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Optimal control of vehicles via adaptive finiteelement methods
K. Kraft1 S. Larsson1 M. Lidberg2
1Department of Mathematical SciencesChalmers University of Technology and Göteborg University
2Department of Applied MechanicsChalmers University of Technology
GMMC scientific advisory board meeting, January 2008
Background
• Optimal control of the dynamics of heavy vehicles.
• Relevant for electronic stability control (ESC) (closed loop).
• Adaptive finite element methods.
Optimal Control of ODEs
Find states y(t) ∈ Rd1 and controls u(t) ∈ R
d2 which fulfill
minimize J (y ,u) = l(y(0), y(T )) +
∫ T
0L(y(t),u(t)) dt
subject to y(t) = f (y(t),u(t)),
I0y(0) = y0, IT y(T ) = yT .
I0,IT are diagonal matrices with zeroes or ones on thediagonals.
Solution strategies
• Direct approach: discretize the equations and solve theoptimization problem.
• Indirect approach: derive the necessary conditions foroptimality, discretize and solve.
• We use the indirect approach.
Solution strategies
• Direct approach: discretize the equations and solve theoptimization problem.
• Indirect approach: derive the necessary conditions foroptimality, discretize and solve.
• We use the indirect approach.
Solution strategies
• Direct approach: discretize the equations and solve theoptimization problem.
• Indirect approach: derive the necessary conditions foroptimality, discretize and solve.
• We use the indirect approach.
Necessary Conditions for an OptimumVariational Calculus
Hamiltonian: H(y , λ,u) = L(y ,u) + λTf (y ,u)The optimal y and u satisfy
y =∂H∂λ
= f (y ,u),
λ = −∂H∂y
=∂L∂y
−( ∂f∂y
)T
λ,
0 =∂H∂u
=∂L∂u
+ λT ∂f∂u,
I0y(0) = y0, IT y(T ) = yT ,
(I − I0)λ(0) = λ0, (I − IT )λ(T ) = λT .
This is a Boundary Value Problem for a system of DAE.
Numerical Methods
• Shooting methods.
• Collocation methods.
• We use the Finite Element Method.
• The finite element method is based on a variationalformulation and piecewise polynomials.
• The finite element method is well suited for mathematicalanalysis.
• Admits error control and adaptive mesh generation basedon a posteriori error estimates.
Numerical Methods
• Shooting methods.
• Collocation methods.
• We use the Finite Element Method.
• The finite element method is based on a variationalformulation and piecewise polynomials.
• The finite element method is well suited for mathematicalanalysis.
• Admits error control and adaptive mesh generation basedon a posteriori error estimates.
Numerical Methods
• Shooting methods.
• Collocation methods.
• We use the Finite Element Method.
• The finite element method is based on a variationalformulation and piecewise polynomials.
• The finite element method is well suited for mathematicalanalysis.
• Admits error control and adaptive mesh generation basedon a posteriori error estimates.
Numerical Methods
• Shooting methods.
• Collocation methods.
• We use the Finite Element Method.
• The finite element method is based on a variationalformulation and piecewise polynomials.
• The finite element method is well suited for mathematicalanalysis.
• Admits error control and adaptive mesh generation basedon a posteriori error estimates.
Numerical Methods
• Shooting methods.
• Collocation methods.
• We use the Finite Element Method.
• The finite element method is based on a variationalformulation and piecewise polynomials.
• The finite element method is well suited for mathematicalanalysis.
• Admits error control and adaptive mesh generation basedon a posteriori error estimates.
2007Results so far
• Using an adaptive FEM to determine the optimal control ofa vehicle during a collision avoidance manoeuvre,Proceedings of SIMS2007
• Application of a standard adaptive finite element method toan optimal control problem from vehicle dynamics.
• Combine x = (y , λ), eliminate u. ODE: x = g(x)
• A posteriori error estimate for an arbitrary linear functional:
|G(x − xh)| ≤N
∑
n=1
Rnωn
2007Results so far
• The dual weighted residuals approach to optimal control ofordinary differential equations, Preprint 2008:2.
• The Dual Weighted Residuals methodology: keep y , λ,u.
• Take advantage of the variational structure of theequations.
• A posteriori error estimate for the cost functional
|J (y ,u) − J (yh,uh)| ≤
N∑
n=1
Rynω
λ
n + Rλ
nωyn + Ru
nωun
2007Results so far
• Implemented the above algorithms in Matlab.
• The implementation is not completely general.
Future Work 1Extend the theory
• Efficient solution of the resulting linear/nonlinear algebraicequations.
• Constraints on controls and states (Pontryagin maxprinciple or slack variables).
• A priori error estimates.
Future Work 2Implementation
• Non-linear problems
• Constraints on controls and states
• More advanced models from vehicle dynamics
Midterm evaluation
• The present 2 publications: licentiate thesis of Karin Kraft.
• One more publication: realistic models from vehicledynamics.
• I hope that we will have demonstrated that adaptive FEM isuseful for optimal control of vehicles.
• (Karin Kraft, PhD 2009, expected.)
Questions, discussion
• Choice of programming language (Matlab, C++, Fortran orother)
• Include constraints on controls and states
• More interaction with GMMC
• Parameter estimation
• Stochastic control
y =
y1
y2
y3
y4
y5
y6
=
vx
vy
rψ
XY
,
y =
VX
VY
rψ
XY
=
a11
a21Vy + a22r + bf1δf + br1δr
a31Vy + a32r + bf2δf + br2δr + rbrake
rVx
Vy
I0y(0) =
2500000
,
J(y ,u) =
∫ T
0
(
Y 2 + r2 + ψ2 + δ2f
)
dt =
∫ T
0
(
‖y‖2Q + ‖u‖2
R
)
dt .
14 6 NUMERICAL EXAMPLES
Lr Lf
FyrFxr vr
αr
δr
Fyf
Fxf
vfαf δf
Ψ
vy
vx
y
x
v
β
Figure 3: The bicycle model, which is used to derive a model of the dynamics of a vehicle. Therectangles represent the wheels of the bicycle, and the dot marks the center of gravity aroundwhich the angular velocity is computed.
0 0.5 110
15
20
25
x−ve
loci
ty [m
/s]
Time [s]0 0.5 1
−0.02
0
0.02
0.04
y−ve
loci
ty [m
/s]
Time [s]
0 0.5 1−0.08
−0.06
−0.04
−0.02
0
angu
lar v
eloc
ity [r
ad/s
]
Time [s]0 5 10 15 20
0
0.005
0.01
0.015
0.02
x−position [m]
y−po
sitio
n [m
]
Figure 4: The optimal states. The last picture shows the optimal track.
