Pseudospectral Methods For Op2mal Control Jus2n Ruths March 27, 2009
Introduc2on Pseudospectral methods arose to find solu2ons to Par2al Differen2al Equa2ons Recently adapted for Op2mal Control Key Ideas (N ~ # of discre2za2ons) Orthogonal basis func2ons leads to spectral accuracy Coefficients decrease faster than 1/N k, for any k Special recursive defini2ons for deriva2ves of orthogonal polynomials lead to wri2ng dynamics as algebraic rela2ons Non uniform node spacing allows for op2mal numerical integra2on through Gauss quadrature
Op2mal Control Controls: State: Endpoint Constraints: Cost Func2on: Dynamics: Path Constraints: Op2mal control addresses deriving control laws for minimiza2on problems Applica2ons wide spread disciplines Indirect methods Calculus of Varia2ons Maximum Principle Difficult to scale up to larger, more complex problems Direct methods Reduce to nonlinear programming problem
Challenges of Op2mal Control Controls: State: unknown func2ons We would like to transform op2mal control problems into constrained op2miza2on problems Endpoint Constraints: Cost Func2on: Dynamics: Path Constraints: Integrate a nonlinear func2on of unknown variables. Only in rare cases is an analy2c expression for this integral possible. Analy2c expressions for evolved dynamics are, in general, not available.
Pseudospectral Conversion Op#mal Control Controls: Interpola2on Pseudospectral Methods Controls: State: Interpola2on State: Endpoint Constraints: Endpoint Constraints: Cost Func2on: Quadrature Cost Func2on: Dynamics: Spectral Deriva2ves Dynamics: Path Constraints: Path Constraints:
Discre2za2on
Discre2za2on
Polynomial Approxima2on Weierstrass (1889) Set of polynomials is a dense subspace in the space of con2nuous func2ons on [ 1,1] with uniform norm Equioscilla2on Theorem & Corollary For posi2ve N, the best approxima2ng polynomial to a con2nuous func2on on [ 1,1] exists, is unique, and interpolates the func2on at N+1 points
Polynomial Approxima2on Given N+1 nodes and con2nuous func2on the N order interpola2ng polynomial is unique, Any interpola2ng polynomial can be represented by the basis of Lagrange polynomials
Lagrange Polynomials Since at the discre2za2on nodes, Now this interpola2on approxima2on is totally specified by the func2on values at the N+1 discre2za2on nodes.
Lagrange Polynomials, N=7
Node Selec2on Polynomial interpola2on is sensi2ve to node selec2on Gauss Lobogo (GL) nodes are close to op2mal Uniform Spaced Nodes GL Spaced Nodes
Op2mal Nodes
Op2mal Nodes Uniform GL N=10 N=16
Orthogonal Polynomials Weight func2on: Inner product: Given a weight func2on, can create orthogonal basis Orthogonalizing non nega2ve powers of t w.r.t. a weight func2on results in a set of orthogonal polynomials
Spectral Approxima2on Given a family of orthogonal polynomials, Use of these orthogonal polynomials leads to spectral accuracy k th coefficient decays faster than k 1/q, for any q Analogous to Fourier series for periodic func2on
Spectral Approxima2on Recursive defini2on of deriva2ve Using orthogonal polynomials, can express deriva2ves as algebraic rela2ons Lagrange basis s2ll benefits from algebraic rela2ons for deriva2ves
Legendre Polynomials, N=7
Gauss Lobago Quadrature Includes endpoints useful for imposing boundary condi2ons Given: Family of orthogonal polynomials: Corresponding weight func2on: N > 0 Nodes are N+1 zeros of: Is exact for (2N 1) order polynomials GL nodes facilitate discre#za#on & numerical integra#on
Spectral Colloca2on Conversion Discrete inner product preserves orthogonality
Spectral vs. Pseudospectral N+1 condi2ons, requiring Spectral approxima2ons Galerkin & Tau: Colloca2on (interpola2on):
Elements of Pseudospectral Methods GL Nodes are op2mal for interpola2on & quadrature Interpola2on GL Nodes Quadraure p N Spectral Nodes are extrema of orthogonal polynomials
Approxima2on, N = 3
Approxima2on, N = 7
Approxima2on, N = 14
Approxima2on, N = 21
Pseudospectral Conversion Op#mal Control Controls: Pseudospectral Methods Controls: State: State: Endpoint Constraints: Endpoint Constraints: Cost Func2on: Cost Func2on: Dynamics: Dynamics: Path Constraints: Path Constraints:
Pseudospectral Preliminaries Physical 2me domain: [0, T] Computa2onal 2me domain: [ 1,1] Pseudospectral Methods Controls: State: Legendre vs. Chebyshev Polynomials Both are Jacobi type polynomials Endpoint Constraints: Cost Func2on: Dynamics: Path Constraints:
Pseudospectral Discre2za2on Pseudospectral Methods Controls: State: Endpoint Constraints: Cost Func2on: Dynamics: Path Constraints: Decision Variables: n x (N+1) m x (N+1)
Pseudospectral Quadrature Note: For Legendre polynomials Pseudospectral Methods Controls: State: Endpoint Constraints: Cost Func2on: Dynamics: Path Constraints: Note: Interpola2on polynomial uses Lagrange polynomials, deriva2ve rule is less obvious
Ex: Single Spin Op2miza2on
Ex: Single Spin Op2miza2on
Ex: Single Spin Opt
Ex: Spin Transfer
Ex: Spin Transfer
Ex: Ensemble Spin Op2miza2on
Discre2za2on & Sampling
Discre2za2on & Sampling
Op2mal Sampling n samples N discre2za2ons
Future Work Bilinear system dynamics Convergence Adapta2ons Op2mal sampling Two dimensional pseudospectral method Selec2ve excita2on pulses
Challenges Difficulty dealing with discon2nui2es Knonng methods Succumbs to locally op2mal solu2ons Computa2onal load Costate discrepancy in Legendre PS at endpts Gauss pseudospectral method
QUESTIONS & COMMENTS