Properness of each element in (5), lim G ik < i,k =,,,n, k i (5) s G ii non-causal delay, following equation in (6) has no pole in right half plane, w

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1 MULTIVARIABLE MD-PID CONTROL SYSTEM DESIGN METHOD AND CONTINUOUS SYSTEM MODELING Takashi Shigemasa and Yasunori Negishi Toshiba Mitsubishi-Electric Industrial Systems Corporation, Tokyo, Japan Abstract In this paper, we propose a multivariable Model Driven (MD)-PID control system which is composed of MD- PID control systems with PD feedback and inverted decoupling, discuss properties of the control system and also introduce a simple continuous system modeling by using CMA-ES and SIMULINK model, numerical example of a TITO process shows the practically effectiveness I INTRODUCTION From the operational point of view, single-loop PID control systems have been widely used for regulatory control systems in Distributed Control System (DCS) even though most industrial processes are basically multivariable and coupling systems In order to control the entire processes optimally, multivariable control methods, such as model predictive control (MPC) and decoupling control are well known For the MPC systems with slow sampling rate, there have been raised some issues by Hugo[, such as difficulties with operation, high maintenance cost, and lack of flexibility, slow regulation for unknown disturbance and low availability of the MPC system Recently inverted decoupling control [, [3, [4, [5 has been attracted attention to instead of normal decoupling control The normal decoupling control is complex to deal with for the anti-reset-windup action and the auto/manual mode, however, the inverted decoupling control is easy to deal with those But stability problem due to using feedback of input signals is remained for the inverted decoupling We have already proposed a Model Driven (MD) PID control[6, [7, [8, [9 which can be obtained a good control performance for a wide class of controlled processes As continuous- system modeling for the process is intuitive and easy to understand for process operators and engineers, we have also tried a continuous- MIMO modeling by using of the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) by Hansen [ and SIMULINK model Where the CMA-ES is a stochastic parameter optimization method of non-linear functions proposed by NHansen and SIMULINK is a trademarks of The MathWorks, Inc We planned an easy handling multivariable control system for DCS systems with good control performance which combine Model-Driven PID control systems with inverted decoupling As we have obtained good results through some simulation studies, in this paper, we propose the new multivariable MD-PID control system for DCS system and a continuous- MIMO modeling by using of the CMA-ES II MULTIVARIABLE MD-PID CONTROL SYSTEM Fig shows a block diagram of the Multivariable MD- PID control system with inverted decoupler D f and PD feedback F diag r SV diag SV filter Gain K cdiag Fig v Q diag Q Filter ~ G diag Model u D f PD feedback F diag Inverted decoupling d Multivariable MD-PID control system G MIMO process where G is a n-inputs and n-outputs controlled process and r, y, v, u and d are n-vectors of set point, process output, internal control output, process input and disturbance, respectively The controlled process G is expressed by a transfer function matrix with n-inputs and n-outputs as in () G G G n G G G n G = () G n G n G nn For the sake of simplicity, by using the Relative Gain Array (RGA) measure introduced by Bristol [, the pairing is set to (y i,u i ) (i =,,,n) Then G diag of a diagonal matrix of G is shown in () y G diag = diag(g,g,,g nn ) () According to the inverted decoupling by Garrido [3, D f becomes as in (3) And (4) shows element-wise expression D f = G diag G I (3) G G G n G G n G G G D f = G n G n G nn G nn In order to operate D f stable, the following three conditions by Garrido et al [3 are necessary (4) 85

