Inversion in optimal control. Principles and examples Nicolas Petit - PowerPoint PPT Presentation

Inversion in optimal control. Principles and examples Nicolas Petit Centre Automatique et Systèmes École 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

Iterating the resolution Bellman’s principle of optimality

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 r : relative degree of z1, zero dynamics, normal form, flatness y stands for Instead of

Comparisons (dim x= n, dim u= 1) 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

Example

• Collocation Software • Easily computed NTG: Mark Milam, Kudah derivatives: B-splines • Analytic gradients Mushambi, • Frontend to NPSOL Richard Murray, CalTech or: Matlab, Optim. Toolbox, Spline toolbox

3 – Three examples � CalTech ducted fan � Missile � Mobile robots

CalTech Ducted Fan (M. Milam) Control variables histories Flat outputs : z1 et z2

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

CalTech Ducted Fan (see M. Milam PhD thesis) open loop closed loop terrain avoidance sequence

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

Controls: α c, β c Data: m(t), T(t) Missile

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 references Numerical 1. 1. U. M. U. M. Ascher, R. M. M. Ascher, R. M. M. Mattheij, Mattheij, and and R. D. R. D. Russell Russell. solution of boundary value problems for ordinary differential equations . Prentice Hall, Inc., Englewood Cliffs, NJ, 1988. Nonlinear Control Systems . Springer, New York, 2nd 2. 2. A. A. Isidori Isidori. edition, 1989. 3. 3. M. Fliess, J. L é vine, P. M M. Fliess, J. L vine, P. Martin, rtin, and nd P. P. Rouchon Rouchon. Flatness and defect of 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 5 th 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 of Technology, 2003. 6. 6. K. K. Graichen Graichen. Feedforward Control Design for Finite-Time Transition Problems of 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 of multivariable systems. To appear ESAIM COCV, 2007.