Research On the Application of Optimal Control Theory On Parameters Optimization of Vehicle Suspension

Transcription

1 JOURNAL OF LOW FREQUENCY NOISE, VIBRAION AND ACIVE CONROL Pages Research O the Applicatio of Optimal Cotrol heory O Parameters Optimizatio of Vehicle Suspesio She Yogju Liu Xiadog Yag Shaopu 3 ( School of Mechaical & Electric Cotrol Egieerig, Beijig Jiaotog Uiversity) sheyogju@sohucom ( Departmet of Automobile Egieerig, Beijig Uiversity of Aeroautics ad Astroautics) (3 Departmet of Mechaical Egieerig, Shijiazhuag Railway Istitute) Received d December 3 ABSRAC Based o optimal cotrol theory, a optimizig algorithm for the parameters of a traditioal passive suspesio system is preseted he method is firstly illustrated through a model of a two-degree-of-freedom vehicle system he the radom excitatio of the road to the tyre is ivestigated, which is obtaied by processig the power spectrum desity of kow road roughess he results of simulatios show that this optimizig algorithm ca improve the performace of suspesio systems i compariso with traditioal passive suspesios, wheever the vehicle is ruig o the siusoidal or the radom surface sated i this paper Moreover, it is easy to modify the existig suspesio system via the cocept put forward i this paper herefore, the proposed method is particularly applicable to o existig suspesio system that eeds modificatio Fially, the possibility of applyig this method to other egieerig fields is poited out Keywords: vehicle suspesio, optimal cotrol, optimizatio, road roughess INRODUCION he suspesio is a importat compoet i a vehicle affectig ot oly the security but also the comfort of passegers I order to improve its the performace the semi-active or active suspesio is ivestigated May of papers have cosidered this subject, ad all kids of cotrol strategies ad dampers have bee applied EI- awwab [] researched a semi-active suspesio with a advaced hydro-peumatic damper, ad the result with the hydro-peumatic damper is better tha that of covetioal suspesio Yoshimura [] described the costructio of a semi-active suspesio of a passeger car usig fuzzy reasoig ad field tests, where the car is described by a oliear system with seve degrees of freedom subject to radom excitatio from a road surface She et al [3] ivestigated a improved suspesio with a magetorheological damper (MR), ad the results showed better performace Ikeaga et al [4] combied a filtered feedback cotrol scheme ad a iput de-couplig trasformatio to a full-vehicle model, where pitch ad roll motio were also icluded Fialho ad Balas [5] preseted a oliear cotroller based o liear parameter-varyig cotrol techiques focusig o maximizig passeger comfort whe the suspesio deflectio is small compared to its structural limit Liberzo et al [6] desig a robust cotroller for the active suspesio of a off-road, high-mobility tracked vehicle usig quatitative feedback theory hough the active suspesio has better comfort tha the semi-active its use presets difficulties due to higher eergy cosumptio, more complicated structure ad lower reliability Based o the above itroductio, this paper presets a revised method for existig passive suspesios he improved parameters are obtaied by comparig the feedback Vol No