2 Properness of each element in (5), lim G ik < i,k =,,,n, k i (5) s G ii non-causal delay, following equation in (6) has no pole in right half plane, which means no unstable pole I D f = (6) By using the inverted decoupling D f, the transfer function matrix from v to y becomes as G diag So the PD feedback F diag in (7) can be designed in order to be matching to (9) and () as first order delay systems with dead proposed by Shigemasa et al [ in elementwisely as shown in appendix The PD feedback F diag is set in (7), F diag = diag(f,f,,f nn ) (7) where a PD feedback of ii loop F ii is expressed as in (8) T f ii s F ii = K f ii i =,,,,,n (8) κ ii T f ii s Matching equations (9) and () are as follows (I G diag F diag ) G diag = G Fdiag (9) G F diag =K F diag (I st F diag ) exp(sl F diag ) () where K F diag, T F diag and L F diag are shown following diagonal matrix K F diag = diag(k F,K F,,K F nn ) () T F diag = diag(t F,T F,,K T nn ) () L F diag = diag(l F,L F,,K L nn ) (3) As the PD loops designed to first order delay systems with dead, main control system is IMC (Internal Model Control) by Morari et al [ So let the gain matrix K cdiag and the internal model G diag are set in (4) and (5) respectively where Λ diag in () and A diag in () are diagonal matrices which work for tuning of each loop Λ diag = diag(λ,λ,,λ nn ) () A diag = diag(α,α,,α nn ) () Based on IMC [, K cdiag, T cdiag and L cdiag are set to as (), (3) and (4) respectively K cdiag = (K F diag ) () T cdiag = T F diag (3) L cdiag = L F diag (4) The process output y, the process input u and the internal control output v are expressed as in (3), (4) and (5) respectively y = (I sλ diag T c diag ) exp(sl C diag) r [I (I sa diag T c diag )(I st c diag ) (I sλ diag T c diag ) exp(sl c diag ) G F diag G diag Gd (5) u = (I D f ) (v F diag y) (6) v = (I st c diag )(I sλ diag T c diag )K cdiag r (I sa diag T c diag )(I sλ diag T c diag ) exp(sl c diag )G diag Gd (7) III TITO PROCESS MODELING Continuous- system modeling for the process is intuitive and easy to understand, we used a very simple continuous- MIMO modeling form input data and output data by using of the CMA-ES and SIMULINK model Outline of the modeling is shown in Fig K cdiag = diag(k c,k c,,k cn ) (4) G diag = (I st c diag ) exp(sl c diag ) (5) where T c diag and L c diag are diagonal matrices as in (6) and (7) respectively T c diag = diag(t c,t c,,t c nn ) (6) L c diag = diag(l c,l c,,l c nn ) (7) Two degrees of freedom Qfilter Q diag of IMC can be set as in (8) Data input CMA-ES Convergence Output of G MVs,PVs Data Simulink Q diag = (I sa diag T c diag )(I st c diag ) (I sλ diag T c diag ) (8) Fig Modeling approach The set point filter SV diag of IMC is set as in (9) SV diag =(I sλ diag T c diag )(I sa diag T c diag ) (9) A TITO (Two Input and Two Output) process [3 tested is shown in (8) 86

3 G = [ 45exp(7s) 5s 539exp(8s) 5s 77exp(8s) 6s 57exp(4s) 6s (8) Firstly we got input data (u,u ) and output data (y,y ) from step responses of the TITO process u u exp( s) exp( s) exp( s) exp( s) ŷ ε ε ŷ dt ε dt ε y y J Fig 5 Comparison of the TITO model responses (Magenta line) by using the converged parameters and the output data(blue line) Fig 3 TITO Block diagram In order to identify the TITO process, a continuous TITO model in SIMULINK model with parameters as shown in Fig 3 are used Fig 4 shows convergence profile of objective function J of IAE in Fig 3 Fig 4 Convergence profile of the objective function J Fig 5 shows comparison of the TITO model responses (Magenta line) by using the converged parameters and the output data(blue line) and shows good convergence The TITO model G converged is shown in (9) Errors on gains, constants and dead s are almost negligible compared to (8) G = [ 45exp(78s) 58s 