H-infinity Vehicle Control Using NonDimensional Perturbation Measures

Transcription

1 H-ininity Vehicle Control sing NonDimensional Perturbation Measures Abstract Robust controller design techniques have been applied to the ield o vehicle control to achieve many dierent perormance measures: robust yaw rate control [1], robust lateral positioning using one [, ] or more [-] driver inputs, robust observer design, and so on. A diiculty with many published approaches is to obtain an adequate description o the model uncertainty. Most bounds on plant requency responses or parameter perturbations are based on ad-hoc limits that rely primarily on the designer s personal choice. he resulting controller design is thereore oten vehicle speciic, and is suitable only or application to a single design vehicle. his work shows that (a) the description o model uncertainty is not a design variable, (b) meaningul bounds on model uncertainty can be obtained rom data and should be sought, and (c) these bounds can be used to generate a single controller suitable or any vehicle. 1. Previous Work and Problem Statement Previous work [] presented a robust lateral-position controller useul or highway driving o any passenger vehicle. he system uncertainty was represented as matrix element perturbations o the system matrices. A Linear Matrix nequality (LM) approach was then used to design a state-eedback lateral position controller robust to expected parameter variation between vehicles. he resulting controller thereore robustly stabilizes all vehicles dynamically described by the bicycle model and which are parametrically bounded by ixed bounds. nortunately, these bounds do not address dynamic uncertainty associated with unmodeled dynamics, disturbances, or measurement noise. An important result o this previous work was the conclusion that a state-eedback controller is not capable o robust lateral vehicle positioning over wide variations in velocity, at least not without some type o gain scheduling. he controller presented in this work seeks to address controller robustness in a more intuitive ramework than previous LM representations. An H-ininity ramework is presented that account or both parametric uncertainty and unmodeled dynamic uncertainty to achieve a robust controller design. A unique aspect o this work is the representation o vehicle dynamics in a nondimensional orm that allows a generalized solution to the lateral control problem. Motivating this nondimensional representation is S. Brennan A. Alleyne* Dept. o Mechanical & ndustrial Engineering niversity o llinois at rbana-champaign 1 W. Green Street, rbana, L 181 *corresponding author 5 the desire to develop controller implementations suitable to any vehicle, not just a particular research vehicle under study. his paper is summarized as ollows: Section presents equations or the linear, lateral vehicle dynamics with a ixed preview distance. hese equations are presented in both dimensional and nondimensional orm, with the nondimensional orm governed by a new set o unitless parameter groupings known as groups. Distributions o these groups deine an average vehicle dynamic. Section deines the robustness criteria required or generalized vehicle control: that a controller must stabilize the average vehicle dynamics in the presence o perturbations that generate the range o published vehicle dynamics. his range is numerically deined by bounding the requency-response dierence between the average vehicle and 5 other published vehicle dynamics. t was again ound that no single controller is capable o robust control over large variations in speed or riction. Section ixes the speed and riction, then develops a robust controller design to demonstrate a single-velocity robust controller implementation. Section 5 then discusses gainscheduling approaches and limitations to robust control. Finally, a conclusion summarizes the primary results.. Vehicle Dynamics Motivating the nondimensional representation is the desire to develop controller implementations suitable to any vehicle. Consequently, vehicle-to-vehicle variation is addressed in a nondimensional ramework that utilizes the Buckingham Pi theorem [7]. he resulting representation accommodates two modeling aspects previously ignored by other researchers: irst, parameter-to-parameter interdependency clearly arises due to common vehicle design; second, the individual parameter distributions show a normal distributions in the nondimensional representation, which speciically deine a mean and standard deviation o vehicle dynamics [, 8]. Application o the Buckingham Pi theorem [7] to the classical vehicle dynamics known as the Bicycle Model yields groupings o parameters that collectively do not have dimensions. hese dimensionless parameters are known as groups, and or the planar bicycle model these parameters are:

