The bang-bang funnel controller Daniel Liberzon and Stephan Trenn - PowerPoint PPT Presentation

The bang-bang funnel controller Daniel Liberzon and Stephan Trenn 49th IEEE Conference on Decision and Control Wednesday, December 15, 2010, 11:20–11:40, Atlanta, USA

Introduction Relative degree one case Relative degree two case Simulations Conclusions Content Introduction 1 Relative degree one case 2 Relative degree two case 3 Simulations 4 Conclusions 5 The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions 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 absolutely continuous The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions 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 absolutely continuous time-varying strict error bound ϕ + ( t ) transient behaviour practical tracking t F ( | e ( t ) | < λ for t >> 0 ) ϕ − ( t ) The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions 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 The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions 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 ˙ The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Switching logic e ( t ) e (0) ϕ + ( t ) t F ϕ − ( t ) u ( t ) = U + u ( t ) = U − u ( t ) = U + The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Switching logic e ( t ) ≤ ϕ − ( t ) e ( t ) > ϕ − ( t ) u ( t ) = U − u ( t ) = U + e ( t ) < ϕ + ( t ) e ( t ) ≥ ϕ + ( t ) Too simple? ⇒ Feasibility assumptions The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions 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 ) The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Main result relative degree one Theorem (Bang-bang funnel controller) 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 The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Content Introduction 1 Relative degree one case 2 Relative degree two case 3 Simulations 4 Conclusions 5 The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Relative degree two Definition (Relative degree two) > 0 � �� � x = F ( x, u ) ˙ ¨ y = f ( y, ˙ y, z ) + g ( y, ˙ y, z ) u ∼ = y = H ( x ) z = h ( y, ˙ ˙ y, z ) Important property u ( t ) << 0 ⇒ y ( t ) << 0 ¨ u ( t ) >> 0 ⇒ y ( t ) >> 0 ¨ The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Feedback loop x = F ( x, u ) ˙ u y y = H ( x ) q e , ˙ e Switching − y ref + logic U − U + Funnels The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions The switching logic e ( t ) decrease e e ( t ) ≤ ϕ d e ( t ) ≤ ϕ d ˙ ˙ − ( t ) − ( t ) ϕ + ( t ) U − U + t F ϕ − ( t ) e ( t ) ≥ ˙ e ( t ) ≥ ˙ ˙ ˙ ϕ + ( t ) ϕ + ( t ) decrease e increase e decrease e e ( t ) ≤ ϕ − ( t ) + ε + e ( t ) ≤ ϕ − ( t ) + ε + e ( t ) ≥ ϕ + ( t ) − ε + e ( t ) ≥ ϕ + ( t ) − ε + e ( t ) ˙ e ( t ) ≥ ϕ d ˙ + ( t ) ϕ − ( t ) ˙ ϕ d + ( t ) t U + U − F d ϕ d − ( t ) e ( t ) ≤ ˙ ˙ ϕ − ( t ) ϕ + ( t ) ˙ increase e The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Feasibility assumptions Funnels F ( ϕ + , ϕ − ) , F d ( ϕ d + , ϕ d − ) Safety distances ε + , ε − > 0 Feasibility of funnels ∀ t ≥ 0 : ε + < ϕ + ( t ) and ε − < ϕ − ( t ) ϕ d ϕ d ∀ t ≥ 0 : + ( t ) > ˙ ϕ − ( t ) and − ( t ) < ˙ ϕ + ( t ) y = f ( y, ˙ ¨ y, z ) + g ( y, ˙ y, z ) u z = h ( y, ˙ ˙ y, z ) Z t := { z ( t ) | z solves ˙ z = h ( y, ˙ y, z ) , z (0) ∈ Z 0 } Choose δ ± > 0 such that ϕ d δ + > max { ˙ − ( t ) , ¨ ϕ − ( t ) } and ϕ d − δ − < min { ˙ + ( t ) , ¨ ϕ + ( t ) } ∀ t ≥ 0 The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Feasibility assumptions Feasibility assumption 1 U − < − δ − + ¨ y ref ( t ) + f ( y t , ˙ y t , z t ) , g ( y t , ˙ y t , z t ) U + > δ + + ¨ y ref ( t ) + f ( y t , ˙ y t , z t ) , g ( y t , ˙ y t , z t ) ∀ t ≥ 0 , ∀ y t ∈ [ y ref ( t ) + ϕ − ( t ) , y ref ( t ) + ϕ + ( t )] , y ref ( t ) + ϕ d y ref ( t ) + ϕ d ∀ ˙ y t ∈ [ ˙ − ( t ) , ˙ + ( t )] , ∀ z t ∈ Z t Feasibility assumption 2 ε + ≥ ( � ϕ d − � + � min { ˙ ϕ + , 0 }� ) 2 2 δ − ε − ≥ ( � ϕ d ϕ − , 0 }� ) 2 + � + � max { ˙ 2 δ + The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Main result relative degree two Theorem (Bang-bang funnel controller) Relative degree two & Funnels & simple switching logic & Feasibility ⇒ Bang-bang funnel controller works: existence and uniqueness of global solution error and its derivative remain within funnels for all time no zeno behaviour The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Content Introduction 1 Relative degree one case 2 Relative degree two case 3 Simulations 4 Conclusions 5 The bang-bang funnel controller Daniel Liberzon and Stephan Trenn

Introduction Relative degree one case Relative degree two case Simulations Conclusions Model of exothermic chemical reactions Model from [Ilchmann & T. 2004]: 340 320 y = br ( z 1 , z 2 , y ) − qy + u, ˙ 300 output y ( t ) y ( t ) y ( t ) y ∗ y ∗ z 1 = c 1 r ( z 1 , z 2 , y ) + d ( z in 280 ˙ 1 − z 1 ) , Funnel Funnel 260 z 2 = c 2 r ( z 1 , z 2 , y ) + d ( z in ˙ 2 − z 2 ) , 240 0 0.5 1 1.5 2 2.5 3 time t b ≥ 0 , q > 0 , c 1 < 0 , c 2 ∈ R , d > 0 , 600 z in 1 / 2 ≥ 0 500 r : R ≥ 0 × R ≥ 0 × R > 0 → R ≥ 0 locally 400 input u ( t ) Lipschitz with r (0 , 0 , y ) = 0 ∀ y > 0 300 y ref = y ∗ > 0 200 0 0.5 1 1.5 2 2.5 3 time t Feasibility assumptions from [IT 2004] imply feasibility for bang-bang funnel controller if ϕ + ( t ) ∈ (0 , y − y ∗ ] , ϕ − ( t ) ∈ ( − y ∗ , 0) , ϕ + ( t ) > − ρ − , ˙ ϕ − ( t ) < ρ + , ˙ The bang-bang funnel controller Daniel Liberzon and Stephan Trenn