LOW ORDER MIMO CONTROLLER DESIGN FOR AN ENGINE DISTURBANCE REJECTION PROBLEM P.Dickinon, A.T.Shenton Department of Engineering, The Univerity of Liverpool, Liverpool L69 3GH, UK Abtract: Thi paper compare low order MIMO controller technique uitable for an engine idle peed problem. One method of mapping the maximum ingular value into parameter plane to create an interactive technique for meeting the nominal performance H norm contraint i dicued. Two other controller technique are applied to the ame multivariable idle peed diturbance rejection problem. The relative merit of each deign are dicued and imulation reult preented. Keyword: Fixed order, Frequency domain, Idle peed, MIMO, Parameter pace, PID.. INTRODUCTION The idle peed problem i a compromie between low engine peed to economie on fuel and the ability to atifactorily reject diturbance (Hrovat and Sun, 22). Torque diturbance due to electrical demand on the alternator, power-teering pump or air conditioning unit can quickly load the engine cauing a reduction in peed which can caue undeirable noie and vibration characteritic for the driver, or poibly engine tall. The problem of regulating peed ha typically required careful control of the air entering the engine by either an electronic throttle or air-bypa valve. For fater repone to diturbance, though at the cot of engine efficiency, the park channel may be ued where thi i retarded from maximum bet torque (MBT) around the idle peed to allow additional control action a neceary. During diturbance the fuel channel i typically ued to maintain the air to fuel ratio (AFR) a cloe to toichiometric a poible. (Ganagopadhyay and Meckl, 2) ugget a different approach to the idle problem for a natural ga automotive engine, where air and fuel are input to control the peed and AFR. Thi formulation of the idle peed problem i intereting ince it allow the park to be kept at MBT and the fuel i then ued to aid in diturbance rejection. Including the fuel i uch away alo avoid having equential SISO loop which can often give le performance or tability. The model in (Ganagopadhyay and Meckl, 2) i difficult to control due to the large amount of interaction, making it poorly uited to SISO loopby-loop control. One major contraint when deigning controller for indutrial application i the order of the controller. Fixed, low order controller are widely implemented in engineering ytem due to their well undertood characteritic, low computational demand and eae of tuning. In automotive application, moreover, validated commercial engine management oftware code ( trategie ) generally require controller to be implemented in fixedorder look-up table format. H theory provide a powerful method for deigning controller, thee are generally of high order and o may not be
implemented directly and then require order reduction. Parameter pace technique on the other hand offer a way of generating low order controller directly. For thee method the controller order may be fixed and o can be made independent of the model order. However for ytem with very complicated dynamic it may become difficult or impoible to meet everal pecification imultaneouly with a controller of a particular order. In thee cae either the fixed order hould be increaed or the pecification hould be relaxed. Frequency domain parameter pace method (Shafiei and Shenton, 997), (Beon and Shenton, 999), and (Beon and Shenton, 2) offer ignificant advantage over other controller deign technique, ince the frequency repone of the plant found from non-parametric identification can be ued directly and a uch irrational ytem including ytem with multiple pure time delay can be handled directly, without the need for rational approximation. Thi i in ditinction to H method where rational repreentation are required. Furthermore, the frequency repone approach i treated uniformly for continuou or dicrete ytem. A further important merit of the thee method over baed technique i that weighting function do not need to be proper or indeed rational. 