2 C L a b α C L z,,, αr (1),. 1 L L m m 5 ml With: m vehicle mass (5.51 kg) z vehicle moment o inertia (.115 kgm ) V vehicle longitudinal velocity (. m/s) a distance rom C.G. to ront axle (.11 m) b distance rom C.G. to rear axle (.191 m) L vehicle length, a + b (.5 m) C α cornering stiness o ront tires (5 N/rad) C αr cornering stiness o rear tires (11 N/rad) he values in parenthesis are quite dierent rom a typical ull-sized vehicle because they correspond to the measured values or a 1/7-scale experimental scale used on the llinois Roadway Simulator, a treadmill/vehicle counterpart to a wind tunnel/airplane testing system. he similarity in dynamics between this vehicle and ull-sized vehicles was proven via the Buckingham-Pi heorem, and was the original motivation or analyzing vehicle dynamics in a nondimensional ramework. Derivation and explanations o the vehicle parameters are given in more detail in [8]. he values o Equation (1) are readily available or many vehicles by utilizing parameters published in the literature or the standard bicycle model; thus ar 5 sets o parameters have been recorded in data set hereater reerred to a V. Each set member will be reerred to as V i. A relative distribution o vehicle parameters or V is shown in Fig. 1, with each vehicle traveling at 8 m/s ( mph). Clearly, a vehicle model described by nondimensional parameters has reduced parametric dimension, rom 8 dimensional parameters to 5 nondimensional parameters. Further, is not independent and can be obtained by deinition: Pi Pi 1 () Pi.1... Fig. 1: Distribution o Pi parameters For this reason, is not plotted. From these distributions, the average parameters are obtained: Pi5 1.7,.15,.175, 5.1 () Note that and are dependent on velocity and cornering stiness, so these average values change with chosen operating speed and road surace, with ratios given by Equation (1). he space description o vehicle dynamics is based on the bicycle model [9] with the state vector [lateral position, lateral velocity, yaw angle, yaw rate]: dy dψ x y ψ () dt dt and ront steering input, u, as the sole control channel. Note that all states are measured with respect to the vehicle s center-o-gravity. he state space representation (in path error coordinates) is: with: dx, dt A x + Buy C x (5) C 1, α, m m m m () A 1 B C a Cα z z z z C + C, ac bc, a C + b C 1 α αr α αr α αr he non-dimensionalization transorms are obtained by a state substitution that normalizes each state with respect to distance, mass, and time. he same approach was used to generate the parameters o Equation (1), and not surprisingly the nondimensional dynamics can be rewritten entirely in terms o the same parameters. Mathematically, this is shown via a variable substitution deined by: x M x * (7) he above state normalizations are combined with a time normalization, t S t*. For the vehicle system, L S, and M is deined as: Giving: diag L 1 M (8) L 1 p1 p1 p, A* 1 B* p p p p +, p, p A similar nondimensionalization can be obtained in the Laplace domain, generating transer unction dynamics solely dependent on the vehicle parameters. (9) 55

3 n the dimensional orm o vehicle dynamics, many authors have noted that additional eedback modiications, such as error preview [1-1] or ull-state eedback [], make the control problem more amenable. n this work, an error preview scheme is utilized, represented by a modiication o the C matrix as presented in [11]. Speciically: [ 1 p ] C. (1) with p being the preview time. n the linear description, the eect o preview is equivalent to adding yaw angle eedback, and thereore preview is simply a special case o state-eedback. n the nondimensional orm, the C* matrix becomes: with the term: [ p L ] C * 1 /. (11) p/ L. (1) as a new variable. o pick the value o p, a wide range o preview values were examined. he perormance eect o increasing preview was primarily to increase the phase margin deiciency caused by the two ree integrators. he robustness eect o increasing preview was to somewhat reduce high-requency model uncertainty. his is not very helpul since the diiculty with vehicle robustness is a problem governed generally by low-requency uncertainty. Rather than include preview as an additional unnecessary design varaible, a ixed preview time o seconds was selected and is used hereater. o complete the vehicle model description, it is useul to add additional scaling transorms to limit the largest control eort, tracking error, and reerence input to all have unity 1-norms. o do this, one uses a variable transormation suggested by [1]. For the vehicle system, reasonable signal norms are: [ ]( [ ]) [ ]( [ ]) [ m] u.175 rad 1 degrees e.15 m.5 scalelanes r.15 Which become in the nondimensional space: [ unitless] [ unitless] (1) u *.175 e (1) e*.18 L r r*.18[ unitless] L With the signals normalized as above, the goal is to maintain an output position within [-1,1], using control inputs bounded by [-1,1], given a reerence input that remains within [-1,1].. Perturbation Description As mentioned earlier, the robust controller problem is not solvable i wide variations in longitudinal velocity or road riction are allowed, both o which are completely captured by variations the value (assuming is a ixed multiple o ). hus, a ixed must be chosen or controller design and or uncertainty representation. For this work was ixed at.5, a value representing a ullsized vehicle on average road surace at 5 mph. Section 5 discusses extending this approach to velocity scheduling. he goal o this section is to numerically bound the model uncertainty caused by variations in parameters. his numerical bound is used aterward or a robust controller design. Whether a controller design is capable o achieving robust perormance depends primarily on the measure o the model perturbation. A very poor method o representing plant uncertainty would be to examine requency-domain perturbations caused by parameter-byparameter variations [] because vehicle parameters are highly interdependent. A principal goal o this work is to use a data-driven approach with an appropriate but simple perturbation representation. A key insight o this work is to utilize the observed variations in parameters in database V to describe the expected variation o any vehicle controller. he advantage o this approach is that each set o parameters in the database are correctly cross-correlated, thus avoiding the conservatism o one-at-a-time variable manipulation. Additionally, a controller design that is robust to the wide variations in vehicles in V should be portable vehicle-tovehicle. From the data in database V, we compare the relative error between the average vehicle and each individual database member. he requency-dependent error, e(jw), between the average plant and the i th plant is given by: (, 1 i,..., 5i ) (, 1,..., 5) G( jw, 1,..., 5) G jw G jw e( jw) (15) Where G(jw) represents the requency response o the vehicle bicycle model dependent on the parameters. A simple multiplicative uncertainty description is used to describe this system variation, represented in block-diagram orm in Fig.. W (s) u Fig. : Multiplicative uncertainty model + + G P (s) G(s) he plot o each plant deviation, calculated rom (15) or each V i in the database V is shown in Fig.. t is clear that the imum multiplicative uncertainty is 5

4 approximately constant over nearly the entire requency range, a result that justiies a multiplicative representation. Mag (db) Fit to pper Bound, W pper Bound on Model Dierences Model Dierences or each ndividual Vehicle Freq. (rad) Fig. : Multiplicative uncertainty, by requency he weight representing system robustness, w, is speciied by itting the upper bound o all observed perturbations. n this case:. s* +.5 w ( s* ) (1).1 s* + 1 he high requency gain was made slightly higher than needed in order to account or possible unmodeled dynamics that are unaccounted or by the bicycle model, such as steering actuator dynamics, vehicle roll and pitch, aerodynamics, etc. Examining Fig. above, the model uncertainty remains below 1 at low requencies, up to some crossover point at w* ~.7 rad/sec*. hereore, the system should be robustly controllable up to this requency. n the case where the requency-dependent bound on the uncertainty, w, is larger than unity, consider the possible plants that must be controlled. From Fig., we see that in the case that w has greater than unit magnitude, a possibility exists that the perturbed plant has zero gain and is or practical purposes uncontrollable. n these situations, no perormance can be realized. Such situations arise at very high velocities or low riction values, or at very low values o. Limiting cases o this phenomenon are discussed in Section 5.. Controller Synthesis With the values ixed as above, the nondimensional transer unction or the nominal system is given by: y s s s ** * +.8 * s* s* s* +. s* + 1. (17) Note that s s* because s has dimensions o 1/t and must also be normalized. With signal normalization as described in Equation (1), the transer unction becomes: y s y s u s s n * * * * * * * +.51 s* s* e* s* s* +. s* + 1. (18) Since the H-ininity system representation does not allow unstable open-loop systems, the double integrator is approximated with poles very close to the jw-axis: ( s* + K) y s s s n * * * * +.51 s * s* +. s* + 1. (19) with K.1. he resulting high DC gain approximates the integrator eect. he H-ininity controller must balance the tradeo between three requency domain criteria: perormance weighting, represented by wp S ; control eort, represented by wu KS ; and model