Essentials of Process Control phần 8 ppsx

424 IVK~~‘IIKEI:: Frequency-Domain Dynamics and Control (a) Sketch Bode and Nyquist plots for this process for a gain K,, = I and a deadtime 1) = 0.5 minutes. (b) Calculate the ultimate frequency o,, and ultimate gain K,, ctnalyticdly for arbi- trary values of gain and deadtime, and confirm your results gt-aphictrl/>~ using the Bode plot from part (a) for the specific numerical values given. (c) Calculate analytically and graphically the value of controller gain that gives a phase margin of 45”. (d) Use a Nichols chart to determine the maximum closedloop log modulus if the gain calculated in part (c) is used. (e) Calculate the TLC settings for a PI controller (K,. = KJ3.2; r/ = 2.2P,,) and generate a Bode plot for GMGc when these controller constants are used. (f) What are the phase margin, gain margin, and maximum closedloop log modulus when this PI controller is used? 11.45. A process has the following openloop transfer function between the controlled variable Y and manipulated variable M: (a) If a proportional analog controller is used, calculate the ultimate gain and ultimate frequency for the numerical values K,, = 2, 70 = 10, and D = I. (6) Sketch Nyquist and Bode plots of G M(iwj. Calculate the value of controller gain that gives a phase margin of 45”. (c) Use a Nichols chart to determine the maximum closedloop log modulus if the controller gain is 3.19. 11.46. A process has an openloop transfer function that is a double integrator. (a) Using a root locus plot, show that a proportional feedback controller cannot pro- duce a closedloop stable system. (b) Using frequency-domain methods, show the same result as in (a). 11.47. The openloop transfer function for a process is KO GM(~) = ~ 7,s + 1 (a) If a proportional-only controller is used, calculate the closedloop servo transfer function Y/pet, expressing the closedloop gain Kc, and closedloop time constant T,I in terms of the openloop gain Ko, the openloop time constant TV,, and the con- troller gain K, Sketch a closedloop log modulus plot for several values of con- troller gain. (b) Repeat part (n) using a proportional-integral feedback controller with reset time r/ = 7,. 11.48. The openloop transfer function G M(.sJ of a process relating the controlled variable Yc,~, and the manipulated variable Mc,s, is a gain K,, = 3 (with units of mA/mA when i (*rr~rwiK f r: FrcqLrcncy-l)ortlairl Analysis of‘C/osctl/oop SysIcrm 425 transmitter and valve gains have been included) and two lirst-order lags in series with time constants 71 = 2 min and 72 = 0.4 min. (n) If a proportional controller is used, sketch a root locus plot. (h) Calculate the controller gain that gives a closedloop damping coefficient of 0.3. (c) What is the closedloop time constant when this gain is used? (d) Make a Bode plot of the openloop system. (e) Using a controller gain of 6.3, calculate the phase margin analytically and graph- ically. 11.49. A deadtime element (D = 0. I min) is added in series with the lags in the process considered in Problem I I .48. (a) Make a Bode plot of the openloop system with a proportional controller and Kc = I. (h) Determine graphically the ultimate frequency and the ultimate gain. (c) Determine graphically the phase margin if a controller gain of 6.3 is used. (d) Determine graphically the maximum closedloop log modulus if a controller gain of 6.3 is used. (e) Calculate the Tyreus-Luyben settings for a PI controller. (.f‘) Determine graphically the phase and gain margins when these settings are used. sfer tant :on- :on- ime able #hen PART FOUR “, ., . Multivariable Processes Perhaps the area of process control that has changed the most drastically in the last two decades is multivariable control. This change was driven by the increasing oc- currence of highly complex and interacting processes. Such processes arise from the design of plants that are subject to rigid product quality specifications, are more energy efficient, have more material integration, and have better environmental per- formance. The tenfold increase in energy prices in the 1970s spurred activity to make chemical and petroleum processes more efficient. The result has been an increasing number of plants with complex interconnections of both material flows and energy exchange. The control engineer must be able to design control systems that give effective control in this multivariable environment. Multivariable systems contain more than one controlled variable and manipulated variable (the type of system we have studied so far). We need to learn a little bit of yet another language! In previous