Mesd Nonlinear Problem Setting 16888 ESD.J7 White Noise Input Science Target Observation Mode Appended LtI System Dynamics Phasing [Ard, Bzd, Cz, Dzal Opto-Structural Plant IA Bo CD DpI z Disturbances J=RMMS WFE W Performances ZCz qzd (RWA, Cryo) Pointing [Aa Ba, Ca Dau Control Actuator (E Noise ACS, FSM Variables: xi IAc B Ce, D Noise J= RSS LOS 11 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
11 Nonlinear Problem Setting Nonlinear Problem Setting White Noise Input “Science Target Observation Mode” Appended LTI System Dynamics d J Disturbances Opto-Structural Plant Control (ACS, FSM) (RWA, Cryo) w u y z Actuator Σ Σ Noise Sensor Noise [Ad,Bd,Cd,Dd] [Ap,Bp,Cp,Dp] [Ac,Bc,Cc,Dc] [Azd, Bzd, Czd, Dzd] Variables: xj J = RSS LOS z,2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Performances Phasing Pointing z,1=RMMS WFE z=Czd qzd
M 16888 esd Problem Statement ESD.J7 Given 9=Aid(,)q+Bed(x )d+Br(*, )/LTI System Dynamics ==Crd(x)a+ De(r,)d+ De(,)r,where j=1,2, And Performance Objectives 1/2 =F(9=1=:2- RMS Find Solutions x. such that J1(x2)=J2mi=1,2,…,n2 Assuming n-2>1 and x, LB sx.uB V j=1, 2, Subject to a numerical tolerance T -,req 100 req o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
12 Problem Statement Problem Statement Given q = A x q + Bzd () x r LTI System Dynamics zd () jjj x d + Bzr ( ) z = Czd () x d + D ( x r , where j = 1, 2,..., np x q j j + Dzd () zr j) And Performance Objectives T T º 1/ 2 § ·1/ 2 J = F ( )z , e.g. J = z z t = E ªz z ¼ = ¨ ¨ T 1 ³ ( )2 z z i , dt¸ RMS 2 ¬ ¸ © 0 ¹ Find Solutions x such that iso J ( x ) ≡ J ∀ i =1, 2,..., n z i iso z req i , , , z − ≥1 and xj LB ≤ xj ≤ x Assuming j,UB ∀ j =1, 2,..., n n z , J x zz ( ) − J ,req τ Subject to a numerical tolerance τ : ≤ = ε J 100 z req , © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
MleSd Bivariate Exhaustive Search(2D)E50. 3 Simple" Start: Bivariate Isoperformance Problem First Algorithm: Exhaustive Search Performance J(x,,x2) coupled with bilinear interpolation Variables x,j=1, 2: n=2 Number of points along j-th axis 2-dimensional Euclidian vector space parameter space X2 n,=A,UB-X;,LB boundary B △x Zoomed Region contours unstable subspace U individual solution point pise JukIMh-I Jni, ,. ILe-Iy k-l contours X1 Can also use standard contouring code like matlab contour. m o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
13 Bivariate Exhaustive Search (2D) Bivariate Exhaustive Search (2D) “Simple” Start: Bivariate Isoperformance Problem First Algorithm: Exhaustive Search Performance J (x x2 ) : z = 1 z 1, coupled with bilinear interpolation Variables x j = 1, 2 : j , n = 2 Number of points along j-th axis: ª x − x j,UB j,LB º x1 x2 n = j « » ∆ » x « Can also use standard contouring code like MATLAB contourc.m © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics
M esd Contour Following(2D) 16888 ESD.J7 Bmpy可)Nmg Taylor series expansion J(x)=J(x)+ Ax+△xH△x+HOT 2 first order ter VJ△x≡0 TJ H: Hessian Step size t: tangential step direction k+1-th isoperformance point =x+△x o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
14 Contour Following (2D) Contour Following (2D) k-th isoperformance point: Taylor series expansion T 1 T J x ( k ( ) = J x ) + ∇J x ∆ + ∆x H x ∆ + H.O.T. k z z z k x x 2 first order term second order term T ∇J ∆ ≡ x 0 ª τ J z k p T α ( ª∂Jz º « » ∂x J 1 ∇= « » z «∂J »z « ¬ ∂x2 ¼ » º 1/ 2 −1 = «2 z req , t H k t k k x k ) » ¬ 100 ¼ H: Hessian −∇J ª0 −1º k z k t =ℜ⋅ = ¬ « 1 0 ¼ »⋅ n αk : Step size k 1 J x − k x k 1 x + k k n t ∇Jz k p z req , k n k t k: tangential ∆ =αk x ⋅ t step direction k+1 k k+1-th isoperformance point: x = x + ∆x © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics