EXPERIMENTAL VALIDATION OF A NUMERICAL CONTROLLER USING CONVEX OPTIMIZATION WITH LINEAR MATRIX INEQUALITIES ON A QUARTER-CAR SUSPENSION SYSTEM

Transcription

1 EXPERIMENAL VALIDAION OF A NUMERICAL CONROLLER USING CONVEX OPIMIZAION WIH LINEAR MARIX INEQUALIIES ON A QUARER-CAR SUSPENSION SYSEM A hei by ROHI HARI CHINALA Submitted to the Office of Graduate Studie of exa A&M Univerity in partial fulfillment of the requirement for the degree of MASER OF SCIENCE Augut 11 Major Subject: Mechanical Engineering

2 EXPERIMENAL VALIDAION OF A NUMERICAL CONROLLER USING CONVEX OPIMIZAION WIH LINEAR MARIX INEQUALIIES ON A QUARER-CAR SUSPENSION SYSEM A hei by ROHI HARI CHINALA Submitted to the Office of Graduate Studie of exa A&M Univerity in partial fulfillment of the requirement for the degree of MASER OF SCIENCE Approved by: Chair of Committee, Committee Member, Head of Department, Won-jong Kim Bryan Ramuen Mehrdad Ehani Denni O Neal Augut 11 Major Subject: Mechanical Engineering

3 iii ABSRAC Experimental Validation of a Numerical Controller Uing Convex Optimization with Linear Matrix Inequalitie on a Quarter-Car Supenion Sytem. (Augut 11) Rohit Hari Chintala, B.E., Birla Intitute of echnology Chair of Adviory Committee: Dr. Won-jong Kim Numerical method of deigning control ytem are currently an active area of reearch. Convex optimization with linear matrix inequalitie (LMI) i one uch method. Control objective like minimizing the H, H norm, limiting the actuating effort to avoid aturation, pole-placement contraint etc., are cat a LMI and an optimal feedback controller i found by making ue of efficient interior-point algorithm. A fulltate feedback controller i deigned and implemented in thi thei uing thi method which then form the bai for deigning a tatic output feedback (SOF) controller. A profile wa generated that relate the change in the SOF control gain matrix required to keep the ame value of the generalized H norm of the tranfer function from the road diturbance to the actuating effort with the change in the prung ma of the quarter-car ytem. he quarter-car ytem make ue of a linear bruhle permanent magnet motor (LBPMM) a an actuator, a linear variable differential tranformer (LVD) and two accelerometer a enor for feedback control and form a platform to tet thee control methodologie.

4 iv For the full-tate feedback controller a performance meaure ( H norm of the 3 tranfer function from road diturbance to prung ma acceleration) of m/ wa achieved enuring that actuator aturation did not occur and that all pole had a minimum damping ratio of.. he SOF controller achieved a performance meaure of m/ enuring that actuator aturation doe not occur. Experimental and imulation reult are provided which demontrate the effectivene of the SOF controller for variou value of the prung ma. A reduction in the peak-to-peak velocity by 73%, 7%, and 71% wa achieved for a prung ma of.4 kg,.8 kg, and 3.4 kg, repectively. For the ame value of the prung ma, a modified lead-lag compenator achieved a reduction of 79%, 77% and, 69%, repectively. A reduction of 76% and 54% in the peak-to-peak velocity wa achieved for a prung ma of 6. kg in imulation by the SOF controller and the modified lead-lag compenator, repectively. he gain of the modified lead-lag compenator need to be recomputed in order to achieve a imilar attenuation a that of the SOF controller when the value of the prung ma i changed. For a prung ma of 3.4 kg and a upenion pring tiffne of 164 N/m the peak-to-peak velocity of the prung ma wa attenuated by 4 %.

5 v ACKNOWLEDGMENS I would incerely like to thank my committee chair Dr. Won-jong Kim for being a mentor and providing valuable inight and guidance through every tep of my reearch. I would alo like to thank the committee member Dr. Bryan Ramuen and Dr. Mehrdad Ehani for their help in completing my reearch. I am grateful to former tudent Mr. Seungho Lee, Mr. Bryan Murphy and Mr. Jutin Allen whoe work formed the bai of my reearch. I would alo like to thank my friend in the reearch lab who made my reearch experience enjoyable. Lat but not the leat, I would like to thank my family for their tremendou upport and love.

6 vi ABLE OF CONENS Page ABSRAC... ACKNOWLEDGMENS... ABLE OF CONENS... LIS OF FIGURES... LIS OF ABLES... iii v vi viii xi CHAPER I.INRODUCION... 1 A. Background and Motivation... 1 B. Contribution of the hei... 5 C. Overview of the hei... 6 II.EXPERIMENAL SEUP... 7 A. Linear Bruhle Permanent Magnet Motor (LBPMM)... 8 B. Linear Variable Differential ranformer (LVD)... 1 C. Accelerometer... 1 D. Sprung Ma, Unprung Ma, Wheel and Supenion Spring III.CONROL DESIGN BASED ON CONVEX OPIMIZAION... 1 A. Full-State Feedback Controller... 1 B. Static Output Feedback Controller... IV. CONROLLER DESIGN AND RESULS... 4 A. Modeling... 4 B. Full-State Feedback Controller... 8 C. Static Output Feedback Controller D. Modified Lead-Lag Controller V.PERFORMANCE ANALYSIS... 51

7 vii CHAPER Page VI.CONCLUSIONS REFERENCES... 6 APPENDIX A APPENDIX B VIA... 75

8 viii LIS OF FIGURES Page Figure 1 Photograph of the quarter-car tet bed with active upenion... Figure Schematic of the quarter-car upenion ytem... 8 Figure 3 Schematic of the control architecture... 8 Figure 4 Schematic diagram of LBPMM... 9 Figure 5 Schematic diagram of the conditioning circuit... 1 Figure 6 Free-body diagram of prung and unprung mae... 4 Figure 7 Deflection of upenion pring with added prung ma... 6 Figure 8 Figure 9 Sprung ma velocity in imulation and experiment ( m =.8 kg; k = 84 N/m)... 7 Unprung ma velocity in imulation and experiment ( m =.8 kg; k = 84 N/m)... 7 Figure 1 LMI region correponding to a damping ratio of ζ... 9 Figure 11 Figure 1 Figure 13 Sprung ma velocity in experiment and imulation for full-tate feedback ( m =.8 kg; k = 84 N/m)... 3 Actuating force calculated by full-tate feedback controller in experiment ( m =.8 kg; k = 84 N/m) Current flow of the full-tate feedback control in experiment ( m =.8 kg; k = 84 N/m)) Figure 14 Poition of the cloed-loop pole for full-tate feedback control Figure 15 Control effort in imulation without contraint uing SOF control ( m =.8 kg; k = 84 N/m)... 38

