DESIGN AND REAL TIME IMPLEMENTATION OF MULTIPLE-MODEL CONTROL SOLUTION FOR SOME CLASSES OF NONLINEAR PROCESSES

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LVII (LXI), Fasc. 1, 2011 SecŃia AUTOMATICĂ şi CALCULATOARE DESIGN AND REAL TIME IMPLEMENTATION OF MULTIPLE-MODEL CONTROL SOLUTION FOR SOME CLASSES OF NONLINEAR PROCESSES BY BOGDAN ILIUłĂ and CIPRIAN LUPU Politehnica University of Bucharest, Faculty of Automatic Control and Computers Science Received: November 11, 2010 Accepted for publication: January 3, 2011 Abstract. Multiple model structure is one of the successful solutions for the real-time control of the nonlinear or multi-regime processes. The use of this structure imposes solving some specific problems, like best algorithm selection or control algorithm switching. The main goal of this paper is to provide a method for switching the algorithms of the multiple-models structure, based on the principles of manual to automatic bumbles transfer. The applicability of the method is proved using a real-time structure with an RST control algorithm. The results are tested on a special designed hardware and software experimental platform. Key words: multiple model, switching algorithm, manual-automatic bumpless transfer, real-time control systems. 2010 Mathematics Subject Classification: 34A34, 93C10. 1. Introduction The essential function of a real-time control system is to preserve the closed-loop performances in case of non-linearity, structural disturbances or Corresponding author; e-mail: cip@ indinf.pub.ro

10 Bogdan IliuŃă and Ciprian Lupu process uncertainties. A valuable way to solve these problems is the multiplemodels or multicontroller structure. The first papers that mentioned the multiple-models structure/system have been reported in the 90s. Balakrishnan and Narendra are among the first authors addressing problems of stability, robustness, switching and designing of this type of structures in their papers (Narendra & Balakrishnan, 1997). Research refinement in this field has brought extensions to multiplemodel control concept. Parametric adaptation procedures Closed-Loop Output Error (Landau & Karimi, 1997), use of Kalman filter representation (Lainiotis & Magill, 1969), the use of neural networks (Balakrishnan, 1996) or the fuzzy systems are some of the important developments. Related to classical control loops, musltiple-model based systems need addressing some supplementary specific aspects: Dimension of multiple-model configuration; Selection of the best algorithm; Control law switching. From the multiple-models control systems viewpoint, two application oriented problems can be highlighted: Class of systems with nonlinear characteristic, which cannot be controlled by a single algorithm; Class of systems with different operating regimes, where different functioning regimes don t allow the use of a uniue algorithm or imposes usage of very complex one with special problems on implementation. As function of the process particularity, several multiple-models structures are proposed (Balakrishnan, 1996). One of the most general architectures is presented in Fig. 1. In Fig. 1, the blocks and variables are as follows: 1. Process physical system to be controlled; 2. Command calculus unit that computes the process control law; 3. Status or position identification system component that provide information about the model control algorithm best matching for the current state of the system; 4. Mod. 1, Mod. 2,, Mod. N previously identified models of different regimes or operating points; 5. Alg. 1, Alg. 2,, Alg. N control algorithms designed for the N models mentioned above; 6. SWITCH mix or switch between the control laws; 7. SELECTOR based on adeuate criteria evaluations, provides information about the most appropriate model for the system s current state; 8. y and y1, y2,, yn output of the process and outputs of the models, respectively;

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 11 9. u output generate by Command calculus block; 10. u and u1, u2, un output of the Command calculus block and outputs of the N control algorithms, respectively; 11. r set point system or reference trajectory; 12. p disturbances of physical process. Fig. 1 General scheme for multiple model structure. As noted above, depending on the process specifics and the approach used to solve the control algorithms switching and/or the best model choice problems, the scheme can be adapted on the situation by adding/eliminating some specific blocks. This paper focuses on the switching problem. 2. Control Algorithms Switching The logic operation of multiple model system structure implies that after finding the best algorithm for the current operating point of the, the next step consists in switching the control algorithm. Two essential conditions must be verified with respect to this operation: To be designed so that no bumps in the applications of the control law are encountered;

