Symbolic Transformations of Dynamic Optimization Problems
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The document discusses symbolic transformations of dynamic optimization problems governed by differential-algebraic equation (DAE) systems, proposing a method to convert these into ordinary differential equation (ODE) problems for easier resolution. It highlights the benefits of reduced complexity and variables, while presenting case studies on optimal vehicle trajectory generation and power plant startup, showcasing improved solution times through this approach. The findings suggest that this transformation is particularly advantageous for models with numerous simple equations.
![System dynamics notation
System dynamics modeled by a differential algebraic equation
(DAE) system of the form
F(t, ˙x(t),x(t), y(t),u(t), p) = 0.
t ∈ [t0,tf ] is time (endpoints free or fixed, but always finite)
x is vector of state variables
y is vector of algebraic variables
u is vector of control variables
p is vector of free parameters
DAE system is assumed to be of index one
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![Objective function and constraints
We want to minimize
tf
t0
L(τ, ˙x(τ),x(τ), y(τ),u(τ), p)dτ
while satisfying the DAE system and the constraints
he(t, ˙x(t),x(t), y(t),u(t), p) = 0,
hi(t, ˙x(t),x(t), y(t),u(t), p) ≤ 0,
∀t ∈ [t0,tf ].
5 / 17](https://image.slidesharecdn.com/modelica2014symbtranfdynoptimfmagnusson-140428081147-phpapp02/85/Symbolic-Transformations-of-Dynamic-Optimization-Problems-6-320.jpg)
![Dynamic optimization problem
The result is the DAE-constrained optimization problem
minimize
tf
t0
L(τ, ˙x(τ),x(τ), y(τ),u(τ), p)dτ,
with respect to t0,tf , ˙x(t),x(t), y(t),u(t), p,
subject to F(t, ˙x(t),x(t), y(t),u(t), p) = 0,
x(t0) = 0,
he(t, ˙x(t),x(t), y(t),u(t), p) = 0,
hi(t, ˙x(t),x(t), y(t),u(t), p) ≤ 0,
∀t ∈ [t0,tf ].
6 / 17](https://image.slidesharecdn.com/modelica2014symbtranfdynoptimfmagnusson-140428081147-phpapp02/85/Symbolic-Transformations-of-Dynamic-Optimization-Problems-7-320.jpg)
![Dynamic optimization problem
The result is the
ODE
DAE-constrained optimization problem
minimize
tf
t0
L(τ, ˙x(τ),x(τ), y(τ),u(τ), p)dτ,
with respect to t0,tf , ˙x(t),x(t), y(t),u(t), p,
subject to F(t, ˙x,x, y,u, p) = 0, ˙x = f (t,x,u, p),
x(t0) = 0,
he(t, ˙x(t),x(t), y(t),u(t), p) = 0,
hi(t, ˙x(t),x(t), y(t),u(t), p) ≤ 0,
∀t ∈ [t0,tf ].
8 / 17](https://image.slidesharecdn.com/modelica2014symbtranfdynoptimfmagnusson-140428081147-phpapp02/85/Symbolic-Transformations-of-Dynamic-Optimization-Problems-9-320.jpg)
![Results
Problem Sol. time [s] Iter. Nbr. of var.
ST–FE
DAE 10.6 112 20880
ODE 5.0 83 15017
ST–WF
DAE 17.6 102 25390
ODE 5.1 77 15017
DT–FE
DAE 152.2 303 39661
ODE 46.0 151 23425
DT–WF
DAE 229.6 364 48681
ODE 116.4 322 23425
CCPP
DAE 5.4 109 23574
ODE 1.4 79 3771
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Symbolic Transformations of Dynamic Optimization Problems
- 1. Symbolic Transformations of Dynamic Optimization Problems Fredrik Magnusson Karl Berntorp, Björn Olofsson, and Johan Åkesson Department of Automatic Control Faculty of Engineering Lund University, Sweden March 12, 2014
- 2. Introduction Optimization problems involving differential-algebraic equation (DAE) systems traditionally solved retaining DAE Consider instead symbolically transforming the DAE into an ODE before solving the problem Will discuss how and why, and present case studies 1 / 17
- 3. Dynamic optimization Optimal control Design optimization Parameter estimation State estimation In practice quite different problems, but solution techniques can be very similar. 2 / 17
- 4. JModelica.org Developed in Lund, Sweden at Modelon AB and Lund University Targets both simulation and optimization Optimica for optimization formulations 3 / 17
