COMPARATIVE STUDY OF ROBUST CONTROL TECHNIQUES FOR OTTO-CYCLE MOTOR CONTROL

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1 ACM Symposim Series in Mechatronics - Vol. - pp Copyright c 28 by ACM COMPARATIVE STUDY OF ROUST CONTROL TECHNIQUES FOR OTTO-CYCLE MOTOR CONTROL Marcos Salazar Francisco, marcos.salazar@member.isa.org Karl Heinz Kienitz, kienitz@ieee.org Institto Tecnológico da Aeronatica ITA Abstract. This paper compares some aspects of the robst control techniqes LQG/LTR and H-. These techniqes are sed to design a mltivariable compensator for an old MW generation Otto-cycle motor. Performance and stability robstness of the control system are evalated for a set of design specifications. Compensators are designed to eliminate rotation distrbances and improve oscillatory response. For both techniqes, many design parameters are modified to compare their effectiveness. Despite system ncertainties, for most choices, the response is satisfactory. Ths, the inflence of the choice of the techniqe and its parameters in a control design for a real application can be compared. Finally, compensator order is redced to simplify its implementation. Keywords: Robst Control, Otto-cycle, LQG, H-. INTRODUCTION A robst system has to work satisfactorily when dynamics differs from the one considered dring design. Despite small alterations in the dynamics, the system shall keep its stability and performance. The prpose of this paper is to stdy some changes in parameters for the LQG/LTR and H- methods, two well known methods for the design of mltivariable robst compensators. These methods aim at a compensator that satisfies a set of specifications for robst stability and performance. The present stdy addresses idle speed compensator design for an old generation MW gasoline Otto-cycle motor (Geering, 996). Sch system is greatly affected by eternal distrbances and parameter changes (Kraoka et al., 99). In the net section, this system will be described and a mathematical model presented. The LQG/LTR Method consists of a recovery process to match the closed-loop response with a desired dynamics. This method is described in section. In section 4, the H- problem is described and its design specifications are chosen. The reslts are shown in section 5. After compensator design, it is possible to redce its order, withot compromising significantly the system response. In section 6, order redction is sed to get a simplified version of the compensator. Section 7 shows reslts obtained with the modified LQG/LTR compensator introdced by Prakash (99). It is shown that the system has a good response for this less well known strctre. 2. MATEMATICAL MODEL Figre shows the block diagram of the motor described in Kienitz (26) based on amgartner et al. (986) and Geering (986). This diagram considers the dynamics of the fel admission valve actator, the flow and the rotation of the engine. Higher-order dynamics are not being considered. The process variable of interest is the engine s rotation ( n). In this model, there are three maniplated variables: the reference for position of the fel valve ( r ), the ignition point ( z ) and the time increment for opening fel valve ( t i ). The distrbance is the variation of the load from the defalt reference ( M L ). For sing state eqations, the following states are being considered: position of the entrance valve ( p ), pressre of the admission pipe ( p S ) and rotation of the engine ( n). The operating point and the parameters of the motor, which were sed herein, can be seen in Tab. and in Fig.. r 5 p z t i M L 7,4,,488 4 / ,6,549 p S n Figre. Linearized model of the motor

2 Table. Operating point of the Motor Rotation n rpm Load M Nm Admission pressre p s,45 ar Ignition point increment z Inkr z Time increment of admission valve movement t i Inkr t i When a step distrbance of 4 Nm is applied in the load (i.e. the load jmps from Nm to 5 Nm) motor rotation decreases from rpm to arond 6 rpm. The response to this distrbance is seen in Fig Rotation [rpm] Figre 2. Response of the system to a step of 4 Nm in load The main task of the compensator is to mitigate the inflence of load distrbances on motor rotation. However, the ignition point and the time increment for opening the fel valve shold retrn to their initial (i.e. pre-transient) vales. Therefore, to eliminate the distrbance, the dynamics of the system shall be increased by adding an integrator at the reference for position of the fel valve. The integrator mst not be added to other inpts, becase these shall retrn to their initial vales after the distrbance is compensated for. As a reslt, Eqation () is obtained based on the model in Fig.. A U d M y C L () Where α p P S U n α r α Z t i 5 5, , 55, 25 A , 44 9, 52 d C 4 [ ] (2) Instead of an integrator control, the se of PI-element that is shown in Eq. () is analyzed. In sch case, only matri of Eq. () is modified. The new matri is given in Eq. (4). α r, 2 () s , 52 (4). LQG/LTR METHOD The nominal system has no pole at the origin. Ths a compensator with an integral component is necessary to remove steady-state error for step distrbances. When an integrator or a PI element is added, the steady-state error is