12 Bogdan IliuŃă and Ciprian Lupu To be (very) fast. Shocks determined by the switching operation cause non-efficient and/or dangerous behaviors. Moreover, a switch determines a slow moving area of action of the control algorithm, which determines at least one performance degradation. These are the main problems to be solved in designing block switching algorithms. From structurally point of view, this block may contain all implementation algorithms or at least the algorithm coefficients. 2.1. Classical Solutions Present solutions (Dumitrache, 2005; Landau et al., 1997) solve more or less this problem and they are based on maintaining in active state all the control algorithms, also called warm state. This supposes that every algorithm receives information about the process output y(k) and set the point value (eventually filtered) r(k), but only the control law u i (k) is applied on the real process, the one chosen by the switching block. This solution does not impose supplementary function logic for the system architecture and, for this reason, the switching time between algorithms is short. The drawback of this approach is that when designing the multi-model structure several supplementary steps are necessary. These supplementary conditions demand the match of the control algorithm outputs in the neighborhood switching zones. The superposition of models identification zones accomplishes this aspect. That can be seen in Fig. 2. As a result of this superposition, the multimodel structure will have an increased number of models. Fig. 2 Superposition of identification zones for two neighbor-models and their corresponding control actions.

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 13 Other approaches (Dussud et al., 2000; Pages et al., 2000) propose the mix of two or more algorithms outputs. The weighting of each control law depends on the distance from the current process operating point and the action zone of each algorithm. Based on this, the switching from an algorithm to another one is done using weighting functions with a continuous evolution in [0 1] intervals. This techniue can be easily implemented using fuzzy approach, An example is presented in Fig. 3. This solution involves solving control gain problems, determined by mixing algorithm outputs. Fig. 3 Algorithms weighting functions for a specified operating position. 2.2. Proposed Solution In this paper, we present a solution that provides very good results for fast processes with nonlinear characteristics. The main idea is that, during the current functioning of multiple-models control systems with N model-algorithm pairs, it is supposed that just one single algorithm is to be maintained active, the good one, and all the other N-1 algorithms rest inactive. The active and inactive states represent automatic, respectively manual, regimes of a control law. The output value of the active algorithm corresponds to the manual control for all the other N-1 inactive algorithms. In the switching situation, when a better Aj algorithm is found, the actual Ai active algorithm is commuted in inactive state, and Aj in active state, respectively. For a bumpless commutation, the manual automatic transfer problems must be solved, and the solution to this is proposed in section 3. The system can be implemented in two variants first - with all inactive algorithms holding on manual regime, or second - just a single operating algorithm (the active one) and activation of the new one after the computation of the currently corresponding manual regime and switching on automatic regime. Both variants have advantages and disadvantages. Choosing

14 Bogdan IliuŃă and Ciprian Lupu one of them necessitates knowledge about the hardware performances of the structure. After a general view, the first variant seems to be more reasonable. In all cases, it is considered that the active algorithm output values represent manual commands for the new selected one. 3. Manual Automatic Bumpless Transfer The implementation practice highlights important problems like manual-to-automatic/automatic-to-manual regime commutations, respectively turning out in/from the control saturation states. Of course, these problems exist in analogical systems and have specific counteracting procedures, which are not applicable on numerical systems. Fig. 4 Proposed solution. The process operation begins on manual regime, this procedure being used as long as the process did not reach the nominal functioning zone. When commutation is done, it is recommended having a very good matching between the set point and process output values. This strategy releases the system of the shocks sent to the actuators. In the following, these facts will be illustrated using an RST control algorithm (Foulloy et al., 2004). 3.1. Practical Considerations About the Real-Time Algorithm Implementation Consider the discrete model of the process: A y k = B u k (1)