9 ix Page Figure 16 Value of generalized H norm H of the tranfer function from g control effort to road diturbance for variou value of α... 4 Figure 17 Figure 18 Figure 19 Sprung-ma velocity in experiment and imulation for SOF control ( m =.8 kg; k = 84 N/m)... 4 Control force calculated by SOF controller in experiment ( m =.8 kg; k = 84 N/m) Current in the three phae coil of the LBPMM for SOF control ( m =.8 kg; k = 84 N/m) Figure (a) K 1 v prung ma fitted with a quadratic curve Figure (b) K v prung ma fitted with a quadratic curve Figure 1 Figure Figure 3 Figure 4 Figure 5 Sprung-ma velocity in experiment and imulation for SOF control ( m = 3.4 kg; k = 164 N/m) Control force calculated by SOF controller in experiment ( m = 3.4 kg; k = 164 N/m) Bode plot of the tranfer function from Fact to x with quarter-car model parameter from able I (y1) and that from Lee [3] (y) 49 Bode plot of the compenated and uncompenated loop tranfer function Sprung-ma velocity with modified lead-lag compenator in experiment and imulation ( m =.8 kg; k = 84 N/m)... 5 Figure 6 Performance of the SOF controller ( m =.8 kg; k = 84 N/m) Figure 7 Performance of the modified lead-lag compenator ( m =.8 kg; k = 84 N/m)... 5 Figure 8 Performance of the SOF controller ( 6. m = kg; 84 k = N/m) in imulation... 53

10 x Page Figure 9 Performance of the modified lead-lag compenator ( m = 6. kg; k = 84 N/m)in imulation Figure 3 Performance of the SOF controller ( m = 3.4 kg; k = 164 N/m) Figure 31 Performance of the modified lead-lag compenator ( m = 3.4 kg; k = 164 N/m)... 56

11 xi LIS OF ABLES Page able I Parameter and correponding value of the quarter-car ytem able II Percentage reduction in prung-ma peak-to-peak velocity... 55

12 1 CHAPER I INRODUCION he function of an active upenion ytem in a vehicle i twofold ride handling that enable the driver to maintain control of the vehicle and ride comfort that minimize the road diturbance being tranmitted to the paenger. hi thei deal with the latter function with the deign and implementation of real-time controller uing convex optimization with LMI. Active upenion ytem could handle ride-comfort requirement better than paive upenion. hey employ an actuator powered by an external ource to minimize the prung ma acceleration. he control algorithm are deigned that calculate the optimum actuating force required to minimize the prungma acceleration making ue of the input data from enor. A. Background and Motivation A quarter-car active upenion ytem wa contructed by Allen [1] to erve a a tet bed for applying variou control methodologie. he variou component of the quarter-car model are hown in Figure 1. It conit of a prung ma and an unprung ma upported on a wheel that rotate on a cam that imulate road diturbance. An LVD and two accelerometer are ued a enor. hi tet bed incorporate the LBPMM developed by Murphy [] a the actuator. Magnetic actuation by thi LBPMM reduce the repone time, enabling the ue of fater controller. hi linear motor hi thei follow the tyle of IEEE ranaction on Control Sytem echnology.

13 eliminate the need for uing rotating machinery making the actuator ideal for ue in active upenion control. Lee [3] deigned four eparate controller uing modified lead-lag, linear-quadratic (LQ) ervo with Kalman filter, fuzzy control, and lide mode control (SMC) and wa able to achieve an attenuation of 78% on the prung ma velocity on the quarter-car tet bed. Figure 1. Photograph of the quarter-car tet bed with active upenion [1] Several claical control methodologie uch a linear quadratic Gauian/loop tranfer recovery (LQG/LR), lead-lag control, etc. depend on intuition and deign iteration. In order to reduce trial and error and alo to make the mot of the recent advance in computational power of computer, reearch in control ytem currently focue on numerical method to come up with an optimal controller that can atify

14 3 variou deign objective. LMI have emerged a a powerful tool in thi regard [4]. Variou problem in control theory can be repreented a LMI which can then be olved by convex optimization [4]. Efficient interior-point algorithm uch a thoe preented in [5] are able to olve the convex optimization problem in polynomial time. he algorithm in [5] wa incorporated in the LMI control toolbox of MALAB [6] where the LMI can be defined and olved. hi toolbox i utilized in thi thei to olve the convex optimization problem. In addition to guaranteed tability of the phyical ytem, the objective of the controller may alo be to achieve everal time- and frequency-domain pecification uch a peak time, ettling time, minimizing H and H norm, pole placement, etc. Each of thee control objective can be cat a LMI [7], [8]. A certain amount of conervatim i introduced into the deign by relating thee LMI with a ingle variable that enable u to etup a convex optimization problem [9] which can then be olved through one of the interior-point algorithm decribed in [5]. hi reearch aim at validating the above control methodology by applying the full-tate feedback controller (FSF) on the quarter-car upenion tet bed [1]. he primary control objective i to minimize the H norm of the tranfer function from the road diturbance to the prung-ma acceleration. Key contraint uch a limiting the actuating force and contraining the placement of pole for optimum dynamic performance are alo incorporated. One of the main advantage that thi method offer i that, the controller need not be redeigned even when there are change in the parameter