3 eliminated. Frthermore it is also necessary to decrease settling time and overshoot. Additionally a main objective of the compensator is to keep stability robstness independently of reasonable modeling errors. The design specifications are shown in the net sections... Stability robstness The stability robstness specification depends on a modeling error assessment. The model given in Eqs. () and (2) describes a nominal system, G nom (s). In this case, a reasonable strctre for the modeling error is shown in Eq. (5). G ( s) [ I E( s)] Gnom( s) (5) Most of modeling errors are de to high-order dynamics neglected in the modeling process, sch as actator dynamics. A first order model of the neglected dynamics is given in Eq. (6). ( Ts) G( s) G nom ( s) ( T2s) ( Ts) (6) Combing Eq. (5) and Eq. (6), the following error eqation reslts: Ts ( Ts) E( s) T2s ( T s) 2 T s ( Ts) (7) Crz (996) gives the condition for stability robstness shown in Eq. (8). [ G(jω)K(jω) ] <, ω ma (8) e m ( ω) where e m (ω) is the pper limit given in Eq. (9) for the modeling error. Tmas e m( w) ma[ E( jω )] ma Tma ma( T,T2, T ) (9) Tmas In this paper, we adopt T ma.2 seconds, i.e. the actator reaches 99% of the desired vale in less time than 5T ma (. seconds). Then, the stability robstness condition is ma,2s ω < (),2s [ G( j ) K( jω) ].2. Rejection of measrement noise Most of the measrement noise is in the freqency of the motor rotation and its harmonics. Crz (996) proposes the following specification to garantee noise rejection: ma [ G(j )K(jω)] α ( ω ) ω () N The motor, whose nominal rotation is rpm, has a rotation freqency of ω 2(/6) rad/s 4,7 rad/s. In this paper, an attenation of -2d in freqencies above rotation freqency was chosen, i.e. N (ω) is eqal to,. This specification is shown in Fig. 4 as a dashed line as well as the specification from Eq. ().

4 .. Target dynamics The LQG/LTR method has two steps. In the first step a target (desired) dynamic is determined. In the second step, a compensator that forces this dynamic nto the actal feedback loop is designed. In Fig., the strctre of the admissible target dynamics is shown. Matrices A and C are sed in this model and the isse is to find a matri H that entails a singlar vale diagram of CΦ(s)H that satisfies the specifications from Eq. () and Eq. (). r(s) H (s)(si A) - C y(s) Figre. Target dynamics The matri H is determined as the soltion of a (fictitios) continos time Kalman filter (Athans, 986). The epression for calclating the filter gain matri H is H ΣC µ T (2) where is the soltion of the algebraic Riccatti eqation AΣ ΣA T T LL T ΣC CΣ µ () The design parameters are the matri L and the nmber >. In this paper, the following choices for L are considered:, a, I and a bi, where a and b are constants, I is a forth order identity matri and is the etended system inpt matri. Figre 4 shows a singlar vale diagram of CΦ(s)H for each choice of L. oth dashed lines are specifications from Eq. () and Eq. (). The first singlar vale diagram is plotted for the system shown in Eq. (2) pls an added integrator. The others are given for the system with an added PI-element of the type shown in Eq. (). The gain vales and the matri L sed in each case can be seen in the legend of Fig. (4). We can see that all graphics have the same shape, ecept for the identity matri. This one decreases the effect of integral control in low freqencies. 2 Singlar Vales (d) Stability robstness Rejection of measrement noise Integrator (2, ) PI (2, ) PI (.2,.9) PI (, I) PI (6,.2.4I) 2 Freqency (rad/sec).4. Loop-Transfer-Recovery Figre 4. Singlar vale diagram of CΦ(s)H for each choice of and matri L In the Loop-Transfer-Recovery process, a compensator is designed to enforce the desired target dynamic. Figre 5 shows the strctre of the controller. At this point matrices A,, C and H are known. In this section, a matri G is determined via the soltion of the Linear-Qadratic Reglator problem (Athans, 986). Eqation (4) presents the state eqation for this strctre.

