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Constrained and Distributed Constrained and Distributed
Optimal ControlOptimal Control
Francesco BorrelliFrancesco Borrelli
University of California
Berkeley, USABerkeley, USA
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ChallengesChallenges
Hybrid Control Design
&
Distributed Control for Large Scale SystemsDistributed Control for Large Scale Systems
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ChallengesChallenges
Hybrid Control DesignSwitched Linear Systems
Constraint Satisfaction
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Pylon
Propeller
Ducted Fan
Fuselage
Organic Air VehicleOrganic Air VehicleDale Swanson et al.
Vanes
At high level: Constrained Switched Linear System
External Switch Selects Mode of Operation
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OAV Autonomous Flight OAV Autonomous Flight
ObjectiveObjectiveFollow given trajectories.
Waypoints= [Time,Space]
ModelModel
Switched Linear – External SwitchSwitched Linear – External Switch
ConstraintsConstraints
Speed and acceleration function of mode
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Vehicle Dynamics ControlVehicle Dynamics Control
A driver aid for atypical road conditions, such as slippery, windy an bumpy roads
yaw
Later
al F
orce
Tir
e F
orc
es
Longitudinal Force
Slip Target
� Traction Control (TC)
� Anti-lock Braking System (ABS)
� Electronic Stabilty Program (ESP)
pitchroll
Fx FyFz
Nonlinear (Piece-wise linear) and Constrained System
Tire SlipMaximum
Acceleration
Slip Target
Zone
Maximum
Cornering
Maximum
Braking
Lateral
Force
Longitudinal
ForceSteer Angle
� Electronic Stabilty Program (ESP)
� Active Front Steering (AFS) systems
� Active Suspension systems
� Active differential systems
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Vehicle Dynamics ControlVehicle Dynamics Control
A driver aid for atypical road conditions, such as slippery, windy an bumpy roads
yaw
Later
al F
orce
Tir
e F
orc
es
Longitudinal Force
Slip Target
� Traction Control (TC)
� Anti-lock Braking System (ABS)
� Electronic Stabilty Program (ESP)
pitchroll
Fx FyFz
Nonlinear (Piece-wise linear) and Constrained System
Tire SlipMaximum
Acceleration
Slip Target
Zone
Maximum
Cornering
Maximum
Braking
Lateral
Force
Longitudinal
ForceSteer Angle
� Electronic Stabilty Program (ESP)
� Active Front Steering (AFS) systems
� Active Suspension systems
� Active differential systems
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Integrated VDC via MPCIntegrated VDC via MPC
ylateral
z yaw
pitch
vertical
longitudinal roll
ψ
θx φ Fx FyF
� Front steering
� Four brakes
� Engine torque
� Active suspensions
� Active differential
MIMO controller integrating
local and global
measurements coming
from GPS, cameras,
infrared and radar
Falcone, Kevizky, Borrelli from 2003 to today
yθx FyF
z
� . Longitudinal, lateral and vertical velocity/accelerations
� Yaw, roll and pitch angles/rates
� Position and velocity in a global frame
Controlling Yaw, Roll, Pitch, Vertical, Lateral and Longitudinal Dynamics via Multiple Input
Enabling path following capabilities Davor Hrovat, Jahan Asgari, Eric Tseng, Mike Fodor
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Chameleon Visual Tracking Chameleon Visual Tracking
ObjectiveObjective
Tracking of a moving prey
ModelModelPTZ camera: Linear
Prey: Linear point massPrey: Linear point mass
ConstraintsConstraints
Pan Tilt and Zoom constraints
Prey in tracking window ∀ unknown bounded
accelerations
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Common Problem FeaturesCommon Problem Features
•• Objective Objective
– Minimization of performance index
•• ModelsModels
– Linear, Uncertain
– Switched-Linear, Uncertain– Switched-Linear, Uncertain
•• ConstraintsConstraints
– States and Inputs
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Solved Problem ~ 40 years agoSolved Problem ~ 40 years ago
•• Objective Objective
– Minimization of performance index
•• ModelsModels
– Linear, Uncertain
– Switched-Linear, Uncertain– Switched-Linear, Uncertain
•• ConstraintsConstraints
– States and Inputs
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•• Objective Objective
– Minimization of performance index
•• ModelsModels
– Linear, Uncertain
– Switched-Linear, Uncertain
Focus of Research ~ 10 years agoFocus of Research ~ 10 years ago
– Switched-Linear, Uncertain
•• ConstraintsConstraints
– States and Inputs
Balluchi, Bemporad, Di Benedetto, Goodwin, Johansen, Johansson, Kerrigan,
Maciejowski, Mayne, Morari, Pappas, Rantzer, Rawlings, Sangiovanni-Vincentelli,
Sastry, Sontag, Tomlin, …. and many others.
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Hybrid Constrained Optimal ControlHybrid Constrained Optimal Control
minU
∑
k=0
N
||Qx(k)||p + ||Ru(k)||p
x(k) ∈ Rn � {0, 1}n , u(k) ∈ Rm � {0, 1}m , U�{u(0), u(1), u(2), . . .}
subj.to x(k+ 1) = Aix(k) +Biu(k) + fiif [x(k), u(k)] ∈ Xi, i = 1, . . ., s
Ex(k) +Lu(k) �M, k = 0, 1, 2, . . .
Borrelli from 1999 to 2004
x(k) ∈ Rn � {0, 1}nb, u(k) ∈ Rm � {0, 1}mb, U�{u(0), u(1), u(2), . . .}
• Understanding solution structures and properties
• Solution computational methods and tools
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Hybrid Constrained Optimal ControlHybrid Constrained Optimal Control
minU
∑
k=0
N
||Qx(k)||p + ||Ru(k)||p
x(k) ∈ Rn � {0, 1}n , u(k) ∈ Rm � {0, 1}m , U�{u(0), u(1), u(2), . . .}
subj.to x(k+ 1) = Aix(k) +Biu(k) + fiif [x(k), u(k)] ∈ Xi, i = 1, . . ., s
Ex(k) +Lu(k) �M, k = 0, 1, 2, . . .
Borrelli from 1999 to 2004
x(k) ∈ Rn � {0, 1}nb, u(k) ∈ Rm � {0, 1}mb, U�{u(0), u(1), u(2), . . .}
• Understanding solution structures and properties
• Solution computational methods and tools
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The solution to the optimal control problem is a time varying
PWA state feedback control law of the form
Characterization of the Solution Characterization of the Solution (p=1,2,(p=1,2,∞∞∞∞∞∞∞∞))Borrelli et al, ACC, 2000
Borrelli et al, AUTOMATICA, 2005
is a partition of the set of feasible states x(k).
• p=1, p=∞∞∞∞:
• p=2:
CRi(k)�{x : Mi(j, k)x � Ki(j, k)}
CRi(k)�{x : x′Li(j, k)x+Mi(j, k)x � Ki(j, k)}
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Hybrid Constrained Optimal ControlHybrid Constrained Optimal Control
minU
∑
k=0
N
||Qx(k)||p + ||Ru(k)||p
x(k) ∈ Rn � {0, 1}n , u(k) ∈ Rm � {0, 1}m , U�{u(0), u(1), u(2), . . .}
subj.to x(k+ 1) = Aix(k) +Biu(k) + fiif [x(k), u(k)] ∈ Xi, i = 1, . . ., s
Ex(k) +Lu(k) �M, k = 0, 1, 2, . . .