15 4 uch a the prung ma, upenion pring tiffne etc. he convex optimization program i rerun with the new parameter value. FSF i rarely achieved in real-world ytem. It become neceary to deign a feedback controller dependent on the available meaurement of the output. Hence, thi reearch alo look into the current numerical method to deign an SOF controller. he problem of finding an SOF controller that achieve the deired performance characteritic or determine that uch a feedback controller doe not exit, i an important open quetion in control theory [1]. Unlike FSF control, the optimization problem for deigning an SOF controller i not convex with the numerical method providing only a uboptimal olution [11]. It i an important control problem, however, a an SOF controller i one of the implet controller to implement ince it require no etimator and jut involve multiplying the output tate by a contant numeric value. A urvey of everal numerical output-feedback control-deign method i preented in [1]. It i poible to claify thee numerical controller into three clae nonlinear programming method, parametric optimization method, and convex programming method [1]. Non-mooth, non-convex optimization method uch a thoe preented in [13], [14] are currently conidered a the mot numerically efficient algorithm for SOF tabilization and cloed-loop performance guarantee [15]. he Levine-Athan primal method [16] and the Levine-Athan dual method [17] obtain a local optimal olution through an iterative procedure involving the olution of nonlinear equation. Optimization method uch a the Min-Max Algorithm [18] and Product Reduction Algorithm [19] try to olve the inequality contraint aociated with SOF

16 5 control through a parametric approach. In thi thei an algorithm baed on [15] i developed and implemented on the quarter-car tet bed. hi method can be claified a a mixture of convex programming method where additional contraint are added to make the SOF problem convex. In thi method, the problem of converting the SOF deign problem into a convex optimization problem i achieved by applying the elimination lemma and introducing lack variable. In thi thei an additional contraint of limiting the actuating effort i applied to thi method and teted for different value of the prung ma. he attenuation of the prung ma acceleration achieved through thi method i then compared with that of the modified lead-lag controller. A method to find a global optimal SOF control matrix i invetigated in [] and [1]. B. Contribution of the hei Multi-objective FSF controller have been deigned in [] and [3] uing convex optimization with LMI. In thi thei the ame principle i ued to deign and implement the FSF controller for the quarter-car tet bed. Uing the FSF control gain matrix an SOF controller i deigned baed on the algorithm in [15]. he main aim of thi thei i to validate thi SOF control algorithm. However, the control algorithm provided in [15] doe not contrain the available actuating effort. he control algorithm i modified o that the SOF controller account for actuator aturation. A relationhip between the prung ma and the entrie of the SOF gain matrix i found. Uing thi relationhip, the modified SOF control algorithm i implemented for three different value of the prung ma. A comparion of thi performance i made with the modified lead-lag controller developed by Lee [3]. With the help of imulation and experimental

17 6 reult it i hown that the gain of the modified lead-lag compenator need to be recomputed in order to achieve a imilar reduction in the peak-to-peak velocity of the prung ma a that of the modified SOF controller. C. Overview of the hei hi thei conit of ix chapter. In the econd chapter, the variou component of the quarter-car tet bed i given. In Chapter III, a background of the control theory ued to deign the full tate feedback and SOF controller i given. Chapter IV contain a decription of the tep-by-tep procedure followed to come up with the controller and alo the experimental and imulation reult. Chapter V conit of the analyi of the obtained reult. A comparion i alo made with the modified lead-lag controller deigned by Lee [3]. Chapter VI preent a ummary of the thei, concluion, and the future work that can be done in thi area.

18 7 CHAPER II EXPERIMENAL SEUP he experimental etup hown in Figure 1 wa developed by Allen [1]. he etup wa built in order to imulate a quarter-car upenion ytem. he quarter-car ytem can be divided into two main part prung ma (the portion of the car that lie above the upenion pring) and the unprung ma (including the unprung ma block and the wheel). A cam driven by a variable-peed electrical drill i ued to imulate the road diturbance. hree enor ( accelerometer and an LVD) are mounted on the etup, which contantly provide real-time data that can be ued for feedback control. An LBPMM i alo mounted on the ytem which erve a a direct-drive linear actuator providing the neceary force to minimize the prung ma vibration. Figure provide a chematic decription of the ytem.. A chematic of the control architecture i hown in Figure 3. he data collected from the enor are ued by the feedback controller which compute the actuating force required to minimize the prung ma vibration. he DS114 digital-ignal-proceing control board from dspace Inc. enable real-time data to be acquired from the enor through 16-bit A/D channel. hrough it 16-bit D/A channel control ignal can be ent to the actuator. he output from the D/A channel i amplified by the three PWM amplifier Model 1A8K from Advanced Motion Control to provide a current in the range of ±6 A to each phae of the LBPMM.

19 8 x M M x u k M u F act k M u x r Figure. Schematic of the quarter car upenion ytem [3] Figure 3. Schematic of the control architecture [3] A brief decription of the individual component ued in the model i decribed below. A. Linear Bruhle Permanent-Magnet Motor A chematic diagram of the LBPMM developed by Murphy [] i hown in Figure 4. he motor work on the principle that a force i induced in the current carrying conductor placed in a magnetic field. he actuator incorporate a new deign where the magnet which are encaed in a bra tube (mover), are arranged in NS-NS SN-SN

20 9 orientation. hi placement of the magnet piece generate a greater magnetic flux denity near the like pole. he tator conit of a three phaed coil. By controlling the amount of current paing through each of thee coil the total force generated by the actuator can be controlled a governed by the commutation law (1) [] given a follow: i () t co z i t γ = C f t i () t 1 3 a 1 o b() 1 3 zd (). in γ1z c (1) In (1), ia(), t ib(), t and ic ( t ) are the current paing through the three-phae coil, z i π the relative diplacement between the mover and the tator, γ 1 = where l i the l pitch of the actuator and f ( t ) i the required force in the axial direction. he contant zd C wa determined experimentally and wa found to be equal to.1383 A/N [1]. Figure 4. Schematic diagram of LBPMM [1]

21 1 B. Linear Variable Differential ranformer (LVD) he LVD (Schaevitz DC-SE 4) i mounted in order to receive real-time data of the relative diplacement between the prung and unprung mae. he input to the LVD (1V) wa provided by the Agilent 3644 A power upply. he output voltage of the LVD had a range between and 5 V. In order to make it compatible with the dspace 114 control board with a voltage wing from 1 to 1V, the output from the LVD wa fed to a ignal conditioning circuit hown in Figure 5 [3]. Figure 5. Schematic diagram of the conditioning circuit [3] C. Accelerometer wo accelerometer (Piezotronic model 355B4) were mounted on the ytem in order to receive real-time data of the prung- and unprung-ma acceleration. he accelerometer have an effective frequency range from 1 Hz to 1 KHz and a enitivity