2 Research o the Applicatio of Optimal Cotrol heory gai based o optimal cotrol theory with the parameters of the passive suspesio, ad i this process the least square method is applied for optimizatio I order to compare the performace of the passive suspesio before ad after improvemet the simulatio of all suspesios, icluded the origial passive, the improved passive ad the active suspesio performed with a siusoidal, road surface, which provides a frequecy respose For a improved compariso oe must iclude the road surface roughess, ad there are two mai methods by which this ca be doe road test, or trasformig the power spectrum desity of the road roughess ito excitatio i a spatial field I 986 the Chiese stadard GB73-86 was issued, which is similar to the Iteratioal stadard ISO/C 8/SCN67, where all kids of road roughess with differet grades are formulated ad classified O the basis of GB73-86 the road roughess is obtaied by the iverse discrete Fourier trasform (IDF) of the power spectrum desity, this ca elimiate the sigificace of frequecy so that it is better tha other methods for road roughess MODELING OF ACIVE SUSPENSION BASED ON OPIMAL CONROL I this paper a stadard quarter vehicle model is discussed, ad the active suspesio is show i Fig Here m, m, k, k, c, u deote the mass of tyre, the mass of vehicle body, the stiffess of tyre, the stiffess of suspesio, the dampig coefficiet of suspesio, ad active cotrol force respectively he equatios of motio are: ( ) + ( - ) + ìm ÇÇ z k z - z c zç zç u í îm ÇÇ z k( z - z ) - k ( z - z ) - c( zç - z ) - u () If u is omitted from Eq (), the cotrol equatios of passive suspesio ca be obtaied I order to obtai the cotrol force, Eq () should be trasformed ito the state equatio: ÇX AX + Bu + Lw, () ad the output equatio: where the state vector is: X x x x x z z Ç z z Ç, ad the output vector is: Y CX + Du, [ ] [ ] 3 4 [ 3 4 ] [ 3 4 ] Y y y y y x xç x xç (3) (4) (5) he above martices are as follows: A æ - è ö k + k c k c - m m m m, k c k c - - m m m m ø C æ - è ö k + k c k c - m m m m, k c k c - - m m m m ø L æ ö æ ö k - m m, B D B w z,, Ç è ø è m ø (6) 54 JOURNAL OF LOW FREQUENCY NOISE, VIBRAION AND ACIVE CONROL

3 She Yogju, Liu Xiadog, ad Yag Shaopu m z k u c m z k z Figure he quarter model of vehicle he cost fuctio is: ò J q ( y - y ) + q y + q y 3 + ru 4 4 dt, [ ] ad the matrix for weightig coefficiet should be: (7) Q æ q -q ö q -q q è q ø 4 (8) Cosiderig Eq (4), oe ca get ò [ ] J X C QCX + u Ru + X Nu dt, (9) where R D QD + r, N C QD () Re-researchig Eq () ad Eq (3), shows that the state equatio i this paper is differet from the stadard state equatio due to Lw Aleksader [7] has ivestigated this kid of problem ad foud that the term of Lw has little effect o the respose of the system because the cotrol force has o effect o w which is the excitatio from road So oe ca get the matrix Riccati equatio based o optimal cotrol theory [8]: - PA + A P PB + N R B P + N + C QC - ( ) ( ) he optimal gai is as follows after P is obtaied from Eq (): () ( 3 4 ) ( ), * * * * K k k k k R B P + N () ad the cotrol force is: * * * * u -KX -k z - k zç - k z - k zç 3 4 (3) 3 OPIMIZAION OF HE PARAMEERS OF HE PASSIVE SUSPENSION I this sectio we ivestigate the optimizatio of the parameters of passive suspesio Supposig aother sprig ad damper are added to coect m ad m, the additioal force will be: ( ) + ( - ) u k z - z c zç zç (4) Vol No

