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1、ed- ves Ped I :h; I nip- the sur- - lag :on- :ient C H A P T E R 9 -” Laplace-Domain Analysis of Advanced Control Systems In the last chapter we used Laplace-domain techniques to study the dynamics and stability of simple closedloop control systems. In this chapter we apply these same methods to mor
2、e complex systems: cascade control, feedforward control, openloop- unstable processes, and processes with inverse response. We also discuss an altema- tive way to look at controller design that is called “model-based” control. The tools used in this chapter are those developed in Chapters 7 and 8. W
3、e use transfer functions to design feedforward controllers or to develop the characteristic equation of the system and to find the location of its roots in the s plane. 9.1 CASCADE CONTROL Cascade control was discussed qualitatively in Section 4.2. It employs two control loops; the secondary (or “sl
4、ave”) loop receives its setpoint from the primary (or “master”) loop. Cascade control is used to improve load rejection and performance by decreasing closedloop time constants. We can apply cascade control to two types of process structures. If the manipu- lated variable affects one variable, which
5、in turn affects a second controlled variable, the structure leads to series cascade control. If the manipulated variable affects both variables directly, the structure leads to parallel cascade. 9.1.1 Series Cascade Figure 9. la shows an openloop process in which two transfer functions Gi and G2 are
6、 connected in series. The manipulated variable A4 enters Gi and produces a change in Yt . The Yt variable then enters G2 and changes Y2. 301 302PARTTWO: Laplace-Dornain Dynamh and Control (a) Openloop process (bI) Conventional feedback control Secondary controller (slave) Primary controller (master)
7、 (c) Series cascade (d) Reduced block diagram (q) Example 9.1 FIGURE 9.1 Series cascade. (a) Openloop process. (b) Conventional feedback control. (c) Series cascade. (d) Reduced block diagram. (e) Example 9.1. CHAPTER 9: Laplace-Domain Analysis of Advanced Control Systems303 Figure 9. lb shows the c
8、onventional feedback control system, where controller senses the controlled variable Y2 and changes the manipulated M. The closedloop characteristic equation for this system was developed ter 8. 1 + G(s,G(s)Gc(s) = 0 a single variable n Chap- cw Figure 9. lc shows a series cascade system. There are
9、now two controllers. The secondary controller Get adjusts 44 to control the secondary variable Yt . The setpoint signal YTt to the Get controller comes from the primary controller; i.e., the output of the primary controller Gc is the setpoint for the Gcr controller. The Gc2 controller setpoint is Yr
10、 . The closedloop characteristic equation for this system is nut the same as that given in Eq. (9.1). To derive it, let us first look at the secondary loop by itself. From the analysis presented in Chapter 8, the equation that describes this closedloop sys- tem is Y, = GGCI set 1 + GGCI Yl So to des
11、ign the secondary controller Gel we use the closedloop characteristic equa- tion 1 + GIGcl = 0 (9.3) Next we look at the controlled output variable Y2. Figure 9. Id shows the reduced block diagram of the system in the conventional form. We can deduce the closed- loop characteristic equation of this
12、system by inspection. 1 +GzGCI(1 +GzGcr)= 0 However, let us derive it rigorously. Y2 = G2YI Substituting for Yt from Eq. (9.2) gives Y2 = G2 GIGCI pet 1 + GGcl 1 But Yyt is the output from the Gc2 controller. Yyt = GC2(Yrr - Y2) Combining Eqs. (9.6) and (9.7) gives Y2 =c,( 1 +C$+2V;t Y+ + WCZ( pz;c,
13、) = G2Gc2JI +“z;c,)Y; - (9.4) (9.5) (9.6) (9.7) Y2) 304PAKTTWO: Laplace-Domain Dynamics and Control f Rearranging gives G2Gc2 (, :$,) +-(1 :c,)i y ii” (9.8) So Eq. (9.4) gives the closedloop characteristic equation of this series cascade sys- tem. A little additional rearrangement leads to a complet
14、ely equivalent form: Y2 = GGGCIGCI 1 + GGciU + GGcd YF (9.9) An alternative and equivalent closedloop characteristic equation is 1 + G,Gc,(l + G2GC2) = 0 (9.10) The roots of this equation dictate the dynamics of the series cascade system. Note that both of the openloop transfer functions are involve
15、d as well as both of the controllers. Equation (9.4) is a little more convenient to use than Eq. (9.10) because we can make conventional root locus plots, varying the gain of the Gc controller, after the parameters of theG1controller have been specified. The tuning procedure for a cascade control sy
