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Inversion in optimal control. Principles and examples
Nicolas PetitCentre Automatique et SystèmesÉcole des Mines de Paris
Knut Graichen – François Chaplais
Outline
1. Receding horizon control (RHC - MPC)
2. Efficient trajectory parameterization
3. Examples
4. Indirect methods
Conclusions and future developments
1- Receding Horizon control
Bellman’s principle of optimality
Iterating the resolution
In the limit
Lyapunov function
Practical issues
2 – Efficient trajectories parameterization
Direct methods: collocation
Collocation (Hargraves-Paris 1987)
Dynamic inversion (Seywald 1994)
Eliminating the control variable
Eliminating the maximum number of variables (Petit, Milam, Murray, NOLCOS 01
y stands for
Instead of
r : relative degree of z1, zero dynamics, normal form, flatness
Comparisons
Full collocation (Hargraves-Paris) : Ο(n+1)
Dynamic Inversion (Seywald) : Ο(n)
(proposed) Inversion : Ο(n+1- r)
Successive derivatives are required (substitutions)
Dedicated software package
(dim x=n, dim u=1)
Example
• Collocation• Easily computed
derivatives: B-splines• Analytic gradients• Frontend to NPSOL
Software
NTG: Mark Milam, Kudah Mushambi, Richard Murray, CalTech or: Matlab, Optim. Toolbox, Spline toolbox
3 – Three examples
CalTech ducted fanMissileMobile robots
CalTech Ducted Fan (M. Milam)
Control variables histories
Flat outputs : z1 et z2
Trajectory optimizationMinimum time transients
« terrain avoidance »
« Half-turn »
open loop
closed loopterrain avoidancesequence
CalTech Ducted Fan (see M. Milam PhD thesis)
Minimum time and terrain avoidance
NTG receding horizon (update every
0.1s)
Missile
Controls: αc, βc
Data: m(t), T(t)
Mobile robots
Mobile robots (Vissière, Petit, Martin, ACC 07)
4 – Indirect methods (Chaplais, Petit, COCV 07)
Solution 1: collocation+inversion
1 unknown, no differential equation
Solution 2: PMP
Two-point boundary value problem
Solution 3: inversion of the adjoint dynamics
Solution 3 (cont.)
Remarkable points of solution 3
• reduction of CPU time
• post-optimal analysis
• increased accuracy
Post-optimal analysis
Numerical analysis of higher-order TPBVPs
Second order example (comparisons against exact solution)
General result
Dealing with input/state constraints (Knut Graichen)
Conclusions• Numerous variables can be eliminated from formulations of optimal control problems
• Direct or indirect methods
• r: relative degree plays a dual role in the adjoint dynamics
• Some constrained cases or singular arcs can be treated
Some references1.
U. M.
Ascher, R. M. M.
Mattheij,
and
R. D.
Russell.
Numerical
solution of
boundary
value
problems
for
ordinary differential equations. Prentice
Hall,
Inc.,
Englewood Cliffs, NJ, 1988.
2.
A.
Isidori.
Nonlinear
Control
Systems. Springer, New York, 2nd
edition, 1989.
3.
M. Fliess, J. Lévine, P. Martin,
and
P.
Rouchon.
Flatness and
defect
of
nonlinear systems:
introductory theory and examples.
Int. J. Control, 61(6):1327–1361, 1995.
4.
N. Petit, M. B.
Milam,
and
R. M. Murray. Inversion
based
constrained trajectory optimization. In 5th IFAC Symposium on Nonlinear
Control
Systems, 2001.
5.
M. Milam.
Real-Time Optimal
Trajectory Generation
for
Constrained Dynamical Systems. PhD thesis. California Institute
of Technology, 2003.
6.
K.
Graichen. Feedforward
Control Design for
Finite-Time
Transition
Problems
of
Nonlinear Systems with
Input
and
Output
Constraints. Doctoral
Thesis, Shaker
Verlag, 2006.
7.
F.
Chaplais and
N. Petit. Inversion in indirect optimal control
of
multivariable systems. To
appear
ESAIM COCV, 2007.