Self-optimizing Control and NCO tracking in the Context of Real-Time Optimization

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1 Proceedings of the 9th International Symposim on Dynamics and Control of Process Systems (DYCOPS 200), Leven, Belgim, Jly 5-7, 200 Mayresh Kothare, Moses Tade, Alain Vande Wower, Ilse Smets (Eds.) WeKeynoteT7. Self-optimizing Control and NCO tracking in the Context of Real-Time Optimization Johannes Jäschke, Sigrd Skogestad Department of Chemical Engineering, NTNU Trondheim, Norway ( Abstract: Thispaperreviewstheroleofself-optimizingcontrol(SOC)andnecessaryconditions of optimality tracking (NCO tracking). We arge that self-optimizing control is not an alternative to real-time optimization (RTO), NCO tracking or model predictive control (MPC), bt is to be seen as complementary. In self-optimizing control we determine controlled variables (CV), that keep the process close to the optimm when a distrbance enters the process. These CVs are controlled at their setpoints sing PID or model predictive controllers. Preferably, the setpoints are kept constant, bt they may also be adjsted sing RTO or NCO tracking. In any case, a good choice of CVs will redce the freqency of setpoint changes by RTO or NCO tracking. When selecting self-optimizing CVs, a set of distrbances has to be assmed, as nexpected distrbances are not rejected in SOC. On the other hand, RTO and NCO tracking adapt the inpts at given sample times withot any assmptions on what distrbances occr. It is only assmed that they occr on a slower time scale than the sampling. Distrbances with highfreqenciesorwhichwhichdonotleadtoasteadystatearenotrejectedoptimally.bysing NCO tracking in the optimization layer and SOC in the control layer below, we demonstrate that the advantages of both methods complement each other. This combination allows fast optimal action for the expected distrbances, while other distrbances are compensated by NCO tracking on a slower time scale. Keywords: Self-optimizing control, Real-time optimization, NCO tracking, Optimal operation,. INTRODUCTION In recent years there has been a plethora of contribtions on optimal operation in literatre. Several approaches and strategies have been independently of each other. Optimal operation methods may be categorized by how the control strategy is determined: Model sed on-line (on-line optimization) Model sed off-line Model not sed at all On-lineoptimizationmakesseofamodelwhichissally pdated sing measrements and which is optimized online in real-time to minimize a predefined cost fnction. This is typically known as real-time optimization (RTO). Intheoff-lineapproach,expensiveonlinecomptationsare avoided, and optimal operation is achieved by designing a smart control strctre. This controlled variable (CV) selection has the objective to transform the economic objectives into control objectives (Morari et al. (980)). A process model is sed to spport decision making in control strctre design, bt it will not be sed for online optimization. Self-optimizing control (Skogestad (2000)) belongs into this category. A third strategy avoids sing a process model at all, bt ses measrements in order to obtain gradient information abot the process. This information is sed to pdate the inpts directly in order to obtain optimal operation. NCO tracking (François et al. (2005)) and extremm seeking control (Krstic and Wang (2000)) represent this category. This paper focses on the two last categories, and is strctred as follows: In the next section we present two methods for implementing optimal operation, NCO tracking (François et al. (2005)) which does not rely on a process model, and the nll-space (Alstad and Skogestad (2007))asarepresentativeofself-optimizingcontrol(sing the model off-line). We discss some properties of selfoptimizingcontrolandncotracking,andtheirrelationto eachother.basedontheresltsweproposetoconsiderthe methods complementary and to se them together. NCO tracking is sed in the RTO layer, while self-optimizing control is sed in the lower, dynamic control layer. To illstrate the presented ideas, we se a dynamic model of a CSTR from Economo and Morari (986) and apply the described methods to it. 2. OPTIMAL OPERATION METHODS In many cases, steady state operation acconts for the largest part of the operating cost, and for these processes significant economical improvements can be achieved by operatingtheplantoptimallyatsteadystate.thereforewe choose to focs on steady state optimization in this work. We formlate the problem of achieving optimal operation as min J(,d) s.t. { plant and constraints () Copyright held by the International Federation of Atomatic Control 593