chapters we have found the perspectives of time (English), Laplace (Russian), and frequency (Chinese) to be useful. Now we must learn some matrix methods and their use in the “state-space” approach to control systems design. Let’s call this state-space method- ology the “Greek” language. The next two chapters are devoted to this subject. Chapter 12 summarizes some useful matrix notation and discusses stability and interaction in multivariable sys- tems. Chapter 13 presents a practical procedure for designing conventional multiloop SISO controllers (the diagonal control structure). It should be emphasized that the area of multivariable control is still in an early stage of development. Many active research programs are under way around the world to study this problem, and every year brings many new developments. The methods and procedures presented in this book should be viewed as a summary of some of the practical tools developed so far. Improved methods will undoubtedly grow from current and future research. 477 I 1 I C HAPTER 12 Matrix Representation and Analysis 12.1 MATRIX REPRESENTATION Many books have been written on matrix notation and linear algebra. Their elegance has great appeal to many mathematically inclined individuals. Many hard-nosed en- gineers, however, are interested not so much in elegance as in useful tools to solve real problems. We attempt in this chapter to weed out most of the chaff, blue smoke, and mirrors. Only those aspects that we have found to have useful engineering ap- plications are summarized here. For a more extensive treatment, the readable book Control System Design: An Introduction to State-Space Methods by Bernard Fried- land (1986, McGraw-Hill, New York) is recommended. We use the symbolism of a double underline (A) for a matrix and a single un- derline (x) for a vector, i.e., a matrix with only one zlumn. This helps us keep track of which-quantities are matrices, which are vectors, and which are scalar terms. We assume that you have had some exposure to matrices so that the standard matrix operations are familiar to you. All you need to remember is how the inverse of a matrix, the determinant of a matrix, and the transpose of a matrix are calculated and how to add, subtract, and multiply matrices. 12.1.1 Matrix Properties A host of matrix properties have been studied by the mathematicians. We discuss only the notions of “eigenvalues” and “singular values” since these are valuable in our design methods for multivariable systems. Eigenvalues is simply another name for the roots of the characteristic equation of the system. Singular values give us a measure of the size of the matrix and an indication of how close it is to be- ing “singular.” A matrix is singular if its determinant is zero. Since the determinant 1’9 430 PAW IWCJK: Multivariable Processes appears in the denominator when the inverse of a matrix is taken [ Eq. ( 12. I )], the inverse will not exist if the matrix is singular. rg-’ = [Cofactor[&]) IT Det A (12.1) = A. Eigenvalues The eigenvalues of a square N x N matrix are the N roots of the scalar equation I I Det[hL - A] = 0 (12.2) where A is a scalar quantity. Since there are N eigenvalues, it is convenient to define the vector A, of length N, that consists of the eigenvalues: Al, AZ, A3, . . . , AN. (12.3) We use the notation that the expression AtA] means the vector of eigenvalues of the - matrix 4. Thus, the eigenvalues of a matriG [i + GM Gc] is written &,+G~ cc]. = =xE== EX A MPLE 12. I. Calculate the eigenvalues of the following matrix A. = A= = (12.4) Det[AL - &] = 0 Det[AZ - A] = Det (A + 2) 0 -2 (A + 4) 1 = (A + 2)(A + 4) - (O)(-2) = = A’ + 6A + 8 = 0 = (A + 2)(A + 4) The roots of this equation are A = -2 and A = -4. Therefore, the eigenvalues of the matrix given in Eq. (12.4) are A, = 2 and A2 = -4. -2 hl$ = -4 [ 1 (12.5) n It might be useful at this point to provide some motivation for defining such seemingly abstract quantities as eigenvalues. Consider a system of N linear ordinary differential equations that model a chemical process. ion 1.2) ine 3) the the !.4) the 3 H ch 1rY CIIAIWK 12, Matrifx Representation and Analysis 431 dX -zz= Ax+Bu dt =- = (12.6) where x = vector of the N state variables of the system A = N X N matrix of constants = B = a different N X A4 matrix of constants z = vector of the M input variables of the system - We show in Section 12. I .3 that the eigenvalues of the A matrix are the roots of the characteristic equation of the system. Thus, the eigenTalues tell us whether the system is stable or unstable, fast or slow, overdamped or underdamped. They are essential