Borrelli from 1999 to 2004
x(k) ∈ Rn � {0, 1}nb, u(k) ∈ Rm � {0, 1}mb, U�{u(0), u(1), u(2), . . .}
• Understanding solution structures and properties
• Solution computational methods and tools
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Computational FlowComputational Flow
Problem Setup ← ← ← ← Invariant set computation
Borrelli et al, JOTA, 2003
Borrelli et al, AUTOMATICA, 2006
Baotic, Borrelli et al, SICON, 2007
Reachability Analysis ← ← ← ← Polyhedral set manipulation
Dynamic Local parametric problems
Solution postprocessing
← ← ← ← Multiparametric LP/QP
u*=fPWA(x)
← ← ← ← LMI and polyhedral set
manipulation
Dynamic
Programming
Google: mpt toolbox
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SummarySummary
• Piecewise affine state feedback control law
• Off-line computation:
Automatic partitioning and control law synthesis
Systematic Model-Based Control Design
MIMO, PWA, Constraints, Logics
Automatic partitioning and control law synthesis
• On-line computation: Lookup Table Evaluation
• Extended to Min-Max Constrained ProblemsBorrelli, Bemporad, Morari, TAC, 2003
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MPC AlgorithmMPC Algorithm
At time t:
subj. to
x(k + 1) = f(x(k), u(k))u(k) ∈ Ux(k) ∈ Xx(0) = x(t)
minUJ(U, x(0))�
∑
k=0
N1
‖Q(x(k) xref)‖p + ‖R(u(k) uref)‖p
U��{u�(t), u�(t+1), . . ., u�(t+N)}
At time t:• Measure (or estimate) the current state x(t)
• Find the optimal input sequence
• Apply only u(t)=u*(t) , and discard u*(t+1), u*(t+2), …
Repeat the same procedure at time t +1
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Important Issues in Important Issues in
Model Predictive ControlModel Predictive ControlEven assuming perfect model, no disturbances:
predicted open-loop trajectories≠
closed-loop trajectories
• FeasibilityOptimization problem may become infeasible at some future time step.Optimization problem may become infeasible at some future time step.
• StabilityClosed-loop stability is not guaranteed.
• Performance
Goal:
What is achieved by repeatedly minimizing
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Feasibility and Stability ConstraintsFeasibility and Stability Constraints
subj. to
x(k + 1) = f(x(k), u(k))u(k) ∈ Ux(k) ∈ Xx(0) = x(t)x(N) ∈ Xf
minUJ(U, x(0))� P(x(N)) +
∑
k=0
N1
‖Qx(k)‖p + ‖Ru(k)‖p
Xf is an Invariant Set
P(x) is a Control Lyapunov Function.
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Chameleon Visual Tracking Chameleon Visual Tracking
ObjectiveObjective
Tracking of a moving prey
ModelModelPTZ camera: Linear
Prey: Linear point massPrey: Linear point mass
ConstraintsConstraints
Pan Tilt and Zoom constraints
Prey in tracking window ∀ unknown bounded
accelerations
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MinMin--Max Predictive ControlMax Predictive Control
xj+1 = A(wj)xj + B(wj)uj + Evj
Fxj + Guj � f
Model:
Uncertainty: Additive vi ∈ V, Polytypic wi ∈ W
Constraints: For all vi ∈ V, wi ∈ W
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Addressing Feasibility: Control Law DesignAddressing Feasibility: Control Law Design
Smooth Pursuit Control Min-Max Predictive Control
y- Tracking Error
Reach set
x- Tracking Error
Scanning Algorithm
Saccade Control MinimumTime Predictive Control
Min-Max Predictive Control
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Robotic Chameleon VideoRobotic Chameleon VideoAvin, Borrelli et al., IROS, 2006
Explicit Min-Max MPC Solved at 50Hz
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Vehicle Dynamics ControlVehicle Dynamics Control
A driver aid for atypical road conditions, such as slippery, windy an bumpy roads
