Numerical algorithms in control Numerical rank revealing - PowerPoint PPT Presentation

AS NLA in CONTROL Zlatko Drmaˇ c Introduction Scaling: examples Numerical algorithms in control Numerical rank revealing Eigenvalues and singular values Zlatko Drmaˇ c Jacobi method Accurate PSVD and University of Zagreb applications Department of Mathematics Concluding remarks Trogir 2011

AS NLA in CONTROL Outline Zlatko Drmaˇ c Introduction Introduction 1 Scaling: examples Numerical rank revealing Scaling: examples 2 Eigenvalues and singular values Numerical rank revealing 3 Jacobi method Accurate PSVD and Eigenvalues and singular values 4 applications Concluding remarks Jacobi method 5 Accurate PSVD and applications 6 7 Concluding remarks

AS NLA in CONTROL . . . .... Zlatko Drmaˇ c Introduction Introduction 1 Problems Problems Machine numbers Examples Machine numbers Consequences Goals Examples Scaling: examples Consequences Numerical Goals rank revealing Eigenvalues and singular values Scaling: examples 2 Jacobi method Accurate Numerical rank revealing 3 PSVD and applications Concluding remarks 4 Eigenvalues and singular values 5 Jacobi method 6 Accurate PSVD and applications

AS NLA in CONTROL LTI systems, control, tasks Zlatko Drmaˇ c Space station, CD player, vehicle suspension system, ... Introduction Problems Machine numbers x ( t ) = A x ( t ) + B u ( t ) , E , A ∈ R n × n , B ∈ R n × m E ˙ Examples Consequences (1) y ( t ) = C x ( t ) + D u ( t ) , C ∈ R p × n , D ∈ R p × m . Goals Scaling: examples x = Ax + Bu + Gr with x ∈ R 6 , Example: ˙ Numerical rank revealing   0 1 0 0 0 0 Eigenvalues kp cp kp cp and singular − mp − 0 0   0 mp mp mp values   0  0 0 0 1 0 0  A = , B = 0 Jacobi method kp cp ks + kp cs + cp   ks cs   1 − −  ms ms ms ms ms ms  Accurate 0   0 0 0 0 0 1 PSVD and − 1 − ks + kt ks cs − cs applications 0 0 mus mus mus mus Concluding remarks G = ( 0 , 0 , 0 , 0 , 0 , k t / m us ) ; r ( t ) = road; u ( t ) = actuator force; x 1 ( t ) = passenger’s vertical displacement. Determine u ( t ) (e.g. u ( t ) = − Kx ( t ) ) to ensure smooth riding on a rough road.

AS NLA in CONTROL Control, introduction, tasks Zlatko Drmaˇ c x ( t ) = A x ( t ) + B u ( t ) , x ( 0 ) = 0 ; Introduction ˙ (2) Problems y ( t ) = C x ( t ) + D u ( t ) . Machine numbers Examples Consequences Apply Laplace transform to get Goals Scaling: y ( s ) = ( C ( sI − A ) − 1 B + D ) u ( s ) examples ˆ ˆ Numerical � �� � rank revealing G ( s ) ≡ transfer function Eigenvalues and singular y ( s ) = G ( s )ˆ u ( s ) . → output behavior ˆ values Of interest is the input − Jacobi method In large scale/real time applications: try to reproduce nearly Accurate the same behavior with a system of smaller dimension PSVD and r ≪ n . Take D = 0. applications Concluding remarks x r ( t ) = A r x r ( t ) + B r u ( t ) � ˙ G r ( s ) = C r ( sI − A r ) − 1 B r y r ( t ) = C r x r ( t ) y ( s ) − ˆ y r ( s ) = ( G ( s ) − G r ( s ))ˆ u ( s ) should be small in some ˆ norm for a class of inputs u ( · ) .

AS NLA in CONTROL Consider n dimensional LTI SISO (more general, XIXO) NLA tasks in control Zlatko Drmaˇ c Introduction x ( t ) = A x ( t ) + bu ( t ) Problems Machine numbers ˙ ⊳ G ( s ) = c ( sI − A ) − 1 b . Examples ⊲ Consequences y ( t ) = cx ( t ) Goals For r < n and r –dimensional V r = R ( V r ) , W r = R ( W r ) with Scaling: examples Numerical � W ⊥ = { 0 } ( ⇔ det ( W T rank revealing r V r ) � = 0) look for V r Eigenvalues and singular V r ∋ v ( t ) = V r x r ( t ) such that ˙ v ( t ) − Av ( t ) − bu ( t ) ⊥ W r . values Jacobi method The reduced output is y r ( t )= cv ( t ) . In the bases V r , W r , Accurate PSVD and applications W T r ( V r ˙ x r ( t ) − AV r x r ( t ) − bu ( t )) = 0 , i.e. Concluding remarks x r ( t ) = A r x r ( t ) + b r u ( t ) ˙ ⊳ G r ( s ) = c r ( sI − A r ) − 1 b r ⊲ y r ( t ) = c r x r ( t ) r A V r , b r = ( W T A r = ( W T r V r ) − 1 W T r V r ) − 1 W T r b , c r = cV r .