4 Research o the Applicatio of Optimal Cotrol heory he aim of the improvemet is to make u as close to u as possible his ca be implemeted by makig ( ) - -( ) - ( ) ( ) - ( - ) * * f z, zç, z, zç u u k + k z k + c zç as small as possible hat is to say, the coefficiets of state variables i f ( z, zç, z, zç ) should be miimized Cosiderig Z spa { z, zç, z, zç } which spas a liear space, u ad u are two curves i this space he the problem could be chaged ito fidig least distace betwee two curves Because the coefficiets of z ad Çz must equal the mius value of the coefficiets of z ad Çz it is difficult to obtai the optimal solutio Accordigly, oe could establish a fuctio by meas of least squares: ( ) + ( 3 - ) + 3( + ) + 4 ( 4 - ) * * * * J l k + k l k k l k c l k c, * * - k - k z k c zç 3 4 (5) (6) where the l (i,,3,4) are weightig coefficiets Usig the least square method oe ca get k ad c : * * lk3 - lk k, l + l * * l4k4 - l3k c l + l 3 4 (7) (8) So, the stiffess ad dampig coefficiet of the revised suspesio are k" k + k, c" c + c (9) If k or c is plus, oe should add suitable sprig or damper O the cotrary, whe k or c is mius, the sprig or damper with smaller a coefficiet should substitute origi passive suspesio 4 RESEARCH ON ROAD SURFACE ROUGHNESS Accordig to much data measured o the road, surface roughess ca be regarded as a statioary Gauss process with ergodicity ad zero mea value It is usual to describe the statistical characteristic of road by power spectrum desity Due to [9], the power spectrum ca be approximated by Eq () G G d ( ) ( ) æ è ö d -W > ø () where spatial frequecy, it deotes the umber of waves i every meter, ad the uit of which is m - ; spatial referece frequecy, it is usually equal to m - ; G d ( ) power spectrum desity with respect to spatial referece frequecy, it is determied by the grade of road, the uit of which is m /m - ; W frequecy idex, it is determied by the igrediet of the power spectrum desity of the road, here W is chose he respose of a vehicle is isesitive to excitatio with very high or very low frequecies So, such excitatio ca be disregarded herefore, oe ca extract a rage of spatial frequecy Here, supposig the upper limit of frequecy eeded is u ad the lower l, we have W G Gd d ( ) ( ) æ - ì u è ö ï í ø ï î otherwise () 56 JOURNAL OF LOW FREQUENCY NOISE, VIBRAION AND ACIVE CONROL

5 She Yogju, Liu Xiadog, ad Yag Shaopu Here u ad l, should that esure the excitatio of the road must cover the atural frequecies of the vehicle system Supposig the speed of the vehicle is v (m/s) ad the spatial frequecy is (m - ), the the excitatio frequecy of road roughess to the tyre is f v () If the rage of atural frequecy of the vehicle is from f l to f u, oe ca get the upper ad lower limit of the power spectrum desity i spatial the domai f fu, u v v (3) Accordig to the Shao theorem the iterval of samplig distace should meet Eq (4) so as to avoid mixture of frequecy D l u (4) Additioally the total samplig distace should satisfy Eq (5) to guaratee the accuracy of the frequecy L (5) So the umber of whole samplig poits ad the resolutio of spatial frequecy are N L, D D l L (6) herefore oe ca get X Nl G æ k N ö ( ),,, L,, D è ø k d k (7) where k kd From Eq (7) the modulus of the discrete Fourier trasform (DF) is obtaied Supposig the phase agle j k could be radomly selected from [ p] so jjk æ Xk Xk e k è N ö,,, L, ø (8) Accordig to the theory of radom vibratio [], the IDF of the discrete sigal x (,,, N ) i the time domai should also have N data But oly the first ( N /) + data will be used whe calculatig the power spectrum desity Now the first ( N /) + data are obtaied I order to obtai x oe should supply the lack of the secod half of data Obviously, whe x (,,, N ) is trasformed ito the frequecy domai by DF, oe ca get N X x e k,,, L N - k N - j k - å p ( ) (9) Accordig to Eq (9) some results should be clear: X is real umber ad X N - x å, ad if x has zero mea value X would be zero; X, ad X N, or X ad X N, et al are mutually cojugated each other Because the maximal frequecy of DF Vol No