- 5. System dynamics notation System dynamics modeled by a differential algebraic equation (DAE) system of the form F(t, ˙x(t),x(t), y(t),u(t), p) = 0. t ∈ [t0,tf ] is time (endpoints free or fixed, but always finite) x is vector of state variables y is vector of algebraic variables u is vector of control variables p is vector of free parameters DAE system is assumed to be of index one 4 / 17
- 6. Objective function and constraints We want to minimize tf t0 L(τ, ˙x(τ),x(τ), y(τ),u(τ), p)dτ while satisfying the DAE system and the constraints he(t, ˙x(t),x(t), y(t),u(t), p) = 0, hi(t, ˙x(t),x(t), y(t),u(t), p) ≤ 0, ∀t ∈ [t0,tf ]. 5 / 17
- 7. Dynamic optimization problem The result is the DAE-constrained optimization problem minimize tf t0 L(τ, ˙x(τ),x(τ), y(τ),u(τ), p)dτ, with respect to t0,tf , ˙x(t),x(t), y(t),u(t), p, subject to F(t, ˙x(t),x(t), y(t),u(t), p) = 0, x(t0) = 0, he(t, ˙x(t),x(t), y(t),u(t), p) = 0, hi(t, ˙x(t),x(t), y(t),u(t), p) ≤ 0, ∀t ∈ [t0,tf ]. 6 / 17
- 8. Symbolic transformation Instead of solving the DAE-constrained optimization problem, transform it to an ODE-constrained problem before solving Achieved by eliminating algebraic variables through causalization Main benefit is reduced number of equations and variables Main drawback is increased equation complexity 7 / 17
- 9. Dynamic optimization problem The result is the ODE DAE-constrained optimization problem minimize tf t0 L(τ, ˙x(τ),x(τ), y(τ),u(τ), p)dτ, with respect to t0,tf , ˙x(t),x(t), y(t),u(t), p, subject to F(t, ˙x,x, y,u, p) = 0, ˙x = f (t,x,u, p), x(t0) = 0, he(t, ˙x(t),x(t), y(t),u(t), p) = 0, hi(t, ˙x(t),x(t), y(t),u(t), p) ≤ 0, ∀t ∈ [t0,tf ]. 8 / 17
- 10. Causalization Main step is DAE causalization Permute variables and equations to get a block-lower triangular (BLT) form of the DAE incidence matrix: ˙x1 y3 y1 ˙x2 y2 F2 ∗ 0 0 0 0 F4 ∗ ∗ ∗ 0 0 F1 0 ∗ ∗ 0 0 F5 0 0 ∗ ∗ 0 F3 ∗ ∗ 0 ∗ ∗ Solve for ˙x and y, symbolically or iteratively 9 / 17
- 11. Causalization cont. We assume that all equation systems can be solved symbolically (no algebraic loops) The result is functions f and such that ˙x = f (t,x,u, p), y = (t,x,u, p). f is used to replace the DAE with an ODE is inlined to eliminate y 10 / 17
- 12. Method properties Less optimization variables, more complex expressions Less sparse system, but minor issue when using e.g. local collocation to solve optimization problem Robustness with respect to trivial algebraic equation modifications Generalization is to only eliminate the algebraic variables which can be solved for symbolically 11 / 17
- 13. Benchmark Comparison of solving the original and transformed problem using direct local collocation Two case studies: optimal vehicle trajectory generation and power plant startup 12 / 17
- 14. Vehicle trajectory generation Generate trajectories minimizing duration of hairpin turn Two chassis models: double and single track (DT & ST) Two tire force models: friction ellipse and weighting functions (FE & WF) 4 different models. Most complex has 21 states, 56 algebraic variables and 3 control variables. Atypical Modelica model; flat implementation 13 / 17
- 15. Power plant startup Optimize startup of combined cycle power plant Model has 10 states, 128 algebraic variables and 1 control variable. Control variable is steam turbine load Typical Modelica model; component-based 14 / 17
- 16. Results Problem Sol. time [s] Iter. Nbr. of var. ST–FE DAE 10.6 112 20880 ODE 5.0 83 15017 ST–WF DAE 17.6 102 25390 ODE 5.1 77 15017 DT–FE DAE 152.2 303 39661 ODE 46.0 151 23425 DT–WF DAE 229.6 364 48681 ODE 116.4 322 23425 CCPP DAE 5.4 109 23574 ODE 1.4 79 3771 15 / 17
- 17. Conclusion The transformation drastically reduces the size of the problem Reduced solution time, between 2 and 4 times for considered cases Especially useful for models involving a lot of simple equations, as is typical for Modelica models Seems to be beneficial also for atypical Modelica models, despite lack of attention outside of the Modelica community 16 / 17
- 18. The end Thank you for listening! The End 17 / 17