Generalized Control Theory pptx

8 Generalized Control Theory 8.1 SERVO BLOCK DIAGRAMS Having discussed individual components of industrial servo drives from an operation point of view and a mathematical descriptive point of view, the next step is to combine these components into a block diagram for the complete servo drive The block diagram is a powerful method of system analysis In the block diagram each component in a servo system can be described by the ratio of its output to input Thus the output of one component is the input to the next component To satisfy an accuracy requirement some form of feedback is required For a velocity servo drive the output is speed Without velocity feedback, the speed will vary greatly with changes in load By providing negative feedback with the use of a tachometer, the feedback voltage is compared with a reference command voltage input by means of a summing junction The difference in the two voltages (command and feedback) is the error voltage For a velocity servo drive the error is finite As load increases the drive slows down, the feedback voltage gets smaller, and the error increases, causing the speed regulator to speed up and maintain the required velocity To represent the servo drive with a block diagram it is necessary to use block diagram algebra In its simplest form the block diagram will have a forward loop block, a feedback loop block, and a summing junction as in Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Figure The output controlled variable (C) ratio to the reference input (R) is B ¼ CH E ¼ C=G (8.1-1) (8.1-2) RÀB¼E (8.1-3) R À CH ¼ C=G R ¼ C=G ỵ CH ẳ C1 ỵ GHị=G C G ẳ R ỵ GH (8.1-4) (8.1-5) (8.1-6) The term G represents the total transfer function for the forward loop The term H represents the total transfer for the feedback loop Normally the G term is made up of the combined transfer functions of several components Using Eq (8.1-6) several examples are given in Figures 2–4 showing the use of block diagram algebra to simplify servo block diagrams, and illustrating their use for a real velocity servo drive having a motor, amplifier, and feedback tachometer To illustrate the use of block diagram algebra for an electric velocity servo drive having a motor, amplifier, and feedback tachometer, Figure presents an example in the frequency domain, where E ¼ error Ka ¼ amplifier gain; V=V Km ¼ motor gain ¼ 1=Ke ; rad=sec=V KTA ¼ tachometer gain, V/rad/sec R ¼ reference command ta ¼ amplifier time constant, sec tm ¼ motor time constant, sec Fig Servo block diagram Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig Servo block diagram algebra Vm ¼ output velocity (controlled velocity), rad/sec The forward loop has the following transfer function: Cjoị Ka Km ẳG ẳ Ejoị ỵ jota ị1 ỵ jotm ị (8.1-7) The open-loop transfer function from which the stability is determined is: Bjoị Ka Km KTA ẳ GH ẳ Ejoị þ jota Þð1 þ jotm Þ (8.1-8) The closed-loop transfer function is: Vmjoị Ka Km 6h ẳ Rjoị ỵ jota ị1 ỵ jotm ị 1ỵ Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Ka Km KTA 1ỵjota ị1ỵjotm ị i (8.1-9) Fig Block diagram algebra which can also be written as Vmjoị Ka Km ẳ Rjoị ỵ jota ị1 ỵ jotm ị ỵ Ka Km KTA ị Vmjoị Ka Km ẳ Rjoị joị2 tm ta þ joðta þ tm Þ þ ð1 þ Ka Km KTA ị Vmjoị Ka Km =1 ỵ Ka Km KTA ị ! ẳ Rjoị joị2 ta ỵtm ị ỵ jo þ K ð1þKatm m KTA Þ ð1þKa Km KTA Þ ta Copyright 2003 by Marcel Dekker, Inc All Rights Reserved (8.1-10) (8.1-11) (8.1-12) Fig Block diagram algebra A general form for a quadratic is: joị2 2d ỵ jo þ o2 om m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Ka Km KTA om ¼ tm ta Fig Electric drive block diagram Copyright 2003 by Marcel Dekker, Inc All Rights Reserved (8.1-13) (8.1-14) To further illustrate the use of block diagram algebra, a hydraulic velocity servo drive can be used as an example in Figure For simplicity the individual drive components are represented by their respective Laplace transfer functions A detailed development of a hydraulic servo drive is presented in Chapter 12 The terms in Figure are: Ga(s) ¼ amplifier transfer function Gv(s) ¼ servo valve transfer function Gm(s) ¼ servo motor transfer function KTA ¼ tachometer constant R(s) ¼ reference input Vm(s) ¼ motor output velocity The forward loop has the transfer function Vmsị ẳ Gasị Gvsị Gmsị ¼ GðsÞ EðsÞ (8.1-15) The open-loop transfer function is BðsÞ ¼ GaðsÞ GvðsÞ GmðsÞ KTA ¼ GH EðsÞ (8.1-16) The closed-loop transfer function is Vmsị Gasị Gvsị Gmsị ẳ Rsị ỵ Gasị Gvsị Gmsị KTA (8.1-17) The open-loop equation by itself is not very useful in determining the system performance, but it can be used to determine the transient performance and the closed-loop performance In the general equation for Fig Hydraulic drive block diagram Copyright 2003 by Marcel Dekker, Inc All Rights Reserved a feedback control system, CðsÞ ẳ Gsị Rsị ỵ Gsị ỵ Gsị ịCsị ¼ GðsÞ RðsÞ (8.1-18) (8.1-19) the transient solution is found by equating ỵ Gsị ẳ (8.1-20) and since Gsị is a transfer function of the form Gsị ẳ K=1 ỵ Tpị (8.1-21) then ỵ Gsị ẳ ỵ Tpị ỵ K ẳ0 ỵ Tpị (8.1-22) or expressed in a general form, ỵ Gsị ẳ s À r1 Þðs À r2 Þ Á Á Á ðs À rn Þ ðs À Þðs À rb Þ Á Á Á ðs À rm Þ (8.1-23) Therefore the transient conditions are determined from the denominator of Eq (8.1-18) For a control system with several components in the term GH, the general equation will have the form GðsÞ HðsÞ ẳ K1 ỵ a1 s ỵ a2 s2 ỵ ỵ an sn ị sn ỵ b1 s ỵ b2 s2 ỵ ỵ bn sn Þ (8.1-24) which can be factored into roots: Gsị Hsị ẳ K1 ỵ aa sị1 ỵ ab sị ỵ ỵ ỵ an ị sn ỵ ba sị1 ỵ bb sị ỵ ỵ ỵ bn sị (8.1-25) The open-loop transfer function can be written directly from the block diagram, which will have the form of Eq (8.1-25), or in a specific example such as Eq (8.1-8) The nature of the system is determined by the power of sn in the denominator of Eq (8.1-25) As an example, consider the type regulator velocity system: sn ¼ s0 ¼ Copyright 2003 by Marcel Dekker, Inc All Rights Reserved The system is a velocity regulator, and a constant error is required to have a constant controlled variable For a type positioning system, sn ¼ s1 ¼ s The system will have zero error for a constant controlled variable For a type acceleration system, sn ¼ s2 The system will have zero error for a constant controlled variable and can maintain a constant velocity with zero error Block diagram algebra can also be used to find the servo error for a velocity servo drive and a positioning servo drive, as in Figures and Fig Velocity loop block diagram Fig Position loop block diagram Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Error for a Velocity Servo Drive E ¼RÀB (8.1-26) B ¼ CH C ¼ EG (8.1-27) (8.1-28) E ¼ R À EGH Eð1 ỵ GHị ẳ R (8.1-29) (8.1-30) E ẳ R=1 ỵ GHÞ (8.1-31) Using Eq (8.1-8) for an electric servo drive, the error is Ejo ¼ Rjo Ka m KTA þ ð1þjotKÞð1þjotm Þ a (8.1-32) For the steady-state case, jo ẳ Therefore, Eẳ R ỵ Ka Km KTA (8.1-33) Error for a Positioning Servo Drive E ¼ yD À y0 (8.1-34) Using Eq (8.1-10) y0jo ¼ Ejo KD Ka Km joẵ1 ỵ jota ị1 ỵ jotm ị ỵ Ka Km KTA Š (8.1-35) Substituting Eq (8.1-35) into Eq (8.1-35) yields Ejo ¼ yDjo À Ejo KD Ka Km joẵ1 ỵ jota ị1 ỵ jotm ị ỵ Ka Km KTA Š (8.1-36) Solving for E yields Ejo Ejo ! ỵ KD Ka Km ẳ yDjo joẵ1 ỵ jota ị1 ỵ jotm ị ỵ Ka Km KTA yDjo joẵ1 ỵ jota ị1 ỵ jotm ị ỵ Ka Km KTA ẳ ỵ KD Ka Km For the steady-state case, jo ¼ and therefore E ¼ Copyright 2003 by Marcel Dekker, Inc All Rights Reserved (8.1-37) (8.1-38) 8.2 FREQUENCY-RESPONSE CHARACTERISTICS AND CONSTRUCTION OF APPROXIMATE (BODE) FREQUENCY CHARTS Having identified a number of servo drive components mathematically with their transfer functions and placed them into system block diagrams, the next step is to examine how the servo block diagram can be used to analyze servo performance All industrial servo drives are connected to a machine of some kind, which is often referred to as the ‘‘servo plant.’’ These industrial machines have inherent nonlinearities and structural resonances that are sinusoidal Therefore the sinusoidal frequency response method is used to analyze the performance of the servo drive block diagrams For the case of sinusoidal analysis the differential operator (p) or the Laplace operator (s) is replaced by jo Frequency-response characteristics are plotted on semilog paper This is to compress the scales to have the important part of the response on an 8.5 11-in sheet of paper The horizontal or frequency scale is logarithmic The ratio of output to input of the system transfer function, referred to as the magnitude in db, or attenuation rate, is plotted on the vertical scale The vertical scale is compressed using decibels Examples of frequency responses for a single time constant, multiple time constants, and a second-order response follow Single Time Constant For a transfer function E0 =Ei ẳ K=1 ỵ jotị (8.2-1) where: K ¼ 10 t1 ¼ 0:01 sec the frequency response is shown in Figure Note that o1 ¼ 1=t1 The attenuation rate for a single time constant is 20 dB per decade From the relation of a change in power level measured in decibels given as 20 log10 ¼ o2 t=o1 t (8.2-2) the attenuation for a 10 to change in frequency is 20 dB A decade is a 10 to change in frequency A gain to decibel conversion chart is given in Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 22 Nichols chart for Eq (8.3.