2. IDLE SPEED ENGINE MODEL A linearied mean value dicrete model of a natural ga engine around the idle operating point i taken from (Ganagopadhyay and Meckl, 2) where alo a novel eigenvalue PI controller deign technique i uggeted. The control input to the plant are throttle, δα (% max ) and fuel, δṁ fi (lbm/h) the diturbance input i torque, T (Nm) and the output are engine peed, δn (rpm) and AFR (δl). The problem i preented a one of diturbance rejection and therefore the objective i to deign a MIMO controller that minimie peed underhoot when a torque load uch a air conditioning or power teering i applied uddenly to the engine. The plant ha ignificant channel interaction and therefore a MIMO controller i uggeted. In (Ganagopadhyay and Meckl, 2) it i demontrated that uing a PI controller adequate performance cannot be achieved uing two SISO Zeigler- Nichol controller, nor with a MIMO method uggeted in (Peltomaa and Koivo, 983). The leanburn natural ga engine plant model wa given a e e 2 K δt G d G u y G 2 G 2 u 2 y 2 G 22 Fig.. Engine model with torque diturbance G (z) = δn δα 2.56z 2.z 2 =.486z.529z 2 G 2 (z) = δn δṁ fi = 6.38z 3.4z 2.28z 3.486z.54z 2 G 2 (z) = δl δα =.64 z.545 G 22 (z) = δl = 2.3 δṁ fi z.537 The tranfer function between the external torque diturbance and the peed i given by: G d (z) = δn δt =.679z.332z 2.486z.529z 2 where the maximum tep diturbance of 3.7 ft lbf (5 Nm) i aumed, correponding to the load of a power-teering pump. The torque diturbance wa found to have negligible effect on the AFR. Figure how the open loop block diagram for the engine model and controller K. Where the torque diturbance i modelled by adding only to the peed channel. The aim of thi problem i to minimie the peed underhoot when a tep torque i applied, without exceive control effort. The fuel channel i fat acting and therefore control effort can be large during tranient, in contrat the electronic throttle in thi example i limited to 4% of maximum throttle per. 3. LOW ORDER CONTROLLER DESIGN In thi ection three controller deign technique are compared for performance and uitability for an engine management ytem. d
3. Parameter Space deign Recently, important MIMO parameter pace reult, uitable for developing computer algebra ytem, have been preented by (Muhler, 22). Parameter pace boundarie are given in the form of equation in the determinant of the Hamilitionian matrice correponding to the continuou H and H 2 equation, and hould thereby lead to ymbolic parameter pace method for rational continuou ytem. An alternative parameter pace method of mapping the ingular value for a range of dicrete frequencie i preented in (Dickinon and Shenton, 26). Since the following method relie only on the frequency repone of the ytem, both dicrete and continuou ytem can be handled uniformly, and irrational ytem with multiple pure time delay handled without additional difficulty. In application to a quare plant (poibly irrational) frequency repone matrix, G with n input and output, let the controller to be deigned be n n rational tranfer function matrix K,... K,n K =....... K n,... K n,n where for a continuou controller each element ha the tructure K ij = b 2 ij 2 b ij b ij a 2ij 2 a ij a ij () The nominal performance requirement for a multivariable ytem i dependent on the ingular value of the primary enitivity function S S = I I G(jω)K(jω) which i haped by a weighting function W S uch that W S (ω)s(jω), ω [; ) The parameter pace technique graphically determine the controller boundarie for the region that meet the weighted primary enitivity function requirement on the maximum ingular value σ[w S (jω)s(jω)] < by finding the olution of σ[s(jω) W S (ω) ] = By chooing two controller parameter at a time region atifying the above requirement can be mapped for a given frequency. Superimpoing a range of dicrete frequencie of interet allow a deigner to graphically elect a pair of controller gain which meet the pecification acro all frequencie. Thi proce i iterative between different plane and controller element until the pecification are met for all frequencie. A caling factor in the weighting function γ i gradually increaed in the iteration. For thi example it wa found that a large improvement could be made over the MIMO controller uggeted in (Ganagopadhyay and Meckl, 2) by uing the weighted primary enitivity function parameter pace method, provided the control effort of the throttle wa monitored with regard to the γ weighting function caling factor. Uing the guideline uggeted in (Skogetad and Potlethwaite, 996) a