Model Predictive Control For Automotive Applications

Department of Signals and SystemsModel Predictive Control forAutomotive ApplicationsPaolo FalconeDipartimento di Ingegneria “EnzoFerrari”Universita’ di Modena e ReggioEmiliaPaolo Falcone (UNIMORE/CHALMERS)Department of Electrical EngineeringChalmers University of TechnologyGöteborg, SwedenAutomatic Control Course – Guest LectureDecember 11th, 20191

Department of Signals and SystemsPredictive Vehicle Dynamics ControlPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20192

Department of Signals and SystemsProblem complexityHow does the logic change if further actuatorsare added?Engine ivedifferentialsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20193

Department of Signals and SystemsGlobal Chassis Control (GCC) problemDriverCommandsCoordinating vehicle actuators in order to controlmultiple dynamicsü Front steeringü Four brakesü Engine torqueü Active suspensionsü Active differentialverticalzyawyGCClongitudinal rolflxFx FFypitch lateralqyzü Longitudinal, lateral and verticalveloctyü Yaw, roll and pitch angles/ratesPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20194

Testing scenario. Autonomous path followingDepartment of Signals and SystemsProblem setup:Control objective: Double lane change Driving on snow/ice, withdifferent entry speedsMinimize angle and lateral distancedeviations from reference trajectoryby changing the front wheel steering angleand the braking at the four wheelsControlling longitudinal, lateral and yaw dynamics by varyingfront steering angle and braking the four wheelsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20195

Department of Signals and SystemsChallenges Highly nonlinear MIMO system with uncertainties– Tires characteristics 6 DOF model– Longitudinal, lateral, vertical, roll, yaw andpitch dynamics Hard constraints– Rate limitations in the actuators, vehiclephysical limits Fast sampling time– Typically 20-50 msPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20196

Model Predictive ControlDepartment of Signals and SystemsMain ingredients Vehicle model Optimization problemPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20197

Department of Signals and SystemsOutline Introduction and motivationsVehicle modelingProblem formulationExperimental resultsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20198

Modeling: bicycle and four wheels modelsDepartment of Signals and SystemsModeling the vehicle motion in an inertial frame subject to lateral,longitudinal and yaw dynamics10 states, 5 inputsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 20199

Department of Signals and SystemsTire modelingStatic tire forcescharacteristicsFc f c (a , s, µ , Fz )Paolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201910

Department of Signals and SystemsTire modelingFzFl 2 cFc2 µFzPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201911

Department of Signals and SystemsOutline Introduction and motivationsVehicle modelingProblem formulationExperimental resultsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201912

NMPC Control designHpCost functionJ (ξ( t ),U ) ηt i,t ηreft i ,ti 1Optimizationproblem Vehicle dynamicsInput constraintsmin J (x t , U )Usubj. tox k 1,t f (x k ,t , uk ,t )h k ,t h(x k ,t )umin uk ,t umaxDepartment of Signals and Systems2Q ut i,t2RU u t ,t ! u t H p ,t Non Linear Programming(NLP) problem Complex NLP solversrequiredDumin Duk ,t DumaxReal time implementation byk t , ! , t H p - 1 ü limiting the number ofiterationsü using short horizonsExperimentally testedConstraints oninput changesReal time testing at low speedPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201913

Department of Signals and SystemsLTV-MPC controllerApproximating the non-linear vehicle model with a Linear Time Varying(LTV) model At, Bt, Ct, Dt.** Kothare and Morari, 1995, Wan and Kothare, 2003At , Bt , Ct , Dt Convex optimizationproblem. QP solvers Easier real-timeimplementation Longer horizonsPerformance and stability issues* due to linear approximation* Falcone et al 2008Paolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201914

Constraints on tire slip angleDepartment of Signals and SystemsStability achieved through ad hoc state and input constraintsa mina maxa min a k ,t a maxk ,tk ,tk t !t H pController performswell up to 21 m/sThe system is still nonlinearPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201915

Stability of the LTV-MPC approachDepartment of Signals and SystemsConsider the discrete time nonlinear system:(1)x (t 1) f (x (t ), u (t ) )x Î Rn u Î RmWe consider the following linear approximation over the horizon N:x (k 1) @ Ak ,tx (k ) Bk ,t u (k ) d k ,tk t ,!t N - 1 f fAk,t ξ (k ) , Bk,t ξ (k ) x u(t0 1) u u(t0 1)ξ0 (k 1) f (ξ0 (k),u(t 1)), ξ0 (k) ξ(t)dk,t ξ0 (k 1) Ak,t ξ0 (k) Bk,t u(t 1)Paolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201916

Department of Signals and SystemsStability of the LTV-MPC approachN 1*N(2) V (ξ(t)) minut ,t , ut N 1,t Qξ2t i,t 2i 1N 12 Rut i,t 2 Pξt N,t22i 0subject to :ξk 1,t At ξk,t Bt uk,t dk,t ,ξk,t X,k t, ,t N 1uk,t U,k t, ,t N 1k t, ,t N 1ξk N ,t X fξt,t ξ(t)(3)Paolo Falcone (UNIMORE/CHALMERS)u( t ) u*t,t (ξ(t))Automatic Control Course – Guest LectureDecember 11th, 201917