539exp(89s) 58s 77exp(87s) 6s 57exp(43s) 67s (9) IV DESIGN OF MULTIVARIABLE MD-PID CONTROL SYSTEM The RGA measure as in (3) shows the pairing of (y,u ) and (y,u ) RGA = G() (G() T ) [ = (3) Based on (3) or (4), the inverted decoupling D f for the TITO process becomes to (3) [ 77 D f = 456s 5sexp(s) s 6sexp(4s) Based on (6), (3) is obtained (3) G G G G = 3435exp(4s) 9543exp(46s) = (3) ( 5s)( 6s) As (33) is obtained from (3), the inverted decoupling D f has no unstable pole as in (33) exp(5s) = < (33) So G diag becomes as in (34) from the pairing information G diag = [ 45exp(7s) 5s 57exp(4s) 6s (34) Table shows the design results of the multivariable MD- PID control system with inverted decoupling Case is using λ ii =, α ii = without PD feedback, Case is using active λ ii, α ii as in Table without PD feedback and Case3 is using λ ii =, α ii = with PD feedback as in Table 87

4 TABLE I SUMMARY OF PARAMETERS DESIGNED FOR G AND G Case Case Case Case Case3 Case3 G G G G G G K c /45 /57 /45 /57 /59 /74 T c L c λ 7 5 α 8 K f 5 3 T f κ K T L r y d u r y d u r y d u r y d u Fig 6 A multivariable MD-PID control system with inverted decoupling for MIMO process (Case and Case3) Fig 6 shows the comparison of the responses of multivariable MD-PID control system of case and case3 In the simulation, set-point r (t) was changed from to at t = while r (t) =, disturbance d (t) was applied from to 5 at t=5sec while d (t) =, set-point r (t) was changed from to at t = sec while r (t) = and disturbance d (t) was applied from to 5 at t = 5sec while d (t) = 5 The case3 using PD feedback shows good control performances, such as quick disturbance regulation and quick set-point tracking Fig 7 shows the comparison of the responses of multivariable MD-PID control system of case and case3 the case3 using PD feedback shows good control performances, such as quick disturbance figuration and quick set-point tracking So the PD feedback is effective way to regulate disturbance quickly and to track to set-point quickly under using inverted decoupling We investigated the influence to the other loop on saturation of process input under the multivariable MD-PID control system using the PD feedback and inverted decoupling at case 3 ARW(anti-reset Windup) operation was set to 4 for Fig 7 A multivariable MD-PID control system with inverted decoupling for MIMO process (Case and Case3) r y d u r y d u Fig 8 Responses of the multivariable MD-PID control system using the inverted decoupling and the PD feedback of case 3 under saturation of process inputs u (t) and for u (t) multivariable MD-PID control system with inverted decoupling and PD feedback can be seen the good control structure that does not affect the other loops by using only normal ARW operation We have already obtained good results for other MIMO processes; such as a 3*3 Tyreus distillation column and a 4*4 HVAC process shown in the paper [ by JGarrido et al, but we would like to omit them because those are the same in the methodological approach V CONCLUSIONS AND FUTURE WORKS The paper introduced a new multivariable model-driven (MD) PID control system using inverted decoupling and PD feedback, the control properties and simple continuous modeling approach by using CMA-ES and SIMULINK model and a TITO process application The multivariable 88

