Draft EE 8235: Lecture 15 1 Lecture 15: Systems with inputs Input - PowerPoint PPT Presentation

Draft EE 8235: Lecture 15 1 Lecture 15: Systems with inputs • Input types ⋆ Additive inputs ⋆ Boundary inputs • Input-output mappings ⋆ Transfer function ⋆ Frequency response ⋆ Impulse response • Abstract evolution equation for boundary control systems ⋆ Objective: bring system into a form that resembles standard formulation • Two point boundary value problems

Draft EE 8235: Lecture 15 2 Additive inputs • Example: diffusion equation on L 2 [ − 1 , 1] with Dirichlet BCs φ t ( x, t ) = φ xx ( x, t ) + u ( x, t ) φ ( x, 0) = φ 0 ( x ) φ ( ± 1 , t ) = 0 • Abstract evolution equation ψ t ( t ) = A ψ ( t ) + u ( t ) d 2 D ( A ) = { f ∈ L 2 [ − 1 , 1] , f ′′ ∈ L 2 [ − 1 , 1] , f ( ± 1) = 0 } A = d x 2 , • Solution � t ψ ( t ) = T ( t ) ψ (0) + T ( t − τ ) u ( τ ) d τ 0 T ( t ) : C 0 -semigroup generated by A

Draft EE 8235: Lecture 15 3 Input-output maps ψ t ( t ) = A ψ ( t ) + B u ( t ) φ ( t ) = C ψ ( t )  A : H ⊃ D ( A ) − → H   • Underlying operators: B : U − → H  C : H − → Y  • Input-output mapping � t φ ( t ) = [ H u ] ( t ) = C T ( t − τ ) B u ( τ ) d τ 0 ⋆ Impulse response H ( t ) = ( C T ( t ) B ) 1 ( t ) ⋆ Transfer function H ( s ) = C ( sI − A ) − 1 B ⋆ Frequency response H (j ω ) = C (j ωI − A ) − 1 B

Draft EE 8235: Lecture 15 4 An example � � φ t ( x, t ) = φ xx ( x, t ) + u ( x, t ) φ ′′ ( x, s ) = s φ ( x, s ) − u ( x, s ) Laplace − − − − − − − − → φ ( ± 1 , t ) = 0 φ ( ± 1 , s ) = 0 transform • Spatial realization of H ( s ) (with ψ 1 = φ , ψ 2 = φ ′ ) � ψ ′ � 0 � � ψ 1 ( x, s ) 1 ( x, s ) � 1 � � 0 � = + u ( x, s ) ψ ′ 2 ( x, s ) s 0 ψ 2 ( x, s ) − 1 0 � � ψ 1 ( x, s ) � 1 � φ ( x, s ) = ψ 2 ( x, s ) � � ψ 1 ( − 1 , s ) � � ψ 1 (1 , s ) � � � � 1 0 0 0 0 = + 0 0 1 0 ψ 2 ( − 1 , s ) ψ 2 (1 , s ) • Two point boundary value problem ψ ′ ( x ) = A ( x ) ψ ( x ) + B ( x ) u ( x ) φ ( x ) = C ( x ) ψ ( x ) 0 = N a ψ ( a ) + N b ψ ( b )

Draft EE 8235: Lecture 15 5 Boundary control • Example: diffusion equation on L 2 [ − 1 , 1]   φ t ( x, t ) = φ xx ( x, t ) + d ( x, t ) φ ′′ ( x, s ) = s φ ( x, s ) − d ( x, s )   Laplace   φ ( − 1 , t ) = u ( t ) − − − − − − − − → φ ( − 1 , s ) = u ( s ) transform   φ (+1 , t ) = 0 φ (+1 , s ) = 0   • Spatial realization of H ( s ) (with ψ 1 = φ , ψ 2 = φ ′ ) � ψ ′ � 0 � � ψ 1 ( x, s ) � � � � 1 ( x, s ) 1 0 = + d ( x, s ) ψ ′ 2 ( x, s ) s 0 ψ 2 ( x, s ) − 1 0 � � ψ 1 ( x, s ) � 1 � φ ( x, s ) = ψ 2 ( x, s ) � � ψ 1 ( − 1 , s ) � � ψ 1 (1 , s ) � � � � � � u ( s ) 1 0 0 0 = + 0 0 0 1 0 ψ 2 ( − 1 , s ) ψ 2 (1 , s ) • Two point boundary value problem ψ ′ ( x ) = A ( x ) ψ ( x ) + B ( x ) d ( x ) φ ( x ) = C ( x ) ψ ( x ) ν = N a ψ ( a ) + N b ψ ( b )