22 11 of 17 mv/g. he accelerometer are powered by a ignal conditioner (PCB model 48A). D. Sprung Ma, Unprung Ma, Wheel and Supenion Spring he parameter which ultimately define the dynamic of the ytem are hown in able 1. he parameter m, m, and k repreent the prung ma, unprung ma and u the upenion pring tiffne repectively. In order to account for the frictional force on the prung ma the damping coefficient c i introduced. he wheel which i covered by a tire made of natural ioprene can be modeled a a pring and a vicou damper with value k w and c w repectively. he value of thee parameter were found by imulation and experimental reult [3]. able I Parameter and correponding value of the quarter-car ytem [3] Parameter Value m.8 kg m u.8 kg k 84 N/m c 17 N-/m c w 15 N-/m k 35 N/m w

23 1 CHAPER III CONROL DESIGN BASED ON CONVEX OPIMIZAION Conider the tate-pace repreentation of a phyical plant L given a follow: x = Ax + B u + B w y = Cx. u z = C x+ Du z = C x+ D u 1 () In (), x i the tate vector, u i the control vector, w i the diturbance vector, z 1 i the performance meaure, z i the contrained output, and y i the meaured output repectively. It i required that a feedback control gain matrix K be deigned o that it not only enure tability of the plant L, but alo guarantee certain performance characteritic. he control effort would then be a hown in (3) a follow: u= KCx. (3) he cloed-loop repreentation of the ytem L i then a hown in (4) a follow: x = Ax+ Bw z cl = C x 1 cl1 z = C x. cl 1 (4) where Acl = A+ Bu KC, Ccl1 = C1+ DKC 1, and Ccl = C + DKC. For the pecial cae of FSF control, C i an identity matrix. A. Full-State Feedback Controller In thi thei the plant L repreent the quarter-car active upenion tet bed. he control objective are preented in thi ubection. he primary objective i to

24 13 deign a controller to minimize the H norm of the tranfer function H( ) from the road diturbance to the performance meaure (prung ma acceleration). he H norm wa elected to erve a a meaure of the ride comfort [9] and can be interpreted a the quare root of the total output energy when a unit impule i given to the ytem. he H norm in the frequency domain i a defined in (5) a follow: H 1 = ( ( )* ( )). tr H j ω H j ω d ω π (5) he quare of the H norm can alo be repreented in term of the tate-pace matrice given in () a follow: ht () = trc ( SC ), (6) cl1 cl1 where S i the olution to AS + SA + BB = (7) cl cl 1 1 and ht () i the impule repone of the tranfer function H( )[8]. he equation given in (6) can be derived by the following method. he tranfer function H( ) i defined a follow: 1 H( ) = Ccl1( I Acl ) B1. (8) he impule repone matrix ht () of the plant i found by taking the invere Laplace tranform of the tranfer function H( ) hown a follow: h t = L C I A B = C e B (9) 1 1 () { ( ) 1} A cl t cl cl cl1 1.

25 14 By Pareval theorem, the econd-order tranfer function norm H() in the frequency domain i equal to the econd-order impule repone norm ht () in the time domain [1]. If h repreent the element in the th ij i row and ht () then the H norm can alo be written a th j column in the matrix = ij = ij ht () h() t dt trhtht { () ()} dt. (1) From (9), (1) can alo be written a At cl A cl t At A t cl1 1 1 cl1 1 1 h() t = tr{ C e B B e C } dt = tr{ C e B B e dtc }. (11) By defining At A t = 1 1 S e BB e dt (1) in (11), we get (6), a S i the controllability gramian which i the unique olution of (7). Hence with the help of (6) and (7), the H norm of H( ) can be repreented in term of the tate-pace matrice in (). In the implementation of the feedback controller, it mut alo be enured that the actuating force mut not exceed the aturation limit of the actuator, the LBPMM in our cae. Failure to do o would reult in exceive current, which might demagnetize the magnet in the actuator. o enure the above time-dependent hard contraint, the generalized H norm H defined in (13) i ued [9]. H repreent the peak g g

26 15 amplitude of the contrained output z when a unit energy input from the diturbance vector w i fed to the ytem and i repreented a follow: H = up z( t) : x() =, t, w( ) d 1. g t τ τ (13) Another contraint that i added while deigning the controller i to retrict the placement of the cloed-loop pole in the complex plane. he location of the pole determine time-domain characteritic uch a ettling time, rie time, etc. Although the pole deep in the left-half complex plane increae tability, thee very fat pole are difficult to model. After identifying the control objective a lited above, the next tep in the deign proce i to repreent them a LMI. An LMI ha the following form Fr () = F+ rf>, m i i= 1 i (14) where F i are contant ymmetric matrice in the real pace matrix which belong to n n and r i a variable m 1 [4] containing the variable r i. he inequality F( x ) > implie that the matrix i poitive definite, i.e. the real part of all the eigen value of F( x) are greater than zero. he LMI a hown in (15) Qr () Sr () > Sr () Rr () (15) i another form of repreenting (14), where Qr ( ), S( r), and R( r) are ymmetric matrice affinely dependent on r [4]. By Schur complement, (16) i equivalent to the following nonlinear convex inequalitie [4]:

27 16 Q( x ) >, 1 Q( x) S( x) R( x) S( x) >. (16) Now we try to repreent the control objective in the form of (15). If we let the H norm be le than or equal to ome poitive valueγ, then the control objective would be equivalent to minimizing the value ofγ. We introduce a variable ymmetric matrix S [5] uch that S > ; AS+ SA + BB 1 1 <. (17) By ubtracting the equation in (7) from the inequality in (17) we get cl cl A ( S S ) + ( S S ) A <. (18) cl cl Lyapunov inequality tate that a ytem repreented by the tate pace equation x a = Ax 1 a i aymptotically table if and only if there exit a ymmetric matrix P uch that A P+ ; P >. (19) 1 PA1 Since Acl i to be tabilized through feedback control, from (18) we can ay that S S mut be poitive definite i.e. From () we get S S >. () tr{ C ( S S ) C } > cl tr{ C S C } < tr{ C SC } cl1 cl1 cl1 cl1 ht () < trc { SC }. cl cl1 cl1 (1) From (1) it i clear that if tr{ Ccl1SCcl1} < γ, then the H norm of H( ) would alo be le than equal toγ. he inequalitie (17) and (1) can now be etup a LMI by

