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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.