Department of Signals and SystemsStability of the LTV-MPC approachTheorem. The system (1) with the control law (2)-(3), where Xf 0,is uniformly asymptotically stable if2222Qξˆt N 1,t Rut N 1,t2*t,t 1 2 Qξ Ru2*t 1,t 1 2N 2 Q ξˆt i,t ξt* i,t 1i 1()22 γwhere:ξˆk 1,t At ξˆk,t Bt u*k,t 1 dk,tk t, ,t N 2N 2γ 2 Q ξˆt i,t ξ*t i ,t 1i 1()2Qξ*t i,t 12ξˆt,t ξ(t)State and input convex constraintsLiu, 1968. Chen and Shaw. 1982. Mayne et al. 2000 Paolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201918

Department of Signals and SystemsWhat does that mean?l ξˆt N 1,t ,ut N 1,t()& ˆ** l(ξt 1,t 1,ut 1,t 1 ) Γ( Q ξt i,t ξt* i,t 1'(Yé At -1êCë t -1é AtêCë tBt -1 ùDt -1 úû)) 2*2Bt ùDt úûxˆk ,tx k*,t -1t -1Paolo Falcone (UNIMORE/CHALMERS)tAutomatic Control Course – Guest Lecturet N -2December 11th, 2019t19

Simulation results at 21 m/sDepartment of Signals and SystemsThe controller is able to stabilize the vehiclewithout any ‘ad hoc’ constraint.Paolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201920

Department of Signals and SystemsOutline Introduction and motivationsVehicle modelingProblem formulationExperimental resultsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201921

Department of Signals and SystemsTesting: Sault St. Marie in Upper Peninsula,MI, USAPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201922

Department of Signals and SystemsSummary Excellent performance Limited tuning effort (less than 10 run 1 hr) Vehicle stabilized up to 70 Km/h on snowy tracks Coordination of steering and braking– Braking is delivered on the “same side” of the steering Front/rear braking distribution– Shifting the braking to the non saturated axle Countersteering– Steering in opposite direction of path following to prevent spinningPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201923

Department of Signals and SystemsSummary Excellent performance Limited tuning effort (less than 10 run 1 hr) Vehicle stabilized up to 70 Km/h on snowy tracks Coordination of steering and braking– Braking is delivered on the “same side” of the steering Front/rear braking distribution– Shifting the braking to the non saturated axle Countersteering– Steering in opposite direction of path following to prevent spinningPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201924

Department of Signals and SystemsSummary Excellent performance Limited tuning effort (less than 10 run 1 hr) Vehicle stabilized up to 70 Km/h on snowy tracks Coordination of steering and braking– Braking is delivered on the “same side” of the steering Front/rear braking distribution– Shifting the braking to the non saturated axle Countersteering– Steering in opposite direction of path following to prevent spinningPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201925

Department of Signals and SystemsTest @ 40 Kph. Steering and braking coordinationPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201926

Department of Signals and SystemsTest @ 40 Kph. Steering and brakingcoordinationLeft side brakingPaolo Falcone (UNIMORE/CHALMERS)Right side brakingAutomatic Control Course – Guest LectureDecember 11th, 201927

Department of Signals and SystemsSummary Excellent performance Limited tuning effort (less than 10 run 1 hr) Vehicle stabilized up to 70 Km/h on snowy tracks Coordination of steering and braking– Braking is delivered on the “same side” of the steering Front/rear braking distribution– Shifting the braking to the non saturated axle Countersteering– Steering in opposite direction of path following to prevent spinningPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201928

Department of Signals and SystemsTest @ 40 Kph. Countersteering manoeuvre.Yawinstabilityinduced bylargeaccelerationPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201929

Department of Signals and SystemsTest @ 40 Kph. Countersteering manoeuvrePaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201930

Test @ 70 Kph. Countersteering manoeuvreDepartment of Signals and SystemsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201931

Department of Signals and SystemsTest @ 70 Kph. Countersteering manoeuvrePaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201932

Department of Signals and SystemsTest @ 70 Kph. Countersteering manoeuvrePaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201933

Experimental resultsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDepartment of Signals and SystemsDecember 11th, 201934

AcknowledgmentsDepartment of Signals and Systems Francesco Borrelli (UC Berkeley) Eric Tseng (Ford) Davor Hrovat (Ford)Paolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201935

Department of Signals and SystemsLong Heavy Vehicles CombinationsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201936

Long Heavy Vehicles CombinationsDepartment of Signals and SystemsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201937

Department of Signals and SystemsPaolo Falcone (UNIMORE/CHALMERS)Automatic Control Course – Guest LectureDecember 11th, 201938

Department of Signals and SystemsControl ObjectivesReducing the yaw rate rearward amplifications r2 /r1and r3 /r1 , by means of the steering angles d 2 , d 3while bounding the steering angles and rates of steering Achieving the control objectives by solving a yaw rate trackingproblem whereG21refr2,refG31refr3,refd11Paolo Falcone (UNIMORE/CHALMERS)Reference models designed inorder to remove resonance peaksin the yaw rates responsesAutomatic Control Course – Guest LectureDecember 11th, 201939