28 17 1 introducing poitive deifinte variable matrice P = S and Q [9]. Uing (15) and (16) we have 1 AP cl 1+ PA 1 cl PB 1 1 < BP 1 1 I ; P1 Ccl1 > Ccl1 Q ; tr( Q) < γ. () he firt matrix inequality enure that the inequality in (18) i atified. From the econd matrix inequality in () we get P 1 > S > (3) and 1 Q Ccl1P Ccl1 > (4) tr( Ccl1SCcl1) < tr( Q) < γ. hu the LMI in () enure that the H norm i le thanγ. It i required that the generalized H norm a decribed in (13) for the plant L mut be le than ome poitive value ν in order to avoid actuator aturation. hi value of ν determine the maximum control effort that can be utilized when a unit energy diturbance enter the ytem. In order to repreent thi objective a an LMI, a Lyapunov function i firt defined a follow [5]: V( x ) = x P x (5) cl where P i a ymmetric poitive definite matrix. Differentiating (5) we get

29 18 d V ( x ) = x Px x Px +. dt (6) Uing (4), (6) can be written a follow: d x AP cl + PA cl PB x 1 V ( x ) = dt w BP 1 w x x x AP cl + PA x cl PB 1 ( ) =. 1 d V x dt w I w w BP I w (7) From (7) the following condition hold true: AP cl + PA cl PB 1 d ( ). < V x ww < BP dt 1 I (8) Integrating (8) we get V( x( τ )) < ww dt. τ Our objective i to enure that for any time τ > the contrained output z be uch that z ( τ) z ( τ) < ν ww dt. hi enure that when a energy diturbance le than unity τ τ ( ww dt < 1) enter the ytem, z doe not exceed ν. From (9) it i ufficient if (9) z( τ ) z( τ) < νv( x( τ)) x ( C C ν P ) x <. cl cl (3) he inequality (3) can be repreented a the following LMI: P C cl > C cl ν I. (31)

30 19 In order to etablih the third contraint that i retricting the location of the cloed loop pole in the complex plan, a convex LMI region i defined a hown below [7]: D = { z C : N + zm + zm } (3) where N i ymmetric matrix, and both N and M are fixed real matrice. If n and ij correpond to general entrie in the N and M matrice, repectively, then the plant L i D table (the pole lie in the LMI region defined by D ) under cloed-loop feedback if and only if there exit a poitive-definite matrix P 3 uch that the following condition i atified [7]: m ij np map mpa (33) ij 3+ ij cl 3+ ji 3 cl <. ij In order to convert thee LMI into a convex optimization problem with the objective of minimizing the value ofγ, we need to equate the matrice P 1, P and P 3 [5]. hi introduce ome conervatim in the deign proce. he following ubtitution are then made in the LMI in (), (6), (9) and (31): X = P 1 Y = K X. f (34) Repreenting the LMI in (), (31) and (33) in term of the tate-pace matrice in (4) and making ue of the ubtitution in (34) we get the following inequalitie:

31 AX + XA + BuY + Y Bu B 1 < B1 I (35) X C X + DY 1 1 > XC1 + XD1 Q ; tr( Q) > γ (36) X XC + Y D CX+ DY ν I > (37) nx ij + mij ( AX+ BY u ) + mji ( XA + Y Bu ) <. (38) ij he LMI (35) (38) are then defined uing the LMI control toolbox in MALAB [6]. he optimization variable are X, Y and γ. he convex optimization program i then run which find the value of thee variable that minimize the value of γ at the ame time enuring that the contraint repreented by the above LMI are not violated. Once the optimal value of X and Y are found, the full-tate feedback controller obtained from the ubtitution in (34). B. Static Output Feedback Controller K can be f Since FSF i not alway available in real-world ytem, the available meaurement mut be ued in order to deign the feedback controller. he convexity in the cae of the FSF optimization problem doe not exit for SOF. No imple ubtitution a in (34) ha been found for tatic output feedback problem [9]. However there exit ome method of obtaining a uboptimal olution for the SOF problem. In thi reearch the algorithm a propoed in [15] ha been ued to deign the SOF controller.

32 1 A parametric approach i ued in [15] wherein by the proce of applying the elimination lemma lack variable are introduced to convert the SOF deign problem to a convex optimization problem. hi enable an iterative procedure baed on LMI optimization to be ued, to converge to a local optimal olution. In thi approach the H norm in (6) i expreed in term of the obervability gramian W a follow: ht () = rc ( SC ) = rbwb ( ). (39) cl1 cl1 1 1 Hence, analogou to the inequality in (17) we introduce a ymmetric matrix W1 uch that the the following inequality i atified [15]: W > ; A W + W A + C C <. (4) 1 cl 1 1 cl cl1 cl1 Expreing the inequality in (4) in term of the tate-pace matrice in (4) we get the following: ( A+ B K C) W + W( A+ BK C) + ( C + C K D )( C + D K C) <. (41) u of 1 1 of 1 of of A linear matrix function M ( W 1) i then defined [15] a follow: A W + W A+ C C WB + C D ( 1) =. B1 W1+ D1 C1 D1 D1 MW (4) With the help of the definition in (4) we can expre the inequality in (41) a follow: I I C K of M( W1 ) <. K of C (43)