2 where is the vector of adjstable inpt variables (e.g. valve opening, pmp speed or the setpoint signal to the reglatory control system), and d is a vector of nknowndistrbanceparameters.fortherestofthepaper we assme that the active constraints have been implemented and that () can be re-written as an nconstrained optimization problem. 2. Self-optimizing control sing the nll space method The idea of self-optimizing control has been formlated by Skogestad (2000): Self-optimizing control is when we can achieve an acceptable loss with constant setpoint vales for the controlled variables (withot the need to re-optimize when distrbances occr). Consider the case when the self-optimizing variable c is a linear combination of measrements y, c = Hy, (2) where H is the constant measrement selection matrix. In the nll-space method (Alstad and Skogestad, 2007), we approximate the original optimization problem () locally by a qadratic optimization problem. Theorem. (Alstad et al. (2009)). Givenasfficientnmber of measrements (n y n +n d ) and no measrement noise, select H in the nll space of the optimal sensitivity matrix F, HF = 0, (3) where F = yopt d. (4) Controlling c = Hy to zero yields locally zero loss from optimal operation. The optimal sensitivity F can be obtained nmerically or calclated sing the shorthand notation J = J/, J d = 2 J/( d) and J = 2 J/ 2 as F = G y J J d +G y d, (5) where we se a linearized process model y = G y + G y d d (Alstad et al., 2009). We sketch the proof of the nll space theorem: In a neighborhood of the nominal point, the optimal change in the measrements can be expressed as y opt (d) y opt (d nom ) = F(d d nom ). (6) The optimal variation in the controlled variables c then becomes c opt (d) c opt (d nom ) = HF(d d nom ), (7) and since H is chosen in left nll space of F, the optimal variation c opt (d) c opt (d nom ) = 0. Another approach is to se the insight that at optimal operation the gradient shold be zero, J = 0 (first order necessary optimality condition). Ths, an ideal selfoptimizing control variable is to select c = J. We show in Appendix A that choosing H in the nll space of F is in fact identical to selecting c = J. 2.2 NCO tracking A different approach, followed by François et al. (2005) is the NCO tracking scheme. It is well known, that the Karsh-Khn-Tcker (KKT) necessary conditions for optimality (NCO) of the plant mst hold at the optimal operating point. If a distrbance enters the process, the control scheme is sed to adapt the inpts stepwise in sch a way, that the NCO are satisfied. We do not present the general NCO tracking procedre (with constraints) here, bt we rather give a derivation of the special case withot constraints, i.e. only the sensitivity seeking directions. Then the optimization problem in consideration is minj(,d), (8) and the necessary condition for optimality are: J () = 0 (9) To achieve optimal operation, we pdate the inpt at each sample time k sing the pdate eqation k+ = k +, (0) ntil (9) is satisfied. To obtain the pdate term, we linearize (9) arond the crrent operating point k : J ( k + ) = J ( k )+J ( k ) () Since we want the pdate to force the sensitivity to zero, we set the left hand side of () to zero and solve for (François et al., 2005). = J ( k )J ( k ) (2) This Newton pdate step is exact for a qadratic approximation of the system (8), in the sense that the NCO (9)are satisfiedafteroneiteration.inpracticewedonotapplythe fll pdate step, becase this may lead to feasibility and convergence problems as the process can move otside the region where the qadratic approximation is valid. To avoid this, the pdate term is mltiplied by some tning parameter β [0], sch that k+ = k +β. To evalate (2) we need the derivative J ( k ) for a given inpt k. In this work it is chosen to make a small pertrbation in the inpt and to rn the process for a given time to estimate the gradient by finite differences. The magnitde of the pertrbation is desired to be small in order not to pset the process excessively. At the same time it has to be larger than the process noise to yield sfficient information abot the descent direction. Since the Hessian J ( k ) is difficlt to obtain, it is often determined once at the nominal operating point. Alternatively, as we choose to do in this work, an approximation of the inverse of the Hessian can be obtained by a BFGS pdate scheme. The NCO tracking algorithm is smmarized in Fig.. This procedre is analog to a Newton(like) method in optimization. In the analogy, the steady state operating periods correspond to fnction evalations in the newton procedre, and the soltion is fond when the NCO hold. 3. RELATIONSHIP BETWEEN SELF-OPTIMIZING CONTROL AND NCO TRACKING Both methods prse the same goal, minimization of the operating cost. However, in NCO tracking, we focs on maniplating the inpt vales (at given sample times) to force the sensitivities to zero. Here it shold be noted that the NCO tracking is more general as described above, since the gradient may be obtained by several different(statistical)methods,schascorrelationmethods Copyright held by the International Federation of Atomatic Control 594

3 Copyright held by the International Federation of Atomatic Control 595