yaw
Later
al F
orce
Tir
e F
orc
es
Longitudinal Force
Slip Target
� Traction Control (TC)
� Anti-lock Braking System (ABS)
� Electronic Stabilty Program (ESP)
pitchroll
Fx FyFz
Nonlinear (Piece-wise linear) and Constrained System
Tire SlipMaximum
Acceleration
Slip Target
Zone
Maximum
Cornering
Maximum
Braking
Lateral
Force
Longitudinal
ForceSteer Angle
� Electronic Stabilty Program (ESP)
� Active Front Steering (AFS) systems
� Active Suspension systems
� Active differential systems
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Traction Control ExperimentTraction Control Experiment2000 Ford Focus, 2.0l 42000 Ford Focus, 2.0l 4--cyl Engine, 5cyl Engine, 5--speed Manual Transspeed Manual Trans
Borrelli et al., IEEE TCST, 2006
“..Traction control on the V-6 test car was just right -- perhaps unique in all the
industry…." USA Today (Oct. 28-2005)
Ford Fusion Production ControllerFord Fusion Production Controller
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Integrated VDC via MPCIntegrated VDC via MPC
ylateral
z yaw
pitch
vertical
longitudinal roll
ψ
θx φ Fx FyF
� Front steering
� Four brakes
� Engine torque
� Active suspensions
� Active differential
MIMO controller integrating
local and global
measurements coming
from GPS, cameras,
infrared and radar
Kevizky, Falcone, Borrelli from 2003 to today
yθx FyF
z
� . Longitudinal, lateral and vertical velocity/accelerations
� Yaw, roll and pitch angles/rates
� Position and velocity in a global frame
Controlling Yaw, Roll, Pitch, Vertical, Lateral and Longitudinal Dynamics via Multiple Input
Enabling path following capabilities Davor Hrovat, Jahan Asgari, Eric Tseng, Mike Fodor
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Vehicle Model Vehicle Model -- 11 States, 6 Inputs11 States, 6 Inputs
States
x
y
&
& Lateral velocity
Longitudinal velocity
Yaw angle
fδ Front steering angle
Inputs
bF FL,FR,RL,RR brakes
τ Desired engine torque
X
Y
ψ
ψ
&
Yaw angle
Yaw rate
Lateral position (I.F.)
Longitudinal position (I.F.)
],,,,,,,,,,[ rrrlfrflXYxyyx ωωωωψψ &&&=
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Pacejka Tire modelPacejka Tire model
Semi-empirical model calibrated on
experimental data
),,,( zFsfF µα=
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Autonomous Vehicle Autonomous Vehicle
Tests and Experimental Tests and Experimental setupsetupSystemSystem• Jaguar X-type
• dSpace rapid prototyping system equipped with a DS1005 processor board Sampling time: 50 ms
• Differential GPS, gyros, lateral accelerometers
ObjectiveObjective
• Minimize angle and lateral distance
deviations from reference trajectory
• Double lane change
• Driving on snow/ice, at
different entry speedsdifferent entry speeds
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Acknowledgements Acknowledgements • University of Minnesota (Minneapolis, USA)
-- Tamas Keviczky, Gary BalasTamas Keviczky, Gary Balas
• Unisannio (Benevento, Italy)
-- Paolo Falcone Paolo Falcone
• Honeywell Labs (Minneapolis,USA)
-- DharmashankarDharmashankar Subramanian, Kingsley Fregene, Datta GodboleSubramanian, Kingsley Fregene, Datta Godbole
-- Sonja Glavaski, Greg Stewart, Tariq SamadSonja Glavaski, Greg Stewart, Tariq Samad
• Ford Research Labs (Dearborn,USA)
-- Jahan Asgari, Eric Tseng, Davor Hrovat Jahan Asgari, Eric Tseng, Davor Hrovat
• Technion (Haifa, Israel)
-- Ofiv Avni, Gadi Katzirinst, Ehud Rivlin, Hector RotsteinOfiv Avni, Gadi Katzirinst, Ehud Rivlin, Hector Rotstein
• ETH (Zurich, Switzerland )
- Mato Baotic,Mato Baotic, Alberto Bemporad, Manfred MorariAlberto Bemporad, Manfred Morari