for the analysis of dynamic systems. B. Singular values The singular values of a matrix are a measure of how close the matrix is to being “singular,” i.e., to having a determinant that is zero. A matrix that is N X N has N singular values. We use the symbol (pi for a singular value. The largest magnitude Cri is called the maximum singular value, and the notation amax is used. The smallest magnitude gi is called the minimum singular value (amin). The ratio of the maximum and minimum singular values is called the “condition number.” The N singular values of a real N X N matrix (i.e., all elements of the matrix are real numbers) are defined as the square root of the eigenvalues of the matrix formed by multiplying the original matrix by its transpose. ai[A] = J h[ATA] i = 1,2, ,N (12.7) = == 12.1. E x A M P L E 12.2. Find the singular values of the A matrix from Example = 2 = [i2 !T4] LT = [-lj2 i4] To get the eigenvalues of this matrix we use Eq. (12.2). Det[A[ - bT4] = 0 8 (A - 16) 1 = 0 = (A - 8)(A - 16) - 64 A’ - 24h + 128 - 64 = 0 = A* - 24A + 64 A, = 20.94 A2 = 3.06 CT] = J%iFi = 4.58 02 = J3.06 = 1.75 (12.8) (12.9) Neither of these singular values is small, so the matrix is not close to being singular. The determinant of A is 8, so it is indeed not singular. n Z-Z 432 PART FOUR: Multivariable Processes EXAMPLE 12.3. Calculate the singular values of the matrix A= -1 1 = [ I 1 -1 Note that the determinant of this matrix is zero, so it is singular. 1 AT = I’ -i = [ 1 ATA = [I’ !J[l’ II] = [_‘, -;“I == Det[AZ - ATA] = 0 = == 2 = = - - - (A - 2) I 0 (A 2)(h 2) 4 A2 - 4A f 4 - 4 = A(A - 4) = 0 A,=0 h2=4 a,=0 a*=2 The singular value of zero tells us that the matrix is singular. n The singular values of a complex matrix are similar to those of a real matrix. The only difference is that we use the conjugate transpose. Oi[A] = = Jh%$j i = 1,2, ,N (12.10) First we calculate the conjugate transpose (the transpose of the matrix with all of the signs of the imaginary parts changed). Then we multiply A by it. Then we cal- culate the eigenvalues. These can be found using Eq. (12.2) f; simple systems. In more realistic problems we use MATLAB or the 1MSL subroutine EIGCC. Note that the product of a complex matrix with its conjugate transpose gives a com- plex matrix (called a “hermitian” matrix) that has real elements on the diagonal and has real eigenvalues. Thus, all the singular values of a complex matrix are real numbers. EXAMPLE 12.4. Calculate the singular values of the complex matrix A = (1 + 4 (1 + 9 = [ (2+i) (1 +i) ACT = t1 +> (2-i) = (1 -i) (1 -i) ACTA = (1-i) (1 +i) (1 ti) = 7 (5 + i) = = (1-i) (l+i) I [ (5 - i) 4 1 Det[Al - ACTA] = 0 = cc 1 0 0 1 I i 7 - (5 - i) (5 + 9 = o 4 II (12.11) n ix. lo) of al- In )te n- i al :a1 1) CIIAIW:.I~ 12. Matrix Representation and Analysis 433 Det (A-7) [ (-5-i) =() (-5 + i) (A - 4) 1 A*-lIA+28-2S-I=A*-llA+2=0 A, = 10.63 A2 = 0.185 uI = 3.29 CT~ = 0.430 Note that the singular values are real. 12.1.2 Transfer Function Representation A. Openloop system Let us first consider an openloop process with iV controlled variables, N ma- nipulated variables, and one load disturbance. The system can be described in the Laplace domain by N equations that give the transfer functions showing how all of the manipulated variables and the load disturbance affect each of the controlled variables through their appropriate transfer functions. YI = G~,,rnl + G~,?rn2 + + GM,,,,mN + GL,L Y2 = GM,, ml + GMz2rn2 + . . . + GM2,,,mN + GL,* L (12.12) YN = GM,, ml + GMN2m2 + . . . + GMNNmN + GL+ L All the variables are in the Laplace domain, as are all of the transfer functions. This set of N equations is very conveniently represented by one matrix equation. Y = G~(,p(s) + &s+(s) - __I (12.13) where Y = vector of N controlled variables Cl = N X N matrix of process openloop transfer functions relating the con- trolled variables and the manipulated variables m = vector of N manipulated variables GL = vector of process openloop transfer functions relating the controlled - variables and the load disturbance Lt,) = load disturbance These relationships are shown pictorially in Fig. 12.1. We use only one load variable in this development to keep things as simple as possible. Clearly, there could be sev- eral load disturbances, which would just appear as additional terms to Eqs. (12.12). Then Lc,r, in Eq. (12.13) becomes a vector, and CL becomes a matrix with N rows and as many columns as there are load disturbances. Since the effects of each of the [...]... vari- umn 3.2) -0. 082 8 98 1 -0. 