Draft EE 8235: Lecture 15 6 Abstract evolution equation for systems with boundary inputs φ t ( x, t ) = φ xx ( x, t ) + d ( x, t ) φ ( − 1 , t ) = u ( t ) φ (+1 , t ) = 0 • Problem: control doesn’t enter additively into the equation • Coordinate transformation ψ ( x, t ) = φ ( x, t ) − f ( x ) u ( t ) ⋆ Choose f ( x ) to obtain homogeneous boundary conditions ψ ( ± 1 , t ) = 0 ⋆ Many possible choices Conditions for selection of f : simple option f ( x ) = 1 − x { f ( − 1) = 1 , f (1) = 0 } − − − − − − − − − − − → 2

Draft EE 8235: Lecture 15 7 • In new coordinates: φ t ( x, t ) = φ xx ( x, t ) + d ( x, t ) φ ( − 1 , t ) = u ( t ) φ (+1 , t ) = 0   � φ ( x,t ) = ψ ( x,t ) + f ( x ) u ( t ) ψ t ( x, t ) + f ( x ) ˙ u ( t ) = ψ xx ( x, t ) + f ′′ ( x ) u ( t ) + d ( x, t ) ψ ( ± 1 , t ) = 0 • New input: v ( t ) = ˙ u ( t ) d � � � � � � � � � � ψ ( t ) A 0 f ′′ ψ ( t ) I − f = + d ( t ) + v ( t ) u ( t ) 0 0 u ( t ) 0 I d t � I f � � � ψ ( t ) φ ( t ) = u ( t ) d 2 d x 2 , D ( A 0 ) = { f ∈ L 2 [ − 1 , 1] , f ′′ ∈ L 2 [ − 1 , 1] , f ( ± 1) = 0 } A 0 =

Draft EE 8235: Lecture 15 8 Two point boundary value problems ψ ′ ( x ) = A ( x ) ψ ( x ) + B ( x ) d ( x ) φ ( x ) = C ( x ) ψ ( x ) ν = N a ψ ( a ) + N b ψ ( b ) • Solution: � x φ ( x ) = C ( x ) Φ( x, a ) ( N a + N b Φ( b, a )) − 1 ν + C ( x ) Φ( x, ξ ) B ( ξ ) d ( ξ ) d ξ − a � b C ( x ) Φ( x, a ) ( N a + N b Φ( b, a )) − 1 N b Φ( b, ξ ) B ( ξ ) d ( ξ ) d ξ a Φ( x, ξ ) : the state transition matrix of A ( x ) dΦ( x, ξ ) = A ( x ) Φ( x, ξ ) , Φ( ξ, ξ ) = I d x For systems with A � = A ( x ) : Φ( x, ξ ) = e A ( x − ξ )

Draft EE 8235: Lecture 15 9 Examples • Heat equation with boundary actuation φ t ( x, t ) = φ xx ( x, t ) φ ( − 1 , t ) = u ( t ) φ (+1 , t ) = 0  � Laplace trasform  � ψ ′ � 0 � � ψ 1 ( x, s ) � � 1 ( x, s ) 1 = ψ ′ 2 ( x, s ) s 0 ψ 2 ( x, s ) 0 � � ψ 1 ( x, s ) � 1 � φ ( x, s ) = ψ 2 ( x, s ) � � ψ 1 ( − 1 , s ) � � ψ 1 (1 , s ) � � � � � � u ( s ) 1 0 0 0 = + 0 0 0 1 0 ψ 2 ( − 1 , s ) ψ 2 (1 , s ) N a + N b e A ( s )( b − a ) � − 1 ν ( s ) φ ( x, s ) = C e A ( s )( x − a ) � sinh ( √ s (1 − x )) sinh (2 √ s ) = u ( s ) � � ∞ 1 − x 2 s � = − s + ( nπ/ 2) 2 v n ( x ) u ( s ) 2 nπ n = 1

Draft EE 8235: Lecture 15 10 • Eigenvalue problem for streamwise constant linearized NS equations � L ψ os = λ os ψ os , ψ os ( ± 1) = ψ ′ os ( ± 1) = 0 Orr-Sommerfeld: S u os = λ os u os − C p ψ os , u os ( ± 1) = 0   � ∆ 2 ψ os � = λ os ∆ ψ os , ψ os ( ± 1) = ψ ′ os ( ± 1) = 0 ∆ u os = λ os u os − j k z U ′ ( y ) ψ os , u os ( ± 1) = 0 Two point boundary value problem for u os : � x ′ � � x 1 ( y, k z ) 1 ( y, k z ) � � 0 1 � � 0 � = + ψ os ( y, k z ) λ os + k 2 x ′ 2 ( y, k z ) 0 x 2 ( y, k z ) − j k z U ′ ( y ) z 0 � � x 1 ( y, k z ) � 1 � u os ( y, k z ) = x 2 ( y, k z ) � � x 1 ( − 1 , k z ) � � x 1 (1 , k z ) � � � � 1 0 0 0 0 = + 0 0 1 0 x 2 ( − 1 , k z ) x 2 (1 , k z )