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 15 where: A( -1 ) and B( -1 ) polynomials are: na = 1 + 1 +... + na A a a nb = 0+ 1 +... + nb B b b b (2) with: n n. For this model, an RST control algorithm is used (Fig. 4): A B S u k + R y k = T y k (3) where: u(k) algorithm output, y(k) process output, y * (k) trajectory or filtered set point. The corresponding polynomials are: ( ) 1 1 n S s s sn S S = 0+ 1 +... + (4) n nr R R = r0 + r1 +... + r (5) n nt T T = t0+ t1 +... + t (6) The closed-loop control representation is given in Fig. 5. Fig. 5 RST algorithm, two freedom-degrees closed-loop canonical form. The control algorithm in e. (3) can be rewritten as follows: ns nr n 1 T u( k) = siu( k i) ri y( k i) + ti y ( k i) s0 i= 1 i= 1 i= 1 (7)

16 Bogdan IliuŃă and Ciprian Lupu n S, n R, n T express the corresponding polynomials degrees and also the memory dimension for the software implementation of the algorithm. For example, if n R = 2, then it should be reserved three memory locations for the process s output: y(k), y(k-1), y(k-2). Respectively, the same rule applies for u(k) and y * (k). When necessary, an imposed trajectory can be generated using a trajectory model generator: m B m y ( k+ 1) = r( k) (8) A with: A m and B m like: 1 m = 1 + m1 +... + mn A a a 1 m = m0+ m1 +... + mn Am n B b b b Bm Am n Bm (9) For the practical implementation, one must be interested in the control algorithm and, eventually, trajectory generator model. An iteration of the continuous monitoring program implies following steps: Data acuisition process; Trajectory computation (if necessary); Control law computation; Sending the controls to the actuators; Process evolution graphical display; Memory updating control algorithm for the new iteration. For example, the control law computation, when n R = n S = n T = 2 and without trajectory generator (y * (k) = r(k)), is like in the following: 0 ( 1 0 1 0 1 ) 1 u( k) = s u( k) r y( k) r y( k 1) + t y ( k) + t y ( k 1) (10) s and e. (11) gives the algorithm s memory actualization for the next iteration: u k 1 = u k ; y k 1 = y k ; y k 1 = y k. (11) 3.1. Manual/Automate Transfer In real functioning, M A transfer is preceded by driving the process in the nominal action zone. To avoid command s switching bumps, one must respect the following two conditions:

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 17 Process output must be perfectly matched with the set point value; Accordingly with the algorithm complexity (function of the degrees of controller polynomials), the complete algorithm memory actualization must be waited of. Neglecting these conditions lead to bumps in the transfer because the control algorithm output value is computed using the actual, but also the past, values of the command, process and set point, respectively. At the same time, there are situations when the perfect matching between process output and set point value is very difficult to be obtained and/or needs very long time. Hence, the application of this procedure becomes impossible in the presence of important disturbances. In this context, since the algorithm output is the manual command set by operator and the process output depends on command, the set point remains the only free variable in the control algorithm computation. Therefore, the proposed solution consists in the modification of the set point value, accordingly with the existent control algorithm, manual command and process output. Memory updating control algorithm is done similarly as in the automatic regime. For practically implementation it is necessary a supplementary memory location for the set point value. From e. (7), it results the expression for the set point value: ns nr n 1 T y ( k) = siu( k i) + ri y( k i) ti y ( k i) t0 i= 0 i= 0 i= 0 (12) When the set point (trajectory) generator e. (8) exists, keeping all the data in correct chronology must be with respect to the following relation: m Am r( k) = y ( k) (13) B System operation scheme is presented in Fig. 6. Fig. 6 Computation of the set point value for imposed manual command.

