Multivariable PID Control Design For Wastewater Systems

Transcription

1 Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- Multivariable PID Control Design For Wastewater Systems N A Wahab, M R Katebi and J Balderud Industrial Control Centre, University of Strathclyde, Glasgow, UK Lisa@eee.strath.ac.uk Abstract his aer investigates the alication of multivariable PID controllers to a wastewater treatment rocess. Four multivariable PID control schemes are investigated. he methods considered are those suggested by Davison, Penttinen- Koivo, Maciejowski and a new method roosed in this aer. All of the methods are suitable for multi-inut multi-outut (MIMO) control loos that exerience loo interaction. he methods, furthermore, requires only simlistic lant models. he erformance of the methods is assessed using a nonlinear benchmark model, and the otimal tuning values are determined using an otimisation method. he simulation results show the significance of the study and lead to the conclusion that the roosed method yields somewhat better results than other methods with resect to decouling caabilities and closed-loo erformance. Keywords: Multivariable PID, Wastewater Systems, Otimisation, controller tuning I. INRODUCION Closed-loo control lays an imortant role in maintaining low oerating costs and adequate effluent water quality for Wastewater reatment Plants (WP). raditionally, scalar PID controllers have been extensively used to control the rocess variables of WPs []. However, due to the inherently multivariable nature of WPs, combined with an increasing demand for a more consistent effluent water quality, scalar PID-based control systems are often no longer sufficient. o meet the current and future demands on effluent water quality multivariable control systems are therefore needed. Existing multivariable control techniques are tyically model-based and require a significant effort and skill to tune. Although the basic rocess of wastewater treatment is similar in most lants, their configuration and inut characteristics are different. herefore, a generic model cannot be used for alications such as model-based control. his combined with the fact that industry accetance for multivariable control techniques generally are low; render the alication of multivariable control techniques cumbersome within the W industry. In an attemt to imrove the W industry accetance of multivariable control techniques, this aer investigates the alication of four different and essentially model-free, multivariable PID control methods to a benchmark model of a WP. he MIMO-PID methods investigated in this aer are those, which require only simle ste or frequency resonse tests to configure and tune. he methods considered are those suggested by Davison [], Penttinen-Koivo [], Maciejowski [4] and a new method roosed in this aer. Common for all of these methods is that the information needed for controller tuning requires only lant ste-tests and/or the determination of the lant frequency resonse at a single frequency. he Davison method introduces decouling at low frequencies by a constant gain comensator, which is the inverse of the lant model at zero frequency. For stable and square linear systems, the method guarantees asymtotic stability and asymtotic tracking for a articular form of disturbances. he Penttinen-Koivo decoules the lant at both low and high frequencies. In Maciejowski s method, the lant is diagonalised at a articular bandwidth frequency to minimize the interaction around the system bandwidth. Based on these existing techniques, a new method is roosed to imrove control erformance while retaining the simlicity of the multiloo strategy in which favour the relevant features required for control of WPs. he model used in the study is based on WP simulation benchmark model develoed by the COS Action 64 & 68 Research Grou. he validated nonlinear model is based on IWA ASM model and the imlementation is carried out in Matlab/Simulink [5]. he aer is organized as follows. In Section II the benchmark model used for assessing the suitability of the different MIMO-PIDs is resented. Linearised models of the benchmark system, which later are used for control design, are derived in Section III using subsace identification techniques. he four different multivariable PID control methods are then described in Section IV. Section V describes the selection of the tuning arameter associated with the MIMO-PIDs. he erformance of the MIMO-PIDs is thereafter assessed in Section VI. Finally, in section VII, conclusions are drawn from the results of the benchmark study. II. HE BENCHMARK PLAN he benchmark lant is comrised of five fully mixed tanks, connected in a series manner as shown in Fig.. In each of the tanks a biological reaction is taking lace that

