180 CHAPTER 6 CLOSED LOOP STUDIES Improvement of closed-loop performance needs proper tuning of controller parameters that requires process model structure and the estimation of respective parameters which are discussed in detail in the previous chapter. The present chapter discusses the methods for the design of multi-loop control strategy and tuning of controller parameters. In order to achieve desired quality, specified output characteristics at the cost of spending optimum inputs one needs to design a controller and run the plant under closed loop so that optimal production is achieved under safe operation. 6.1 INTRODUCTION Controllers for MIMO systems can be either multi-loop (Controllers are designed only for diagonal elements of process models of transfer function matrix) or multivariable (Controllers are designed for all the elements of the MIMO transfer function matrix). Multi-loop control scheme has an edge over multivariable as the former can work even if a single-loop fails. In the presence of interactions between input/ output the process needs to be decoupled and then multi-loop controllers can be designed. When interaction effects produce a significant deterioration in control system performance, decoupling control has to be considered. Thus multi-loop or multivariable controller involves the objective of maintaining several controlled variables at independent set points.
181 Many researchers have worked on design of multi-loop controllers for MIMO systems. Luyben (1986) presented a simple practical approach to the problem of finding reasonable controller settings for N SISO controllers in N th order which is a typical industrial multivariable process. Loh et al (1993) proposed the autotuning of multi-loop PI controller. This procedure is the combination of sequential loop closing and relay tuning often used for tuning single PI/ PID controllers. Huang et al (1993) proposed and implemented PID controllers with the form of no proportional kick. Shen et al (1994) proposed multivariable automatic tuning that performs the identificationdesign procedure in a sequential manner which discusses the concept of multivariable autotuner and the underlying theory for sequential design and sequential identification employed in autotuning procedure. Palmor et al (1995) proposed the automatic tuning of decentralized PID controllers for TITO processes. Huang et al (2000) derived tuning rule from inverse based PID controllers. Toh et al (2002) proposed a methodology for autotuning decentralized proportional-integral-derivative (PID) controllers for multivariable systems. Lee et al (2004) proposed a tuning method for multiloop PID controllers by extending the generalized IMC PID tuning method for SISO systems. Liu et al (2005) proposed an analytical multi-loop controller design for industrial and chemical 2-by-2 processes with time delays. Vu et al (2008) proposed a new analytical method based on direct synthesis approach for design of multi-loop PID controllers. This method is aimed to achieve desired closed loop response for MIMO systems with multiple time delays. Lin et al (2009) proposed a systematic procedure to design multivariable controllers that have options for selective decoupling of different structures (e.g. full or partial decoupler). Vu et al (2010) proposed a novel method for independent design of multi-loop PI/ PID controllers. The idea of an effective open-loop transfer function (EOTF) is introduced to decompose multi-loop control system into a set of equivalent independent single-loops. Veronesi and Visoli (2011) proposed a new automatic tuning
182 technique for multi-loop PID controllers applied to MIMO systems. Jeng et al (2011) proposed the methods including model identification, controller monitoring and controller retuning which in turn combined to develop an intelligent control systems. Rajapandiyan and Chidambaram (2012) proposed a method for the independent design of PI/ PID controllers based on equivalent transfer function (ETF) model of individual loops and simplified decoupler matrix. 6.2 DIFFERENT CONTROL DESIGN STRATEGIES MIMO systems came into use in chemical industries as the processes were redesigned to improve efficiency. Multivariable control involves the objective of maintaining several controlled variables at independent set points. Interaction between inputs and output cause a manipulated variable to affect more than one controlled variable. The various control schemes studied here are decentralized, centralized and decoupled systems. In decentralized structure diagonal controllers are used hence, result in system having n controllers whereas in the centralized control systems having n x n controllers. In decoupled systems the process interactions are decoupled before they actually reach and affect the processes. 6.2.1 Centralized Structure Centralized control scheme is a full multivariable controller where the controller matrix is not a mere diagonal one. The decentralized control scheme is preferred over the centralized control scheme mainly because the control system has only n manipulated signals controlling n output variables, and the operator can easily understand the control loops. However, the design methods of such decentralized controllers require first pairing of input-output variables whereas tuning of controllers requires trial and error steps. The centralized control system requires n x n controllers for controlling n output
183 variables using n manipulated variables. While calculating the control action using computer, the problem of requiring n x n controllers does not exist. The advantage of centralized controller is easy to tune even with the knowledge of steady state gain matrix alone thereby multivariable PI controllers can be easily designed. For the centralized structure, Internal model control-proportional integral tuning is adopted based on the studies and recommendations of Reddy et al (1997) on tuning of centralized PI controllers for a Multi-stage flash desalination plant using Davison, Maciejowski and Tanttu-Lieslehto methods. The IMC-PID tuning relations are used in tuning the controller. When a first order system is in the form as follows: k e Dp s p s 1 p, the PI controller settings are K C k l p (6.1) (6.2) I p where l max 1/ 0.7 D,0.2 p p These tuning relations are derived by comparing IMC control with the conventional PID controller and thereby formulating the equations to determine the proportional gain and integral time.