5 z Az H y Gz ( e Cz) (4) r(s) e(s) Controller: K(s) -I z(s) H (s)(si A) - -G C ( s) α Z t i Distrbance d(s) M L Motor with integrator or PI element Rotation y(s) n Figre 5. Compensator strctre We shold solve the algebraic Riccatti eqation in Eq. (5), and then compte the matri G by sing Eq. (6). T T T KA A K C C K K ρ ρ (5) T G K (6) ρ As approimates to zero, the closed loop transfer fnction of the system tends to the closed loop transfer fnction of target dynamics shown in Fig. (Athans, 986). This happens whenever the system to be controlled does not have nonminimm phase zeros. In Fig. 6, the target freqency response is given for L and 2. The freqency response of the recovered system is also plotted for, and for,. It is clearly seen that lower lead to better recovery. Singlar Vales (d) H- METHOD - Stability robstness specification -4 Rejection of measrement noise specification Target dynamics -5 Recovered systems for p, Recovered systems for p, -6 2 Freqency (rad/sec) Figre 6. Singlar vale diagrams for the target and the recovered systems for, and,. To se H- design, the system mst be redrawn as shown in Fig. 7 (Levine and Reichert, 99). The related eqations are given in Eq. (7). (s) y w P(s) z K(s) v Figre 7. General strctre of H- problem, inclding modeling error

6 w P P v z P( s ) 2 w P P (7) 2 22 In the H- design, the controller K is determined sch that the infinity norm of the transfer fnction from inpt to otpt, F(P, K) in Eq. (8), is minimized (Maciejowski, 989). P2 K( I P22 K ) P2 z F( P,K )w F( P,K ) P (8) A diagram for matri P(s) is shown in Fig. 8. The inpts w, w 2 e w are respectively represented by (providing for (s)), d(s) (distrbance) and n(s) (measrement noise). The otpts z e z 2 represent the inpt to the ncertainty block (s) and the otpt of the system y(s). Then, when minimizing F(P,K), the inflence of, d(s) and n(s) in the otpt of the system is minimized. Despite modeling errors (s), this garantees the stability of the system (Skogestad and Postlethwaite, 996). For adjsting system performance, the weights W a, W b e W c were added as shown in Fig. 8. w w 2 d w n W a W b W c d z y (s) z 2 y(s) (si A) - C v e(s) Figre 8. Diagram of matri P(s) The weight W a is chosen eqal to the pper limit of the modeling error. This pper limit was determined in section. and is applied here, i.e. Eqation () with T ma eqal to.2 is sed to compte the weight W a. This eqation for W a is given in Eq. (9) where γ is a design constant. In this case, the error (s) is nknown bt limited by (s). W a γ Tma s T s ma [ I ] [ I ] γ s s 5 (9) The weight W b is chosen as a low-pass filter fnction given in Eq. (2) where γ 2 and b are design constants. W b 2b γ s b (2) The last weight W c is chosen as a high-pass filter that stops freqencies lower than the ctoff freqency. The ctoff freqency is adjsted considering the measrement noise spectrm. It was seen in section.2 that most of the measred noise is in the rotation freqency (4,7 rad/s) and its harmonics. This reslts in a choice of W c according to Eq. (2) where γ and b are design constants. W c γ ( s b ) γ ( s b ) s c s 4 7, (2) Using Eq. (9), Eq. (2) and Eq. (2), the etended system P(s) is described by the state eqation in Eq. (2) with the following matrices. C 4 f f 2 C 4 ( s ) A ( s ) w d( s ) z A 5I y( s ) n( s ) C2 d b 4 7, [ ] I C D D D [ γ ] D [ ] γ γ 5I γ 2b γ 4 7, γ 2 2 b 22 2 (22)