-2) measure of the machine’s ability to make rapid changes in velocity during machining A simple example can be stated to illustrate this point If it is required to mill a rectangular pocket in a piece of metal, the machine should be capable of milling into a corner at the desired feed, change direction (90 degrees), and have a square corner as a result This is also about as severe a condition possible for a servo to meet In some respects this is an indication of the servo performance capabilities Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 23 Closed-loop frequency response for Eq (8.3-2) The question then arises, how we measure this capability, what criterion we have to compare with? Feedback control system designers have two ways to measure performance These measuring techniques are called the frequency response and the transient step response The Frequency Response What does this frequency response tell us? To understand the relevance of servo loop bandwidth to servo performance it is necessary to define what is meant by servo bandwidth ‘‘Bandwidth’’ is a term describing the frequencyresponse characteristics of a servo drive In its simplest definition, the frequency response is a measure of how well an output of a servo follows the input as a sinusoidal input driving frequency increases At low frequencies the output of a servo faithfully follows the input, so the ratio of output/ input is This is often referred to as the flat part of the response At high frequencies, the output of a servo no longer follows the input The higher the servo bandwidth, the greater the capability of the servo to follow rapid changes in the commanded input Copyright 2003 by Marcel Dekker, Inc All Rights Reserved As the frequency increases beyond the flat part of the response, the servo output begins to lag behind the input signal There is a phase lag and an amplitude attenuation between the output and input When the phase lag reaches 180 degrees the servo is unstable Thus it is a requirement to establish some performance specification to state how much phase lag is acceptable before instability occurs Most industrial axis servo drives use a velocity servo inside a positioning servo system It is critical that the position servo drive have an open-loop phase lag less than 180 degrees to avoid instability of the servo As an index of performance, a maximum of 135 degrees phase lag has been used as a criterion This means there are 45 degrees phase lag left before instability occurs, and this 45 degrees is referred to as the phase margin before the servo goes unstable Thus, the total acceptable phase lag of the internal velocity loop and other components cannot be allowed more than 135 degrees phase lag In any positioning servo system with an internal velocity servo there will be a required integration of the velocity output of the velocity servo to the position output of the position loop This integration component takes place in the servo motor and is measured by the position transducer (often a resolver) Any integration component has a fixed 90 degrees phase lag at all frequencies Thus the total allowable phase lag of 135 degrees for the velocity servo drive and integration is now reduced to 45 degrees (1358–908) for the closed velocity loop servo Therefore the useful velocity servo bandwidth occurs at a frequency where the closed velocity servo loop has a phase lag of 45 degrees (often stated as the 45-degree phase shift frequency) Defining Position-Loop Gain A term used to define the performance criteria of a position servo is referred to as the ‘‘position-loop gain’’ or ‘‘velocity constant.’’ This gain is actually the open-position loop gain The open-loop gain is a parameter that represents the product of the individual gains in the position servo loop without the feedback being closed Without the feedback being closed, a position command will cause the servo drive to have motion The ratio of output/input will therefore be in units of velocity/position As an example, if the input command has the units of inches, the open-position loop output will be inches per second Thus the open-position loop gain, Kn , is in units of (inches per second/inches) or 1/seconds A position-loop gain, in units of 1/ seconds, does not have a practical meaning to many machine control people Copyright 2003 by Marcel Dekker, Inc All Rights Reserved working in industry By making a units conversion of Kv ¼ 1=sec660 sec=min61 in:=1000 mils ¼ 1=sec60:06 in:=min=mil or (inches per minute/mil)616:67 ¼ 1=sec Then the open-position loop gain expressed as some velocity (ipm) divided by distance (mils) has a practical significance The Transient Step Function Response It is also possible to measure the performance of a positioning servo by its response to a step change in input The system block diagram for a positioning system can be represented with a single block with the overall gain Kv, as shown in Figure 24 The gain of the system is then called the velocity constant Kv by definition A step function response (typically a second-order response) can be plotted for a command change in position versus time as shown in Figure 25 The step function response can vary from one extreme of self-sustained oscillations (curve a) to an overdamped system (curve d) Curve c represents a system with critical damping (damping factor ¼ 1), which is considered an ideal performance for a positioning servo drive, and curve b represents a system with approximately one overshoot The transient response of a velocity servo drive can also be used to observe the change in velocity versus time for a step change in input The response will be characteristically like Figure 25 except the controlled variable will be velocity instead of position Velocity servo drives are characteristically adjusted to have a time response of one overshoot Servo Analysis and the Machine Designer Two things, system ‘‘stability’’ and ‘‘accuracy,’’ determine the performance of the positioning system The accuracy of a positioning system is directly related to the drive resolution (see Section 12.4) Each component in the Fig 24 Simplified block diagram for a position loop Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 25 Step function response positioning system contributes to the overall gain Thus the power amplifier, drive motor, gearing, and feedback have individual gains that contribute to the overall system gain when they are cascaded together The important point as far as the mechanical designer is concerned is the mechanical design of the machine parts, drive screw, and gearing as related to the inertias involved System performance or stability is partially dependent on keeping the total reflected inertia to the motor shaft as small as practical (see Section 12.2) It is quite possible the resultant total inertia and the drive resonant frequencies will not make it possible to get the required positioning system performance and some mechanical redesign may be necessary 8.5 SERVO COMPENSATION Once the components of an industrial servo drive have been selected and the block diagram constructed, it is necessary to make sure the drive will be stable Thus it will be necessary to compensate the servo drive This section Copyright 2003 by Marcel Dekker, Inc All Rights Reserved discusses some of the accepted methods of compensating a servo drive to make it stable A stability criterion for servo drives is that the open-loop difference in phase shift of the output referred to the input, for a sinusoidal input, should not be more than 135 degrees (for 45 degrees phase margin with the output lagging the input) If a servo drive motor (actuator) and its associated amplifier (without servo compensation) are the main components of the servo drive, closing the servo drive loop through the feedback will most likely result in an unstable servo drive because of excessive phase lag Thus additional components must be used to make the servo drive stable The addition of these components is referred to as compensating the servo, equalizing the servo, or synthesizing the servo Using Bode frequency response plots for an uncompensated electric velocity servo drive shown in Figure 26a, the output lags the input at 600 rad/sec or 95 Hz This can