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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 1. 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) R
B¼E (8.1-3) R
CH ¼ C=G R ¼ C=G þ CH ¼ Cð1 þ GHÞ=G C G ¼ R 1 þ 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 5 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. 1 Servo block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved Fig. 2 Servo block diagram algebra. Vm ¼ output velocity (controlled velocity), rad/sec The forward loop has the following transfer function: CðjoÞ Ka Km ¼G ¼ EðjoÞ ð1 þ jota Þð1 þ jotm Þ (8.1-7) The open-loop transfer function from which the stability is determined is: BðjoÞ Ka Km KTA ¼ GH ¼ EðjoÞ ð1 þ jota Þð1 þ jotm Þ (8.1-8) The closed-loop transfer function is: VmðjoÞ Ka Km 6h ¼ RðjoÞ ð1 þ jota Þð1 þ jotm Þ 1þ Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved 1 Ka Km KTA ð1þjota Þð1þjotm Þ i (8.1-9) Fig. 3 Block diagram algebra. which can also be written as VmðjoÞ Ka Km ¼ RðjoÞ ð1 þ jota Þð1 þ jotm Þ þ ðKa Km KTA Þ VmðjoÞ Ka Km ¼ RðjoÞ ðjoÞ2 tm ta þ joðta þ tm Þ þ ð1 þ Ka Km KTA Þ VmðjoÞ Ka Km =ð1 þ Ka Km KTA Þ ¼ RðjoÞ ðjoÞ2 ðta þtm Þ 1þKa Km KTA þ ð1þK K K Þ jo þ 1 a m TA ð tm ta Þ Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved (8.1-10) (8.1-11) (8.1-12) Fig. 4 Block diagram algebra. A general form for a quadratic is: ðjoÞ2 2d þ jo þ 1 o2m om sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Ka Km KTA om ¼ tm ta Fig. 5 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 6. 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 6 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 VmðsÞ ¼ GaðsÞ GvðsÞ GmðsÞ 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 VmðsÞ GaðsÞ GvðsÞ GmðsÞ ¼ RðsÞ 1 þ GaðsÞ GvðsÞ GmðsÞ 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. 6 Hydraulic drive block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved a feedback control system, CðsÞ ¼ GðsÞ RðsÞ 1 þ GðsÞ ð1 þ GðsÞ ÞCðsÞ ¼ GðsÞ RðsÞ (8.1-18) (8.1-19) the transient solution is found by equating 1 þ GðsÞ ¼ 0 (8.1-20) and since GðsÞ is a transfer function of the form GðsÞ ¼ K=ð1 þ TpÞ (8.1-21) then 1 þ GðsÞ ¼ ð1 þ TpÞ þ K ¼0 ð1 þ TpÞ (8.1-22) or expressed in a general form, 1 þ GðsÞ ¼ ðs
r1 Þðs
r2 Þ    ðs
rn Þ ðs
ra Þðs
rb Þ    ðs
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 ¼ K1 ð1 þ a1 s þ a2 s2 þ    þ an sn Þ sn ð1 þ b1 s þ b2 s2 þ    þ bn sn Þ (8.1-24) which can be factored into roots: GðsÞ HðsÞ ¼ K1 ð1 þ aa sÞð1 þ ab sÞ þ    þ ð1 þ an Þ sn ð1 þ ba sÞð1 þ bb sÞ þ    þ ð1 þ bn sÞ (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 0 regulator velocity system: sn ¼ s0 ¼ 1 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 1 positioning system, sn ¼ s1 ¼ s The system will have zero error for a constant controlled variable. For a type 2 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 7 and 8. Fig. 7 Velocity loop block diagram. Fig. 8 Position loop block diagram. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved Error for a Velocity Servo Drive E ¼R
B (8.1-26) B ¼ CH C ¼ EG (8.1-27) (8.1-28) E ¼ R
EGH Eð1 þ GHÞ ¼ R (8.1-29) (8.1-30) E ¼ R=ð1 þ GHÞ (8.1-31) Using Eq. (8.1-8) for an electric servo drive, the error is Ejo ¼ Rjo Ka Km KTA 1 þ ð1þjot a Þð1þjotm Þ (8.1-32) For the steady-state case, jo ¼ 0. Therefore, E¼ R 1 þ Ka Km KTA (8.1-33) Error for a Positioning Servo Drive E ¼ yD
y0 (8.1-34) Using Eq. (8.1-10) y0jo ¼ Ejo KD Ka Km jo½ð1 þ jota Þð1 þ jotm Þ þ Ka Km KTA  (8.1-35) Substituting Eq. (8.1-35) into Eq. (8.1-35) yields Ejo KD Ka Km jo½ð1 þ jota Þð1 þ jotm Þ þ Ka Km KTA  (8.1-36) Solving for E yields  1 þ KD Ka Km Ejo ¼ yDjo jo½ð1 þ jota Þð1 þ jotm Þ þ Ka Km KTA  yDjo jo½ð1 þ jota Þð1 þ jotm Þ þ Ka Km KTA  Ejo ¼ 1 þ KD Ka Km (8.1-37) Ejo ¼ yDjo
For the steady-state case, jo ¼ 0 and therefore E ¼ 0. Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved (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 6 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 ¼ K=ð1 þ jotÞ (8.2-1) where: K ¼ 10 t1 ¼ 0:01 sec the frequency response is shown in Figure 9. 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 1 change in frequency is 20 dB. A decade is a 10 to 1 change in frequency. A gain to decibel conversion chart is given in Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved