Indexes of Performance

9 Indexes of Performance The state of the art of the machine industry has experienced a rapidly accelerating evolution since the outset of the first industrial revolution. Modern machines are required to perform to varying requirements of accuracy, productivity, and surface finish requirements. It is the purpose of this discussion to establish some guidelines of machine system design to attain the desired performance requirements of accuracy, surface finish, etc. Modern industrial machines are considered as a total system comprised of the control, drive, and machine. There are two kinds of performance criteria. One, referred to as indexes of performance (I.P.), relates to specifying servo drive stability in relation to design parameters. The second, performance characteristics, relates to parameters such as drive stiffness, resolution, maximum acceleration, and the effects of friction. To some extent the performance characteristics are a function or the end result of the indexes of performance that are selected. 9.1 DEFINITION OF INDEXES OF PERFORMANCE FOR SERVO DRIVES During World War II, the growth of feedback control theory was given a stimulus that has continued for five decades. As the state of the art improved, control system performance could be predicted with increased accuracy. However, as the state of the art in feedback control theory Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved increased in reliability, it also increased in complexity. There is a need to specify performance in relation to design parameters. An accepted technique to relate performance, I.P., is an attempt to optimize performance using constraints of design criteria. Some I.P.s in common use are: A. From the frequency response: 1. Phase margin 2. Gain margin 3. Gain margin versus control loop bandwidth B. From the step response: 1. Percent overshoot 2. Rise time 3. Delay time 4. Settling time These I.P.s are given for the frequency response and step response because these two methods of analysis are also related to diagnostic techniques to measure actual performance of a feedback control system. There are numerous types of feedback control applications. The I.P. to be discussed are limited to hydraulic and electric feed drives used with position control systems. The frequency-response characteristics of these drives are considered as the analytical tool in specifying I.P.s. Frequency-response characteristics describe how well the output of a given control system will follow the input variations as a function of frequency. In brief, the frequency response describes the dynamic characteristics of the drive. In addition, required compensation can be determined from the frequency-response character-istics; required compensation is an important part of the synthesis of feedback control drive systems. The following is a description of the various indexes of performance. ‘‘Phase margin’’ is the amount of phase shift remaining between the output controlled variable and the input reference at the crossover frequency (Kv of Figure 1) before 1808 of phase shift occurs. (Note that phase shift is a negative angle.) Feedback control systems with 180 degrees phase shift at the open-loop crossover frequency are unstable. Therefore, phase margin is a measure of how much additional phase shift can be tolerated before instability will occur. As an I.P., the phase margin at the crossover frequency should be 458 or more. Values of phase margin less than 458 will result in an oscillatory condition that will have increasing oscillation with less phase margin until self-sustained oscillation will occur at a phase margin of 08 (see Section 8.4). Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved Fig. 1 Open-position loop frequency response for a hydraulic drive with an internal tachometer loop. ‘‘Gain margin’’ is the amount the position-loop gain can be raised to produce a phase margin of zero or a phase shift of 1808. As an I.P. the gain margin should be no less than 2.0 (or 6dB). In Figure 1 the gain margin is shown as 6dB or a gain of 2. Gain margin versus control loop bandwidth is the most important of the I.P.s. This I.P. relates the gain margin to the attainable servo bandwidth. The bandwidth of the position loop, velocity loop, and hydraulic or mechanical resonance have a relationship that is the basis for system stability. Hydraulic and electric drives are next considered separately. These I.P.s are summarized in Eq. (9.1-1) to (9.1-6). They are derived in Section 9.2. Hydraulic Drives 1. Hydraulic drive with position loop only: Minimum allowable hydraulic resonance ¼ oh ¼ 100rad=sec Kv ¼ oh (9.1-1) Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved 2. Hydraulic drive with tachometer loop: Minimum allowable hydraulic resonance ¼ oh ¼ 200rad=sec. oc ¼ velocity loop bandwidth ¼ 1oh 1 1 v 2 c 6 h Electric Drives Electric drive with tachometer loop: ome ¼ mechanical time constant ¼ t1e ¼ o1 o1 ¼ lead compensation oc ¼ velocity loop bandwidth ¼ oe Kv ¼ 2oc (9.1-2) (9.1-3) (9.1-4) (9.1-5) (9.1-6) 9.2 INDEXES OF PERFORMANCE FOR ELECTRIC AND HYDRAULIC DRIVES I.P.s were defined at the beginning of this section. Of the available I.P.s, gain margin versus control loop bandwidth is the most important index. The I.P. for hydraulic drives is considered first. Hydraulic Drive without a Tachometer Loop The first hydraulic feed drive considered is the simplest case of a position loop (single servo loop). Bandwidth for a stable servo drive is a function of the hydraulic damping factor, illustrated with the block diagram for this servo loop shown in Figure 2. The position feedback uses a resolver for this example. Since the valve response ðovÞ is usually greater than the other system responses, it is neglected in the open-loop response represented by the transfer of Eq. (9.2-1). BðsÞ KAKfb ðsÞ D s s2 þ 2dhs þ1 h h (9.2-1) Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved Fig. 2 Hydraulic servo-drive block diagram without a tachometer loop. Let KAKfb ¼ Kv (9.2-2) m sffiffiffiffiffiffiffiffiffiffiffiffi 2bD2 h JTVc Also, for sinusoidal analysis s ¼ jo. BðjoÞ Kv Kv EðjoÞ jo ðjoÞ2 þ 2dhjo þ 1 jo 1 o2 þ 2jd o h h h (9.2-3) The critical frequency occurs at o ¼ oh: BðjoÞ Kv Kv EðjoÞ joh 1 o2 þ 2jdh oh 2dhoh h (9.2-4) This denotes an amplitude of Kv=2dhoh at a phase angle of 1808. For a gain margin of 2 at o ¼ oh, BðohÞ < 6dB (9.2-5) which can be observed from Figure 3, since the height of the attenuation peak at the hydraulic resonance should not be higher than 0dB for the system to be stable. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved ... - tailieumienphi.vn