be ascertained from the open-loop response at dB gain with approximately 170 degrees phase lag At 170 degrees phase lag, the drive will be oscillatory; as seen in the transient response of Figure 26b At 180 degrees phase lag the drive will be unstable Therefore some type of compensation must be added to make the drive stable Fig 26a Bode diagram for an uncompensated electric drive Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 26b Transient response of the uncompensated servo of Figure 26a Compensation in servo drives can be accomplished with analog circuits or digital algorithms The compensation is usually put in the forward loop amplifier These compensating elements are described using frequency-response representations of the actual analog or digital algorithms The first type of classical compensating element is referred to as lead and lag compensation A typical lead compensation has the transfer function of K t1 s ỵ 1ị t2 s ỵ 1ị t1 > t2 (8.5-1) where a lag term is used to put a limit on the differential effect of the compensation Without this limit the differential response would excite unwanted higher frequencies (noise) A frequency plot of a lead and lag compensation is shown in Figure 27 Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 27 Bode diagram for lead compensation The second type of compensation used is the lag and lead compensation, which has a transfer function of K t1 s ỵ 1ị t2 s ỵ 1ị t2 > t1 (8.5-2) The lag compensation has the frequency-response characteristic shown in Figure 28 Also note that these compensations often are identified by the lower frequency term Thus, a lag and lead network would be called a lag and the lead and lag network would be called a lead These lead and lag types of servo compensation can be used to modify the frequency of a servo to make it stable It is desired to decrease the phase lag of the servo output relative to the input for a sinusoidal input A typical example is shown in Figure 29a where the servo of Figure 26a had a lag/lead (shorthand for lag and lead) compensation added to the servo amplifier It should be noted that the lead term in the numerator of the amplifier compensation is the same as one of the motor frequencies in the denominator and thus they will cancel each other With the lag term at o ¼ 0.2 rad/sec, the phase lag will be about 90 degrees when the open-loop response magnitude is equal to dB Since the servo would be unstable for a Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 28 Bode diagram for lag compensation 180 degrees phase lag, the 90 degrees phase lag of Figure 29a leaves a margin of 90 degrees for a stable drive The classical use of lead/lag circuits has been with analog operational amplifiers to compensate servo drives Another servo compensation method is the use of proportional, integral, and differential (PID) algorithms This type of servo compensation is used widely with digital control techniques A PID servo block diagram with the associated approximate (Bode) frequency response is shown in Figure 30 The differential part of the compensation should be used with caution since the amplitude response increases with frequency, which may amplify some undesirable noise In general, many servo drives use proportional plus integral (PI) type compensation The servo block diagram and approximate (Bode) frequency responses are shown in Figures 31 and 32 The foregoing types of servo compensation have an important application with velocity or type servo drives For good servo regulation and minimum servo error, it is desirable to have a large gain at low frequencies With lag/lead compensation (Figure 28), the low-frequency gain should be as high as practical to maintain good servo regulation It is also important to reduce this gain with increasing frequency Therefore the lag term o2 in Figure 28 should start to reduce the gain starting at a low Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 29a Bode diagram for a compensated electric drive frequency such as 0.1 rad/sec Reducing the gain as frequency increases will minimize the excitation of resonant frequencies in the servo drive In addition, the lead term of Figure 28 will provide some phase lead, which will reduce the overall phase lag of the open-loop response of the servo drive The application of lag/lead compensation is shown in Figure 29a, with a stable transient response