KUL guest presentation Florin Stoican Norwegian University of - PowerPoint PPT Presentation

KUL guest presentation Florin Stoican Norwegian University of Science and Technology (NTNU) - Department of Engineering Cybernetics Tuesday 3 rd July, 2012

Outline Set theoretic elements 1 Fault tolerant control based on set-theoretic methods 2 Description of non-convex regions 3 Remarks upon the structure of explicit MPC 4

Outline Set theoretic elements 1 Families of sets Invariance notions Zonotope applications Other issues Fault tolerant control based on set-theoretic methods 2 Description of non-convex regions 3 Remarks upon the structure of explicit MPC 4

Set theoretic elements Families of sets Families of sets – generalities Various families of sets in control: Issues to be considered: ellipsoids ( Kurzhanski˘ ı and Vályi [1997]) flexibility of the polytopes/zonotopes (Motzkin et al. [1959]) representation (B/L)MIs (Nesterov and Nemirovsky [1994]) numerical star-shaped sets (Rubinov and Yagubov [1986]) implementation 1 . 2 0 . 8 6 1 0 . 6 0 . 8 4 0 . 6 0 . 4 0 . 4 2 0 . 2 0 . 2 x 2 0 x 2 0 x 2 0 − 0 . 2 − 2 − 0 . 2 − 0 . 4 − 0 . 6 − 4 − 0 . 4 − 0 . 8 − 0 . 6 − 6 − 1 − 1 . 2 − 0 . 8 − 1 . 2 − 1 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 6 − 4 − 2 0 2 4 6 x 1 x 1 x 1 x T Qx ≤ γ Kern ( S ) � = ∅ G ( x ) ≤ 0 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 1 / 38

Set theoretic elements Families of sets Families of sets – generalities Various families of sets in control: Issues to be considered: ellipsoids ( Kurzhanski˘ ı and Vályi [1997]) flexibility of the polytopes/zonotopes (Motzkin et al. [1959]) representation (B/L)MIs (Nesterov and Nemirovsky [1994]) numerical star-shaped sets (Rubinov and Yagubov [1986]) implementation 2 . 5 0 . 8 6 2 0 . 6 4 1 . 5 0 . 4 2 1 0 . 2 x 2 0 x 2 0 . 5 x 2 0 − 2 0 − 0 . 2 − 0 . 5 − 4 − 0 . 4 − 1 − 0 . 6 − 6 − 1 . 5 − 0 . 8 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 − 2 . 5 − 2 − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 1 . 5 − 6 − 4 − 2 0 2 4 6 x 1 x 1 x 1 � A 0 + x i A i ≻ 0 Kern ( S ) � = ∅ G ( x ) ≤ 0 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 1 / 38

Set theoretic elements Families of sets Families of sets – polyhedral/zonotopic sets (more “structured”) Best compromise: polytopic(zonotopic) sets 6 Polyhedral sets: 4 dual representation 2 half-space: x 2 0 − 2 h i x ≤ k i , i = 1 . . . N h − 4 − 6 vertex: − 6 − 4 − 2 0 2 4 6 x 1 � � 6 α i v i , α i ≥ 0 , α i = 1 , i = 1 . . . N v 4 i i 2 efficient algorithms for set containment x 2 0 problems ( Gritzmann and Klee [1994]) − 2 can approximate any convex shape − 4 (Bronstein [2008]) − 6 − 6 − 4 − 2 0 2 4 6 x 1 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 2 / 38

Set theoretic elements Families of sets Families of sets – polyhedral/zonotopic sets (more “structured”) Best compromise: polytopic(zonotopic) sets Zonotopic sets: obtained as 10 hypercube projection Minkowski sum of generators 5 additional representation 2 generator form: 4 0 2 0 � − 2 − 4 − 2 λ i g i , | λ i | ≤ 1 , i = 1 . . . N g 1 . 5 i compact representation Fukuda: 1 0 . 5 � � d − 1 � � N g � N g − 1 x 2 0 N h = 2 · , N v = 2 n − 1 i − 0 . 5 i = 0 − 1 limited to symmetric objects − 1 . 5 − 1 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x 1 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 2 / 38

Set theoretic elements Invariance notions Invariance notions Consider a system in R n x + = f ( x , δ ) with disturbances bounded by the set ∆ ⊂ R n . Definition (RPI set) A set Ω is called robust positive invariant (RPI) iff f (Ω , ∆) ⊆ Ω . The minimal RPI set (which is contained in all the RPI sets) can be defined as: k →∞ f ( k ) ( 0 , ∆) . Ω ∞ = f ( f ( . . . , ∆) , ∆) = lim � �� � ∞ iterations Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 3 / 38

Set theoretic elements Invariance notions Invariance notions Consider a LTI system in R n x + = Ax + B δ with A a Schur matrix and disturbances bounded by the set ∆ ⊂ R n . Definition (RPI set) A set Ω is called robust positive invariant (RPI) iff A Ω ⊕ B ∆ ⊆ Ω . The minimal RPI set (which is contained in all the RPI sets) can be defined as: ∞ � A i B ∆ . Ω ∞ = i = 0 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 3 / 38