2 Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- hels to degrade the biodegradable matter of the wastewater. hus, as the wastewater influent flow is assed through the series of tanks its cleanliness rogressively increases. he effluent from the tank system is connected to a clarifier, which is finalizes the cleaning rocess of the waste water (see Fig. ). he biological rocesses that are taking lace in the tanks are modelled by the IAWQ Activated Sludge Model No (ASM) [6], whilst the model used for the clarifier follows (akács et al.,99).[7] In a wastewater treatment system of the tye shown in Fig., the biodegradable matter of the wastewater is rogressively degraded in the tank reactor system by microorganisms that consume dissolved oxygen (DO). he rocess of degrading the biodegradable matter can be divided into two stages: nitrification and denitrification. A sufficiently high concentration of DO is needed to satisfy the nitrification rocess in the aerobic tanks, while a too high DO level will unfavourably affect the denitrification rocess in the anoxic tanks. A control scheme that can adequately maintain the balance of the DO levels in the system during set oint manoeuvring and during influence from external disturbances is consequently essential for an efficient and effective cleaning of the wastewater. Influent ANX ANX AER AER4 Internal recirculation AER5 Recirculated sludge Fig.. Layout of benchmark lant Wastage Effluent III. LINEAR MODELS IDENIFICAION he controller design techniques described in this aer can be successfully alied in ractice using only very simlistic linear rocess models of the lant. he simlicity of these models means that they tyically can be derived directly from data obtained from lant ste-tests. More detailed linear models, if they are available, can also be used, and will often imrove the control design by roviding additional insight into the dynamic behaviour of the rocess. he investigations carried out in this aer emloys linear models of the WP rocess for control design. he motivation for using linear models in this instance is to gain additional insight into the dynamic behaviour of the WP rocess and to allow for a more recise determination of the best controller tuning arameters for each of the control techniques investigated, where the latter will subsequently enable a more objective comarison of the control techniques. he linear rocess models of the WP rocess were obtained using subsace identification techniques. he use of these techniques is almost as straightforward as carrying out ste or frequency resonse tests. he algorithm emloyed was N4SID [8], which exhibit robust numerical roerties and relatively low comutational comlexity. Perturbing the benchmark lant inuts using PRBS signals, whilst recording the resonse on the lant oututs generated data for model identification. During the identification exeriments the amlitude and frequencies of excitation signals were selected such as to maximise the information within the bandwidth of each reactor. he lant oututs of interest were the DO concentrations in aerated tank, 4 and 5. he maniulated inuts of interest were the three air flow rates (K La) to the tanks. Disturbances, in the form of variations of the influent flow rate and influent ammonium concentration, were also considered. he identified model in dry weather conditions was simulated and the studied controller was alied to nonlinear model. he models obtained from the identification exeriments were in discrete form which then converts to the following structure: x& ( t) Fx( t) + G u( t) + G d( t) () y( t) Cx( t) () where y(t) is the outut vector, u(t) is the inut vector, d(t) the measurable disturbance vector and x(t) is a state vector. F, G, G d and C are matrices of aroriate dimensions. he system transfer function is defined as: G s C si F G () ( ) ( ) he identified model given in discrete form was oen-loo stable as for a given oles (.77 ± j.5,.9987,.9699 ) and also controllable and observable. he resulting model had 4 states. o study the loo interactions, the steady state Relative Gain Array, RGA(), Bristol (996)[9] was calculated as follows: G( s) C( si F) G G() CF G (4) d ( ) ( ) RGA() CF G o CF G (5) where o indicates the element-by-element roduct. For the studied influent disturbance, the steady state RGA was calculated as: RGA () (6) Negative off diagonal elements indicate that the corresonding variables should be aired along the diagonal elements. Hence, the DOs in the tanks are controlled by their airflow rates as exected. Big values in the steady state RGA entries indicate strong interactions between the loos.