5 MD-PID control system with inverted decoupling and PD feedback show good control performance Data driven tuning approach without modeling for MD-PID control system was presented by Shigemasa et al [6, [7 As the multivariable MD-PID control system can work at DCS with with fast control period, a large scale multivariable control system can be developed at DCS in the bottom-up style APPENDIX By applying the partial model matching approach by Kitamori [4, [5 to (7) (), (35) for ii loop can be obtained G ii F ii = ( T iis)exp(l ii s) (35) where G ii can be expanded as denominator series of G ii as in (36), G ii = g ii g ii s g ii s g ii3 s 3 g ii4 s 4 (36) F ii and ( T ii s)exp(l ii s)/ can be expanded as in (37) and (38) by taking a well-known Taylor series expansion method for each transfer functions F ii = K Fii K Fii T Fii ( κ ii )s K Fii T Fiiκ ii ( κ ii )s K Fii T 3 Fiiκ ii( κ ii )s 3 (37) ( T ii s)exp(l ii s) = T ii L ii s T iil ii L ii / s T iil ii / L3 ii /6 s 3 (38) By substituting (36), (37) and (38) to (35) and matching the low-order coefficients at powers of s between both side, nonlinear equations as shown in (39) (43) are obtained = g ii K Fii (39) T ii L ii = g ii K Fii T Fii ( κ ii ) (4) T ii L ii L ii / = g ii K Fii T Fii( κ ii )κ ii (4) REFERENCES [ Hugo, A, Limitations of Model Predictive Controllers Hydrocarbon Processing, 83-88, [ FG Shinskey, Feedback Controllers for the Process Industries, McGraw-Hill Publishing, New York, 994 [3 Garrido, J, V asquez, F, and Morilla, F, An extended approach of inverted decoupling Journal of Process Control,, 55-68, [4 Garrido, J, V asquez, F, and Morilla, F () Inverted Generalized Decoupling for TITO processes, Proceedings of the 8th World Congress The International Federation of Automatic Control Milano (Italy) August 8 - September, [5 Tore Hagglund, Signal Filtering in PID Control, IFAC Conference on Advances in PID Control PID Brescia (Italy), March 8-3, [6 T Shigemasa, M Yukitomo, R Kuwata, A Model-Driven PID Control System and its Case Studies, Proceedings of the IEEE International Conference on Control Applications,Glasgow, pp , [7 Shigemasa T, MYukitomo, YBaba, THattori and RKuwata, Model- Driven PID Controller, INTERMAC, Joint Technical Conference Tokyo [8 ShigemasaT, MYukitomo, Model-Driven PID Control System, its properties and multivariable application, Proceedings of 8th Annual IEEE Advanced Process Control Applications for Industry Workshop, Vancouver, Canada, 4 [9 ShigemasaT,MYukitomo,YBaba and FKojima, On tuning approach for a Model-Driven PID Control Systems, Proceedings of the nd International Symposium on Advanced Control of Industrial Processes, Seoul, 5 [ NHansen, CMA-ES informations hansen/ [ ShigemasaT, YNegishi and YBaba, A TDOF PID CONTROL SYSTEM DESIGN BY REFERRING TO THE MD-PID CONTROL SYSTEM AND ITS SENSITIVITIES, European Control Conference, Zurich, 3 [ M Morari and E Zafiriou Robust Process Control, Prentice-Hall, IncEnglewood Cliffs, NJ,989 [3 Rivera, DE and KS Jun, An Integrated Identification and Control Design Methodology for Multivariable Process System Applications, IEEE Control Systems, Special Issue on Process Control,, No 3, pp 5-37, June, [4 T Kitamori, A design method for control systems based upon partial knowledge about controlled processes, Trans of Society of Instrument and Control Engineers(In Japanese),Vol5, No4, pp , 979 [5 T Kitamori, Partial Model Matching Method conformable to Physical and Engineering Actualities, Preprints of st IFAC Symposium on System Structure and Control, [6 ShigemasaT, YNegishi and YBaba, From FRIT of a PD feedback control system to process modelling and control system design, th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, Caen, France, 3 [7 ShigemasaT, YNegishi and YBaba, FRIT for a PD feedback loop from the results to process modelling and control system design, Trans of Society of Instrument and Control Engineers (In Japanese), Vol49, No7, pp 5-9, 3 [8 ShigemasaT, YNegishi and YBaba, Multivariable MD-PID control system with Inverted Decoupler and PD feedbacks, Trans of Society of Instrument and Control Engineers (In Japanese), Vol49, No9, pp , 3 T ii L ii / L3 ii /6 = g ii3 K Fii T 3 Fii( κ ii )κ ii (4) T ii L 3 ii /6 L4 ii /4 = g ii4 K Fii T 4 Fii( κ ii )κ 3 ii (43) By solving these equations of (39) (43), optimal PD feedback parameters K Fii,T Fii and the first order delay system with dead,t ii and L ii can be obtained 89