Nonlinear infinite-horizon control using generalized Lyapunov - PowerPoint PPT Presentation

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Nonlinear infinite-horizon control using generalized Lyapunov equations Tobias Breiten Karl Kunisch (KFU, Graz), Laurent Pfeiffer (INRIA, Paris) Workshop on “New trends in PDE constrained optimization” Special Semester on Optimization 2019 - RICAM Linz October 14, 2019

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Linear quadratic control problems Consider a linear system y ( t ) = Ay ( t ) + Bu ( t ) , ˙ y (0) = y 0 ∈ Y , y obs ( t ) = Cy ( t ) Hilbert spaces U , Y , Z , A generator of an analytic C 0 -semigroup e At on Y , control operator B ∈ L ( U , Y ), s.t. ( A , B ) is stabilizable, output operator C ∈ L ( Y , Z ), s.t. ( A , C ) is detectable. Let us focus on the infinite-horizon control problem � ∞ 1 Z + β 2 � y obs � 2 2 � u � 2 u ∈ L 2 (0 , ∞ ; U ) J ( y 0 , u ) = min U d t . 0 u = − 1 Optimal feedback ¯ β B ∗ Π¯ y by algebraic Riccati equation A ∗ Π + Π A − 1 β Π BB ∗ Π + C ∗ C = 0 .

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Linear quadratic control problems Consider a linear system y ( t ) = Ay ( t ) + Bu ( t ) , ˙ y (0) = y 0 ∈ Y , y obs ( t ) = Cy ( t ) Hilbert spaces U , Y , Z , A generator of an analytic C 0 -semigroup e At on Y , control operator B ∈ L ( U , Y ), s.t. ( A , B ) is stabilizable, output operator C ∈ L ( Y , Z ), s.t. ( A , C ) is detectable. Let us focus on the infinite-horizon control problem � ∞ 1 Z + β 2 � y obs � 2 2 � u � 2 u ∈ L 2 (0 , ∞ ; U ) J ( y 0 , u ) = min U d t . 0 u = − 1 Optimal feedback ¯ β B ∗ Π¯ y by algebraic Riccati equation A ∗ Π + Π A − 1 β Π BB ∗ Π + C ∗ C = 0 .

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING The ARE: feedback, optimal cost Well-known : ∃ ! stabilizing Π = Π ∗ � 0 ∈ L ( Y ) s.t. ∀ y 1 , y 2 ∈ D ( A ): � Ay 1 , Π y 2 � Y + � Π y 1 , Ay 2 � Y + � Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U = 0 . u J ( y 0 , u ) = 1 Minimal value function V ( y 0 ) := min 2 � y 0 , Π y 0 � Y . Note : for B ∈ L ( U , [ D ( A ∗ )] ′ ) not obvious that B ∗ Π ∈ L ( Y , U ) Alternative interpretation for A π := A − 1 β BB ∗ Π T ( A π y 1 , y 2 ) + T ( y 1 , A π y 2 ) = R ( y 1 , y 2 ) , (1) where T ( y 1 , y 2 ) := � y 1 , Π y 2 � Y , R ( y 1 , y 2 ) := −� Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U .

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING The ARE: feedback, optimal cost Well-known : ∃ ! stabilizing Π = Π ∗ � 0 ∈ L ( Y ) s.t. ∀ y 1 , y 2 ∈ D ( A ): � Ay 1 , Π y 2 � Y + � Π y 1 , Ay 2 � Y + � Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U = 0 . u J ( y 0 , u ) = 1 Minimal value function V ( y 0 ) := min 2 � y 0 , Π y 0 � Y . Note : for B ∈ L ( U , [ D ( A ∗ )] ′ ) not obvious that B ∗ Π ∈ L ( Y , U ) Alternative interpretation for A π := A − 1 β BB ∗ Π T ( A π y 1 , y 2 ) + T ( y 1 , A π y 2 ) = R ( y 1 , y 2 ) , (1) where T ( y 1 , y 2 ) := � y 1 , Π y 2 � Y , R ( y 1 , y 2 ) := −� Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U .

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING The ARE: feedback, optimal cost Well-known : ∃ ! stabilizing Π = Π ∗ � 0 ∈ L ( Y ) s.t. ∀ y 1 , y 2 ∈ D ( A ): � Ay 1 , Π y 2 � Y + � Π y 1 , Ay 2 � Y + � Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U = 0 . u J ( y 0 , u ) = 1 Minimal value function V ( y 0 ) := min 2 � y 0 , Π y 0 � Y . Note : for B ∈ L ( U , [ D ( A ∗ )] ′ ) not obvious that B ∗ Π ∈ L ( Y , U ) Alternative interpretation for A π := A − 1 β BB ∗ Π T ( A π y 1 , y 2 ) + T ( y 1 , A π y 2 ) = R ( y 1 , y 2 ) , (1) where T ( y 1 , y 2 ) := � y 1 , Π y 2 � Y , R ( y 1 , y 2 ) := −� Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U .