16、stem is to tune the secondary con- troller first and then tune the primary controller with the secondary controller on au- tomatic. As for the types of controller used, we often use a proportional controller in the secondary loop. Since it has only one tuning parameter, it is easy to tune. There is
17、no need for integral action in the secondary controller because we donlt care if there is offset in this loop. If we use a PI primary controller, the offset in the primary loop will be eliminated, which is our control objective. EXAMPLE 9.1. Consider the process with a series cascade control system
18、sketched in Fig. 9.le. A typical example is a secondary loop in which the flow rate of condensate from a flooded reboiler is the manipulated variable M, the secondary variable is the flow rate of steam to the reboiler, and the primary variable is the temperature in a distillation column. We assume t
19、hat the secondary controller Gctand the primary controller Cc2 are both proportional only. In this example GCl= KI cc2 = K2 G, = 1 c;s + l)(S + 1) G2 = -ii- 5s + 1 Conventionalcontrol.First we look at a conventional single proportional controller (K,) that manipulates M to control YFl.The closedloop
20、 characteristic equation is 1 + ($s + l)(S + I)(% + 1) Kc = 0 ;.s + 8s -t +s + 1 + K, = 0(9.12) To solve for the ultimate gain and ultimate frequency, we substitute io for .i. UIAPN:$: Laplace-Domain Analysis of Advanced Control Systems305 10) hat x-s. zan the on- au- r in Lere e if =Y d in sate flo
21、w tion Gc2 -iiw7 - 80 + LJw + I + K,. = 0 (9.13) (-8 + I + K,.) + i( +J - ;u3) = 0 + io Solving the two equations simultaneously for the two unknowns gives K =?t? u 5 and w, = Designing the secondary (slave) loop. We pick a closedloop damping coefficient spec- ification for the secondary loop of 0.7
22、07 and calculate the required value of Ki. The closedloop characteristic equation for the slave loop is 1 -t K, 1 - 0 = ls2+l+1+K (is + l)(s + 1) -*2 I (9.14) Solving for the closedloop roots gives s=-$tiiJm(9.15) To have a damping coefficient of 0.707, the roots must lie on a radial line whose an-
23、gle with the real axis is arccos(0.707) = 45”. See Fig. 9.2. On this line the real and imaginary parts of the roots are equal. So for a closedloop damping coefficient of 0.707 ;=+Jm 3 K,=J4(9.16) Now the closedloop relationship between Y1 and Ypt is 1/5 Y, = GIGI 1 + G ys,t (9.17) (9.18) Designing t
24、he primary (master) loop. The closedloop characteristic equation for the master loop is l+-(l:g)= l+()(s2+js+p)=0 (9.19) 5s3 + 16s2 + ys + ; + ;K2 = 0(9.20) Solving for the ultimate gain K, and ultimate frequency w, by substituting iw for s gives K, = 30.8co, = ,/5.1 = 2.26 It is useful to compare t
25、hese values with those found for a single conventional control loop, K, = 19.8 and w,= 1.61. We can see that cascade control results in higher con- troller gain and a smaller closedloop time constant (the reciprocal of the frequency). Therefore, the system will show faster response with cascade cont
26、rol than with a single loop. Figure 9.2b gives a root locus plot for the primary controller with the secondary controller gain set at i. Two of the loci start at the complex poles s = - $ 5 ii that come from the clo;edloop secondary loop. The other curve starts at the pole s = - i. ? 306fvwr Two: La
27、place-Domain Dynamics and Control Im Kc=0 -2-1 (a) Root locus for secondary loop J K2=0 X,=0 (6) Root ldcus for primary loop Im Is plane - Re f IKu= 30.8 s plane 1 - Re FIGURE 9.2 (n) Root locus for secondary loop. (b) Root locus for primary loop. CHAPTER Y: Laplace-Domain Analysis of Advanced Contr
28、olSystems307 9.1.2 Parallel Cascade Figure 9.3 shows a process where the manipulated variable affects the two con- trolled variables Yt and Y2 in parallel. An important example is in distillation col- umn control where reflux flow affects both distillate composition and a tray temper- ature. The pro
29、cess has a parallel structure, and this leads to a parallel cascade control system. If only a single controller Gc is used to control Yz by manipulating M, the closedloop characteristic equation is the conventional 1 + Gi- ve iat G- bY $I- bY “g CIIAIT:.K 9: Laplace-Domain Analysis of Advanced Contr
30、ol Systems 309 Feedforward control is probably used more in chemical engineering systems than in any other field of engineering. Our systems are often slow-moving, nonlinear, and multivariable, and contain appreciable deadtime. All these characteristics make life miserable for feedback controllers.
31、Feedforward controllers can handle all these with relative ease as long as the disturbances can be measured and the dynamics of the process are known. 9.2.1 Linear Feedforward Control A block diagram of ,a simple openloop process is sketched in Fig. 9.4. The load disturbance LQJ and the manipulated
32、variable Mts,affect the controlled variable YQJ. A conventional feedback control system is shown in Fig. 9.4b. The error signal I?(,) is fed into a feedback controller Gccs)that changes the manipulated variable MC,). Figure 9.4 shows the feedforward control system. The load disturbance L+) still ent
33、ers the process through the GLqs)precess transfer function. The load disturbance is also fed into a feedforward control device that has a transfer function GF(). The feedforward controller detects changes in the load Lt, and adjusts the manipulated variable Mt,). Thus, the transfer function of a fee
34、dforward controller is a relationship between a manipulated variable and a disturbance variable (usually a load change). G A4 F(s) = z = 0 ( manipulated variable disturbance 1 (9.26) (4 Y constant To design a feedforward controller, that is, to find GF(), we must know both GL() and GM(). The objecti
35、ve of most feedforward controllers is to hold the controlled variable constant at its steady-state value. Therefore, the change or perturbation in Yes) should be zero. The output Yc,) is given by the equation Y(s) = G(s, = !(” + j2 5 K,m Again, this controller is physically unrealizable because the
36、order of the numerator is greater than the order of the denominator. We would have to modify our specified S, to make this controller realizable.? This type of controller design has been around for many years. The “pole place- ment” methods used in aerospace systems employ the same basic idea: the c
37、ontroller is designed to position the poles of the closedloop transfer function at the desired lo- cation in the s plane. This is exactly what we do when we specify theclosedloop time constant in Eq. (3.63). 9.5.2 Internal Model Control Garcia and Moral-i (lnd. Eq. Chcm. Pt-oce.ss Des. Dev. 21: 308,
38、 1982) have used a similar approach in developing “internal model control” (IMC). The method gives the control engineer a different perspective on the controller design problem. The basic idea of IMC is to use a model of the openloop process GMM(.J transfer function in such a way that the selection
39、of the specified closedloop response yields a physically realizable feedback controller. Figure 9.11 gives the IMC structure. The model of the process GM() is run in parallel with the actual process. The output of the model Y is subtracted from the actual output of the process Y, and this signal is
40、fed back into the controller GIMC(). If our model is perfect (GM= GM), this signal is the effect of load disturbance on the output (since we have subtracted the effect of the manipulated variable M). Thus, we are “inferring” the load disturbance without having to measure it. This signal is - 1 Model
41、 (a) Basic structure r - - - - -_-_ 1 :+a -, y= 1-_- -JTraditional G, (b) Reduced structure FIGURE 9.11 IMC. la es he in IlY in he .s)* on JS, is CIIAITIK c): Lapluce-Donwin Analysis of Advanced Control Systcrns 329 YI, the output of the process load transfer function, and is equal to GQJLQ). We kno
42、w from our studies of feedforward control Eq. (9.28) that if we change the manipulated variable Mts) by the relationship -CL M(s) = - ( 1GbJl (s) L(s) (9.72) we get perfect control of the the output Ycsj. This tells us that if we could set the controller 1 GIMCW = - G(s) (9.73) . we would get perfec
43、t control for load disturbances. In addition, this choice.of GIMC() gives perfect control for setpoint disturbances: the total transfer function between the setpoint Ysetand Y is simply unity. Thus, the ideal controller is the inverse of the plant. We use this notion again in Chapter 13 when we cons
44、ider multivariable processes. However, there are two practical problems with this ideal choice of the feed- back controller GIMC(). First, it assumes that the model is perfect. More important, it assumes that the inverse of the plant model GM()is physically realizable. This is almost never true sinc
45、e most plants have deadtime or numerator polynomials that are of lower order than denominator polynomials. So if we cannot attain perfect control, what do we do? From the IMC perspective, we simply break up the controller transfer function GIMC() into two parts. The first part is the inverse of GM(+
46、The second part, which Garcia and Morari call a “filter,” is chosen to make the total GIMC() physically realizable. As we will show, this second part turns out to be the closedloop servo transfer function that we defined as Sts) in Eq. (9.64). Referring to Fig. 9.11 and assuming that GM = GM, we see
47、 that Y = GLL + GM.; + (1 - S(,)GL(,)L(,) (9.74) (9.75) (9.76) (9.77) So the closedloop servo transfer function St, must be chosen such that G1h.f) is physically realizable. EXAMPLE 9.8. Lets take the process studied in Example 9.7. The process openloop transfer function is 330 r+v(.r) = 2 L(s) (s +
48、 I)(% + l)($ + 1) “ 332PARTTWO Laplace-Domain Dynamics and Control nf J Derive the feedforward controller transfer function that will keep the process output Yfsj constant with load changes Lcs). ;j -2 -ii 9.2. Repeat Problem 9.1 with 9.3. The transfer functions of a binary distillation column betwe
49、en distillate composition xg and feed rate F, reflux rate R, and feed composition z are3 -=.7 XD K-DF. XD KZeeDZS XD KRe-DR” .-.-I=-=-= F(TFS + 1* 2 (TzS + I)*R7/ m-gzm -ST Kc,) _ e-i(wf+arg Gc;w,) 2i Therefore, Jk-# = IG(iw) 1 sin(ot + arg G(i,) (10.26) (10.27) We have proved what we set out to prove: (1) the magnitude ratio MR is the absolute value of Gts) with s set equal to io, and (2) the phase angle is the argument of G(,J with s set equal to io. 10.3 REPRESENTATION Three different kinds