083 55 48 - 0 3 7 2 8 142 -0 .89 15611 -0.2523673 -0.0002 581 0.0270092 0.01 787 41 0.0 089 766 -0.0391 486 0.1473 784 -0.6 482 796 II (13.4) -0.6 482 796 -0.4463671 -0.245045 1 -0.1 182 182 0 0.052061 = a-ix f the tray nple (13.3) K, = - - UCVT = I/= = 13.1) 459 1 0.7191619 -0.69 484 26 -0.69 484 26 -0.7191619 I (13.5) (13.6) The largest element in the first column in U is -0 .89 15611, which... 0.0134723 0.23 787 52 2.4223 120 5. 783 780 0 1.6 581 630 0.02594 78 -0.1 586 702 -0.10 689 00 -0.05 386 32 The entries in this table are the elements in the steady-state gain matrix of the column K,, which has nine rows and two columns (13.2) CIIAITER lot so nsive aphs ! conIn, its : feed 13: Design of Controllers for Multivariable Processes Now K,, is decomposed into the product of three matrices - -0.00159 68 -0.0361514... ~(.&(.y)l (12.41) The contour of F,,, is plotted in the F plane The number of encirclements of the origin made by this plot is equal to the difference between the number of zeros and the number of poles of Fts, in the right half of the s plane If the process is openloop stable, none of the transfer functions in &ts, has any poles in the right half of the s plane And the feedback controllers in Gee,, are... we take the ijth element of the product of the Kp and [KpwllT matrices EXAMPLE~~.~~ UseEq.(l2.57)tocalculatealloftheelementsoftheWoodandBerry column K P - Kp-’ = = 12 .8 [ 6.6 - 18. 9 - 19.4 1 1 -19.4 18. 9 - 6 6 1 2 8I -123. 58 = ’ 2.01 This is the same result we obtained using Eq (12.56) P21 = (6.6) ( -y5, I = -1.01 Note that the sum of the elements in each row is 1 The sum of the elements in each column... use of the RGA in deciding how to pair variables is not an effective tool for process control applications Likewise, the use of the RGA in deciding what control structure (choice ot S9) CI~AI~~I:K 12: Matrix Representation anti Analysis manipulated and controlled variables) is best is not effective What is important is the ability of the control system to keep the process at setpoint in the face of. .. the development of control structures for these processes Because of their widespread use in real industrial applications, conventional diagonal control structures are discussed These systems, which are also called decentralized control, consist of multiloop SK0 controllers with one controlled variable paired with one manipulated variable The major idea in this chapter is that these SISO controllers should... right half of the s plane Clearly, the poles of Fc,, are the poles of GM~~,G~(~) Thus, if the process is openloop stable, the F,,, function has no poles in= right half of the s plane So the number of encirclements of the origin made by the F’c,~, function is equal to the number of zeros in the right half of the s plane Thus the Nyquist stability criterion for a multivariable openloop-stable process is:... 5s t 1 7.09s t 1 46.2e-9.4” 10.9s + 1 0 .87 (11.61s + l)e-’ (3 .89 s + 1)( 18. 8s + 1) (j&-‘.65 GlLlI.S) = 8. 15s + 1 0.66 KP = = 1.11 -34. 68 ( 12.59) - 0 6 1 -0.0049 -2.36 -0.012 46.2 0 .87 Table 12.2 gives a MATLAB program using Eq (12.57) to calculate the elements of the 3 X 3 RGA matrix 1.96 RGA = i -0.67 -0.29 -0.66 1 .89 -0.23 -0.30 -0.22 i 1.52 Note that the sums of the elements in all rows and all columns... SELECTION OF CONTROLLED VARIABLES 13.2.1 Engineering Judgment Engineering judgment is the principal tool for deciding what variables to control A good understanding of the process leads in most cases to a logical choice of what needs to be controlled Considerations such as economics, safety, constraints, and the availability and reliability of sensors must be factored into this decision We must control. .. boilup V Ii 1 - 18. 9 R -19.4 v ( 12. 48) If the pairing had been reversed, the steady-state gain matrix would be - 18. 9 -19.4 I 12 .8 V 6.6 IC R I (12.49) and the NI for this pairing would be NI = -J 60 Det[Kpl n ;=, KP,,, = (- 18. 9X6.6) - (12.8X-19.4) = -o 991 (- 18. 9)(6.6) Therefore, the pairing of xg with V and XB with R gives a closedloop system that is a “integrally unstable” for any controller tuning . of zeros and the number of poles of Fts, in the right half of the s plane. If the process is openloop stable, none of the transfer functions in &ts, has any poles in the right half of. (12.6) where x = vector of the N state variables of the system A = N X N matrix of constants = B = a different N X A4 matrix of constants z = vector of the M input variables of the system - We. we can tune each controller independently. As we will soon see, the dy- namics and stability of this multivariable closedloop process depend on the settings of all controllers. Controller structures