18 Bogdan IliuŃă and Ciprian Lupu Concluding, this solution proposes the computation of that set point value that determines, accordingly to the algorithm history and process output, a control eual to the manual command applied by the operator. At the time instant of the M A switching, there are no gaps in the control algorithm memory that could determine bumps. An eventually mismatching between the set point and process output is considered as a simple change of set point value. Moreover, this solution can be successfully used in cases of command limitation. The only inconvenient of this solution is represented by the necessary big computation power when approaching high order systems, which is not, however, a problem nowadays. 4. Experimental Results We have evaluated the achieved performances of the multi-model control structure using a hardware and software experimental platform. The software was developed using LabWindows/CVI from National Instruments (www.ni.com). In Fig. 7, one can see a positioning control system. The main goal is the vertical control of the ball position. Fig. 7 Process experimental platform and cdaq data acuisition devices. Fig. 8 Nonlinear diagram of the process.

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 19 The nonlinear relation between the position Y, [%] and actuator command U, [%] is presented in Fig. 8. One considers three operating points P 1, P 2, and P 3 on the plant nonlinear diagram (Fig. 8). Three different models are identified like: M 1 (0-33%), M 2 (33-55%) and M 3 (55-100%). These will be the zones for corresponding algorithms. Accordingly to the models-algorithms matching zones (Fig. 2) we have identified the models M 1, M 2 and M 3, as being appropriated to the following intervals (0-40%), (30-60%), (50-100%) respectively. For a sampling period T e = 0.2 sec, the least-suares identification method from Adaptec/WinPIM ( * * * Méthodologie WinPIM+TR, 2004) platform identifies the next models: M M M B 1 1 = 0.05234+ 0.11072 2 A 1 1.36318 0.51135 1 = + B 2 2 = 0.02009 0.02549 1 2 A = + 1 1.22793 + 0.40819 2 B 3 3 = 0.021550 0.03885 1 2 A = + 1 1.22911 + 0.412510 3 In this case, we have computed three corresponding RST algorithms using * a pole placement procedure from Adaptech/WinREG ( * * Méthodologie WinPIM+TR, 2004) platform. The same nominal performances are imposed to all systems, through a second order system, defined by the dynamics ω 0 = 1.25, ξ = 1.2 (tracking performances) and ω 0 = 2, ξ = 0.8 (disturbance rejection performances) respectively, keeping the same sampling period as for identification. All of these algorithms control the process in only their corresponding zones. 2 3 R1 = 2.215432 0.801662.927648 + 1.089446 2 3 S1 = 1 0.441128 0.322980 0.235892 2 3 T1 = 6.1327120.354176 + 6.090028.292996 2 3 R2 = 6.180115.431282 5.150584 + 2.460812 2 3 S2 = 1 0.584581 0.261751 0.153668 2 3 T2 = 21.939447 37.041509 + 21.786749 4.625625

20 Bogdan IliuŃă and Ciprian Lupu 2 3 R3 = 4.640871.107348 3.863951 + 1.884269 2 3 S3 = 1 0.559253 0.263288 0.177459 2 3 T3 = 16.556291 27.952848 + 16.441060 3.490662 To verify the proposed switching algorithm (Fig. 4), we designed and implemented in LabWindowsCVI (www.ni.com) a multiple model controller real-time software application which can be connected to the real process. In the figures bellow you can see the user interface and the results. Fig. 9 Switching between first and second controllers using the classical method. Fig. 10 Switching between first and second controllers using the proposed method. Figs. 9 and 10 present comparative evolution for first to second controller switching (pink represents controller output). In first situation (Fig. 9)

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 21 where classic switching solution ( warm controller) was applied, controller s output and the process s output have an important bump evolution. Fig. 11 Switching between second and third controllers using the classical method. Fig. 12 Switching between second and third controllers using the proposed method. The same test, shown in Figs. 11 and 12, is made for comparative evolution for second to third controller switching (green represents controller output). This proves that the proposed solution (Fig. 12) has small/no bumps when switching. 5. Conclusions The method was successfully tested on a hardware and software experimental application with nonlinear characteristic, using a 3 multimodel/controller real-time software application. Moreover, the first variant (with all algorithms active) of the approach was implemented, ensuring the fast switching (one step) between algorithms.

