Draft EE 8235: Lecture 23 1 Lecture 23: Optimal control of distributed systems • Linear Quadratic Regulator (LQR) ⋆ Linear: plant ⋆ Quadratic: performance index ⋆ Infinite horizon problem ⋆ Algebraic Riccati Equation (ARE) • Spatially invariant systems ⋆ LQR: also spatially invariant ⋆ Feedback gains decay exponentially with spatial distance • Examples ⋆ Distributed control ⋆ Boundary control
Draft EE 8235: Lecture 23 2 Linear Quadratic Regulator � ∞ � � minimize J = � ψ ( t ) , Q ψ ( t ) � + � u ( t ) , R u ( t ) � d t 0 subject to ψ t ( t ) = A ψ ( t ) + B u ( t ) , ψ (0) ∈ H • Finite dimensional problems ⋆ Optimal controller determined by − K ψ ( t ) u ( t ) = R − 1 B T P K = ⋆ P = P ∗ – non-negative solution to ARE A ∗ P + P A + Q − P B R − 1 B ∗ P = 0 ⋆ ARE – quadratic equation in the elements of P
Draft EE 8235: Lecture 23 3 • Infinite dimensional problems ⋆ Optimal controller determined by u ( t ) = −K ψ ( t ) R − 1 B † P K = ⋆ P = P † – bounded non-negative operator that solves ARE � � � B † P ψ 1 , R − 1 B † P ψ 2 � 1 1 2 ψ 1 , Q 2 ψ 2 �A ψ 1 , P ψ 2 � + �P ψ 1 , A ψ 2 � + Q − = 0 ψ 1 , ψ 2 ∈ D ( A ) ⋆ ARE – operator-valued equation in the unknown P
Draft EE 8235: Lecture 23 4 An example • Mass-spring system on a line � � � � � � � � ˙ 0 0 p I p = + u v ˙ T 0 v I − 2 1 0 0 1 − 2 1 0 ∼ T 0 1 − 2 1 − 2 0 0 1 In class: use Matlab to illustrate structure of optimal feedback gains
Draft EE 8235: Lecture 23 5 Structure of optimal solution log 10 ( | K p | ) : K p : diag ( K p ) : K p (25 , :) :
Draft EE 8235: Lecture 23 6 ✲ G 0 ✲ G 1 ✲ G 2 ✲ ✛ ✛ ✛ ✛ ✻ ✻ ✻ ❄ ❄ ❄ K u 1 ( t ) ∗ ∗ ∗ ∗ p 1 ( t ) ∗ ∗ ∗ ∗ v 1 ( t ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ u 2 ( t ) p 2 ( t ) v 2 ( t ) = − − ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ u 3 ( t ) p 3 ( t ) v 3 ( t ) u 4 ( t ) ∗ ∗ ∗ ∗ p 4 ( t ) ∗ ∗ ∗ ∗ v 4 ( t ) � �� � � �� � K p K v • Observations: ⋆ LQR – centralized controller ⋆ Diagonals almost constant (modulo edges) ⋆ Off-diagonal decay of centralized gain
Draft EE 8235: Lecture 23 7 Spatially invariant systems ψ t ( x, t ) = [ A ψ ( · , t ) ] ( x ) + [ B u ( · , t ) ] ( x ) spatial coordinate: x ∈ G translation invariant operators: A , B S PATIAL F OURIER TRANSFORM ˙ ˆ A ( κ ) ˆ ˆ ψ ( κ, t ) + ˆ B ( κ ) ˆ ψ ( κ, t ) = u ( κ, t ) spatial frequency: κ ∈ ˆ G A ( κ ) , ˆ ˆ B ( κ ) multiplication operators: R reals G R S Z Z N Z integers ˆ S unit circle G R Z S Z N integers modulo N Z N
Draft EE 8235: Lecture 23 8 • Partial Differential Equations ⋆ Constant coefficients + Infinite spatial extent ψ t ( x, t ) = ψ xx ( x, t ) + u ( x, t ) , x ∈ R � Fourier transform ˙ ψ ( κ, t ) = − κ 2 ˆ ˆ u ( κ, t ) , κ ∈ R ψ ( κ, t ) + ˆ ⋆ Constant coefficients + Periodic domain ψ t ( x, t ) = ψ xx ( x, t ) + u ( x, t ) , x ∈ S � Fourier series ˙ ψ ( κ, t ) = − κ 2 ˆ ˆ u ( κ, t ) , κ ∈ Z ψ ( κ, t ) + ˆ
Draft EE 8235: Lecture 23 9 • Spatially discrete systems (Interconnected ODEs) ⋆ Constant coefficients + Infinite lattices � � � � 0 1 0 ˙ ψ ( x, t ) = ψ ( x, t ) + u ( x, t ) , x ∈ Z S − 1 − 2 + S 1 0 1 � Z -transform evaluated at z = e j κ � � � � 0 1 0 ˙ ˆ ˆ ψ ( κ, t ) = ψ ( κ, t ) + u ( κ, t ) , κ ∈ S ˆ 2 (cos κ − 1) 0 1
Draft EE 8235: Lecture 23 10 ⋆ Constant coefficients + Circular lattices Example: Mass-spring system on a circle � � � � 0 1 0 ˙ ψ ( x, t ) = ψ ( x, t ) + u ( x, t ) , x ∈ Z N S − 1 − 2 + S 1 0 1 � discrete Fourier transform � � � � 0 1 0 ˙ ˆ ˆ u ( κ, t ) , κ ∈ Z N ψ ( κ, t ) = ψ ( κ, t ) + ˆ � � cos 2 π κ 2 − 1 0 1 N
Draft EE 8235: Lecture 23 11 LQR for spatially invariant system over Z N � ∞ � � ψ ∗ ( t ) Q ψ ( t ) + u ∗ ( t ) R u ( t ) minimize J = d t 0 ˙ subject to ψ ( t ) = A ψ ( t ) + B u ( t ) • Circulant matrices: A , B , Q , R ⋆ Jointly unitarily diagonalizable by DFT Matrix V ˙ ˆ A d ˆ ψ ( t ) = ψ ( t ) + B d ˆ u ( t ) � � ˆ = V A V ∗ = diag A ( κ ) A d ψ ∗ Q ψ ψ ∗ Q d ˆ ˆ = ψ ⋆ Entries into ARE – diagonal matrices d P d + P d A d + Q d − P d B d R − 1 A ∗ d B ∗ d P d = 0 � A ∗ ( κ ) ˆ ˆ P ( κ ) + ˆ P ( κ ) ˆ A ( κ ) + ˆ Q ( κ ) − ˆ P ( κ ) ˆ B ( κ ) ˆ R − 1 ( κ ) ˆ B ∗ ( κ ) ˆ P ( κ ) = 0 , κ ∈ Z N
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