uncertainty, represented by w. hese three design goals are represented approximately by minimization o the stacked H-ininity norm o [1]: wp S. () N w KS u w While w was deined in the previous section, the remaining weights, w P and w, represent design variables. For this problem, these weights were chosen to be : w w P ( s* ) ( s* ) (1/ M P s* + w ) ( s* + w A ) BP BP (1/ M s* + wb ) ( s* + w A ) B P (1) () Where M i is the high requency gain, A i is the steady-state gain, and w Bi is the approximate crossover bandwidth. he perormance weighting unction was chosen with M P 1.5, with A P.1, and w BP.7 rad/sec*. he control weighting was chosen with M 1/1, A 1, and w B 1 rad/sec*. he H-ininity controlle r is obtained using standard robust synthesis routines, which solve the control problem by iterating through possible controller representations seeking to minimize the norm o Equation (). A solution was ound with a norm o.878, with a controller given by: ( s* + 7. ) ( s* + 1) ( s * +.98 s* +.85) (*) s Es (*) s* s*+1. s * s* +.8 s* +.1 s* + 1.9E- s* E s* s* +.777E s * + 5.E s * + 7.9E () where E(s*) is the error between the reerence signal and previewed eedback. Examination reveals that this controller is acting to perorm to pseudo-inversion o the plant and control weightings, a act that will sharply limit the perormance due to the uncertainty o the system. he H-ininity controller synthesis with the previous weights 57

5 achieved the ollowing loop shapes, which show that all speciications were met:.18.1 Desired Simulated 1.1 Magnitude [normalized] /wp S 1/wi SK 1/wu Measured ime Frequency [rad/sec] Fig. : Controller loop shapes Note that the above controller is designed in nondimensional time and space. For implementation in true space, one must multiply by the correct time and spatial actors. n the vehicle case, the spatial correction is obtained rom (1) and is a constant: u * Correction e*. L he temporal correction is obtained by dividing all pole locations by the time actor S used in Equation (7). Experimental testing o this H-ininity controller was conducted in both simulation and experimental platorms. he simulation was necessary to represent the ull possible range o vehicle plants, while the experimental vehicle is used to introduce real-world plant variations including nonlinearities, unmodeled dynamics, and disturbances that are otherwise ignored in a simulation study. For the experimental vehicle to operate at a ixed value o.5, it was driven at a speed o. m/s. his speed approximates an average ull-sized vehicle at a speed o 5 mph. he vehicle responses rom the experimental vehicle are shown in Fig. 7 above as the vehicle attempts to track a square wave. he H-ininity controller is very underdamped and low-gain, a result that should be expected in consideration o essentially single-state eedback combined with severe robustness constraints. However, there was no system identiication outside o measurement o basic parameters: vehicle mass, vehicle length, vehicle velocity, and tire cornering stiness. he controller was designed to be robust enough to operate nearly any vehicle, so the eectiveness o the controller on this arbitrary vehicle without identiication is not surprising. Such a result greatly validates a methodology o utilizing nondimensional robustness measures to achieve very general controllers. Fig. 5: Experimental closed loop step responses n terms o the tuning variables or the controller, the key actor limiting the perormance is the choice o perormance bandwidth, w BP, in Equation (1). mproved bandwidth can be obtained, but signiicant improvement is obtained only by violating the norm criteria o Equation (). For instance, increasing w BP to.5 gives a gamma value o 1.1, a value o w BP o 1 gives a gamma o 1.5. Further, plots o the controller loop shapes or this w BP show possible violation o robustness constraints. his classic tradeo between robustness and perormance is well known, and the exact choice o values depends on the intent o the control designer. 5. Extensions to Velocity Scheduling he previous result was presented or a ixed value, which necessarily corresponds to a ixed velocity and cornering stiness (representative o ixed road riction). n the case where some type o velocity gain scheduling is desired, note that velocity enters the nondimensional orm o the bicycle model through the parameters o and. For a particular vehicle, V i, we may ix the ratio o these two parameters, so that is a constant unction o without loss o generality. Because riction and velocity enter the dynamic equations through the same parameter, we conclude that velocity gain scheduling must be used cautiously, because appropriate scheduling o the requires very good knowledge o the road riction. he duality o riction scheduling and velocity scheduling is discussed in more detail in []. At each velocity, the controller must still be robust to model variation, and thereore bounds on the model uncertainty must be recalculated at each operating condition. With a ixed preview o seconds, the upper bounds on uncertainty were calculated or many values, and are shown in the igure below: 58