184 6.2.2 Decentralized Structure In spite of developments in advanced controller synthesis for multivariable controllers, decentralized controller remain popular in industries because of the following reasons: 1. Decentralized controllers are easy to implement. 2. They are easy for operators to understand. 3. The operators can easily retune the controllers for change in process conditions. 4. Some manipulated variables may fail. Tolerances of such failures are easily incorporated into the design of decentralized controllers than the full controllers. 5. The control system can be bought gradually into service during process start up and taken gradually out of service during shut down. steps: The design of decentralized control system consists of two main Step 1 Step 2 control structure selection design of SISO controller for each loop. In decentralized control of multivariable systems, the system is decomposed into number of subsystems and individual controllers are designed for each subsystem.
185 For tuning of controller, Biggest Log Modulus Tuning (BLT) method is used (Luyben, 1986), which is an extension of Multivariable Nyquist Criterion and gives satisfactory response. Detuning factor F (typical values are said to vary between 2 and 5) is chosen so that closed-loop log modulus L max cm >= 2n, w L cm = 20log (6.3) 1+w ( p c) w = - 1+ det I + G G (6.4) where G C is n x n diagonal matrix of PI controller transfer functions, G p is n x n matrix containing process transfer functions relating n controlled variables to n manipulated variables. Now the PI controller parameters are given as: K Ci K = CiZ-N (6.5) F Ii F IiZ-N (6.6) where i stands for individual transfer function component, K CiZ-N and IiZ-N are Ziegler-Nichols tuning parameters. These parameters are calculated from the system perturbed in closed loop by relay of amplitude h, reaches a limit cycle whose amplitude a and period of oscillation P u. These parameters are correlated with the ultimate gain (K u ) and frequency ( u) by the following relationships: K u 4h a (6.7)
186 u 2 P u (6.8) Detuning factor F determines the stability of each loop. The larger the value of F more stable the system is but the set point and load responses is sluggish. This method yield settings give a reasonable compromise between stability and performance in multivariable systems. The decentralized scheme is more advantageous in fact that the system remains stable even when one controller goes down and easier to tune because of less number of tuning parameters. However, pairing analysis needs to be done as n! pairings between input/ output. 6.2.3 Decoupled Structure This structure has additional elements called decouplers to compensate for interaction phenomenon. When RGA shows strong interaction then a decoupler is designed. However, decouplers are designed only for order less than 3 as the design procedure becomes more complex as the order increases. The BLT (Luyben, 1986) procedure of tuning the decentralized structure follows the generalized way for all n x n systems as mentioned above. The centralized controllers are tuned using IMC-PI tuning relations which are appropriately selected for first order and second order systems. The decoupled structure adopts the various methods like partial, static and dynamic decoupling to produce the best results. The design equations for general decoupler for n x n systems are conveniently summarized using matrix notations, defined as follows:
187 Transfer function matrix, G G s G s 11 1n G s G s n1 nn Decoupler matrix, D D s D s 11 1n D s D s n1 nn Diagonal matrix of decoupler, H H 11... 0 s H... 22... s H 0... 33 s u u 1... u n ; M M... M 1 n ; y y 1... y n Manipulated variable (new) Manipulated variable (old) Output variable For a decoupled multivariable system output can be written as: where, y GM (6.9) M=Du (6.10) Equation (6.10) becomes: y GDu (6.11) Equation (6.11) becomes: y Hu (6.12) GD=H (6.13)
