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Inversion in optimal control. Principles and examples Nicolas Petit Centre Automatique et Systmes cole des Mines de Paris Knut Graichen Franois Chaplais Outline 1. Receding horizon control (RHC - MPC) 2. Efficient trajectory

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Inversion in optimal control. Principles and examples

Nicolas Petit Centre Automatique et Systèmes École des Mines de Paris

Knut Graichen – François Chaplais

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Outline

  • 1. Receding horizon control (RHC - MPC)
  • 2. Efficient trajectory parameterization
  • 3. Examples
  • 4. Indirect methods

Conclusions and future developments

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1- Receding Horizon control

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Bellman’s principle of optimality Iterating the resolution

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In the limit

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Lyapunov function

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Practical issues

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2 – Efficient trajectories parameterization

Direct methods: collocation

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Collocation (Hargraves-Paris 1987) Dynamic inversion (Seywald 1994)

Eliminating the control variable

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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

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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)

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Example

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  • Collocation
  • Easily computed

derivatives: B-splines

  • Analytic gradients
  • Frontend to NPSOL

Software NTG: Mark Milam, Kudah Mushambi, Richard Murray, CalTech

  • r: Matlab, Optim. Toolbox,

Spline toolbox

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3 – Three examples

CalTech ducted fan Missile Mobile robots

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CalTech Ducted Fan (M. Milam)

Control variables histories Flat outputs : z1 et z2

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Trajectory optimization

Minimum time transients « terrain avoidance » « Half-turn »

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  • pen loop

closed loop terrain avoidance sequence

CalTech Ducted Fan (see M. Milam

PhD thesis)

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Minimum time and terrain avoidance NTG receding horizon (update every 0.1s)

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Missile

Controls: αc, βc

Data: m(t), T(t)

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Mobile robots

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Mobile robots (Vissière, Petit, Martin, ACC 07)

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4 – Indirect methods (Chaplais, Petit, COCV 07)

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Solution 1: collocation+ inversion

1 unknown, no differential equation

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Solution 2: PMP

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Two-point boundary value problem

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Solution 3: inversion of the adjoint dynamics

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Solution 3 (cont.)

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Remarkable points of solution 3

  • reduction of CPU time
  • post-optimal analysis
  • increased accuracy
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Post-optimal analysis

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Numerical analysis of higher-order TPBVPs

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Second order example (comparisons against exact solution)

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General result

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Dealing with input/state constraints

(Knut Graichen)

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Conclusions

  • Numerous variables can be

eliminated from formulations of

  • ptimal 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

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Some references

1. 1.

  • U. M.
  • U. M.

Ascher, R. M. M. Ascher, R. M. M. Mattheij, Mattheij, and and

  • R. D.
  • R. D.

Russell Russell. Numerical solution of boundary value problems for

  • rdinary differential equations.

Prentice Hall, Inc., Englewood Cliffs, NJ, 1988.

2. 2.

A. A. Isidori Isidori. Nonlinear Control

  • Systems. Springer, New York, 2nd

edition, 1989.

3. 3.

  • M. Fliess, J. L
  • M. Fliess, J. Lévine, P. M

vine, P. Martin, rtin, and nd P. P. Rouchon Rouchon. Flatness and defect

  • f

nonlinear systems: introductory theory and examples.

  • Int. J. Control, 61(6):1327–1361, 1995.

4. 4.

  • N. Petit, M. B.
  • N. Petit, M. B.

Milam, Milam, and and

  • R. M. Murray

. M. Murray. Inversion based constrained trajectory optimization. In 5th IFAC Symposium on Nonlinear Control Systems, 2001.

5. 5.

  • M. Milam.
  • M. Milam.

Real-Time Optimal Trajectory Generation for Constrained Dynamical Systems. . PhD thesis. California Institute

  • f Technology, 2003.

6. 6.

K. K. Graichen

  • Graichen. Feedforward

Control Design for Finite-Time Transition Problems

  • f

Nonlinear Systems with Input and Output

  • Constraints. Doctoral

Thesis, Shaker Verlag, 2006.

7. 7.

F. F. Chaplais and Chaplais and

  • N. Petit

. Petit. Inversion in indirect optimal control

  • f

multivariable systems. To appear ESAIM COCV, 2007.