Funnel control in mechatronics: An overview

Transcription

1 Funnel control in mechatronics: An overview Position funnel control of stiff industrial servo-systems C.M. Hackl 1, A.G. Hofmann 2 and R.M. Kennel 1 1 Institute for Electrical Drive Systems and Power Electronics (EAL), Technische Universität München (TUM), Germany 2 Institute for Power Electronics and Electrical Drives (ISEA), RTWH Aachen University, Germany 5th IEEE Conference on Decision and Control and European Control Conference Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

2 Outline 1 Motivation: High-gain control 2 Funnel control (relative-degree-two systems) Admissible system class Control objective Control law 3 Application: Position control of stiff industrial servo-systems 4 Conclusion Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

3 Motivation: High-gain control controller k(y +k D ẏ) u F 2 (s) = system (s+5) (s 1) 2 (s+1) y ẏ k D structural properties of F 2 (s) relative degree (pole excess): r = 2 positive high-frequency gain (lim s s 2 F 1 (s) = 1) minimum-phase (numerator is Hurwitz) imaginary axis Root-locus ( poles, zeros) k D = k D = 1/ real axis Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

4 Funnel control (r = 2): Admissible system class Only structural system knowledge necessary: relative degree two positive (or known sign of) high-frequency gain stable zero-dynamics (minimum-phase in the LTI case) (derivative feedback admissible) Example: LTI SISO system of form with d dt x(t) = Ax(t)+bu(t), x() = x R n y(t) = c x(t) c b = and c Ab (relative degree two) c Ab > (positive high-frequency gain) ([ ]) sin A b det c for all s C with R(s) (minimum-phase) Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

5 Funnel control (r = 2): Control objective tracking with prescribed transient accuracy of y ref ( ) C 1 L (R ;R) ψ () ψ 1 () ψ 1 (t) ψ (t) e() e(t) ψ ( ) λ 1 e( ) ė(t) t t λ ė( ) funnel ψ () ė() ψ 1 () with tracking error e(t) = y ref (t) y(t) ψ 1 ( ) funnel boundary design: ψi( ) absolutely continuous and ψ i(t) λ i > for all t and i {,1}; e() < ψ() and ė() < ψ 1() ψ1(t) d ψ(t)+δ for all t dt Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

6 Funnel control (r = 2): Control law ψ () ψ (t) ψ 1 () e() e(t) ψ ( ) λ 1 e( ) λ t ė( ) t funnel ψ () ė() ψ 1 () ψ 1(t) d dt ψ(t) + δ ė(t) ψ 1 ( ) funnel controller (with derivative feedback) ( u(t) = k (t) 2 e(t)+ k ) 1(t) k (t)ė(t), e(t) = y ref (t) y(t) with time-varying gains k (t) = s (t) ψ (t) e(t) and k 1 (t) = s 1 (t) ψ 1 (t) ė(t) scaling functions s i ( ) C L (R ;[m i, )) where m i >, i {,1} Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

7 Application: Stiff industrial servo-systems drive (m M ) load (m L ) Θ {}}{ drive torque m M ( ) C(R ;R) [Nm] (control input) load torque m L ( ) L (R ;R) [Nm] (disturbance) inertia Θ > [ kgm 2] state x = (φ, ω) : position φ [rad], speed ω [rad/s] friction (on drive & load side) unbounded viscous friction with coefficients ν1,ν 2 [Nms/rad] bounded dynamic friction F1,F 2 : C(R ;R) L (R ;R) [Nm] gear with ratio g r R\{} [1] (neglecting dynamics & backlash) signals available for feedback: position φ and speed ω = φ (deteriorated by n m ( ) W 2, (R ;R) & ṅ m ( ), resp.) Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

8 Application: Model of stiff industrial servo-systems LTI SISO system with input disturbance and nonlinear perturbation d dt =:x(t) ({}} ){ φ(t) ω(t) = =:A {[ }}{ =:b ] (φ(t) ) ({}}){ =:u(t) 1 ( {}}{ + ν1+ν2/g2 r 1 m ω(t) M (t) (F 1 ω)(t) ) Θ Θ ) (ml ω ( 1 (t)+(f 2 g r )(t) ) g rθ y(t) = ( 1 ) x(t), (φ(), }{{} ω()) = (φ, ω ) =:c Structural properties of model: c b = and c Ab = 1/Θ > relative degree two and positive high-frequency gain det ([ sin A b c ]) for all s C with R(s) (unperturbed system is minimum-phase) and m L ( ) L (R ;R), F 1,F 2 : C(R ;R) L (R ;R) (bounded disturbance & perturbations) stable zero-dynamics Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

9 Application: Position control implementation implementation in xpc target C 1 y ref = φ ref ẏ ref = ω ref e ė funnel controller u laboratory setup with sensor(s) φ ω ẏ = ω +ṅ m y = φ+n m ṅ m n m Reference 2 Load 12 1 [rad], [rad/s] 1 [Nm] φ ref ( ) ω ref ( ) m L( ) Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

10 Application: Position control measurement results Position error Speed error 2 e( ) ±ψ ( ) 1 5 ė( ) ±ψ 1( ) [rad] [rad/s] 2 5 Gains k ( ) k 1( ) Torque [Nm] m M( ) m L( ) Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

11 Conclusion funnel control applicable for position control for application only structural proporties must be checked system identification or parameter estimation is not required time-varying gains (not monotone) tracking with prescribed transient accuracy independent of system parameters (robustness) Some more results (not presented) funnel control applicable for speed control steady state accuracy is feasible in conjunction with PI-like extension funnel control is also applicable for speed and position control of elastic servo-systems (two mass-systems) position funnel control of rigid revolute joint robotic manipulators is feasible (if inertia matrix is known) funnel control in presence of actuator saturation is feasible (conservative feasibility condition must be satisfied) Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12

12 References C. M. Hackl. High-gain adaptive position control. International Journal of Control, 84(1): , 211. C. M. Hackl, N. Hopfe, A. Ilchmann, M. Mueller, and S. Trenn. Funnel control for systems with relative degree two. accepted for publication in SIAM Journal on Control and Optimization (preprint available at the author), 21. A. Ilchmann, E. P. Ryan, and C. J. Sangwin. Tracking with prescribed transient behaviour. ESAIM: Control, Optimisation and Calculus of Variations, 7: , 22. Christoph Hackl ( ) Funnel control in mechatronics: An overview December 15, / 12