Draft EE 8235: Lecture 16 1 Lecture 16: Controllability and - PowerPoint PPT Presentation

Draft EE 8235: Lecture 16 1 Lecture 16: Controllability and observability • Controllability ⋆ Ability to steer state • Observability ⋆ Ability to estimate state • Topics: ⋆ Connections and differences with finite-dimensional case ⋆ Exact vs. approximate controllability/observability ⋆ Conditions for controllability/observability ⋆ Gramians ⋆ Operator Lyapunov equations

Draft EE 8235: Lecture 16 2 An example • Diffusion equation on L 2 [ − 1 , 1] with point actuation and sensing ψ t ( x, t ) = ψ xx ( x, t ) + b ( x ) u ( t ) � 1 φ ( t ) = c ( x ) ψ ( x, t ) d x − 1 ψ ( x, 0) = ψ 0 ( x ) ψ ( ± 1 , t ) = 0 Control and sensing points x c and x s 1 b ( x ) = 2 ǫ 1 [ x c − ǫ, x c + ǫ ] ( x ) 1 c ( x ) = 2 δ 1 [ x s − δ, x s + δ ] ( x ) � 1 , x ∈ [ a, b ] 1 [ a, b ] ( x ) = 0 , otherwise

Draft EE 8235: Lecture 16 3 Controllability operator and Gramian ψ t ( t ) = A ψ ( t ) + B u ( t ) A : H ⊃ D ( A ) − → H B : U − → H • Controllability operator R t : L 2 ([0 , t ]; U ) − → H � t ψ ( t ) = [ R t u ] ( t ) = T ( t − τ ) B u ( τ ) d τ 0 ⋆ Adjoint � � ( τ ) = B † T † ( t − τ ) , τ ∈ [0 , t ] R † t ψ • Controllability Gramian � t T ( τ ) B B † T † ( τ ) d τ P t = R t R † t = 0

Draft EE 8235: Lecture 16 4 Exact vs. approximate controllability • Exact controllability on [0 , t ] range ( R t ) = H ⋆ rarely satisfied by infinite-dimensional systems ⋆ never satisfied for systems with finite-dimensional U • Approximate controllability on [0 , t ] range ( R t ) = H ⋆ reasonable notion of controllability for infinite-dimensional systems ⋆ easily checkable conditions for Riesz-spectral systems approximate controllability on [0 , t ] � P t > 0 ⇔ {� ψ, P t ψ � > 0 , for all 0 � = ψ ∈ H } or � � � B † T † ( τ ) ψ = 0 on [0 , t ] ⇒ ψ = 0 � R † ⇔ = 0 null t

Draft EE 8235: Lecture 16 5 Observability operator and Gramian ψ t ( t ) = A ψ ( t ) φ ( t ) = C ψ ( t ) A : H ⊃ D ( A ) − → H C : H − → Y • Observability operator O t : H − → L 2 ([0 , t ]; Y ) φ ( t ) = [ O t ψ (0) ] ( t ) = C T ( t ) ψ (0) ⋆ Adjoint � t � � T † ( τ ) C † φ ( τ ) d τ O † ( t ) = t φ 0 • Observability Gramian � t T † ( τ ) C † C T ( τ ) d τ V t = O † t O t = 0

Draft EE 8235: Lecture 16 6 Exact vs. approximate observability • Exact observability on [0 , t ] ⋆ O t one-to-one and O − 1 bounded on the range of O t t • Approximate observability on [0 , t ] ⋆ null ( O t ) = 0 • ( A , · , C ) approximately obsv on [0 , t ] ⇔ ( A † , C † , · ) approximately ctrb on [0 , t ] approximate observability on [0 , t ] � V t > 0 ⇔ {� ψ, V t ψ � > 0 , for all 0 � = ψ ∈ H } or null ( O t ) = 0 ⇔ {C T ( τ ) ψ = 0 on [0 , t ] ⇒ ψ = 0 }

Draft EE 8235: Lecture 16 7 Infinite horizon Gramians • Exponentially stable C 0 -semigroup T ( t ) �T ( t ) � ≤ M e − αt ∃ M, α > 0 ⇒ • Extended (i.e., infinite horizon) Gramians � ∞ T ( τ ) B B † T † ( τ ) d τ R ∞ R † P = = ∞ 0 � ∞ T † ( τ ) C † C T ( τ ) d τ O † V = ∞ O ∞ = 0 • Approximate controllability � � R † P > 0 ⇔ = 0 null ∞ • Approximate observability V > 0 ⇔ null ( O ∞ ) = 0

Draft EE 8235: Lecture 16 8 Lyapunov equations Controllability Gramian P – unique self-adjoint solution to: � � � � � � � A † � A † ψ 1 , P ψ 2 P ψ 1 , A † ψ 2 B † ψ 1 , B † ψ 2 + = − for ψ 1 , ψ 2 ∈ D � � A † � � A † � ⊂ D ( A ) and A P ψ + P A † ψ = − B B † ψ P D for ψ ∈ D Observability Gramian V – unique self-adjoint solution to: �A ψ 1 , V ψ 2 � + �V ψ 1 , A ψ 2 � = − �C ψ 1 , C ψ 2 � for ψ 1 , ψ 2 ∈ D ( A ) � � A † � and A † V ψ + V A ψ = − C † C ψ V D ( A ) ⊂ D for ψ ∈ D ( A )

Draft EE 8235: Lecture 16 9 Controllability of Riesz-spectral systems m � ψ t ( x, t ) = [ A ψ ( · , t )] ( x ) + b i ( x ) u i ( t ) i = 1 ⇔ modal controllability approximate controllability A − Riesz-spectral operator with e-pair { ( λ n , v n ) } n ∈ N e-functions of A † s.t. � w n , v m � = δ nm { w n } n ∈ N − ∞ � [ A f ] ( x ) = λ n v n ( x ) � w n , f � n = 1   � �� � w n , b 1 � � w n , b m � �� ⇔ · · · approximate controllability = 1 rank • Necessary condition for controllability ⋆ Number of controls ≥ maximal multiplicity of e-vectors of A

Draft EE 8235: Lecture 16 10 Example (to be done in class) • Diffusion equation on L 2 [ − 1 , 1] with Dirichlet BCs ψ t ( x, t ) = ψ xx ( x, t ) + b ( x ) u ( t ) ψ ( x, 0) = ψ 0 ( x ) ψ ( ± 1 , t ) = 0 Diagonal coordinate form � nπ � 2 α n ( t ) = − α n ( t ) + � v n , b � u ( t ) , n ∈ N ˙ 2 � �� � b n ⇔ { b n � = 0 , for all n ∈ N } approximate/modal controllability