Non-identifier based adaptive control in mechatronics 3.1 High-gain adaptive stabilization and & 3.2 High-gain adaptive tracking

Transcription

1 Non-identifier based adaptive control in mechatronics 3.1 High-gain adaptive stabilization and & 3. High-gain adaptive tracking Christoph Hackl Munich School of Engineering (MSE) Research group Control of renewable energy systems (CRES) Lecture & tutorial C. Hackl Non-identifier based adaptive control in mechatronics 1/15

2 Introduction Schedule (tentative) Date Content Introduction and. Non-identifier based speed control (relative-degree-one case).1 High-gain adaptive stabilization High-gain adaptive tracking (using internal models) and.3 Adaptive λ-tracking control and funnel control Tutorials for Lectures canceled (Christi Himmelfahrt) Practical course (relative-degree-one case) Practical course (relative-degree-one case) [contd.] Practical course (relative-degree-one case) [contd.] &.3 Proofs of high-gain adaptive stabilization & funnel control Applications: Speed control of electrical drives (and some new results) 3. Non-identifier based adaptive position control (relative-degree-two case) High-gain adaptive stabilization and & 3. High-gain adaptive tracking Adaptive λ-tracking control and funnel control (with derivative feedback) Practical course (relative-degree-two case) Application: Position funnel control of servo-systems & industrial robots Conclusions & exam revision C. Hackl Non-identifier based adaptive control in mechatronics /15

3 Outline 3 Non-identifier based adaptive position control 3.1 Motivation 3. High-gain adaptive control with derivative feedback 3.3 High-gain adaptive tracking with internal model C. Hackl Non-identifier based adaptive control in mechatronics 3/15

4 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 3/15

5 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 3/15

6 Motivation: Position control of stiff servo-systems Stiff servo-system (1MS) Drive (m M ) Θ Load (m L ) hkkkkkkkikkkkkkkj φ, ω 1{g r m L ν ω g r ` F ω g r 1MS ν 1 ω ` F 1 ω u actuator ω φ 9 k A 1{Θ 1{g m M r satûa 9ω φ φ{g r u A ω m φ m 9n m sensor(s) n m C. Hackl Non-identifier based adaptive control in mechatronics 4/15

7 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 4/15

8 Motivation: Position control of stiff servo-systems Structural properties of stiff servo-system (1MS) d dt :xptq hkkkikkkj ˆφptq ωptq hkkkkkkkkkikkkkkkkkkj :A «0 1 0 ν1`ν {g r ˆ Θ 0 1 Θ ff ˆφptq ωptq ` hkikj :b ˆ 0 k AΘ sat pua `uptq ` ua ptq pf 1 ωqptq ` 1 g r `ml ptq ` pf ω g r qptq yptq `1 loomoon 0 xptq, pφp0q, ωp0qq :c J c J b 0 and c J Ab k A {Θ ą 0 ñ relative degree two and positive high-frequency gain! ı sin A b det 0 for all s P C with Rpsq ě 0 c J 0 J pφ 0, ω 0 q J ñ unperturbed system is minimum-phase! and m L p q P L 8 pr ě0 ; Rq, F 1, F : CpR ě0 ; Rq Ñ L 8 pr ě0 ; Rq (bounded disturbance & perturbations) ñ input-to-state stable zero-dynamics!, /. /- (1MS φ ) C. Hackl Non-identifier based adaptive control in mechatronics 5/15

9 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 5/15

10 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 5/15

11 High-gain adaptive control with derivative feedback Obstacle of higher relative degrees r=1 Ipsq r= α 1 π α 1 π Ipsq Rpsq Rpsq α 3π r=3 Ipsq r=4 α π α 1 π 3 α 3π 4 Ipsq α 1 π 4 α 3 5π 3 Rpsq α 3 5π 4 α 4 7π 4 Rpsq C. Hackl Non-identifier based adaptive control in mechatronics 6/15

12 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 6/15

13 High-gain control with derivative feedback Root locus controller k py `k D 9yq u F psq system ps`5q ps 1q ps`1q y 9y k D structural properties of F psq relative degree (pole excess): r positive high-frequency gain (lim sñ8 s F 1 psq 1) minimum-phase (numerator is Hurwitz) Root-locus (ˆ poles, zeros) imaginary axis k D = 0 k D = 1/ real axis C. Hackl Non-identifier based adaptive control in mechatronics 7/15

14 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 7/15

15 High-gain adaptive control with derivative feedback System class S lin Definition (see Definition.8 in [1]) A system of form 9xptq A xptq ` b uptq, yptq c J xptq is of Class S lin if, and only if, the following hold: n P N, xp0q x 0 P R n, up q P L 1 locpr ě0 ; Rq pa, b, cq P R nˆn ˆ R n ˆ R n. (1) (S lin -sp 1 ) the relative degree is two and the sign of the high-frequency gain is known, i.e. c J b 0, γ 0 : c J Ab 0 and signpγ 0 q is known; (S lin -sp ) it is minimum-phase, s P C ě0 : det sin A b c J 0 j 0, and (S lin -sp 3 ) the (regulated) output yp q and its derivative 9yp q are available for feedback. * C. Hackl Non-identifier based adaptive control in mechatronics 8/15