33 It i poible to apply the elimination lemma to the above inequality. Elimination lemma tate that if a ymmetric matrix given then there exit a matrix n n k n Q E and matrice B E K E j k j n and be C E uch that the et of inequalitie i equivalent to the inequality In (45), B E and B Q B < and C Q C < (44) E E E E E E Q + C K B + B K C <. (45) E E E E E E E C E are the orthogonal complement of B E and C E, repectively. Uing the property of the elimination lemma decribed above, the inequality in (43) can be expreed a the inequality in (46) a follow: 1 1 of of 1 MW ( ) + E K C I + K C I E <. (46) he matrix E 1 i written in a block matrix form introducing two new variable F a follow: F and E F 1. = F (47) Uing (47), the inequality in (46) can now be written a follow: FK ofc F C Kof F C Kof F MW ( 1) + FKof C F + <. F F (48) By making ue of the ubtitution in (49) and (5) hown below: FF K 1 = f R = FK of (49) (5) the inequality in (48) can be written a follow:

34 3 Kf RC Kf F C R Kf C R MW ( 1) + +. < RC F F Kf F (51) he above inequality ha four optimization variable K, R, W, and F. Since the matrix entrie in the above inequality i nonlinear in thee variable, an iterative procedure called the coordinate decent-type algorithm a decribed below i et up to olve the optimization problem [15]. a. Coordinate Decent-ype Algorithm [15] f 1) A table full-tate feedback matrix ) reating i K f form the tarting point of the algorithm. i K f a a contant matrix, a convex optimization problem i etup with the objective of minimizing the H norm ht () a defined in (39) ubject to the LMI in (46). he matrice F and R correponding to thi optimal value ( h 1 ) of ht () i kept contant. 3) With the above value of F and R the convex optimization problem of minimizing the H norm i etup again treating K and W a the optimization variable. he f optimal value of the H norm obtained in thi tep i equal to h. 4) he value of iteration i i increaed. he optimal value of K obtained in Step 3 f form the new K to be ued in Step. he iteration are repeated until a atifactory i f value of h1 h i obtained. When the iteration ceae, the optimum value of R and K are utilized to obtain the SOF controller by uing the ubtitution in (47). of

35 4 CHAPER IV CONROLLER DESIGN AND RESULS A. Modeling he firt tep in the deign proce i to model the ytem dynamic. In Figure 6, x and xu are the prung- and unprung-ma diplacement, repectively. Fact i the actuating force provided by the LBPMM. he pring tiffne k and the tire i modeled to have a tiffne of kw and a damping coefficient of c w. A friction force act even on the prung ma. In order to repreent the quarter-car tet bed a a linear ytem, thi friction force i accounted for by introducing a damping coefficient c which act only on the prung ma. he free-body diagram in Figure 6 i ued to generate the equation of motion. ( x ) k x u Fact x x u m Mm u ( x ) k x u cx Fact ( ) c ( x x ) k x x w u r w u r Figure 6. Free-body diagram of prung and unprung mae follow: he equation of motion derived from the free body diagram in Figure are a

36 5 ( ) mx = k x x cx + F (5) u act ( ) ( ) ( ). mx = kx x k x x c x x F (53) u u u w u r w u r act he tate of the quarter-car tet bed are elected to be a follow: [ x x x x ]. (54) u u In term of the tate defined in (54), the quarter-car model can be repreented a a tatepace ytem defined in () a follow: 1 x () t 1 x() t u() k k c x t xu () t xr () t 1 = + x() t m m m x() t xr () t + m kw c w xu() t k ( k+ kw ) c x () 1 w u t mu m u m m u mu m u u F act () t x () t k k xu () t 1 z1 = + Fact ( t) m () m x t m x u () t x () t x () t 1 z = + F ( t). [ ] u x () t Fmax x u () t act (55) Once the dynamic of the ytem have been derived, it i required to find the parameter uch a the prung and unprung mae, upenion pring tiffne, etc. which govern them. he value of the prung and unprung mae were meaured to be

37 6.8 kg and.8 kg, repectively. In order to determine the value of the pring tiffne k, additional mae on the prung ma were placed and the deflection of the pring from the mean poition were noted. Figure 7 how the correponding data point. We ee that up to a diplacement of around 1.5 cm, the diplacement i directly proportional to the added mae. By uing the Hooke law in thi region, the pring tiffne k wa then determined to be 84 N/m. he value of the damping coefficient c, cw and the tire tiffne k w hown in able I in Chapter II, Subection D were etimated through a proce of trial and error uch that there wa a good match between experiment and imulation. 3 Diplacement (cm) X:.75 Y:.9 data point quadratic fit Ma (kg) Figure 7. Deflection of upenion pring with added prung ma Figure 8 and 9 how the prung and unprung ma velocitie repectively, when a road diturbance in the form of a inuoid with a frequency of. Hz wa applied in imulation and experiment. he figure how a good match between

38 7 imulation and experiment indicating a good etimation of the ytem parameter. A jerking action occur when the tire pae through the top poition of the cam. he effect of thi action i greater on the unprung ma a the tiffne of the tire i greater than that of the upenion pring. Hence a mooth profile of the unprung-ma velocity i not obtained cauing a dicrepancy between experiment and imulation reult. Sprung Ma Velocity (m/) Experiment Simulation ime () Figure 8. Sprung ma velocity in imulation and experiment ( m =.8 kg; k = 84 N/m) Unprung Ma Velocity (m/) Experiment Simulation ime () Figure 9. Unprung ma velocity in imulation and experiment m =.8 kg; k = 84 N/m) (

39 8 B. Full-State Feedback Controller Once the ytem i completely modeled and the ytem parameter are identified, it i now poible to deign the control ytem. Following the tep provided in Chapter III of the thei for the deign of the FSF controller, the objective of the control ytem are firt defined. he primary objective i to minimize the H from the input road diturbance to the prung ma acceleration. he next objective i a performance contraint where the actuation force i to be limited to 5 N. Exceeding thi force for a prolonged time may caue exce current to pa through the coil which might damage the magnet of the LBPMM. Another contraint that can be added to the deign of the control ytem i to force the pole to lie in a deired portion of the - plane. hi enable u to control certain performance characteritic uch a damping ratio. In order to achieve thi deign pecification, the dominant cloed loop pole mut lie in the haded region of Figure 1 where the value of θ i a hown in (56) and ζ i the deired damping ratio expreed a follow: θ 1 = co ( ζ ). (56) he LMI region correponding to the haded area in Figure 1 can be expreed a in (3) where the matrice N and M are a follow [7]: N = ; M inθ coθ =. coθ inθ (57)