uitable primary enitivity weighting function wa elected. For good tracking repone when ubject to diturbance high gain at low frequencie are deirable. To limit the number of iteration required a low frequency breakpoint i alo conveniently included in the weighting function. The election for the hape of the weighting wa accordingly choen a W S = γ.2.5 I 2 For proper comparion with the Eigenvalue baed technique of (Ganagopadhyay and Meckl, 2) only a pure MIMO PI parameter pace controller i deigned here. Therefore, each of the elemental controller of eqn ha coefficient b 2ij =, a 2ij =, a ij = and a ij = with b ij and b ij (K P and K I ) determined in the parameter pace. The parameter pace deign mut begin with an initial table tart controller. In thi cae thi could be b = b = for all element. However the parameter pace method i alo an excellent tuning method for exiting controller. For example the parameter pace method could be ued to re-tune coefficient of a controller after order reduction. Therefore to demontrate thi merit the initial controller ued here will be the Zeigler-Nichol SISO controller which wa hown to be inadequate for thi problem...2 K ZN () =.4.5 Since the problem i concerned with a phyically dicrete event ytem (four-troke engine cycle), only frequencie up to the Nyquit frequencie need to be conidered. A total of 25 dicrete frequencie logarithmically paced from. to 3.4 rad/ were ued for the deign. Starting with γ =.2 four iteration to γ =.7 were neceary
b b plane.5 b...2.3.4.5 4 3 2 b Fig. 2. Parameter plane for γ =.7 for K 2 before the required performance wa achieved without exceeding the control effort contraint on the throttle. Further increaing γ wa found to lead to better performance but exceive throttle control effort. An example plane for two of the parameter i hown in Figure 2. The reulting parameter pace controller wa found to be.623.82 K PS () =.76.2296.649.748 2.998.48 3.2 Eigenvalue MIMO PI controller deign In the multivariable PI controller deign method uggeted by the author of the idle peed model developed, the proportional gain are elected to enure good diturbance rejection by inpection of the eigenvalue. The integral gain are choen to decouple the plant at teady tate and enure good tracking by looking at the cloed loop eigenvalue. The uggeted dicrete deigned controller for thi method ha a continuou equivalent.37.9 K EIG () =.6.98 3.3 H controller deign.2.98.64.235 A higher order controller i alo deigned with the two algebraic Equation (ARE) method uing the Matlab Robut Control toolbox to invetigate the poible performance benefit of variou order controller. When deigning uch a H controller for thi problem it wa found that the primary enitivity function alone wa Speed (rpm).5.5.5 2 2.5 5th order 6th order 7th order 8th order th (full order) 3 2 4 6 8 Time () Fig. 3. Time repone for reduced order controller not ufficient to achieve acceptable performance whilt meeting the control effort contraint on the throttle, and conequently the control effort enitivity wa alo ued. For acceptable performance, a econd order weighting function, equation (2), wa required to get both good peed and ettling time. The control effort, equation (3), wa bounded at all frequencie for the throttle and le everely on the fuel by the weighting function: and W S = γ 2 5 5 2 3. I 2 (2) W U = [ ].5.5 (3) Thee weighting reult in an th order controller. The order can be reduced without ignificant lo in performance uing a balanced model truncation on the normalized coprime factor to an 8th order controller. Further reduction i poible but the performance rapidly decreae. Figure 3 compare the full order controller with a number of reduced order controller. It can be een that there i ignificant lo in performance below 8th order where the 5th order controller i very ocillatory. 4th order and below were found to be untable, thi trend wa true for all the common reduction method found in the Matlab Robut Control toolbox (Bala and et al, 25). 4. RESULTS In thi ection the three controller deign technique are dicued. Each controller wa converted to it dicrete form uing a Tutin bilinear tranformation and ued to compare time repone reult from a imulation. Time repone reult comparing the parameter pace and deigned MIMO controller to the MIMO controller uggeted by (Ganagopadhyay and Meckl, 2) are preented in Figure 4, 5, 6