AS NLA in CONTROL Numerical tasks Zlatko Drmaˇ c � � ∞ • � G � H 2 = −∞ | G ( i ω ) | 2 d ω ; 1 Introduction 2 π Problems Machine numbers • min � G − G r � H 2 , G r stable of order r Examples Consequences • Let G r be a local minimizer with simple poles at ˜ Goals λ i , i = 1 , . . . , r . Then at σ i = − ˜ λ i : G r ( σ i ) = G ( σ i ) , Scaling: examples G ′ r ( σ i ) = G ′ ( σ i ) , i = 1 , . . . , r . Numerical rank revealing • Hermite interpolation by V r = Span (( σ i I − A ) − 1 b ) r i = 1 , Eigenvalues W r = Span (( σ i I − A T ) − 1 c T ) r and singular i = 1 . values • Solving linear systems for V r and W r . Reduce to Jacobi method generalized upper Hessenberg form Accurate PSVD and applications � � � � � � � � � � � � � � � Q T EZ = , Q T AZ = , Q T b = Concluding 0 � � � � � � � remarks � � � � � 0 � � � 0 and work on ( E , A , b , c ) ≡ ( Q T EZ , Q T AZ , Q T b , cZ ) is efficient. Simpler if E = I , Z = Q .

AS NLA in CONTROL Numerical tasks Zlatko Drmaˇ c Introduction Generate many interesting and challenging problems. Problems Machine numbers • Simple questions, difficult answers: Compute the Examples Consequences transfer function G ( ζ ) = C ( ζ E − A ) − 1 B for many Goals complex values of ζ . Here n can be large. Scaling: examples • By changing the state space coordinates, x ( t ) = T ˆ x ( t ) , Numerical rank revealing the new representation is, e.g. for E = I , given with Eigenvalues A , ˆ B , ˆ C , ˆ D ) = ( T − 1 AT , T − 1 B , CT , D ) . Find T such that (ˆ and singular values the new representation reveals structural properties of Jacobi method the system. Various canonical forms. Accurate • Solve Lyapunov equation AH + HA T + BB T = 0. Solve PSVD and applications Riccati eqn: XA + A T X + Q − XSX = 0. Many other Concluding remarks types of matrix equations. • Find invariant subspace that corresponds to specified eigenvalues.

AS NLA in CONTROL ... algorithms, software Zlatko Drmaˇ c Introduction Problems • Solve eigenvalue and singular value problems. Machine numbers Examples Consequences • Given A with eigenvalues λ 1 , . . . , λ n and B , find K such Goals that A − BK has prescribed eigenvalues α 1 , . . . , α n . Scaling: examples • Pressure from applications to deliver accurate solutions Numerical rank revealing quickly. Computing environments changing rapidly. Eigenvalues and singular • Users from applied sciences and engineering – usually values not interested in math details, just solutions, software. Jacobi method • Pure mathematicians not interested because the Accurate PSVD and problems are "trivial", non–fundamental or just too applications Concluding messy. remarks • And we have high performance computers. So, why is this difficult?

AS NLA in CONTROL Yes, have computer, but .. Zlatko Drmaˇ c Machine (floating–point) numbers F ⊂ Q . Introduction Problems f = ± m · 2 e , e = − 126 : 127 , m = 1 . z 1 . . . z 23 . Machine numbers Examples Consequences � Goals { + Infinity , − Infinity , NaN } F = F Scaling: examples Machine arithmetic ⊕ , ⊖ , ⊙ , ⊘ . Numerical rank revealing • F finite, 2 32 (single), 2 64 (double); 0 . 1 �∈ F ; Eigenvalues • a ⊕ b ≡ FL ( a + b ) = ( a + b )( 1 + ǫ a , b ) , and singular values Jacobi method | ǫ a , b | ≤ u ≡ eps = round-off ≈ 10 − 8 . Accurate PSVD and applications • In general, ( a ⊕ b ) ⊕ c � = a ⊕ ( b ⊕ c ) , Concluding ( a ⊙ b ) ⊙ c � = a ⊙ ( b ⊙ c ) ; x ⊕ y ⊕ z =?? remarks • 1 ⊕ 10 − 9 = 1; x = y �⇔ x − y = 0; 10 − 30 ⊙ 10 − 30 = 0; • Finite speed, finite memory. • Faster = ⇒ more mess per second.

AS NLA in CONTROL Example using MATLAB, Zlatko Drmaˇ c eps ≈ 2 . 2 · 10 − 16 Introduction Problems � a c � Machine numbers � � ∈ R m × 2 , = computed ( X T X ) , X = x y Examples Consequences c b Goals Scaling: examples � 1 � � 1 � 3 x = 5 · 10 153 , y = 10 ( cos ∠ ( x , y ) = Numerical Let √ . ) . rank revealing 1 2 10 Eigenvalues and singular Test the orthogonality of x and y , values c Jacobi method cos ∠ ( x , y ) ≡ √ ≤ ǫ Accurate ab PSVD and applications Concluding ( c / sqrt(a*b) <= eps ) = 1 , remarks ( (c / sqrt(a)) / sqrt(b) <= eps ) = 0 , ( c <= sqrt(a*b) * eps ) = 1 , ( c <= sqrt(a)*sqrt(b) * eps ) = 0 .

AS NLA in CONTROL Example using MATLAB, Zlatko Drmaˇ c eps ≈ 2 . 2 · 10 − 16 Introduction Problems Machine numbers Examples Consequences Goals � 1 � 1 � � Scaling: x = 5 · 10 − 153 , y = 10 − 16 Let examples − 1 1 Numerical rank revealing Then Eigenvalues and singular values ( c / sqrt(a*b) <= eps ) = 0 , Jacobi method ( (c / sqrt(a)) / sqrt(b) <= eps ) = 1 , Accurate PSVD and applications ( c <= sqrt(a*b) * eps ) = 0 , Concluding ( c <= sqrt(a)*sqrt(b) * eps ) = 1 . remarks