7 A w C D D v z 2 2 C D D (2) To apply H- design, some conditions mst be satisfied. First, (A, 2 ) mst be controllable and (A, C 2 ) observable. It is also necessary that the matrices in Eq. (24) have a fll rank for any freqency. These conditions garantee the eistence of the H- compensator. In section 2, an integral or PI control was added to eliminate the steady-state error. ecase this new pole in the origin of comple aes, the matrices in Eq. (24) do not have fll rank. To solve this problem, the imaginary ais was shifted a distance of d (see Kienitz and Yoneyama, 99 and references therein). After the design of the compensator, the imaginary ais is shifted back again. This reslts in sboptimality and the distance d becomes a new design parameter. A jwi C D 2 2 A jwi C2 D 2 (24) For performance tning, the design parameters, 2,, b and d are sed. The following vales were fond to be appropriate:, 2 5,5 b,2 d,5 (25) These parameters were sed to compte the matrices in Eq. (22). Thereafter the compensator was fond sing the algorithm given by Safonov et al. (989). This algorithm initially finds a sboptimm compensator that satisfies Eq. (26) for some vale of. Then, the procedre is repeated redcing ntil the optimal compensator is fond. F ( P,K ) < γ (26) 5. RESULTS Figre 9 shows the step-distrbance responses reslting from the se of each compensator. The distrbance is a step of 4 Nm in motor load. It is observed that, ecept for response for L I, all responses are correcting the distrbance effect retrning to nominal rotation in reasonable time. Figre presents the variation of position of the fel valve ( r ) for some compensators. We can see that this variable assmes a new operating point becase of the distrbance. The variation of the ignition point ( z ) and the time increment for opening fel valve ( t i ) for each compensator are illstrated in Fig. and in Fig. 2. These variables retrn to zero after the effect of the distrbance is mitigated, as reqired in the problem formlation. Ecept for the case L I, all controlled systems have better settling time than the response in Fig. 2 and a PI control prodces a compensator with less oscillatory amplitde than an integral control, besides the least steady-state error. In the contet of LQG/LTR method, comparing all responses, the best one happens when matri L is eqal to,2,4 I, less settling time and less amplitde of transient than the others Rotation [rpm] LQG/LTR: Integrator (2, ) LQG/LTR: PI (2, ) LQG/LTR: PI (.2,.9) LQG/LTR: PI (, I) LQG/LTR: PI (6,.2.4I) H-Infinity Figre 9. Step-distrbance responses for each compensator

8 2.5 2 LQG/LTR: Integrator (2, ) LQG/LTR: PI (2, ) LQG/LTR: PI (.2,.9) LQG/LTR: PI (, I) LQG/LTR: PI (6,.2.4I) H-Infinity Degree Figre. Variation of position of the fel valve ( p ) for each compensator LQG/LTR: Integrator (2, ) LQG/LTR: PI (2, ) LQG/LTR: PI (.2,.9) LQG/LTR: PI (, I) LQG/LTR: PI (6,.2.4I) H-Infinity Degree Figre. Variation of the ignition point ( z ) for each compensator Increment LQG/LTR: Integrator (2, ) LQG/LTR: PI (2, ) LQG/LTR: PI (.2,.9) LQG/LTR: PI (, I) LQG/LTR: PI (6,.2.4I) H-Infinity Figre 2. Time increment for opening fel valve ( t i ) for each compensator

9 6. ORDER REDUCTION The LQG/LTR design prodces a compensator with the same order as the etended system. Ths the third-order system etended with an integral or PI element prodces a forth-order LQG/LTR compensator. This reslts in an overall fifth-order controller. To have a simplified implementation, the order of controller can be redced. This was done sing a Matlab implementation of Schr balanced trncation. To assess the effect of order redction, the best compensator was chosen, i.e. the compensator for L,2,4 I. Dring order redction one state of the compensator was removed. The H- design prodces a compensator with the order of P(s), i.e. order 9 in this paper. Ths, the order of the compensator inclding the integrator or the PI element is ten. To have a simplified implementation, the order of compensator was redced sing the same algorithm mentioned before, Schr balanced trncation. The redced compensator is of forth order. Figre 4 shows the otpt for a step distrbance of 4 Nm in the load. The system shows satisfactory performance in both cases. Rotation [rpm] LQG/LTR 9 H-Infinity Figre. Step-distrbance responses for redced and non-redced compensator, sing both methods. 7. NEW STRUCTURE FOR LTR In Prakash (99), an alternate control strctre was introdced for LQG/LTR. Figre 4 shows this strctre. e(s) z(s) H (s)(si A) - -G (s) Figre 4. Prakash s strctre To compte the gain G, the process is the same that was sed in section. The control transfer fnction still works inverting the dynamics of the system in Eq. (). Eqation (27) presents the state eqation for this new strctre. z Az He( t) y Gz (27) R otation [rpm ] Defalt strctre 9 Defalt Strctre Prakash`s Strctre 88-2 Prakash's Strctre Freqency (rad/sec) Figre 5. Step-distrbance responses and singlar vale diagram of the recovered system (,) for defalt strctre and for Prakash s strctre.

