The bang-bang funnel controller Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany Arbeitstreffen SPP 1305 “Event based control”, M¨ unchen 1. Oktober 2012
Introduction Relative degree one case Higher relative degree Content Introduction 1 Relative degree one case 2 Higher relative degree 3 Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Feedback loop x = F ( x , u ) ˙ u y y = H ( x ) q Switching e + − y ref logic U − U + Funnel Reference signal y ref : R ≥ 0 → R suficiently smooth Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree The funnel Control objective Error e := y − y ref evolves within funnel F = F ( ϕ − , ϕ + ) := { ( t , e ) | ϕ − ( t ) ≤ e ≤ ϕ + ( t ) } where ϕ ± : R ≥ 0 → R sufficiently smooth time-varying strict error bound ϕ + ( t ) transient behaviour practical tracking t F ( | e ( t ) | < λ for t >> 0) ϕ − ( t ) Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree The bang-bang funnel controller Continuous Funnel Controller: Introduced by Ilchmann et al. in 2002 New approach Achieve control objectives with bang-bang control, i.e. u ( t ) ∈ { U − , U + } x = F ( x , u ) ˙ u y y = H ( x ) q Switching e − y ref + logic U − U + Funnel Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Relative degree one Definition (Relative degree one) > 0 � �� � x = F ( x , u ) ˙ y = f ( y , z ) + ˙ g ( y , z ) u ∼ = y = H ( x ) z = h ( y , z ) ˙ Structural assumption f , g , h can be unknown feasibility assumption (later) in terms of f , g , h and funnel Important property u ( t ) << 0 ⇒ y ( t ) << 0 ˙ u ( t ) >> 0 ⇒ y ( t ) >> 0 ˙ Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Switching logic e ( t ) e (0) ϕ + ( t ) t F ϕ − ( t ) u ( t ) = U + u ( t ) = U − u ( t ) = U + Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Feasibility assumptions y = f ( y , z ) + g ( y , z ) u , ˙ y 0 ∈ R z 0 ∈ Z 0 ⊆ R n − 1 z = h ( y , z ) , ˙ z : [0 , t ] → R n − 1 solves ˙ � z = h ( y , z ) for some � � z 0 ∈ Z 0 and for some y : [0 , t ] → R � � Z t := z ( t ) . � with ϕ − ( τ ) ≤ y ( τ ) − y ref ( τ ) ≤ ϕ + ( τ ) � � � ∀ τ ∈ [0 , t ] Feasibility assumption U − < ˙ ϕ + ( t ) + ˙ y ref ( t ) − f ( y ref ( t ) + ϕ + ( t ) , z t ) g ( y ref ( t ) + ϕ + ( t ) , z t ) ∀ t ≥ 0 ∀ z t ∈ Z t : U + > ˙ ϕ − ( t ) + ˙ y ref ( t ) − f ( y ref ( t ) + ϕ − ( t ) , z t ) g ( y ref ( t ) + ϕ − ( t ) , z t ) Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Main result relative degree one Theorem (Bang-bang funnel controller, Liberzon & T. 2010) Relative degree one & Funnel & simple switching logic & Feasibility ⇒ Bang-bang funnel controller works: existence and uniqueness of global solution error remains within funnel for all time no zeno behaviour Necessary knowledge: for controller implementation: relative degree (one) signals: error e ( t ) and funnel boundaries ϕ ± ( t ) to check feasibility: bounds on zero dynamics bounds on f and g bounds on y ref and ˙ y ref bounds on the funnel Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Content Introduction 1 Relative degree one case 2 Higher relative degree 3 Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Relative degree r Definition (Relative degree r ) > 0 � �� � = y ( r ) = f ( y , ˙ x = F ( x , u ) ˙ y , . . . , y ( r − 1) , z ) + g ( y , . . . , y ( r − 1) , z ) u ∼ y = H ( x ) y , . . . , y ( r − 1) , z ) z = h ( y , ˙ ˙ Essential property y ( r ) ( t ) << 0 u ( t ) << 0 ⇒ y ( r ) ( t ) >> 0 u ( t ) >> 0 ⇒ Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Hirachical structure of switching logic x = F ( x , u ) ˙ u y y = H ( x ) q Switching e − y ref + logic U − U + Funnels d d d d d t d t d t d t e · · · e ˙ e ( r − 2) e ( r − 1) q 1 q 2 q r − 2 q r − 1 · · · B r − 2 B r − 1 q B 0 B 1 ψ 1 ψ 2 ψ r − 2 ψ r − 1 ϕ + ϕ + r − 1 ( t ) r − 2 ( t ) ϕ + 1 ( t ) ϕ + 0 ( t ) F 1 F r − 2 F r − 1 F 0 ϕ − 0 ( t ) ϕ − 1 ( t ) ϕ − r − 2 ( t ) ϕ − r − 1 ( t ) Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Desired behaviour of block B i ϕ + i ( t ) e ( i ) ( t ) ϕ + i ( t ) − ε + i ≤ ∆+ i F i λ + i max { ψ i ( t ) , λ + i } t min { ψ i ( t ) , − λ − i } − λ − i min { ψ i ( t ) , − λ − i } ≤ ∆ − i ϕ − i ( t ) + ε − i ϕ − i ( t ) ≤ ∆ − i q i ( t ) = true q i ( t ) = false q i ( t ) = true Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Definition of the swichting logic e ( i ) Goal of block B i : � make e ( i ) smaller q i q i +1 q i = true ⇒ than min { ψ i , − λ − B i i } , � make e ( i ) bigger ψ i ψ i +1 q i = false ⇒ than max { ψ i , λ + i } ϕ + i , ϕ − i , ε + i , ε − i , λ + i , λ − i q 1 = true q 1 = false e ( i ) ( t ) ≤ ϕ − i ( t ) + ε − e ( i ) ( t ) ≤ max { ψ i ( t ) , λ + i } + ε + i i q i +1 = true q i +1 = false q i +1 = true q i +1 = false ψ i +1 = ˙ ϕ − ϕ + ψ i +1 = ˙ ψ i +1 = ˙ ψ i ψ i +1 = ˙ ψ i i i e ( t ) ≥ min { ψ i ( t ) , − λ − i } − ε + e ( t ) ≥ ϕ + i ( t ) − ε + i i Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Illustration of switching logic ≤ ∆ − i +1 ϕ + i ( t ) e ( i ) (0) d t e ( i ) ≤ − λ − d ϕ + i ( t ) − ε + i +1 i F i λ + i max { ψ i ( t ) , λ + i } t min { ψ i ( t ) , − λ − i } − λ − i min { ψ i ( t ) , − λ − i } ϕ − i ( t ) + ε − ≤ ∆ + i ϕ − i ( t ) i ≤ ∆ − i q i +1 ( t ) = true q i +1 ( t ) = false q i +1 ( t ) = true q i +1 ( t ) = false q i +1 ( t ) = true q i +1 ( t ) = false q i +1 ( t ) = true q i ( t ) = true q i ( t ) = false q i ( t ) = true Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
Introduction Relative degree one case Higher relative degree Main result Theorem (Bang-bang funnel controller works, Liberzon & T. 2012) Feasibility assumptions: structural assumptions relative degree r smoothness and boundedness of y ref funnels feasible initial error values contained within funnels sufficently smooth funnel boundaries funnel boundaries large enough settling times and safety distance compatible U + and U − large enough ⇒ bang-bang funnel controller works. Theorem (Feasibility) Mild assumptions on F 0 + BIBO of zero dynamics + boundedness of y ref ⇒ feasibility assumption satisfiable with sufficiently large U + and U − Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller
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