follow the feedback signal as well as possible, in many cases the cyclical error will affect the control-system response.

Stochastic or nondeterministic errors are those errors that cannot be predicted. The most common example of stochastic error is high-frequency noise. High-frequency noise can be generated by electronic amplification of low-level signals and by conducted or transmitted electrical noise commonly known as electromagnetic interference (EMI). High-frequency noise in sensors can be attenuated by the use of electrical filters; however, such filters restrict the response rate of the sensor as discussed above. Designers usually work hard to minimize the presence of electrical noise, but as with cyclical error, some noise will always remain. Filtering is usually a practical cure for such noise; it can have minimal negative effect on the control system if the frequency content is high enough so that the filter affects only frequency ranges well above where phase lag is a concern in the application.

The end effect of sensor error on the control system depends on the error type. Limited responsiveness commonly introduces phase lag in the control system, reducing margins of stability. Noise makes the system unnecessarily active and may reduce the perceived value of the system or keep the system from meeting a specification. Deterministic errors corrupt the system output. Because control systems are designed to follow the feedback signal (including its deterministic errors) as well as possible, deterministic errors will carry through, at least in part, to the control-system response.

17.1.2 Tuning the P/PI Loop

This section will discuss a tuning procedure for the P/PI loop. The discussion starts with Experiment 17A, which is shown in Figure 17.3. A waveform generator, through a DSA, commands velocity, which is integrated in a sum block to form position command. This structure is a simple profile generation; more practical profile generation will be discussed at the end of this chapter. The loop is the P/PI structure of Figure 17.1. A Live Relay “Position Loop” is provided to change the configuration between position and velocity control so the velocity loop can be tuned separately. The sample rate is 4 kHz, and the current loop is an 800-Hz two-pole low-pass filter. Unlike the transfer function, the model does include the effects of sampling and digital differentiation, though their effects will be insignificant in these experiments.

17.1.2.1 Tuning the PI Velocity Loop

Begin tuning by configuring the system as a velocity controller. In Experiment 17A, double-click on the Live Relay “Position Loop” so that “OFF” is displayed. As with standard PI control, set the command as a square wave with amplitude small enough to avoid saturation; in this experiment you will need to double-click on the waveform generator “Cmd” and change the waveform to “Square.” As discussed in Chapter 6, set the integral gain (KVI) to zero and raise the proportional gain (KVP) no higher than the high-frequency effects (current loop, sample rate, resonance, feedback resolution) allow without overshoot. Then raise the integral for about 5% overshoot.

In Experiment 17A, the value of KVP has been set to 0.72. This produces a velocity loop bandwidth of about 75 Hz, which is typical for motion applications; this is shown in Figure 17.4. Many motion systems use higher gains, and velocity loop bandwidths of 200 Hz are common; in fact, KVP in the system of Experiment 17A could have been doubled without inducing overshoot. However, the value of 0.72 provides performance more common in motion systems, which is often limited by resolution and mechanical resonance. This value will be used as the starting point for all three methods, allowing the clearest comparisons. KVI is set to 20, which induces about 4% overshoot.

Figure 17.4. From Experiment 17A: Bode plot of velocity loop command response.

The guidelines here are general, and there is considerable variation in the needs of different applications. Accordingly, be prepared to modify this procedure. For example, the value of integral is set low to minimize overshoot. The primary disadvantage of low integral gain is the loss of low-frequency stiffness. For applications where low-frequency disturbance response is important or where overshoot is less of a concern, the integral gain may be set to higher values. On the other hand, the integral may be zeroed in applications that must respond to aggressive commands without overshoot.

17.1.2.2 Tuning the P Position Loop

The position loop is tuned after the velocity loop. Start by configuring the system as a position loop. In Experiment 17A, the Live Relay “Position Loop” should display “ON”; double-click on it if necessary. For point-to-point applications, apply the highest-acceleration command that will be used in the application. The waveform generator in Experiment 17A defaults to a trapezoidal command that reaches 50 RPM in about 10 msec (acceleration = 5000 RPM/sec). Always ensure that the system can follow the command. Commanding acceleration too great for the system peak torque will produce invalid results. As always in tuning, avoid saturating the current controller. The command in Experiment 17A requires about 1 A of current, well below the 15-A saturation level set in the PI controller.

The position loop gain, KPP, is usually set to the largest value that will not generate overshoot. Overshoot is normally unacceptable because it implies that the load backs up at the end of the move. Normally, this causes a significant loss of accuracy, especially when backing up a load connected to a gear box or lead screw results in lost motion because of backlash or compliance. The result here is that KPP is set to 140; this is shown in Figure 17.5a. For comparison, Figure 17.5b shows the overshoot that comes from setting KPP to 200, about 40% too high.

Figure 17.5. From Experiment 17A: response with (a) KPP = 140 and (b) KPP = 200.

Settling time is an important measure of trapezoidal response. It is commonly measured as the time between the end of the command to when the system comes to rest, as shown in Figure 17.5a. “Rest” implies that the position error is small enough for the system to be “in position,” a measure that varies greatly between applications. It is common to minimize the total move time (TMOVE + TSETTLE in Figure 17.5a) to maximize machine productivity. Typically, mechanical design and motor size determine the minimum TMOVE; TSETTLE is directly affected by tuning. Of course, higher loop gains reduce settling time, assuming adequate margins of stability.

The responsiveness of motion systems can also be measured with Bode plots, although this is less common in industry. The bandwidth is PF(s)/PC(s), which is equal to VF(s)/VC(s). This equality can be seen by multiplying PF(s)/PC(s) by s/s = 1 and recalling that s × P(s) = V(s). Velocity signals are often used because many DSAs require an analog representation of input signals, something difficult to do with position signals. The Bode plots in this chapter will rely on the ratio of velocities. Figure 17.6 shows the Bode plot for both position (“Position Loop” = ON) and velocity (“Position Loop” = OFF) operation. The bandwidth of the position loop is 35 Hz, a little less than half that of the velocity loop, which is typical.

Figure 17.6. From Experiment 17A: Bode plots of position and velocity loop command response.