6 Mag (db) Pi.1 Pi. Pi. Pi.5 Pi.7 Pi 1 Pi 1.5 ncreasing Velocity Freq. (rad) Max. allowable uncertainty or easibility Fig. : Eect o velocity on uncertainty Because the uncertainty cannot be larger than unity, there is clearly some limiting value below which robust vehicle lateral control is not achievable due to increasing system uncertainty. he limiting case corresponds to:, CRCAL.7. () From this inormation and Equation (1), one may backcalculate the speed/cornering-stiness relationship at which generalized robust control is no longer easible, which we call the critical speed or robustness: L Cα CR,. (5).7 m Note that the vehicle mass, m, and length, L, are easy to measure and remain approximately ixed or a given vehicle. For the vehicles in the database, V, the value o C,R ranges between extremes o 15 and 5 m/s ( to 1 mph) because o dierences in reported road riction. he majority o vehicles in V have critical robustness speeds o 5 m/s (about 55 mph). he critical robustness speed has interesting implications or vehicle control, as other researchers have pointed out that signiicant modiications to control approaches are needed to achieve high-speed lateral control [1]. n the MMO control case, the limiting case o Equation () could be calculated using the imum singular values o the uncertainty model. n either case, we must conclude that severe limits exist to robust, high-speed vehicle control algorithms.. Conclusions he controller presented in this work addressed controller robustness in a generalized nondimensional ramework that brings insight to the easibility o a robust controller design. By parameterizing plant uncertainty nondimensionally, normal distributions were obtained o the plant parameters that deined an average plant. Measured dierences between a vehicle database and an average plant motivated a multiplicative uncertainty description. An H-ininity methodology is then presented that utilizes a stacked sensitivity approach. While this approach achieves robust control, it is only by sacriicing perormance. Discussion is given to extensions o the work to velocity scheduling, and a limiting speed/riction relationship is encountered that deines operating regions where robust control is and is not achievable. Reerences [1] J. Ackermann, 199 Bode Prize Lecture: Robust Control Prevents Car Skidding, in EEE Conerence on Decision and Control, June 199, Reprinted in EEE Control Systems Magazine, June 1997, 199, pp. -1. [] agawa,., e.a. A Robust Active Front Wheel Steering System Considering the Limits o Vehicle Lateral Force, presented at Proceedings o AVEC'9, Aachen, 199. [] S. Brennan and A. Alleyne, Robust Stabilization o a Generalized Vehicle, a Nondimensional Approach, presented at 5th nternational Symposium on Advanced Vehicle Control (AVEC), niversity o Michigan, Ann- Arbor,. [] S. Horiuchi, N. Yuhara, and A. akei, wo Degree o Freedom H-ininity Controller Synthesis or Active Four Wheel Steering Vehicles, Vehicle System Dynamics Supplement, vol. 5, pp. 75-9, 199. [5] E. Ono, K. akanami, N. wama, Y. Hayashi, Y. Hirano, and Y. Satoh, Vehicle ntegrated Control or Steering and raction Systems by Mu-Synthesis, Automatica, vol., pp , 199. [] J. Sun, A. W. Olbrot, and M. P. Polis, Robust Stabilizations and Robust Perormance sing Model Reerence Control and Modeling Error Compensation, EEE ransactions on Automatic Control, vol. 9, pp. -5, 199. [7] E. Buckingham, On Physically Similar Systems; llustrations o the use o dimensional equations, Physical Review, vol., pp. 5-7, 191. [8] S. Brennan and A. Alleyne, sing a Scale estbed: Controller Design and Evaluation, in EEE Control Systems Magazine, vol. 1, 1, pp [9] A. Alleyne, A Comparison o Alternative Obstacle Avoidance Strategies or Vehicle Control, Vehicle System Dynamics, vol. 7, pp. 71-9, [1] S. Patwardhan, H.-S. an, and J. Guldner, A General Framework or Automatic Steering Control: System Analysis, presented at Proceedings o the American Control Conerence, Albuquerque, NM, [11] H. Peng and M. omizuka, Preview Control or Vehicle Lateral Guidance in Highway Automation, ASME Journal o Dynamic Systems, Measurement and Control, vol. 115, pp. 79-8, 199. [1] J. Guldner, H. S. an, and S. Patwardhan, Analysis o Automatic Steering Control or Highway Vehicles with Look-Down Lateral Reerence Systems, Vehicle System Dynamics, vol., pp. -9, 199. [1] S. Skogestad and. Postlethwaite, Multivariable Feedback Control, Analysis and Design. Chichester: Wiley,