3 Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- Loos (, ) have the strongest interactions followed by loos (, ) and (, ). Since the controller design methods investigated in this aer emloys decouling at secific frequency oints, it is useful to examine the dynamic RGA and use the resulting information to decoule the system at frequency oints with highest interactions. he DRGA is defined as: scalar arameter, ε, is the controller s single tuning arameter. Since the integral feedback roortional to the inverse of the lant dynamics at zero frequency, the Davison method is exected to rovide good decouling characteristics at low frequencies. DRGA( s) G( s) o G( s) (7) he frequency lots both for magnitude and hase of the DRGA for the dry influent rate is shown in Fig. -a and -b, resectively: Phase (deg) Magnitude (db) Frequency (rad/day) Frequency (rad/day) - -4 Fig.-a DRGA gains for dry influent rate he DRGA demonstrates that the interactions occur mainly at frequencies about a decade below the oen loo bandwidth. he low frequency decouling is therefore most likely to decentralise the control system and minimised the effect of interactions. IV. CONROLLER DESIGN SRAEGIES he control structures and tuning methods associated with the control techniques investigated in this aer are briefly described below. Davison Method: he multivariable PID design method suggested by Davison uses only integral action. he control law is given by: u( s) Ki e( s), (8) s where K i ε G () is the integral feedback gain, G() is zero frequency gain of the oen loo transfer function matrix, G(s), and where e(s) denote the control error. he Fig.-b DRGA hase for dry influent rate Penttinen Koivo Method he design method roosed by Penttinen and Koivo is slightly more advanced than the Davison method. A roortional term has been added to the control law, giving: u( s) Kce( s) + Ki e( s) (9) s where, K ρ( CG ) and K i ε G (). he controllers c roosed by Davison Penttinen-Koivo are similar in the sense that the integral feedback gains of both controllers are linearly related to the inverse of the lant dynamics at zero frequency, and both controllers are therefore exected to rovide good control-loo decouling characteristics at low frequencies. Unlike the Davison controller, the Penttinen-Koivo controller also includes roortional control action, where the feedback gain is linearly related to the inverse of the lant dynamics at high frequencies. herefore, by following the same line of reasoning as above the Penttinen-Koivo controller is exected to exhibit good decouling characteristics also at high frequencies. he term CG corresonds to the initial degree of sloe at each outut in resonse to a unit ste inut: CG y&, L y&, m M O M ym, y & L & m, m ()

4 Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- where m is the number of lant inuts and oututs and y& i, j is the degree of sloe at outut, i, in resonse to a ste inut at inut, j. hat the roduct CG indeed corresonds to the inverse of the lant dynamics at high frequencies can be shown by alying a Laurent series exansion of the transfer function, eq. : CG CFG CF G G( s) () s s s herefore at high frequencies, the K i /s terms are negligible comared to K c so that G(s) CG /s and G(s)K c I/s, thus giving the following closed-loo transfer function: H ( s) L I + GK GK M O M L H n( s) ( ) for large s () he controller has two scalar tuning arameters, ρ and ε, which resectively determine the controller s roortional and integral action. Maciejowski Method he PID design method roosed by Maciejowski builds on the work carried out by Penttinen and Koivo. However, in Maciejowski s controller the roortional and integral feedback gains remain are equal and linearly related to the inverse of the lant dynamics at a articular design frequency, w b ( K ρg ( ), and K ε G ( ) ). he c jw b i jw b controller tuning arameters ρ and ε are scalars. G Since the evaluation of ( jw b ) tyically will yield a comlex matrix, it is in ractice often necessary to emloy a G real aroximation of ( jw b ). his can be achieved by solving the following otimisation roblem: J ( K, Θ) Θ diag( θ,..., θn) jθ jθ [ G( jwb ) K e ] [ G( jwb ) K e ], () By aroriately selecting the matrix K (such that it