.5.5.5.4.3 Speed (rpm).5.5 2 Fuel (lbm/hr).2.. 2.5.2 3.3 3.5 2 3 4 5 6 Fig. 4. Speed repone to a torque diturbance.4 2 3 4 5 6 Fig. 7. Fuel repone due to a torque diturbance.2 4 3.5.8 3 AFR.6.4.2.2 Speed (rpm) 2.5 2.5.5.4 2 3 4 5 6 Fig. 5. AFR repone to a torque diturbance 2 3 4 5 6 Fig. 8. Speed repone to a tep demand.9.8.8.6.7.4 Throttle (% max ).6.5.4 Fuel (lbm/hr).2.3.2.2..4 2 3 4 5 6 Fig. 6. Throttle repone due to a torque diturbance and 7, 8, 9 and. Figure 4, 5, 6 and 7 how the reult of the three controller for a tep torque diturbance. A mall decreae in the maximum overhoot, ignificantly better ettling time and a much le ocillatory repone were found from the higher order algebraic and low order parameter pace controller. The improvement in performance can alo be een in figure 8 and 9 which how the repone of the two output to a tep change of in the peed demand. The reduced 8th order ARE controller wa found to have the mot robutne, indicated on the.6 2 3 4 5 6 Fig. 9. AFR repone to a tep demand change in peed plot of primary enitivity function in Figure. The parameter pace method wa found to give imilar robutne to the PI deign baed on the eigenvalue, although with a lower abolute maximum value. However, none of the controller were deigned pecifically for robutne. Without uffering le robutne, the performance improvement of the parameter pace deign over the uggeted eigenvalue-deigned PI controller i ubtantial for the ame order controller. The ARE controller ha very imilar level of performance to the parameter pace deign, with the benefit
Singular value (db) 3 2 2 3 4 5 2 Frequency (rad/) Fig.. Primary enitivity function of each controller of additional robutne, however it wa demontrated that it order could not be reduced below 5th. 5. CONCLUSIONS The technique preented in thi paper are for achieving nominal performance for multivariable problem. For MIMO problem there are only a limited number of technique that can produce fixed low order controller, which are eential for ue in indutrial application uch a engine management ytem. A parameter pace technique defining controller region atifying the H norm wa ued uccefully on a MIMO problem and the merit of the method were hown in imulation. The method i well adapted to both continuou and dicrete ytem for producing low order controller. The idle peed diturbance rejection problem howed that mapping jut the primary enitivity function give large improvement over exiting low order controller deign method, with performance cloe to full order algebraic olution. Beon, V. and A.T. Shenton (2). Interactive parameter pace deign method for robut performance of MISO control ytem. IEEE Tranaction on Automatic Control 45(), 97 924. Dickinon, P. and A.T. Shenton (26). Interactive controller deign by a mixed - parameter pace method for MIMO ytem. Submitted to the International Journal of Robut and Nonlinear Control. Ganagopadhyay, A. and P. Meckl (2). Multivariable PI tuning for diturbance rejection and application to engine idle peed control imulation. International Journal of Control 74(), 33 4. Hrovat, D. and J. Sun (22). Model and control methodologie for IC engine idle peed control deign. Control Engineering Practice, 279 29. Muhler, M. (22). Mapping MIMO control ytem pecification into parameter pace. Proceeding of the 4t IEEE Conference on Deciion and Control pp. 4527 4532. Peltomaa, A. and H.N. Koivo (983). Tuning of a multivariable dicrete time PI controller for unknown ytem. International Journal of Control 38, 735 745. Shafiei, Z. and A.T. Shenton (997). Frequencydomain deign of PID controller for table and untable ytem with time delay. Automatica 33(2), 2223 2232. Skogetad, S. and I. Potlethwaite (996). Multivariable Feedback Control - Analyi and Deign. Wiley. New Jerey. ACKNOWLEDGMENTS The author would like to thank the Engineering and Phyical Science Reearch Council (EPSRC) for their financial upport. REFERENCES Bala, G. and R. Chiang et al (25). Robut Control Toolbox - Uer Guide. Mathwork. Natick, MA. Beon, V. and A.T. Shenton (999). An interactive parameter-pace method for robut performance in mixed enitivity problem. IEEE Tranaction on Automatic Control 44(6), 272 276.