22 Bogdan IliuŃă and Ciprian Lupu With regards to the results obtained in the paper, the switching method can be successfully recommended in multi-model real-time control structures for fast processes. Acknowledgements. This work was supported by CNCSIS IDEI Research Program of Romanian Research, Development and Integration National Plan II, Grant no. 1044/2007. REFERENCES * * * Méthodologie WinPIM+TR pour l'identification et la validation d'un modèle, les calculs, simulations et tests d'un correcteurs numériue PID/RST. Format electronic, 2004. * * * www.ni.com (Softwaresection LabWindowsCVI) Balakrishnan J., Control System Design Using Multiple Models, Switching and Tuning. Ph. D. Diss., University of Yale, USA, 1996. Dumitrache I., Ingineria reglării automate. Politehnica Press, Bucureşti, 2005. Dussud M., Galichet S., Foulloy L., Fuzzy Supervision for Continuous Casting Mold Level Control. IFAC, 2000. Foulloy L., Popescu D., Tanguy G.D., Modelisation, identification et commande des systemes. Edit. Academiei Române, Bucureşti, 2004. Landau I.D., Karimi A., Recursive Algorithm for Identification in Closed Loop: a Unified Approach and Evaluation. Automatica, Vol. 33, 8, 1499523, 1997. Landau I.D., Lozano R., M' Saad M., Adaptive Control. Springer Verlag, London, 1997. Lupu C., Popescu D., Petrescu C., Ticlea Al., Dimon C., Udrea A., Irimia B., Multiple- Model Design and Switching Solution for Nonlinear Processes Control. ISC'08, The 6th Annual Industrial Simulation Conference, 09-11 June, Lyon, France, 71 76, 2008. Lainiotis D.G., Magill D.T, Recursive Algorithm for the Calculation of the Adaptive Kalman Filter Weighting Coefficients. IEEE Transactions on Automatic Control, 14, 2, 215 218, April 1969. Narendra K.S., Balakrishnan J., Adaptive Control Using Multiple Models. IEEE Transactions on Automatic Control, Vol. 42, 2, February, 171 187, 1997. Pages O., Mouille P., Caron B., Multi-Model Control by Applying a Symbolic Fuzzy Switches. IFAC, 2000. PROIECTAREA ŞI IMPLEMENTAREA ÎN TIMP REAL A SISTEMELOR MULTIMODEL PENTRU CONDUCEREA UNOR CLASE DE SISTEME NELINIARE (Rezumat) Structurile de conducere multimodel sunt una din soluńiile de succes pentru conducerea sistemelor neliniare sau cu regimuri multiple. Din punctul de vedere al proiectării, fańă de sistemele clasice, aceste sisteme necesită în mod suplimentar

Bul. Inst. Polit. Iaşi, t. LVII (LXI), f. 1, 2011 23 rezolvarea unor probleme specifice, cum sunt cele legate de alegerea celui mai bun model/algoritm şi/sau comutarea algoritmilor. Unul din principalele scopuri ale articolului este de a prezenta şi valida o modalitate de comutare a algoritmului de reglare bazată pe comutarea manul-automat fără şocuri. Aplicabilitatea metodei propuse este demonstrată pe o structură de conducere în timp real în care algoritmii structurii de conducere multimodel sunt de tip RST (cu două grade de libertate în urmărirea referinńei şi în reglare). O altă secńiune importantă a lucrării este prezentarea implementării soluńiei de conducere propusă pe o arhitectură hardware cdaq şi software dezvoltată în mediul LabWindowsCVI de la NaŃional Instruments. Procesul neliniar, Ńintă, este o instalańie de laborator în care se urmăreşte reglare pe verticală a pozińiei unei bile aflată într-un curent de aer, al cărui debit este controlat.