4 for the NCO tracking layer are the setpoints for the selfoptimizing control layer. 4. Model 4. SIMULATIONS To illstrate the statements above, we present simlation reslts for a dynamic CSTR with a feed stream F containing mainly the component an and a reversible chemical reaction A B, Fig. 3. The process model is taken from F CA,in C B,in T i Variable Vale Unit Description F holdp min Flow rate C 5000 s Arrhenis factor C s Arrhenis factor 2 C p 000 cal kg K Heat capacity E 0 4 cal mol Activation energy E cal mol Activation energy 2 R.987 cal mol K Ideal gas constant T i K Inpt,in mol/l Conc. A in feed C B,in 0 mol/l Conc. B in feed H rx 5000 cal mol Heat of reaction ρ kg/l Density τ min Time constant Table 3. Nominal vales for the CSTR model.4 Fig. 3. Schematic diagram of a CSTR T C B Economo and Morari (986), and the dynamics of the system are described by following set of eqations: Distrbances.2 CAin CBin d = dt τ (,in ) r (3) dc B = dt τ (C B,in C B )+r (4) dt dt = τ (T i T)+ H rx r ρc p (5) The reaction rate r is defined by r = k k 2 C B (6) where k = C e E RT andk 2 = C 2 e E 2 RT. (7) This process has one maniplated inpt (), the jacket temperatre T i. The expected distrbances enter the process as variations in the feed concentrations,in,c B,in, and the measred variables are, C B, T, and the inpt T i. The objective is to maximize the profit fnction which is a trade-off between heating cost and income from selling prodct B: P = [p CB C B (p Ti T i ) 2 ], (8) Here p CB is the price of the desired prodct B and p Ti is the cost for heating the reactor. The parameter vales are Parameter Vale p CB p Ti Table 2. Objective fnction parameters given in table 2, and the nominal operation vales for all variables are listed in table Simlations First, we control the process for the expected distrbances sing direct NCO tracking. Next we se the nll space 0 Fig. 4. Distrbance trajectories,in,c b,in methodandcomparetheresltswithdirectncotracking. Aftercomparingbothcontrolstrctresforannexpected distrbance we finally combine the methods as in Fig.2. The expected distrbance scenario is given in Fig. 4. After 500 mintes at the nominal vale, the concentration,in varies sinsoidal before retrning to the nominal vale. Then ramp distrbances are introdced, followed by large step distrbances. At 0 mintes, the concentration C B,in makes a step change of 0.3 mole/l. The non-steady state periods (sinsoid and ramp) are inclded to test how the controller behaves in this case, which is likely to happen in reality. Note that strictly speaking, the gradient is not defined, as the process is not at steady state. Direct NCO tracking To obtain the gradient information, the inpt T i is pertrbed with a step of size K. Starting with a positive vale, the sign is altered every forth NCO iteration. Changing the sign of the pertrbation was fond to give better overall performance of the NCO procedre. No steady state detection is implemented in the NCO tracking procedre. Instead, a step test is sed to determine the approximate time for the system to settle down to a new steady state. At the nominal point, the system has a time constant of less than two mintes for an inpt step of T i = 5 K. To let the system settle down far from the nominal point, where the system dynamics are different, a sample time of 0 mintes is chosen for the direct NCO tracking procedre. The step size parameter β is set to. Fig. 5 shows the concentration and temperatre trajectories for the NCO tracking procedre. The control strategy enables stable control. It is frthermore fond that the Copyright held by the International Federation of Atomatic Control 596

5 step distrbances are very well handled. Since the method, C B [mole/l] Reactor temp. [K] C B Fig. 5. NCO tracking, concentrations and temperatre assmes steady state after 0 mintes, and ses the reslts at each sample time for calclating the inpt pdate, it has difficlties handling distrbances which do not lead to a steady state (sinsoidal and ramp). However, the controller manages to keep the system stable dring these periods. It is fond that the performance of NCO tracking algorithm is very sensitive to the tning parameter β, the sample time, and the timing and kind of the distrbance. Self-optimizing control sing the nll-space method Next, the process is controlled sing the nll space method from section 2.. Since we have one inpt and 2 distrbances to compensate for, we need three measrements for the invariant variable combination. We choose two concentrations and the reactor temperatre, so y = [ C B T ] T. We optimize the steady state system at the nominal operating point and then introdce small pertrbations in the distrbance variables d = [,in C B,in ] T. After reoptimizing we calclate F = yopt d = [ ] (9) Then H = [ ], and HF = 0. Using a PID controller, the self-optimizing variable c = Hy is controlled to zero. The concentration and temperatre trajectories with self-optimizing control are plotted in Fig. 6. Comparing inpts and profit for NCO tracking and selfoptimizing control The inpt sage for the two cases described above is qite differently, Fig. 7. While the NCO tracking procedre needs large inpt variations to control the