Non-identifier based adaptive control in mechatronics€¦ · Non-identiﬁer based adaptive...

Non-identifier based adaptive control in mechatronics3.3 Adaptive λ-tracking control and funnel control (with derivative feedback)

Christoph Hackl

Munich School of Engineering (MSE)Research group “Control of renewable energy systems (CRES)”

www.cres.mse.tum.de

Lecture & tutorial24.06.2016

C. Hackl | 24.06.2016 | Non-identifier based adaptive control in mechatronics 1/20

IntroductionSchedule (tentative)

Date Content

15.04.2016 1. Introduction and2. Non-identifier based speed control (relative-degree-one case)2.1 High-gain adaptive stabilization

22.04.2016 2.2 High-gain adaptive tracking (using internal models) and2.3 Adaptive λ-tracking control and funnel control

29.04.2016 Tutorials for Lectures 2.1-2.306.05.2016 canceled (Christi Himmelfahrt)13.05.2016 2.4 Practical course (relative-degree-one case)20.05.2016 2.4 Practical course (relative-degree-one case) [contd.]27.05.2016 2.4 Practical course (relative-degree-one case) [contd.]03.06.2016 2.1 & 2.3 Proofs of high-gain adaptive stabilization & funnel control10.06.2016 2.5 Applications: Speed control of electrical drives (and some new results)

3. Non-identifier based adaptive position control (relative-degree-two case)17.06.2016 3.1 High-gain adaptive stabilization and & 3.2 High-gain adaptive tracking24.06.2016 3.3 Adaptive λ-tracking control and funnel control (with derivative feedback)01.07.2016 3.4 Practical course (relative-degree-two case)08.07.2016 3.5 Application: Position funnel control of servo-systems & industrial robots15.07.2016 4. Conclusions & exam revision


Outline

3 Non-identifier based adaptive position control3.3 Adaptive λ-tracking control and funnel control (with derivative feedback)


Outline


MotivationAdaptive λ-tracking control with derivative feedbackFunnel control with derivative feedback


MotivationNonlinear model of stiffly coupled servo-system

Drive (mM ) Load (mL)

φ, ω

Θhkkkkkkkikkkkkkkj

ddt

“:xptqhkkkikkkj

ˆ

φptqωptq

˙

“

“:Ahkkkkkkkkkkkikkkkkkkkkkkj

«

0 1

0 ´ν1`ν2{g

2r

Θ

ff

ˆ

φptqωptq

˙

`

“:bhkkikkj

ˆ

0kAΘ

˙

satpuA

`

uptq ` uAptq˘

´

ˆ

01Θ

˙

´

pF1ωqptq `1gr

`

mLptq ` pF2ωgrqptq

˘

¯

yptq “`

1 0˘

loomoon

“:cJ

xptq, pφp0q, ωp0qqJ“ pφ0, ω0q

J

,

/

/

/

/

/

/

/

/

/

/

/

.

/

/

/

/

/

/

/

/

/

/

/

-

(1MSφ)

whereΘ ą 0, gr P Rzt0u, ν1, ν2 ě 0, uA, kA ą 0, uAp¨q, mLp¨q P L8pRě0;Rq,

and @ i P t1, 2u : Fi : CpRě0;Rq Ñ L8pRě0;Rq.

+

(1MSφ-Data)


MotivationGain drift: High-gain adaptive control with noise & disturbances

time t [s]0 10 20 30 40 50

−0.5

0

0.5

1

1.5

Zoom

0 501.16

1.17

1.18

(a) due to noise nmp¨q PW2,8pRě0;Rq

(udp¨q “ 0): kp¨q, yp¨q.

time t [s]0 10 20 30 40 50

−1

0

1

2

3

4

(b) due to disturbanceudp¨q P L8pRě0;Rq(nmp¨q “ 0): kp¨q and yp¨q.

System

:yptq “ uptq ` udptqymptq “ yptq ` nmptq

,pyp0q, 9yp0qq “ p1, 0q, udp¨q P L8pRě0;Rq,nmp¨q PW2,8

pRě0;Rqunder control of

uptq “ ´kptq2yptq ´ 2kptq 9yptq where 9kptq “ exp

ˆ

´kptq

100

˙

‖ˆ

yptq9yptq

˙

‖2, kp0q “ 1.


MotivationTypical position reference (qualitative)

Time t [s]

Reference

&D

eriv

ativ

es

0 2 4 6 8−1

−0, 5

0

0, 5

1

1, 5

2

yrefp¨q, 9yrefp¨q, :yrefp¨q

ùñ A single internal model does not exist!


MotivationMore realistic scenarios

‚ Motivation 1: high-gain adaptive controllers not robust (gain drift due tonoisy measurements or external disturbances)

‚ Motivation 2: System class S lin2 is restrictive (e.g. (1MSφ) R S lin

2 )˝ admissible systems are linear˝ no (external) disturbances (possibly piecewise continuous!)˝ no (nonlinear) perturbations

‚ Motivation 3: reference tracking not considered (only stabilization, or forcertain reference signals, asymptotic tracking with internal models)

Hence, we need‚ a broader system class ùñ S2

‚ a wider class of admissible reference signals ùñW2,8pRě0;Rq (see

Tutorial)‚ more robust controllers (for real world application, noise being admissible)


Definition (see Definition 1.7)Let n,m P N, h ě 0, pA, b, cq P Rnˆn ˆ Rn ˆ Rn and BT P Rnˆm. A system of form

9xptq “ Axptq ` b`

uptq ` udptq˘

`BT

`

pTxqptq ` dptq˘

yptq “ cJxptq, x|r´h,0s “ x

0p¨q P C

`

r´h, 0s; Rn˘

,

.