16 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 8/15

17 High-gain adaptive control with derivative feedback Theorem Theorem (see Theorem.36 in [1]) Consider a system of class S lin described by (1). The high-gain adaptive controller uptq signpc J Abq kptq yptq ` q 1 kptq 9yptq where 9kptq q expp q 3 kptqq pyptq, 9yptqq J, kp0q k 0 with design parameters q 1, q, q 3 ą 0 and k 0 ą 0 applied to (1) yields a closed-loop initial-value problem with the following properties:,. - (HG ) (i) there exists a unique and maximal solution px, kq : r0, T q Ñ R n ˆ R ě0, T P p0, 8s (ii) the solution is global, i.e. T 8 (iii) all signals are bounded, i.e. xp q P L 8 pr ě0 ; R n q and kp q P L 8 pr ě0 ; R ą0 q (iv) lim tñ8 9 kptq 0 and limtñ8 xptq 0 n. C. Hackl Non-identifier based adaptive control in mechatronics 9/15

18 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 9/15

19 High-gain adaptive control with derivative feedback Influence of design parameter q 1 :yptq uptq, ˆyp0q 9yp0q ˆ1 P R. () 0 1, 5 0, 5 0 0, 5 1, Time t [s] (a) output yp q for different designs Time t [s] (b) gain kp q for different designs. Figure: Simulation results for closed-loop system (), (HG ) with k 0 q q 3 1 and q 1 P t 0.1, 1,, 5u. C. Hackl Non-identifier based adaptive control in mechatronics 10/15

20 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 10/15

21 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 10/15

22 High-gain adaptive tracking with internal model Admissible reference class (revisited) Y ref y ref e 9y ref 9e high-gain v adaptive internal u model controller augmented system of class S lin system of class S lin y 9y where # Y ref : y ref p q P C 8 pr ě0 ; Rq ˇ ˇ DIM ` d dt yref p q 0, D IM P Rrss, monic with RpD IM q Ă C ě0 +. (3) is the admissible reference class. C. Hackl Non-identifier based adaptive control in mechatronics 11/15

23 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 11/15

24 High-gain adaptive tracking with internal model Augmented system (serial interconnection) Y ref y ref e 9y ref 9e high-gain v adaptive internal u model controller augmented system of class S lin system of class S lin y 9y d dt ˆxptq looomooon pxptq :x S ptq yptq «ff A bpc J O pˆn A p looooooooomooooooooon :A S PR pn`pqˆpn`pq c J, 0 J p loooooomoooooon :c J S PR1ˆpn`pq ˆxptq ˆxγ0 b ` pxptq p loomoon b ˆxptq, pxptq :b S PR n`p augmented system has relative degree r (as system of S lin )? ˆxp0q x 0, vptq, pxp0q px 0 P R nˆp loomoon /. :x 0 S (4) high-frequency gain γ S 0 γ 0 xγ 0 ðñ signpγ S 0 q signpγ 0 q? augmented system still minimum-phase? C. Hackl Non-identifier based adaptive control in mechatronics 1/15 /-

25 High-gain adaptive tracking with internal model Theorem (see Corollary.41 in [1]) Consider a system of class S lin given by (1) and some arbitrary y ref p q P Y ref with known D IM P Rrss as in (3). Under identical presuppositions as in Theorem.40, the high-gain adaptive (tracking) controller vptq signpc J Abq kptq eptq ` q 1 kptq 9eptq 9kptq q expp q 3 kptqq peptq, 9eptqq J, kp0q k 0 where eptq y ref ptq yptq,. - (5) with design parameters q 1, q, q 3, k 0 ą 0 applied to the serial interconnection (4) yields, for any initial value x 0 S P R nˆp, a closed-loop initial-value problem (5), (4) with the following properties (i) there exists a unique and maximal solution px S, kq : r0, T q Ñ R n`p ˆ R ě0, T P p0, 8s; (ii) the solution is global, i.e. T 8; (iii) the gain is bounded, i.e. kp q P L 8 pr ě0 ; R ą0 q; (iv) the tracking error vanishes asymptotically, i.e. lim tñ8 eptq lim tñ8 y ref ptq yptq 0; (v) the state does not grow faster than the reference, i.e. D M ą t ě 0: x S ptq ď Mp1 ` max spr0,ts y ref ptq q. (vi) lim tñ8 9eptq lim tñ8 9y ref ptq 9yptq 0. C. Hackl Non-identifier based adaptive control in mechatronics 13/15

26 Outline 3 Non-identifier based adaptive position control 3.1 Motivation Position control of stiff servo-systems Structural properties of stiff servo-system 3. High-gain adaptive control with derivative feedback Obstacle of higher relative degrees Root locus System class S lin Theorem Influence of design parameter q High-gain adaptive tracking with internal model Admissible reference class (revisited) Augmented system (serial interconnection) Simulation results C. Hackl Non-identifier based adaptive control in mechatronics 13/15

27 High-gain adaptive tracking with internal model Simulation results y ref : R ě0 Ñ R, t ÞÑ y ref ptq : t ` sinptq y ref psq 1 s ` 1 s ` 1. Output Errors Time t [s] (a) top: output yp q (blue) and reference y ref p q (dashed red); bottom: error ep q (blue) and derivative 9ep q (dashed red) Controls Gain Time t [s] (b) top: controller output vp q (blue) and control action up q (dashed red); bottom: gain kp q (blue) C. Hackl Non-identifier based adaptive control in mechatronics 14/15

28 References I [1] C. M. Hackl, Contributions to High-gain Adaptive Control in Mechatronics. PhD thesis, Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, Technische Universität München (TUM), Germany, 01. C. Hackl Non-identifier based adaptive control in mechatronics 15/15