40 9 Imaginary Axi θ Real Axi Figure 1. LMI region correponding to a damping ratio of ζ Once the control objective have been identified they are cat a LMI. he LMI correponding to the control objective defined above are rewritten a follow: AX + XA + BuY + Y Bu B 1 < B1 I (58) X XC + XD CX 1 + DY 1 Q 1 1 > ; tr( Q) > γ (59) X XC + Y D CX+ DY ν I > (6) nx ij + mij ( AX+ BY u ) + mji ( XA + Y Bu ) <. (61) he tate-pace matrice (defined in ()) correponding to the quarter-car model are obtained from (55). he matrice X, Y, and Q and the variable γ now need to be optimized uch that the above inequality contraint are met and the value of γ i minimized. hee variable mut firt be defined in order to olve the convex ij

41 3 optimization problem. For the cae of the quarter-car model, X i a ymmetric 4 4 matrix containing 1 variable, Y i a 1 4 matrix containing 4 variable and Q i a 1 1 matrix. Hence along with γ a total of 16 variable are preent. In order to completely define the LMI, we mut aign value to ν and ζ. he value of ν determine the limit on the actuating effort when a unit energy diturbance enter the ytem. In order to reduce conervativene, a good knowledge of the road diturbance i eential. In the cae of the quarter-car tet bed the road diturbance can be modeled a w =.15in(.5 t). (6) Even with the knowledge of the road diturbance, the only way the value of ν can be elected i to deign the controller tarting with an arbitrary value and checking the actuating effort applied in imulation when the road diturbance decribed in (6) i applied to the model. he greater the value of ν, the greater i the actuating effort applied. After trial and error, a value of 5, wa elected. hi correpond to a maximum actuating effort of 13 N when the road diturbance in (6) i applied in imulation. Although, a maximum force of 5 N can be applied, a maller value of 13 N wa choen in imulation to account for the modeling uncertaintie. In order to atify the LMI given in (61) all the dominant and non-dominant pole mut lie in the LMI region hown in Figure 1. Hence, it i not poible to contrain the location of elective pole. With many value of ζ teted, it wa oberved that increaing the damping ratio alo increaed the H. Hence, a mall value of.

42 31 wa elected for ζ. Correponding to thee value of ζ, the matrice N and M were computed uing (57) and are hown below: N = ; M = (63) Uing the value of the matrice N and M in (64), the LMI in (6) i written a follow:.9798( AX + BuY ) ( XA + Y Bu).( AX + BuY ) +.( XA + Y Bu). <.( AX + BuY ).( XA + Y Bu).9798( AX + BuY ) ( XA + Y Bu) (64) Once the LMI variable and inequalitie are defined, the convex optimization program B.1 in Appendix B i run with the help of the LMI control toolbox in MALAB [6]. After optimization, the value of γ and the matrice X and Y were found to be a hown in (65), (66) and (67): X = (65) [ ] 6 Y = (66) he full tate feedback controller ubtitution in (35): 3 γ = m/. (67) K hown below i then obtained by uing the f [ ] K = (68) f 3 hu thi method wa able to achieve a performance meaure H of m/ correponding to a generalized H norm H of the tranfer function from z g to the

43 3 road diturbance of 93 N. he FSF controller obtained in (68) through the proce of convex optimization i now teted on the quarter-car tet bed. Due to the preence of a jerking action when the wheel move on top of the cam, we ee pike in the acceleration of the prung ma. Hence to give a better idea of the performance of the control ytem, we can alo look at it effect on the velocity of the prung ma. Since the magnitude of the acceleration and velocity differ by a factor of ω, when the diturbance i in the form of a pure inuoid of frequency ω, the percentage reduction in the acceleration of the prung ma i equal to it percentage reduction in the velocity. 1 Controller OFF Controller ON Sprung Ma Velocity (m/) Experiment Simulation ime () Figure 11. Sprung-ma velocity in experiment and imulation for full-tate feedback m =.8 kg; k = 84 N/m) ( Figure 11 how the prung ma velocity in imulation and experiment, with the controller in the on and off poition when a inuoidal input diturbance of frequency.3 Hz wa applied to the ytem. From the figure, ignificant reduction in the peak-to-

44 33 peak velocity of the prung ma can be oberved along with a good match between the imulation and experimental reult. Controller OFF Controller ON 1 Force (N) ime () Figure 1. Actuating force calculated by the full-tate feedback controller in experiment m =.8 kg; k = 84 N/m) ( he control force calculated by the full-tate feedback controller during the experiment i hown in Figure 1. No current pae through the actuator when the controller i off. he non-zero offet when the controller i off It can be een that the force doe not exceed the aturation limit of 5 N, thu atifying one of the control objective. Figure 13 how the current paing through the three phae of the LBPMM during the experimental run. It can be een that none of the three-phae current paing through the LBPMM exceed 6 A. Figure 14 how the location of the cloed-loop pole of the quarter-car tet bed with the full-tate feedback control which are equal to the eigen value of the matrix A cl. he value obtained were

45 ± j and.367 ± j.65. It i evident from the figure that none of the pole lie in the right hand plane, hence the ytem i table. he pole alo lie in the LMI region correponding to a damping ratio of ζ =.. Although all the control objective were atified, the full potential of the FSF controller could not be achieved a the data from the unprung-ma accelerometer wa not accurate due to the jerking motion experienced by the unprung ma. In order to olve thi problem where the data from the unprung-ma accelerometer wa not reliable, an SOF controller i deigned baed on the algorithm in [15]. 4 Controller OFF Controller ON Current (A) - -4 Phae C Phae B Phae A ime () Figure 13. Current flow of the full-tate feedback control in experiment m =.8 kg; k = 84 N/m) (

46 35 5 Imaginary Axi θ Real Axi Figure 14. Poition of the cloed-loop pole for full-tate feedback control C. Static Output Feedback Controller In order to deign the SOF controller the procedure decribed in Chapter III i adopted. Firt, the linear matrix variable function M ( W 1) i defined a in (4) uing the tate pace matrice correponding to the quarter-car model in (55). Once, thi function ha been defined the LMI a expreed in (48) can be defined in order to run the coordinate-decent algorithm decribed in Chapter III. hi LMI i rewritten in (69) a follow: KfiRC Kf F C R Kfi C R MW ( 1) + +. < RC F F Kfi F (69) It can be een from the above inequality that the matrice K, R, F, and W 1 are the variable that need to be optimized uch that it not only minimize the value of the trace of the matrix BWBbut alo atifie the inequality contraint in (69). he term of fi