6 Research o the Applicatio of Optimal Cotrol heory is ( N /)Df, the Eq (3) should be satisfied so as to avoid mixture of frequecies f u < N N f D t D (3) his will result i X( N / ) Based o the above results oe could supply the lack of data ad X k (k,,, N ) could be obtaied rasformig X k by IDF oe ca obtai the road roughess X N - j pk å N Xke (,,, L N - ) N (3) Because the above process for obtaiig the road roughess is the iverse oe for calculatig power spectrum desity, this esures the road surface roughess obtaied is completely cosistet with the requested results he a uiversal example is show Supposig the lower limit f l of atural frequecy is 5 Hz, the upper oe f u is 3 Hz, ad the velocity rage of vehicle is 36 km/h-8 km/h ( m/s-5 m/s), the requested lower ad upper limit of frequecy of the road roughess i the spatial domai are f 5 - fu 3 - m, u 3 m v 5 v (3) his shows that researchig the road roughess i -3 - m of spatial frequecy would iclude the atural frequecy of vehicle ad this could reflect the real circumstace I order to avoid mixture of frequecies i samplig, the followig equatios should be satisfied D l 667 m, L m 6 u Here cosiderig Dl m ad L 496 m oe could obtai that N ( L/ D) 496 ad the iterval of spatial frequecy is D (/L) 44 m - Accordig to GB/ G d ( o ) for road grade C is 56 mm /m - Based o the above statemet oe ca get the G d (k) ad subsequetly the X k (k,,,48) After supplyig the lack of data oe ca get x (,,,, 495) by IDF ad amely the road surface roughess Fig is the curve of the road surface roughess for grade C, where the abscissa is the course ad the ordiate is the road roughess Fig 3 is the power spectrum desity obtaied from the road roughess, where the abscissa is spatial frequecy ad the ordiate is the power spectrum desity, ad both of them are expressed i logarithmic form to show the result more distictly Obviously, the power spectrum desity of Fig is cosistet with requiremets l (33) he road roughess (mm) he course (m) C Figure he road roughess for grade C 58 JOURNAL OF LOW FREQUENCY NOISE, VIBRAION AND ACIVE CONROL

7 She Yogju, Liu Xiadog, ad Yag Shaopu mm /m PSD of Fig + he requested PSD Figure 3 he compariso of power spectrum desity (PSD) 5 HE RESULS OF SIMULAION I this sectio we compare the performace of the improved passive suspesio with the origial passive suspesio, ad the oe of active suspesios is also show Selectig a group of parameters for the suspesio system: m 4 kg, m 36 kg, k 6 4 N/m, k 6 5 N/m, c 5 N s/m he aim of the improvemets is to reduce the acceleratio of the vehicle body ad ot reduce security After a lot of calculatio the weightig coefficiets for J were selected as follows: r, q, 6 3, q, q he weightig coefficiets for J are: l l 3, l, l 4 5 Based o the Schur decompositio algorithm, the optimal gais are: * * * * k 359, k 3, k - 44, k (34) herefore the added stiffess of the sprig ad the dampig coefficiet are: k N / m, c - 546N s/ m (35) It is show that oe should chage the sprig ad damper ito oes which have lower stiffess or dampig coefficiet Firstly we ivestigate the respose of the vehicle to siusoidal excitatio Here the amplitude is chose as 5 m, but it has o effect o the qualitative relatio betwee the three suspesios however large the amplitude is he results is show i Fig 4 to Fig 6, where the abscissa is the frequecy of excitatio ad the ordiate 4 the origial suspesio 3 the improved suspesio the active suspesio m /s / Hz Figure 4 he compariso of acceleratio of vehicle body Vol No

8 Research o the Applicatio of Optimal Cotrol heory 6 the origial suspesio the improved suspesio 4 the active suspesio 5 N / Hz Figure 5 he compariso of dyamic tyre load is the mea square value of respose icludig the acceleratio of the vehicle body, the dyamic tyre load ad the deformatio of the suspesio We the researched the performace of suspesio whe the road roughess preseted i this paper is used I table I the respose of the three suspesios to the road roughess for grade A, B, C, D ad E are listed, ad the acceleratio of vehicle body, the dyamic tyre load ad the deformatio of suspesio are all obtaied, where the suspesio I deotes the origi passive suspesio, while the suspesio II is for the improved passive suspesio ad suspesio III for active cotrol suspesio he acceleratio of vehicle body for the three suspesios i time domai for grade B are show i Fig 7 to Fig 9 From the results of the simulatio, icludig Fig 4 to Fig 9 ad the table I, the coclusios are listed as follows: l he improved passive suspesio has less acceleratio of the vehicle body tha the origial suspesio except i a short rage of low frequecies, which could ehace the comfort for passegers remarkably 6 the origial suspesio the improved suspesio 4 the active suspesio -5 m / Hz Figure 6 he compariso of deformatio of the suspesio 6 JOURNAL OF LOW FREQUENCY NOISE, VIBRAION AND ACIVE CONROL