minimises the above otimisation roblem) the roduct of G(jw b) and K will be as close to the identity matrix as ossible at the design frequency, and therefore rovide good control-loo decouling characteristics around this frequency. he Proosed method Maciejowski s control design technique has many tractable roerties and an intuitive control structure. Initial benchmark results also indicated that the controller was very effective for the control roblem osed by the WP benchmark roblem. However, since Maciejowski s control design technique involves lant frequency analysis exeriments (to obtain the rocess model) it is redicted that industry accetance for the technique will be low. his aer therefore rooses a new control design technique that retains some of the roerties that makes the Maciejowski controller tractable, but eliminates the need frequency analysis. he roosed control design technique assumes the following control structure: u( s) ρke( s) + ε K e( s) (4) s where, K αg() + ( α) CG (5) he roortional and integral feedback gain of the roosed controller is a blend between the inverse of the lant dynamics at zero frequency and the inverse of the lant dynamics at high frequency. hus, rovided the lant has low-ass frequency characteristics, a good aroximation of G ( jw b ) can be obtained by aroriately selecting the additional controller tuning arameter, α. V. SELECION OF HE UNING CONSANS o allow for an objective comarison of the erformance achieved by the multivariable controllers investigated in this aer the tuning arameters of each of the controllers has been selected such that the following enalty function was minimised: ~ J x( t) Qx ~ ( t) + u& ( t) Ru& ( t) (6) he weighting matrices, Q and R, were non-negative definite symmetric matrices, tuned such that adequate closed loo erformance was obtained. It was assumed that the rocess dynamics and controller states could be described using, x &% ( t) Ax % ( t) + Bu( t) (7) y( t) Hx% ( t) (8) where ~ x ( t) [ x( t) v( t) ] and where v(t) denoted the controller integrator states. Under these assumtions the multivariable PID control laws could be exressed using: u( t) Kx% ( t) (9) ( ) u& ( t) Kx% &( t) K Ax% ( t) + Bu( t) () where K [K c K i ]. hen, by substituting eq.() in eq.(6) the following was obtained: (& ) (& ) J x% ( t) Qx% ( t) + x% ( t) K RK x% ( t) () hen by letting, ( ) ( ) Q + A BK K RK A BK M ( F BK ) P + P( F BK) he enalty function could be written as, () 4

5 Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- J x % ( t) Mx % ( t) x % () Px % () () where P denote the solution to the continuous-time Lyaunov equation. hus, for each of the PID control schemes, the controller arameters,θ was selected such that the matrix norm of P was minimised, i.e.: min P Θ (4) In each of the control design cases, the above roblem was solved using a numerical otimisation method. VI. SIMULAION AND RESULS Prior to benchmarking of the controller s the controller tuning arameters were determined by solving the otimisation roblem defined by eq. 4. A linearised model was used for tuning uroses. For the Davison method the otimisation, eq. 4, roduced ε 95, which yielded the following control matrix K i (5) For the Penttinen-Koivo method the otimisation resulted in ρ 66 and ε 5, thus giving the following controller matrices: K c (6) K i (7) he selection of bandwidth frequency in Maciejowski is of imortant for the minimisation of the cross-couling. he DRGA can be used to find the frequency where the highest interactions occur. However, in this instance the bandwidth frequency was selected by the otimiser, which found that 5 rad/day roduced the best results. At this frequency, j j.4.5 j. K G(( j5)) j j j9.8 (8).6 j.. j j4.8 which can be aroximated by, K G(( j5)) (9)..4.4 he remaining controller arameters, ρ and ε were selected (by the otimiser) as: ρ 4 and ε 67. he otimal selection of controller tuning arameters for the combined method (the roosed method) was found to be, ρ, ε and α.995. For α.995, the controller gain matrix is given by: K () he four PID design methods, described reviously, were successfully alied to the COS simulation benchmark. he dynamic (dry) influent flow conditions has been utilised to assess the each