Set theoretic elements Invariance notions Invariance notions – exemplification RPI set mRPI set 8 8 6 A Ω ⊕ B ∆ 6 Ω ∞ = A Ω ∞ ⊕ B ∆ 4 4 2 2 x 2 0 x 2 0 − 2 − 2 − 4 − 4 − 6 Ω − 6 Ω − 8 − 8 − 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 − 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 x 1 x 1 ∞ � A Ω ⊕ B ∆ ⊆ Ω A i B ∆ Ω ∞ = i = 0 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 3 / 38

Set theoretic elements Zonotope applications Ultimate bounds Theorem (Ultimate bounds – Kofman et al. [2007]) For system x + = Ax + B δ with the Jordan decomposition A = V Λ V − 1 � � � ≤ ¯ and assuming that � δ δ we have that the set Ω UB ( ǫ ) is RPI. 8 | x | ≤ | V | b Particularities: 6 explicit linear formulations 4 2 “good” approximation of the mRPI x 2 0 set − 2 can be extended to various − 4 degenerate cases ( Haimovich et al. − 6 | V − 1 x | ≤ b − 8 [2008], Kofman et al. [2008]) − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 6 x 1 � � x : | V − 1 x | ≤ ( I − | Λ | ) − 1 | V − 1 B | ¯ Ω UB ( ǫ ) = δ + ǫ Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 4 / 38

Set theoretic elements Zonotope applications Ultimate bounds Theorem (Ultimate bounds – Kofman et al. [2007]) For system x + = Ax + B δ with the Jordan decomposition A = V Λ V − 1 � � � ≤ ¯ and assuming that � δ δ we have that the set Ω UB ( ǫ ) is RPI. 3 2 ∈ ∆ 1 , | δ 1 | ≤ ¯ δ 1 δ 1 ⇒ x 2 0 ∈ ∆ 2 , | δ 2 | ≤ ¯ δ 2 δ − 1 − 2 − 3 − 4 − 3 − 2 − 1 0 1 2 3 4 x 1 Sets with the same bounding box will give the same UBI set for a given dynamic. Improvement ( Stoican et al. [2011a]): use zonotopic sets for describing the disturbance. Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 4 / 38

Set theoretic elements Zonotope applications QP problem with zonotopic bounds 1 2 u T Hu + x T min 0 Fu u Gu ≤ W + Ex 0 . s.t. z y x 1 2 λ T ˜ 0 ˜ H λ + x T min F λ 1 2 ( z a ) T H a z a + x T λ 0 F a z a min | λ | ≤ 1 . s.t. z a , z b | V a z a + V b z b | ≤ 1 . s.t. Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 5 / 38

Set theoretic elements Zonotope applications QP problem with zonotopic bounds 1 2 u T Hu + x T min 0 Fu u Gu ≤ W + Ex 0 . s.t. z y x 1 2 λ T ˜ 0 ˜ H λ + x T min F λ 1 2 ( z a ) T H a z a + x T λ 0 F a z a min | λ | ≤ 1 . s.t. z a , z b | V a z a + V b z b | ≤ 1 . s.t. Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 5 / 38

Set theoretic elements Other issues Other set theoretic topics 3 2 set separation between sets 1 x 2 through a separating hyperplane 0 through a barrier function − 1 − 2 − 6 − 4 − 2 0 2 4 6 8 10 12 x 1 10 8 6 4 2 upper bound for the inclusion time x 2 0 − 2 particular bounds for a given attractive set − 4 − 6 − 8 − 10 − 10 − 8 − 6 − 4 − 2 0 2 4 6 8 10 x 1 1 . 5 1 RPI description for particular dynamics 0 . 5 x 2 0 switched/with delay − 0 . 5 cyclic invariance − 1 − 1 . 5 − 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5 x 1 Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 6 / 38

Outline Set theoretic elements 1 Fault tolerant control based on set-theoretic methods 2 Problem statement FDI mechanism RC strategies Extensions Description of non-convex regions 3 Remarks upon the structure of explicit MPC 4

Fault tolerant control based on set-theoretic methods The need for FTC in control applications Bhopal chemical spill Flight 1862 crash (~4000 casualties) (43 casualties) Fukushima meltdown BP oil spill (~40 km exclusion zone) (~60000 barrels/day) Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 7 / 38

Fault tolerant control based on set-theoretic methods Fault tolerant control requirements 2 0 x 2 − 2 − 4 − 6 − 4 − 2 0 2 x 1 50 40 45 35 25 30 20 10 15 4 − 5 5 0 t Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 8 / 38

Fault tolerant control based on set-theoretic methods FTC generalities Legend u = inputs w = disturbances r = references v = noise Fault Detection z = tracking error and Isolation (FDI) Actuator System Sensor w v Faults Faults Faults Control Reconfigurable r u System z (Reference) Feedforward Actuators Sensors Governor Controller - Reconfiguration Mechanism Reconfigurable Feedback Controller FTC characterization FDI directions passive (robust control) stochastic (Kalman filters, active (adaptive control) sensor fusion) FDI and RC blocks artificial intelligence link and reciprocal influences set theoretic methods between FDI and RC Tuesday 3rd July, 2012 Florin Stoican KUL guest presentation 9 / 38