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING The ARE: feedback, optimal cost Well-known : ∃ ! stabilizing Π = Π ∗ � 0 ∈ L ( Y ) s.t. ∀ y 1 , y 2 ∈ D ( A ): � Ay 1 , Π y 2 � Y + � Π y 1 , Ay 2 � Y + � Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U = 0 . u J ( y 0 , u ) = 1 Minimal value function V ( y 0 ) := min 2 � y 0 , Π y 0 � Y . Note : for B ∈ L ( U , [ D ( A ∗ )] ′ ) not obvious that B ∗ Π ∈ L ( Y , U ) Alternative interpretation for A π := A − 1 β BB ∗ Π T ( A π y 1 , y 2 ) + T ( y 1 , A π y 2 ) = R ( y 1 , y 2 ) , (1) where T ( y 1 , y 2 ) := � y 1 , Π y 2 � Y , R ( y 1 , y 2 ) := −� Cy 1 , Cy 2 � Z − 1 β � B ∗ Π y 1 , B ∗ Π y 2 � U .

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Outline of this talk Consider the multidimensional analogue of (1) � k T k ( y 1 , . . . , y i − 1 , A π y i , y i +1 , . . . , y k ) = R k ( y 1 , . . . , y k ) i =1 where ( y 1 , . . . , y k ) ∈ D ( A ) k and R k ∈ M ( D ( A ) k , R ). Questions: Why should we consider these equations? Which theoretical results do we have for these equations? How should we treat these equations numerically?

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Outline of this talk Consider the multidimensional analogue of (1) � k T k ( y 1 , . . . , y i − 1 , A π y i , y i +1 , . . . , y k ) = R k ( y 1 , . . . , y k ) i =1 where ( y 1 , . . . , y k ) ∈ D ( A ) k and R k ∈ M ( D ( A ) k , R ). Questions: Why should we consider these equations? Which theoretical results do we have for these equations? How should we treat these equations numerically?

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Outline of this talk Consider the multidimensional analogue of (1) � k T k ( y 1 , . . . , y i − 1 , A π y i , y i +1 , . . . , y k ) = R k ( y 1 , . . . , y k ) i =1 where ( y 1 , . . . , y k ) ∈ D ( A ) k and R k ∈ M ( D ( A ) k , R ). Questions: Why should we consider these equations? Which theoretical results do we have for these equations? How should we treat these equations numerically?

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Outline of this talk Consider the multidimensional analogue of (1) � k T k ( y 1 , . . . , y i − 1 , A π y i , y i +1 , . . . , y k ) = R k ( y 1 , . . . , y k ) i =1 where ( y 1 , . . . , y k ) ∈ D ( A ) k and R k ∈ M ( D ( A ) k , R ). Questions: Why should we consider these equations? Which theoretical results do we have for these equations? How should we treat these equations numerically?

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Nonlinear ∞ -horizon control: R n Consider y ( t ) = Ay ( t ) + f ( y ( t )) + Bu ( t ) , ˙ y (0) = y 0 , y obs ( t ) = Cy ( t ) , A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , f : R n → R n , f (0) = 0 Associated minimal value function � ∞ � ∞ 1 � y obs ( t ) � 2 d t + β � u ( t ) � 2 d t . V ( y 0 ) = inf 2 2 u ∈ L 2 (0 , ∞ ; R m ) 0 0 If V sufficiently smooth, the Hamilton-Jacobi-Bellman equation ( Ay + f ( y )) ⊤ ∇V ( y ) + 1 2 � Cy � 2 − 1 2 β � B ⊤ ∇V ( y ) � 2 = 0 y ) = − 1 β B ⊤ ∇V (¯ yields optimal feedback law ¯ u (¯ y ).

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Taylor expansions – basic idea Idea: Expand V around 0 as follows V ( y ) = V (0) + D V (0)( y ) + 1 2! D 2 V (0)( y , y ) + 1 3! D 3 V (0)( y , y , y ) + . . . and approximate optimal feedback law via d � u d ( y ) = − 1 1 ( k − 1)! B ⊤ D j V (0)( · , y , . . . , y ) . β k =2 Finite-dimensional case: [Aguilar,Al’brekht,Cebuhar,Costanza,Garrard,Krener,Lukes,... ] Infinite-dimensional case: [Thevenet/Buchot/Raymond,B./Kunisch/Pfeiffer]

INSTITUTE OF MATHEMATICS Nonlinear infinite-horizon control using generalized Lyapunov equations AND SCIENTIFIC COMPUTING Taylor expansions – basic idea Idea: Expand V around 0 as follows V ( y ) = V (0) + D V (0)( y ) + 1 2! D 2 V (0)( y , y ) + 1 3! D 3 V (0)( y , y , y ) + . . . and approximate optimal feedback law via d � u d ( y ) = − 1 1 ( k − 1)! B ⊤ D j V (0)( · , y , . . . , y ) . β k =2 Finite-dimensional case: [Aguilar,Al’brekht,Cebuhar,Costanza,Garrard,Krener,Lukes,... ] Infinite-dimensional case: [Thevenet/Buchot/Raymond,B./Kunisch/Pfeiffer]