Plant tuning: a robust Lyapunov approach Franco Blanchini 1 , - PowerPoint PPT Presentation

Plant tuning: a robust Lyapunov approach Franco Blanchini 1 , Gianfranco Fenu 2 , Giulia Giordano 3 , Felice Andrea Pellegrino 2 1 Dipartimento di Scienze Matematiche, Informatiche e Fisiche University of Udine, Italy; 2 Dipartimento di Ingegneria e Architettura University of Trieste, Italy 3 Linnaeus Center and Department of Automatic Control University of Lund, Sweden January 10, 2017 pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Uumidity-temperature regulation T ω h I T = φ ( I , ω ) , ψ ( I , ω ) , h = pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Uumidity-temperature regulation We know that pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Uumidity-temperature regulation We know that the temperature T is a decreasing function of the fan speed ω and an increasing function of the current I ; pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Uumidity-temperature regulation We know that the temperature T is a decreasing function of the fan speed ω and an increasing function of the current I ; the relative humidity h is a decreasing function of both the fan speed ω and the current I . pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Uumidity-temperature regulation We know that the temperature T is a decreasing function of the fan speed ω and an increasing function of the current I ; the relative humidity h is a decreasing function of both the fan speed ω and the current I . Available information: � + m 1 � ∂ T ∂ T � � − m 2 ∂ I ∂ω ∈ , ∂ h ∂ h − m 3 − m 4 ∂ I ∂ω where m 1 , m 2 , m 3 , m 4 ∈ [ ε , µ ]. pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Plant tuning without a model u y g TUNER pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Plant tuning without a model u y g TUNER � � r r ∂ g ∑ ∑ α i M i , α i = 1 , α i ≥ 0 ∂ u ∈ M = M = i =1 i =1 pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Problem formulation pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Problem formulation Problem Given the static plant y = g ( u ) , where g : R q → R p , p ≤ q, assume that, for some (unknown) ¯ u, 0 = g (¯ u ) . and that the following inclusion holds: � ∂ g � G u . = ∈ M . ∂ u where M is a polytopic set. Find a dynamic algorithm such that y ( t ) → 0 and u ( t ) → ¯ u pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Main result pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Main result Assumption Robust non–singularity. Any matrix in the polytope M is right invertible. pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Main result Assumption Robust non–singularity. Any matrix in the polytope M is right invertible. Theorem Under Assumption 2, Problem 1 can be solved with a control scheme of the form u ( t ) ˙ = v ( t ) , v ( t ) = Φ( y ( t )) , with added input variable v ( t ) . pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

A no–solution Observation The dynamics of the output y can be described by y = G u ˙ ˙ u = G u v pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

A no–solution Observation The dynamics of the output y can be described by y = G u ˙ ˙ u = G u v This would be a driftless system if G u were known. pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

A no–solution Observation The dynamics of the output y can be described by y = G u ˙ ˙ u = G u v This would be a driftless system if G u were known. Take v = − G − 1 u y, then y = − y ˙ and y ( t ) → 0 exponentially (Newton’s method). pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

A no–solution Observation The dynamics of the output y can be described by y = G u ˙ ˙ u = G u v This would be a driftless system if G u were known. Take v = − G − 1 u y, then y = − y ˙ and y ( t ) → 0 exponentially (Newton’s method). u ( t ) converges to ¯ u such that G (¯ u ) = 0 . pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Cheating and regretting .... pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Cheating and regretting .... Consider the Lyapunov like function V ( y ) = 1 2 y ⊤ y , V = y ⊤ ˙ ˙ y = y ⊤ G u ˙ u = y ⊤ G u v . pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Cheating and regretting .... Consider the Lyapunov like function V ( y ) = 1 2 y ⊤ y , V = y ⊤ ˙ ˙ y = y ⊤ G u ˙ u = y ⊤ G u v . Take the “fake” control v = − γ ( y ) G − 1 u y V = − γ ( y ) y ⊤ y < 0 , ˙ for y � = 0 ( ∗ ) pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Cheating and regretting .... Consider the Lyapunov like function V ( y ) = 1 2 y ⊤ y , V = y ⊤ ˙ ˙ y = y ⊤ G u ˙ u = y ⊤ G u v . Take the “fake” control v = − γ ( y ) G − 1 u y V = − γ ( y ) y ⊤ y < 0 , ˙ for y � = 0 ( ∗ ) Claim There exists � v ( y , G u ) � = � γ ( y ) G − 1 u y � ≤ ξ such that (*) holds. pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control) : chooses � v � ≤ ξ to minimize ˙ V ; pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control) : chooses � v � ≤ ξ to minimize ˙ V ; Bad player (misfortune) : chooses G u ∈ M to maximize ˙ V . pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control) : chooses � v � ≤ ξ to minimize ˙ V ; Bad player (misfortune) : chooses G u ∈ M to maximize ˙ V . µ + = min G u ∈ M y ⊤ G u v � v �≤ ξ max pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control) : chooses � v � ≤ ξ to minimize ˙ V ; Bad player (misfortune) : chooses G u ∈ M to maximize ˙ V . µ + = min G u ∈ M y ⊤ G u v � v �≤ ξ max µ − = max � v �≤ ξ y ⊤ G u v G u ∈ M min pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control) : chooses � v � ≤ ξ to minimize ˙ V ; Bad player (misfortune) : chooses G u ∈ M to maximize ˙ V . µ + = min G u ∈ M y ⊤ G u v � v �≤ ξ max µ − = max � v �≤ ξ y ⊤ G u v G u ∈ M min This game has a saddle–point µ − = µ + = y ⊤ M ∗ ( y ) v ∗ ( y ) pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Sketch of proof p = q Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control) : chooses � v � ≤ ξ to minimize ˙ V ; Bad player (misfortune) : chooses G u ∈ M to maximize ˙ V . µ + = min G u ∈ M y ⊤ G u v � v �≤ ξ max µ − = max � v �≤ ξ y ⊤ G u v G u ∈ M min This game has a saddle–point µ − = µ + = y ⊤ M ∗ ( y ) v ∗ ( y ) Then v = Φ( y ) = v ∗ ( y ) pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Facts pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach

Facts The computation of v ∗ requires solving a convex optimization problem on–line with domain M . pollo F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino Plant tuning: a robust Lyapunov approach