process, the inpt sage of the self-optimizing control strctre is very moderate and smooth. Comparing the profits, Fig. 8, shows that both systems performqitesimilartoeachotheratsteadystateperiods, bt for distrbances, where no steady state is reached within one sample time, NCO tracking is not performing as goodas theself-optimizing control policysingthenllspace method. Using NCO tracking as RTO and self-optimizing control in the lower layer If it can be garanteed that the Concentrations SOC Reactor temp. [K] Fig. 6. SOC, concentrations and temperatre Inpt Ti NCO tracking 395 C B self optimizing control Fig. 7. Inpt sage for SOC and NCO tracking profit NCO profit SOC profit Fig. 8. Profit for SOC and NCO tracking distrbancesinthefeedconcentrationaretheonlyonesentering the process, then sing only self-optimizing control issfficient,andartolayerisnotnecessary.however,the sitation changes for distrbances not anticipated in the control strctre design. Consider a positive step change in the activation energy E 2 of 0% after 3200 min. This distrbance hinders the reverse reaction. Comparing the profits sing the two control strctres, Fig. 9, shows that the self-optimizing control system cannot exploit the improved conditions cased by the nexpected distrbance. Adapting the self-optimizing control setpoints by RTO or NCO can solve this problem, and at the same time redce RTO/NCO sample time. In Fig. 0 the instantaneos profit for direct NCO tracking (sample time: 0 min) and the combined system with a sample time of 25 min is shown. The combined system operates smoother than the Copyright held by the International Federation of Atomatic Control 597

6 profit NCO profit SOC profit Fig. 9. Profit, direct NCO tracking and SOC. Note: nexpected distrbance occrs after 3200 min profit Combined System NCO tracking 0 Fig. 0. Profit, combined NCO/SOC and direct NCO tracking pre NCO system while giving similar performance. In particlar, the inpt is sed less aggressive, Fig.. inpt NCO tracking NCO tracking and SOC combined Fig.. Inpt, combined NCO/SOC and direct NCO tracking UsingonlineRTOtheperformancecoldbeimprovedeven frther becase the setpoints move directly to the optimal vales instead of iteratively approaching them. 5. CONCLUSION De to the limitations of self-optimizing control it can generally not replace an RTO system. However, combining RTO (NCO tracking) and self-optimizing control overcomes the limitations of the local natre of self-optimizing control. At the same time we have fast, dynamic steady state optimal control for the expected distrbances. Since almost every RTO system has a dynamic control system in the layer below, sing a self-optimizing control strctreinthelower layer,improves performanceandcan significantly redce the RTO pdates. For NCO tracking this means less pertrbations for gradient estimation. For an online RTO, this means more time for complex, time intensive, comptations. REFERENCES Alstad, V. and Skogestad, S. (2007). Nll space method for selecting optimal measrement combinations as controlled variables. Ind. Eng. Chem. Res., 46, Alstad, V., Skogestad, S., and Hori, E. (2009). Optimal measrement combinations as controlled variables. Jornal of Process Control, 9(), Economo, C.G. and Morari, M. (986). Internal model control. 5. extension to nonlinear systems. Ind. Eng. Chem. Process Des. Dev., 25, François, G., Srinivasan, B., and Bonvin, D. (2005). Use of measrements for enforcing the necessary conditions of optimality in the presence of constraints and ncertainty. Jornal of Process Control, 5(6), doi:doi: 0.06/j.jprocont Krstic, M. and Wang, H.H. (2000). Stability of extremm seeking feedback for general nonlinear dynamic systems. Atomatica, 36(4), doi:doi: 0.06/S (99) Morari, M., Stephanopolos, G., and Arkn, Y. (980). Stdies in the synthesis of control strctres for chemical processes. part i: formlation of the problem. process decomposition and the classification of the control task. analsysis of the optimizing control strctres. AIChE Jornal, 26(2), Skogestad, S. (2000). Plantwide control: The search for the self-optimizing control strctre. Jornal of Process Control, 0, Appendix A. CONNECTION BETWEEN THE GRADIENT AND THE NULL SPACE METHOD Consider the nconstrained optimization problem [ ][ J J d J d J dd d min J(,d) = min [ T d T ] ]. (A.) In Alstad et al. (2009) it is shown, that if the nmber of measrements is eqal to the sm of inpts and distrbances,n y = n +n d, and the measrement map y = G y [ d] T, is invertible, then we can express H as H = M n [ J 2 J /2 J J d ][ Gy ], (A.2) (A.3) where M n can be freely chosen and is sally set to the identity matrix, M n = I. In this case, however, we choose to set M n = J 2. This gives H = [J J d ][ G y ], (A.4) Calclating the gradient of problem (A.), [ J = [J J d ], (A.5) d] and inserting (A.2), we obtain J = [J J d ][ G y ] y = Hy = c. (A.6) Comparing with (A.4), we see that the nll space method is identical to controlling the gradient. Copyright held by the International Federation of Atomatic Control 598