The bang-bang funnel controller: time delays and case study Stephan Trenn (joint work with Daniel Liberzon, UIUC) Technomathematics group, University of Kaiserslautern, Germany 12th European Control Conference (ECC’13) Thursday, 18.07.2013, ThA5.3, 10:20
Introduction Time delays in feedback loop Simulations Content Introduction 1 Time delays in feedback loop 2 Simulations 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Control setup x = F ( x , u ) ˙ u y y = H ( x ) q Switching e y ref − logic U − U + Funnel Goal: Tracking with prespecified error bounds for uncertain system with only two control values Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations 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 > 0 time-varying strict error bound transient behaviour ϕ + ( t ) practical tracking ( | e ( t ) | < λ for t >> 0) t F proposed by Ilchmann ϕ − ( t ) et al. 2002 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations The switching logic (CDC 2010) e ( t ) decrease e e ( t ) ≤ ϕ − e ( t ) ≤ ϕ − ˙ ˙ 1 ( t ) 1 ( t ) ϕ + 0 ( t ) U − U + t F 0 ϕ − 0 ( t ) ϕ + ϕ + e ( t ) ≥ ˙ e ( t ) ≥ ˙ ˙ ˙ 0 ( t ) 0 ( t ) decrease e increase e decrease e 0 ( t ) + ε + 0 ( t ) + ε + e ( t ) ≥ ϕ + e ( t ) ≥ ϕ + 0 ( t ) − ε + 0 ( t ) − ε + e ( t ) ˙ e ( t ) ≤ ϕ − e ( t ) ≤ ϕ − 0 0 0 0 e ( t ) ≥ ϕ + ˙ 1 ( t ) ϕ − ˙ 0 ( t ) ϕ + 1 ( t ) t U + U − F 1 ϕ − 1 ( t ) e ( t ) ≤ ˙ ˙ 0 ( t ) ϕ − ϕ + ˙ 0 ( t ) increase e Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Theoretical result Structural assumption and feasibility u ( t ) << 0 ⇒ y ( t ) << 0 ¨ Relative degree two: u ( t ) >> 0 ⇒ y ( t ) >> 0 ¨ feasibility of funnels input values large enough Theorem (CDC 2010) 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 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations More realistic scenario x = F ( x ) + G ( x ) u ˙ u y y = H ( x ) Switching τ q τ e y ref − logic q ( t − τ q ) q ( t ) e ( t − τ e ) e ( t ) e ( t − τ e ) ˙ e ( t ) ˙ U − U + Funnel Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Adjusted switching logic and new feasibility assumption e ( t ) Same switching logic Apart from introduction of safety distance ε ± 1 also for the derivative ϕ + 0 ( t ) funnel t F 0 ϕ − 0 ( t ) New feasibility assumption Bounding the time delay τ e + τ q in q 1 = true q 1 = false q 1 = true decrease e increase e decrease e e ( t ) ˙ terms of safety distances ε ± 1 and ε ± 0 . Theorem ϕ − ϕ + ˙ 0 ( t ) 1 ( t ) Bang-bang funnel controller also t F 1 works in the presence of sufficiently ϕ − 1 ( t ) small time delays. ϕ + ˙ 0 ( t ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Physical background for simulation � 0 x ( t ) = [ 0 1 � � � ˙ 0 0 ] x ( t ) + u ( t ) + u L ( t ) − ( Tx 2 )( t ) , γ � 1 0 � y ( t ) = x ( t ) , x 1 : angle of the rotary machine x 2 = ˙ x 1 : angular velocity u L : unknown load torque T : C ( R ≥ 0 → R ) → L ∞ loc ( R p → R ) friction operator Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Tracking of given reference trajectory y ( t ) 40 35 30 25 20 15 10 5 0 t − 5 0 5 10 15 20 25 30 35 40 Feasibility conditions too conservative simulation carried out with U ± = ± 2425 Nm much larger than technical possible ( ± 22 Nm ) switching frequency (about 10 4 Hz ) too high Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Heuristic improvement Underlying problem good long-time accuracy ⇒ small safety distance large error-tolerance ⇒ need large safety distance Use time-varying safety distances works very well in simulations switching logic remains the same formal proof even more technical and not carried out yet Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
Introduction Time delays in feedback loop Simulations Summary Introduced new controller design: Bang-bang funnel controller Design only depends on relative degree extremely simple Feasibility assumptions U + , U − must be large enough in terms of bounds on systems dynamics better performance ⇒ larger values for U + , U − Tolerates time delays Higher relative degree (not presented here) Switching logic remains simple (hierarchically) Feasibility assumptions remain similar Switching frequency increase significantly for details see: Liberzon & Trenn , IEEE TAC 2013 (to appear) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany The bang-bang funnel controller: time delays and case study
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