controller s ability to resond to set-oint changes and demonstrated controller s erformance with resect to disturbance rejection. able shows a summary of the results obtained for setoint tracking and disturbance rejection using the four control strategies. Fig. shows the closed-loo dynamic erformance statistics. ABLE : DYNAMIC PERFORMANCE OS(%) s (min) SSQ DO Dav e-5 DO4 Dav e-4 DO5 Dav e-4 DO PK e-5 DO4 PK e-4 DO5 PK e-4 DO Mac e-5 DO4 Mac e-4 DO5 Mac e-4 DO Com e-6 DO4 Com.88 7.e-5 DO5 Com e-4 he closed-loo resonse for a set-oint change occurs at 8, and days in DO, DO 4 and DO 5, resectively with the maniulated variable changes in K La, K La4 and K La5 is shown in Fig.4. Notice how the controller tries to comensate for the set-oint changes and disturbances roagated through the system DODav Min. Max. Mean St.Dev DO4Dav DO5Dav DOPK DO4PK DO5PK DOMac DO4Mac DO5Mac DOCom DO4Com DO5Com Fig.. Dry weather erformance statistic 5

6 Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- DO (mg/l) DO 4 (mg/l) DO 5 (mg/l) Dav ison ref erence Penttinen-Koiv o Maciejowski Combined ime (days) KLa (/hr) KLa4 (/hr) KLa5 (/hr) ime (days) Fig. 4. Set oint changes and disturbance rejections Fig. 5 shows a close examination of the different tuning method alied to the DO in reactor when a set oint is given at day. he roosed method exhibited somewhat faster resonses than the controllers from the other design techniques. It rovides slightly faster resonses with only small overshoots to set-oint changes and with minimal settling time as shown in able. he method of Davison is much too oscillatory with tyically large overshoot. Although the erformance of Maciejowski is satisfactory, it uses the more time-consuming sequential identification rocedure for obtaining its tuning constant. he closed-loo resonse for a set-oint change in Penttinen-Koivo is fine. Considering the de-couling caabilities and the trade-off between loo erformance and robustness, the roosed tuning method is most favourable. DO (mg/l) KLa (/h) Reference Penttinen-Koivo Maciejow ski Proosed Davison ime (days) Fig. 5.Controlled and maniulated variables in aeration his aer rooses a new multivariable PID tuning method. he alication of the roosed method and three other multivariable PID methods to a wastewater rocess is assessed. All of these methods require information only from simle ste or frequency tests. Many other test (for rain and constant influent), not reorted for sace reasons, indicate that the roosed combined roduces sensible results. he methods are based on decouling the system at different frequency oints. he aer rooses DRGA to find the best frequency oint for decouling. It was also roosed to fine-tune the controllers using an otimisation rocedure. Finally, extensive simulation studies carried out on a nonlinear model demonstrated that the roosed method gives the best erformance. ACKNOWLEDGMEN he work has been suorted financially by Malaysian Government and Malaysia, University of echnology. his suort is gratefully acknowledged. REFERENCES [] Carl-Fredrik Lindberg, Control and Estimation Strategies Alied to the Activated Sludge Process, PhD thesis, Usala University (997). [] Davison, E., Multivariable uning Regulator. IEEE ransaction on Automatic Control, Volume, Number, (976). [] Penttinen, J. and Koivo, N.H. (98), Multivariable tuning regulators for unknown systems, Automatica, Volume 6, [4] Maciejowski, J. M., Multivariable Feedback Design. st edition, Addison Weslew, Wokingham, England. [5] Co, J., (). COS Action 64- he COS Simulation Benchmark: Descrition and Simulation Manual. Euroean Commission Euroean cooeration in the field of scientific and tech. research. [6] Henze, M., C.P.L.Grady Jr., W.Gujer, G.v.R.Marais and.matsuo (987). Activated sludge model no.. Scientific and echnical Reort No.. IAWQ. [7] akács, I.,G.G.Patry and D.Nolasco (99). A dynamic model of the clarification-thickening rocess. Water Research 5(), 6-7. [8] Van Overschee, P. and B.De Moor (994), N4SID: subsace algorithms for the identification of combined deterministic-stochastic systems. Automatica, [9] Bristol, E.H. (996). On a new measure of interaction for multivariable rocess control. IEEE rans. On AutoControl, AC-, -4. VII. CONCLUSION 6