10 Simlation reslts sing this strctre and the same step distrbance sed in section 5 are smmarized in Fig. 5. It also shows the singlar vale diagram of the recovered system for this strctre and for the defalt strctre, sing,. We can see that the new response is almost the same as the response of the defalt LQG/LTR strctre. It was fond in this application that the new strctre leads to a stable compensator and a smaller LTR-Gain for recovery, corroborating the reslts pblished by Prakash (99). 8. CONCLUSIONS This work analyzed the LQG/LTR and H- methods in an idle speed compensator design for an old generation MW gasoline Otto-cycle motor. In this design, we evalated the performance and stability robstness of the control system in the light of two simple bt realistic design specifications. To satisfy the performance reqirements in both methods, we attempt to maimize the closed-loop bandwidth withot violating robstness stability reqirements and withot violating restriction of measrement noise rejection. The inflence of load distrbances on motor rotation was mitigated inclding an integrator at the reference for position of the fel valve. However, the integrator was not added to other inpts, becase these shold retrn to their pre-transient vales. Instead of an integrator control, we also tried the se of a PI element for the same prpose. We verified that, in this case, we reach less oscillatory response and less settling time than sing a simple integrator. In the contet of the LQG/LTR method, the target dynamic with best performance (least transient amplitde and least settling time) was obtained for L,2,4 I. When we compare the transient amplitde and the settling time in H- and LQG/LTR methods, we observe that H- design prodces a compensator with better performance than LQG/LTR design. On the other hand, the H- compensator is more comple than the LQG/LTR compensator. After controller redction this difference in compleity is significantly redced. We obtained a third order compensator from the LQG/LTR design and a forth order one from the H- design. Prakash s strctre is less well known; it also led to a system with satisfactory response. This strctre garantees a stable compensator. The new strctre also garantees a smaller LTR-Gain for recovery that helps to avoid rnning into nonlinear actator regions. Finally, both methods show qalities and difficlties to design a controller that satisfy all robstness and performance reqirements. 9. REFERENCES Athans, M., 986, A Ttorial on LQG/LTR Method, Proceedings of American Control Conference, Seattle, WA, USA, pp amgartner, C.E. et al., 986, "Robst mltivariable idle speed control", Proceedings of the 986 American Control Conference. Crz, J. J., 996, Controle Robsto Mltivariável, Edsp, S. Palo, razil, 68 p. Geering, H.P., 986, "Entwrf robster Regler mit Hilfe con Singlarwerten: Anwendng af Atomobilmotoren", GMA-ericht Nr., Robste Regelng, pp Geering, H.P., 996, Regelngstechnik, Springer, erlin, Germany, 8p. Kienitz, K.H., 26, Private Commnication, ITA, São José dos Campos, razil. Kienitz, K.H. and Yoneyama, T., 99, A Robst Controller for Inslin Pmps ase ased on H-Infinity Theory, IEEE Transactions on iomedical Engineering, Vol. 4, No., pp. -7. Kraoka, H., Ohka, N., Ohba, M., Hosoe, S. and Zhang, F., 99, Application of H-Infinity Design to Atomotive Fel Control, IEEE Control Systems Magazine, Vol., No., pp Levine, W.S. and Reichert, R.T., 99, An Introdction to H Control System Design, Proceedings of the 29 th Conference on Decision and Control, Vol.6, Honoll, HI, USA, pp Maciejowski, J.M., 989, Mltivariable Feedback Design, Addison-Wesley Pblishing, Wokingham, England, 48 p. Prakash, R., 99, Target Feedback Loop/Loop Transfer Recovery (TFL/LTR) Robst Control Design Procedres, Proceedings of the 29th Conference on Decision and Control, Vol., Honoll, HI, USA, pp Safonov, M. G., Limebeer, D. J. N. and Chiang, R. Y., 989, "Simplifying the H-Infinity Theory via Loop Shifting, Matri Pencil and Descriptor Concepts", Internacional Jornal of Control, Vol.5, No. 6, pp Skogestad, S. and Postlethwaite, I., 996, Mltivariable Feedback Control. John Wiley & Sons, England, 572 p.. RESPONSIILITY NOTICE The athors are the only responsible for the printed material inclded in this paper.