188-1 (or) D=G H (6.14) which defines the decoupler. For 2-by-2 system, equations are derived for decouplers by taking that loop as well as other interacting loops into account. The scope of the discussion is the decentralized PI controller for MIMO systems and the closed loop responses for set point tracking using BLT tuning and IMC-PID Laurent series. The illustrations of decentralized PI controllers are given in the following section using BLT tuning. 6.3 BIGGEST LOG MODULUS TUNING Luyben proposed BLT tuning involves the following four steps: Step 1 : Calculate the Ziegler-Nichols settings for each individual loop. The ultimate gain and ultimate frequency of each diagonal transfer function G jj (s) are calculated in the classical way. To do numerically, a value of frequency, is guessed. The phase angle is calculated and the frequency is varied to find the point where the Nyquist plot of G jj crosses the negative real axis (the phase angle is -180 degrees). The frequency where it occurs is u. The reciprocal of the real part of G jj is the ultimate gain. Step 2 : Detuning factor F is assumed which is always greater than 1. Typical values are between 1.5 and 4. The gain of all feedback controllers K Ci are calculated by assuming Ziegler- Nichols gain K ZNi by the factor F. K Ci KZNi F
189 where, K K ui ZNi 2.2 Then all feedback controller reset times Ii are calculated by multiplying the Z-N reset times ZNi by the same factor F. Ii ZNi F where Pui ZNi 1.2 The F factor can be considered as detuning factor which is applied to all loops. Larger the value of F more stable the system will be but more sluggish will be the set point and load responses. The method yield settings that give a reasonable compromise between stability and performance in multivariable systems. Step 3 : Using the guessed value of F and the resulting controller settings, a multivariable nyquist plot of scalar function W i 1 det I GM i B is made. The point (-1, 0) is closer to this contour then the system is closer to instability. Therefore the quantity w 1+w will be similar to closed loop G servo transfer function for SISO loop M B 1+ GMB. Based on intuition and empirical grounds, multivariable closed loop log modulus L cm is defined. cm 20log w L 1 w The peak in the plot of L cm over the entire frequency range is the biggest log modulus L max cm.
190 Step 4 : The F factor is varied until L max cm is equal to 2N. Where N is the order of the system. For N=1 in SISO case, the familiar +2dB max closed loop log modulus criterion is obtained. For a 2-by-2 system, a +4dB value of L max cm is used and for 3-by-3 system a +6dB; hence forth. An example given here is 2-by-2 MIMO system (WB column) with transfer function matrix as in Equation (2.1). Step 1: The ultimate gain and ultimate frequency for loop 1 and loop 2 are K u1 = 2.112, P u1 = 3.9 and K u2 = -0.418, P u2 = 11.04. Using Ziegler- Nichols settings the following controllers are obtained for loop 1 and loop 2 that are K C1,ZN 11,ZN = 3.25 and K C2,ZN = 0.19, 12,ZN = 9.2. Step 2: Assuming, F = 2.55. Step 3: The F factor is varied until L max cm is equal to 4 for 2-by-2 MIMO systems. This empirically determined BLT criterion is tested on ten multivariable distillation columns, varying from 2-by-2 system upto 4-by-4 MIMO systems (Luyben, 1986). 6.4 IMC PID LAURENT TUNING IMC-PI controller parameters are derived by equating the closed loop response to desired closed-loop response involving user defined tuning in process transfer function. Controller synthesis procedure using desired closed loop response involves synthesis of coefficient terms s 0, s -1 and s 1 in
191 PID parallel structure are discussed briefly by Panda (2009) in section 4.3.1. Thus, the IMC-PI controller parameters for FOPDT processes are computed using the following expressions: K C D 2 I p ; I p p p p k D D 6.5 RESULTS AND DISCUSSION The parameters of multi-loop controllers using IMC and BLT design for MIMO processes are listed in the following Table 6.1.