9 She Yogju, Liu Xiadog, ad Yag Shaopu able I he mea square value of respose to the road roughess Acceleratio of vehicle body Dyamic tyre load Deformatio of suspesio Grade (m s -4 ) ( 4 N ) ( -6 m ) A suspesio I suspesio suspesio B suspesio suspesio suspesio C suspesio suspesio suspesio D suspesio suspesio suspesio E suspesio suspesio suspesio he differece betwee the dyamic tyre load of the improved suspesio ad that of the origial suspesio is isigificat, ad they are ot as good as the active suspesio So the improved suspesio would apparetly ot shorte tyre life 3 Whe the excitatio of the road is smaller tha 4 Hz the improved suspesio has bigger deformatios of the suspesio tha the origial suspesio his has o effect o the vehicle because the deformatio does ot exceed desig limits Whe larger tha 4 Hz they have almost the same deformatio of suspesio 4 he aim of the improvemets is to reduce the acceleratio of vehicle body so as to ehace comfort ad it this bee attaied, especially betwee 4 Hz ad 8 Hz, the most sesitive rage of frequecy to passegers 4 m / s t / s Figure 7 he acceleratio of vehicle body for the origial suspesio Vol No 4 3 6

10 Research o the Applicatio of Optimal Cotrol heory 4 m / s - Figure t / s he acceleratio of the vehicle body for the improved suspesio 4 m / s - Figure t / s he acceleratio of the vehicle body for the active suspesio 6 CONCLUSIONS I this paper a optimizig algorithm is preseted which could be used to revise the origial suspesio or desig ew suspesio based o the origial parameters A ew method describig the road surface roughess is also show he results of simulatio verify the validity of this optimizig method Moreover, this optimizig algorithm could be exteded to other fields where the structure eeds revisio or a ew structure is to be desiged ACKNOWLEDGEMENS his research is supported i part by the Natioal Natural Sciece Foudatio of Chia (No76) ad i part by the Fazhou Foudatio of Beijig Uiversity of Aeroautics ad Astroautics (No33) REFERENCES Ali M Abd EI-awwab Advaced Hydro-Peumatic Semi-Active Suspesio System Joural of Low Frequecy Noise, vibratio ad Active Cotrol Vol, No 4, pp 93-3, Yoshimura Semi-Active Suspesio of Passeger Cars Usig Fuzzy Reasoig ad the Field estig Iteratioal Joural of Vehicle Desig, Vol 9, No, pp 5-66, She Yogju, Yag Shaopu, Liu Xiadog Studies o Improved ype of Semi-Active Car Suspesio with Magetorheological Damper Joural of Vibratio, Measuremet ad Diagosis Vol, No 4, pp 53-57, 6 JOURNAL OF LOW FREQUENCY NOISE, VIBRAION AND ACIVE CONROL

11 She Yogju, Liu Xiadog, ad Yag Shaopu 4 S Ikeaga, F L Lewis, J Campos, L Davis Active Suspesio Cotrol of Groud Vehicle Based o a Full-Vehicle Model Proceedigs of the America Cotrol Coferece, Vol 6, pp 49-44, 5 I J Fialho, G J Balas Desig of Noliear Cotrollers for Active Vehicle Suspesios Usig Parameter-Varyig Cotrol Sythesis Vehicle System Dyamics, Vol 33, No 5, pp 35-37, 6 A Liberzo, D Rubistei, P O Gutma Active Suspesio for Sigle Wheel Statio of Off-Road rack Vehicle Iteratioal Joural of Robust ad Noliear Cotrol, Vol, No, pp , 7 H Aleksader, Optimal liear preview cotrol of active suspesio Vehicle System Dyamics, Vol, pp 67-95, 99 8 D K Aad, R B Zmood, Itroductio to Cotrol Systems, 3 rd editio, Butterworth-Heiema Ltd, Li Maocheg, Zhao Jihai, GB/73-986, Chiese Stadard Press, 987 D E Newlad, A itroductio to radom vibratios ad spectral aalysis, Logma: Logma Group Ltd, 984 Vol No