47 36 the matrice in the inequality in (69) which contain the product of the variable are nonlinear. Since each term of the matrice in the inequality mut be affine in order to run the convex optimization problem, it i run in two part. In the firt part, the matrice R, F, and W 1 are treated a variable. For the quarter-car model, W1 i a 4 4 poitive definite ymmetric matrix containing 1 variable element, R i a 1 matrix containing variable element and F i a 1 1matrix. he full-tate feedback controller K i f taken a K fi in the firt iteration of the coordinate-decent algorithm. In the econd part of the algorithm the matrice K and W fi 1 form the variable. he matrix K i defined a a 1 4matrix containing 4 variable. he optimum value of fi R and F obtained a a reult of running the convex optimization program in the firt part of the algorithm are ued in the econd part. he optimization program i run again to get new value K andw fi 1. Several iteration of the algorithm are run until the value of the trace of the matrix BWBin ucceive iteration doe not differ by more than a deired value λ. he program to run thi algorithm i written in the LMI toolbox of MALAB and hown in Appendix B. where the value of λ i taken to be 1. he value of the matrice R, F, and W1 obtained at the end of the iteration are hown below: R = [ ] ; F =.14 (7)

48 W1 = (71) Making ue of the ubtitution in (5) the following SOF controller K of i obtained: K = [ ]. (7) of Figure 15 how the exerted actuating force when the SOF controller deigned above i ued in imulation when the input road diturbance i equal to that in (6). he figure clearly how that the actuating effort would exceed the allowable value of 5 N in imulation. Alo while performing the experiment, the controller i turned on after the cam reache the deired frequency of rotation (around.5 Hz). A a reult, becaue of the large gain the initial control effort would be very large a i evident from the pike in the magnitude of force in Figure 15. Hence, thi controller cannot be implemented to the experimental etup. In order to olve the above problem, a multi-objective coordinate-decent algorithm [15] i ued. In order to achieve thi multi-objective control, it i required that the inequalitie expreed in (8) and (31) (which enure that the performance output z doe not exceed a certain value equal to the quare root of ν when a unit energy diturbance i the input into the ytem) be expreed in the form of the LMI in (48). K of he inequalitie (8) and (31) when expanded in term of the SOF control matrix and the tate-pace matrice in (), we obtain the following inequalitie:

49 38 ( A+ B K C) W + W ( A + C K B ) + BB < u of of u 1 1 I I BuKofC M( W ) < C K of B u (73) 1 3 ( ) P C + DKof C ( C + DK of C ) > ν I I C K of M( W3 ) > K of C (74) where AW + WA + BB 1 1 W MW ( ) = and W 1 1 W3 CC CD ν ν MW ( 3) =. 1 1 DC DD ν ν Force (N) ime () Figure 15. Control effort in imulation without contraint uing SOF control ( m =.8 kg; k = 84 N/m)) Applying the elimination lemma to (73) (74), we obtain the following inequalitie:

50 39 I MW ( ) + E CKof Bu I + E < BK u ofc (75) CK ' of MW ( 3) + E 3 Kof C I + E3 <. I (76) We ee that although the inequalitie in (48), (75), and (76) are of the imilar form, the coordinate-decent algorithm cannot be applied becaue of the nonlinear contraint W 1 W 3 =. Hence the method of introducing lack variable by applying the elimination lemma in the cae of multi-objective control lead u to the initial problem of nonlinearity we tried to olve. In order to olve thi problem, we make ue of the fact that for a ingle-input, ingle-output ytem, the generalized H norm a decribed in (13) i equivalent to the following equation [4]: P cl cl 4 cl cl H = inf{ B W B : W A + A W + C C < }. g (77) Following the ame procedure a above (introducing lack variable by applying elimination lemma) we obtain the inequality C K MW ( 4) + E4[ KC I] + E4' < I (78) where A W4 + W4A+ CC W4B1+ CD MW ( 4) =. By equating W4 and W 1, and E4 BP DC DD and E1 the coordinate decent algorithm can be run which eek to minimize both the H norm of the tranfer function H( ) and alo the generalized H norm H g of the

51 4 tranfer function from the road diturbance to the control effort. However equating W4 and W1 would mean that both the norm would be forced to be equal. Hence a weighing factor α i introduced on the contrained output z (control effort). he coordinate decent algorithm i now run to minimize H (48) and (78) for different value of α. atifying the inequality contraint of 1 α Weighing factor X: 194. Y: H (N) Figure 16. Value of generalized H norm H g of the tranfer function from control effort to road diturbance for variou value of α. g Figure 16 how the value of H for different value of α. We ee that the g effect of the invere of α i imilar to the effect of ν in the full-tate feedback control deign. he greater i the value ofα, the maller i the available control effort. he value of α wa o choen by ome trial and error uch that the control effort in imulation did not exceed 14 N. hi reulted in a value of H equal to 194. N. g

52 41 he coordinate-decent algorithm i run with the value ofα = 9.. he optimum value of the matrice R, F, and W1 obtained for thi value of α are hown in (79) and (8) a follow: R = [ ] ; F =.68 (79) W1 = W4 = Making ue of the ubtitution in (5) the SOF controller K of (8) hown in (81) i obtained a follow: of [ ] giving a value of177.7 m/ for the H norm of H( ). K = (81) Figure 17 how the effect of the SOF controller in imulation and experiment. We ee that there i a very good match between the experimental and imulation reult and the prung-ma velocity i coniderably attenuated when the controller i turned on. When the controller i turned off, we ee that the magnitude of the prung-ma velocity i maller in the downward direction in experiment. When running the imulation it wa aumed that the pring tiffne remain contant and i independent of it deflection. However, the phyical pring that we have exhibit variable tiffne a i een in Figure 7. Hence, when the pring i compreed (i.e, when the prung ma i below it equilibrium poition) it tiffne increae and the prung ma i unable to reach the