192 Table 6.1 Parameters of multi-loop PI controllers for MIMO processes S.no Process BLT design IMC design 1 WB 0.375-0.075 2 VL -1.07 1.97 3 WW 27.4-13.3 4 TS -16.6 70.6 5 OR 1.28-0.251 2.24 6 T1-14.3 0.602-0.21 7 T4-7.04-2.2-0.114 K C I K C 8.29 23.6 7.1 2.58 41.4 52.9 20.6 80.1 19.3 21.1 7.78 5.6 6.98 173 11.3 23.2 24.2 0.3102-0.0975-1.2391 1.0044 27.5573-14.4122-4.0008 8.5810 1.5862-0.2972 3.3905-2.4507 0.3303-0.2243-2.4507-2.4040-0.2834 17.2327 15.3156 I 7.3602 9.458 62.5 37.3565 6.5389 57.233 7.349 5.6806 6.9280 68.3547 2.4088 39.1936 68.3547 4.0055 11.9384 8 DL -0.084-5.16 0.305 0.529 33 15.5 17 11.2-0.118-7.26 0.429 0.743 23.5 11 12.1 7.94 9 A1 0.945 1.22 0.491 0.837 174 18 17.8 66.9 0.3896 1.2820 0.8924 0.1059 47.8713 32.8688 16.7250 34.3824 10 A2 0.393 0.495 0.31 0.927 145 31 31 93.8 0.1461 0.6651 0.8765 0.1194 28.7163 45.7664 19.0727 35.6131
193 To evaluate the output control performance, it is considered a unit setpoint change of all control loops one by one and the integral square error ISE ( e i y i r i ) used to evaluate the control performance. ISE e dt (6.15) 0 2 i The simulation results and ISE values are given in Figures 6.1 to 6.10. The results show that IMC design provides better performance than BLT design. Figure 6.1 Step response and ISE values of multi-loop PI controllers for WB column (solid line: IMC design; dashed line: BLT design)
194 Figure 6.2 Step response and ISE values of multi-loop PI controllers for VL column (solid line: IMC design; dashed line: BLT design)
195 Figure 6.3 Step response and ISE values of multi-loop PI controllers for WW column (solid line: IMC design; dashed line: BLT design)
196 Figure 6.4 Step response and ISE values of multi-loop PI controllers for TS column (solid line: IMC design; dashed line: BLT design)
Figure 6.5 Step response and ISE values of multi-loop PI controllers for OR column (solid line: IMC design; dashed line: BLT design) 19
Figure 6.6 Step response and ISE values of multi-loop PI controllers for T1 column (solid line: BLT design)
Figure 6.7 Step response and ISE values of multi-loop PI controllers for T4 column (solid line: BLT design)
Figure 6.8 Step response and ISE values of multi-loop PI controllers for DL column (solid line: BLT design)
Figure 6.9 line: BLT design) Step response and ISE values of multi-loop PI controllers for A1 column (solid
Figure 6.10 Step response and ISE values of multi-loop PI controllers for A2 column (solid line: BLT design)
203 6.6 OPTIMAL CONTROL DESIGN The main objective of the present work is to capture the disturbance dynamics thereby measure the interaction in terms of area under the closed loop undesired response. The goal is to minimize the interaction by using medium-scale algorithm with termination tolerances for step and objective function in the order of 0.001. The optimization gives the solution for proportional and integral gains (K C, K I ) of the controller after 73 function evaluations. 6.6.1 Parameter Optimization The performance of closed-loop system is measured with single scalar quantity performance index. Performance of each loop needs to be better or improved at the cost of spending optimal/ minimum inputs. Thus the problem is to formulate performance criteria that will lead to find optimal solution of manipulated inputs. It needs to determine control configuration selection and the free parameters of controller that optimizes the performance index. A linear 2-by-2 MIMO process is considered to formulate performance criteria. The optimal controller parameters found from the solution will be used to retune the loops so as to minimize loss through undesirable response and unnecessary disturbance through undesired loops. A commonly used performance criterion is the area under the regulatory response which is given by: J y t dt (6.16) 0 2 This criterion has good mathematical track ability properties which is acceptable in practice as a measure of system performance. Smaller the
204 value J results in small overshoot in the system. Since, the integration is carried out ov all error lasting for long time and thus results in small settling time. Also, a finite J implies that the steady state error is zero. Therefore, a more realistic performance index is of the form: J y t dt subject to the following constraint on control signalu t, max 0 2 u t M for some constant M M determined by the linear range of plant. 2 u t is a measure of instantaneous rate of energy expenditure. To minimize energy expenditure: 0 2 u t dt. To replace the performance criterion the following quadratic performance index as: 2 2 J e t u t dt (6.17) 0 To allow greater generality, a real positive constant inserted to obtain the performance criterion J. can be 2 2 J e t u t dt (6.18) 0 By adjusting the weighting factor, one can weight the relative importance of the system error and the expenditure in energy. Increasing the by giving sufficient weight to control effort, the amplitude of the control