shown in Figure 29b No servo compensation results in a servo drive that will be oscillatory or unstable (Figure 26b) In addition, the use of high open-loop gain increases the drive stiffness, since the stiffness is proportional to the product of the velocity servo loop gains and the position-loop gain (see Section 12.3) Using PI compensation or PID compensation has the same application as lag/lead compensation It should be noted in Figures 30, 31, and 32 that the integral or lag component of the compensation has no lowfrequency gain limit This results in a servo drive with even higher drive stiffness, and improved servo regulation In commercial servo drives that use digital-type PID servo compensation, the values of Kp (proportional compensation), Ki (integral compensation), and Kd (differential compensation) are usually dimensionless numbers These numbers differ from one manufacturer to another depending on what manufacturer supplies the digital compensation circuitry Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 29b Transient response of the compensated servo of Figure 29a The actual units of the proportional gain ðKp Þ should be A/rpm The actual units of the integral gain ðKi Þ should be A/sec/rpm Since the numerical values assigned to Kp and Ki are different from one manufacturer to another, it is useful to the servo engineer to be able to relate the compensation values of one manufacturer to another in their actual units To illustrate the use of actual compensation values, three servo drive manufacturers are considered The first drive to be considered is a Fanuc brushless DC design using a model 20S/3000 motor The actual proportional compensation can be calculated as follows: Kp ẳ 26prad=secị6min =60secị6260:7626p6206 Jm ỵ JL ị A=rpmị KT Jm ẳ 0:17 kg-cm=sec2 ị motor inertiaị Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 30 PID compensation JL ẳ 0:17 kg-cm=sec2 ị load inertiaị KT ẳ ðkg-cm=AÞ ðmotor torque constantÞ ! ! I Ki Kp s ỵ Ki Ki Kp ẳ ẳ Kp ỵ sỵ1 ¼ V2 s s s Ki Kp Ki t1 ¼ o¼ Ki Kp "K p # ! I Kp Ki jo ỵ ẳ Ki ỵ ẳ Kp jo ! ? ¼ Ki V2 Ki jo jo Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Fig 31 PI compensation Fig 32 Varying Kp compensation Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Kp ẳ 26p 0:17 ỵ 0:17ị 6267626p6206 ẳ 0:894 A=rpmị 60 The actual integral gain can be calculated as follows: Ki ẳ 26prad=secị6min=60 secị626p620ị2 Jm ỵ JL ị KT A=sec=rpmị Ki 26p 0:17 ỵ 0:17ị 626p620ị2 ẳ 80:4 Amp=sec=rpmị 60 The second drive to be considered is an Allen-Bradley brushless DC servo Allen-Bradley uses per unit values for the proportional and integral compensation values The actual proportional compensation can be calculated as follows: Kp ẳ Kp;pu Rated current Aị ẳ A=rpmị 1000 ðrpmÞ The actual integral compensation can be calculated as follows: Ki ẳ Ki;pu Rated current A=secị ẳ A=sec=rpmị 1000 rpmị For example, for the 1326-ABC3E motor: Gain ẳ 30% Kp;pu ¼ 16 Ki;pu ¼ 1482 Rated current ¼ 49 A Kp ¼ 16649=1000 ¼ 0:78 A=rpm Ki ¼ 1482649=1000 ¼ 72:62 A=sec=rpm The third drive to consider is an Indramat vector-controlled induction motor design The actual proportional compensation can be calculated as follows: P ẳ %Pgain 6Proportional68:85 ẳ lb-in.-minị Copyright 2003 by Marcel Dekker, Inc All Rights Reserved Pðlb-in.-minÞ 26p radị ẳ A=rpmị KT lb-in:=Aị revị I ẳ %Igain 6Integral68:85 ẳ lb-in.-min=secị Ki ẳ Ki ẳ Ilb-in.-minị 26pradị ẳ A=sec=rpmị KT lb-in:=Asecị revị For example, for the 2AD180D motor: Proportional ¼ 5:32 N-m-min Integral ¼ 379:8 N-m=sec %Pgain ¼ 0:8 %Igain ¼ 0:4 KT ¼ 17:5 lb-in:=A motor torque constantị P ẳ 0:865:3268:85 ẳ 37:67 lb-in.-min 37:67 626p ¼ 13:52 A=rpm Kp ¼ 17:5 I ¼ 0:46379:868:85 ¼ 1344 lb-in.-min 1344 626p ¼ 482:48 A=sec=rpm Ki ¼ 17:5 Copyright 2003 by Marcel Dekker, Inc All Rights Reserved ... constant error is required to have a constant controlled variable For a type positioning system, sn ¼ s1 ¼ s The system will have zero error for a constant controlled variable For a type acceleration... by Marcel Dekker, Inc All Rights Reserved Fig Servo block diagram algebra Vm ¼ output velocity (controlled velocity), rad/sec The forward loop has the following transfer function: CðjoÞ Ka Km... Hydraulic drive block diagram Copyright 2003 by Marcel Dekker, Inc All Rights Reserved a feedback control system, Csị ẳ Gsị Rsị ỵ Gsị ỵ Gsị ịCsị ẳ Gsị Rsị (8.1-18) (8.1-19) the transient solution