205 signal which minimizes the overall performance index which may be kept within practical bounds although at the expenses of increased system error. In this work, the main intention is to show the amount of interaction obtainable in area calculation of closed loop undesirable/ regulatory responses. For a general MIMO systems cost is minimized by changing the input control signal, u. The control pairing which needs the least area to fulfill its control targets will be the most efficient control pairing. the following steps: The design approach based on parameter optimization consists of Step 0 : Compute the performance index (J) as a function of free parameters k 1, k 2 k n, of the system with fixed configuration. J J k1, k2... k n (6.19) Step 1 : Determine the solution set k i of the equations J k i 0; i 1,2...n (6.20) Equation (6.17) gives the necessary conditions for J to be minimum. From the solution set of these equations, find the subset that satisfies the sufficient conditions which requires that hessian matrix given is positive definite. 2 2 2 J J J... k k k k k 2 1 1 2 1 n 2 2 2 J J J... H k k k k 2 2 1 k2 2 n.... 2 2 2 J J J... k k k k k 2 n 1 n 2 n
206 Since, 2 2 J J k k k k i j j i the matrix H is always symmetric. Step 2 : If there are two or more sets of k i satisfying the necessary and sufficient conditions for minimization of J, then compute the corresponding J for each set. The set which gives the smallest J is optimum set. In this work, the performance of the control system can be adequately specified in terms of settling time, overshoot and steady state error. Thus, the performance index can be chosen as J k 1 (settling time) + k 2 (overshoot) + k 3 (steady state error) (6.21) In this work, the performance index which includes the undesirable system characteristics and in addition good mathematical track ability are presented. The performance indexes often involve integrating closed loop regulatory response when the system is subjected to a standard disturbance such as a step. The system whose design minimizes the selected performance index on controller configuration is the optimal. 6.6.2 Monitoring of Closed Loop Undesired Responses and Redesign of Controllers The optimization toolbox routine offers a choice of algorithms. For constrained minimization, minimax, goal attainment and semi-infinite optimization, variations of sequential quadratic programming are used. Nonlinear least squares problem uses the gauss-newton and levenbergmarquardt methods. To optimize the control parameters in simulink model optsim.mdl (This model can be found in the optimization toolbox optim directory) which
207 includes nonlinear process plant (Equation 2.11) modeled as a simulink block diagram. The problem is to design a feedback control law that tracks unit step input to the system. One way to solve this problem is to minimize the error between the output and the input signal. The variables are parameters of PI controllers. The routine lsqnonlin is used to perform a least squares fit. The function tracklsq must run the simulation. The simulation can be run either in the base workspace or in current workspace. To run the simulation in optsim, the variables K P, K I, a 1 and a 2 (a 1 and a 2 are variables in the plant block) must be defined. K P and K I are the variables that are optimizing here. The simulation is performed using fixed-step fifth-order method to 100 seconds. in Table 6.2: The controller settings before and after optimization are tabulated Table 6.2 Controller parameters and area under undesired response for coupled tanks system Controller parameters Before optimization After optimization k c 28.9520 41.9656 I 1.4476 0.0215 Area Before optimization After optimization 2793.2 2655.2 With these controller settings, the closed loop undesired responses before and after optimization is shown in Figure 6.11.
208 Figure 6.11 Closed loop undesired responses 6.6.3 Performance measure Control effort calculated using this objective function is optimum that saved energy. Thus, for the coupled tanks system gain optimum is calculated as: Gain optimum = 2793.2 2655.2 *100 4.95% 2793.2 Optimum gain in this coupled tanks system is 5% when the disturbance is 10%, thus the control effort is calculated using this objective function which is found to be optimum that saved energy. Hence, the optimized control signal has saved 5% of utility in this particular coupled tanks system.
209 The performance is measured using standard deviation and variances are shown in Table 6.3. Table 6.3 Performance measure Performance measure Before optimization After optimization Standard deviation 0.2397 0.1368 Variance 0.0574 0.0187 6.7 SUMMARY In this chapter multi-loop PI controllers are designed and tuned for all 10 distillation columns for the cited examples. The control effort and performance of closed-loop for all cases are evaluated here. It is found that IMC-PI control tuning rule produces lower IAE with smoother responses. Thus, it can be concluded that IMC-PI works better than BLT method.