-

(1)

with disturbances ud : r´h,8q Ñ R and d : r´h,8q Ñ Rm, operatorT : Cpr´h,8q;Rnq Ñ L8pRě0;R

mq, control input u : Rě0 Ñ R and regulated output

yp¨q, is of Class S2 if, and only if, the following hold:(S2-sp1) the relative degree is two and the sign of the high-frequency gain is known, i.e.

cJb “ 0, c

JBT “ 0

Jm, γ0 :“ c

JAb ‰ 0 and signpγ0q known;

(S2-sp2) the unperturbed system is minimum-phase (see (S1-sp2));

(S2-sp3) the operator is of class T and globally bounded (see (S1-sp3));

(S2-sp4) the disturbances are bounded (see (S1-sp4));

(S2-sp5) feedback of the regulated output and its derivative is admissible, i.e. yp¨q and 9yp¨q areavailable for feedback.


Outline




Adaptive λ-tracking control (see [1, Theorem 3.13])

ep¨q “ yrefp¨q ´ yp¨qep0q

λ

´λ

¨ ¨ ¨

λ-strip

time t rss

‚ Controller & gain adaption

uptq “ kptq2eptq ` q1kptq 9eptq where (LT2)

9kptq “ q2 exp p´q3q4kptqq dλ

˜∥∥∥∥∥˜

eptq9eptqkptq

¸∥∥∥∥∥¸q4

‚ with tuning parameters kp0q “ k0, q1, q2, q3 ą 0, q4 ě 2.C. Hackl | 24.06.2016 | Non-identifier based adaptive control in mechatronics 9/20

Adaptive λ-tracking control (see [1, Theorem 3.13])

ep¨q “ yrefp¨q ´ yp¨qep0q

λ

´λ

¨ ¨ ¨

λ-strip

time t rss

‚ tracking with prescribed asymptotic accuracy, i.e. for all λ ą 0, tracking error eptqand 9eptq{kptq asymptotically converge into the “λ-strip”, i.e.

limtÑ8

dist´

‖peptq, 9eptq{kptqq‖, r0, λs¯

“ 0;

‚ state variable is bounded, i.e. xp¨q P L8pRě0;Rnq;

‚ control action is bounded, i.e. up¨q P L8pRě0;Rq.C. Hackl | 24.06.2016 | Non-identifier based adaptive control in mechatronics 10/20

Outline




Funnel control with derivative feedbackMotivation for funnel control

‚ Motivation 1: Adaptive λ-tracking (and high-gain adaptive) control exhibita non-decreasing gain and so, as time tends to infinity, it is likely thate.g. noise sensitivity permanently exceeds an acceptable level (if gainadaption is not stopped).

‚ Motivation 2: Adaptive λ-tracking control assures tracking with prescribedasymptotic accuracy, however statements on the transient accuracy arenot possible; e.g.

˝ albeit bounded large overshoots might occur and˝ the λ-strip is not reached in finite time (in general)

‚ Motivation 3: Input saturations are not yet considered (e.g. actuatorsaturation of electrical drives) [in next lecture].


Funnel control with derivative feedbackControl objective

‚ ‘tracking with prescribed transient behavior’ for yrefp¨q PW2,8pRě0;Rq, i.e.

@ t ě 0: |eptq| “ |yrefptq ´ yptq| ă ψ0ptq and | 9eptq| ă ψ1ptq

ep¨q

ep0q ψ0p¨q

ψ0p0q

´ψ0p0q

eptq

ψ0ptq

´λ0 9ep¨q9ep0q

ψ1p¨q

ψ1p0q

´ψ1p0q

9eptq

ψ1ptq

λ1

tt TIME t rss

funnel

‚ funnel boundary

B2 :“

#

pψ0, ψ1q : Rě0 Ñ R2

ˇ

ˇ

ˇ

ˇ

ˇ

piq @ i P t0, 1u D ci ą 0: ψip¨q PW1,8pRě0, rci,8qq,

piiq D δ ą 0 for a.a. t ě 0: ψ1ptq ě ´ddt ψ0ptq ` δ

+

(2)C. Hackl | 24.06.2016 | Non-identifier based adaptive control in mechatronics 12/20

Funnel examples

ExamplesFor Λ0 ě λ0 ą 0, TL, TE ą 0 rss and λ1 ą 0 rss define

1.ˆ

ψ0

ψ1

˙

: Rě0 Ñ R2, t ÞÑ

ˆ

ψ0ptqψ1ptq

˙

:“

ˆ

max

Λ0 ´ t{TL, λ0

(

1{TL ` λ1

˙

2.ˆ

ψ0

ψ1

˙

: Rě0 Ñ R2, t ÞÑ

ˆ

ψ0ptqψ1ptq

˙

:“

¨

˝

pΛ0 ´ λ0q exp´

´ tTE

¯

` λ0

Λ0´λ0TE

exp´

´ tTE

¯

` λ1

˛

‚.

Zeit t [s]

0 2 4 6 8 100

2

4

6

8

10

ψ0(·)

ψ1(·)

Zeit t [s]

0 2 4 6 8 100

2

4

6

8

10

ψ0(·)

ψ1(·)


Funnel control with derivative feedbackFunnel controller (variant 1, see [2])

ep¨q

ep0q ψ0p¨q

ψ0p0q

´ψ0p0q

eptq

ψ0ptq

´λ0 9ep¨q9ep0q

ψ1p¨q

ψ1p0q

´ψ1p0q

9eptq

ψ1ptq

λ1

tt TIME t rss

funnel

uptq “ k0ptq2eptq

loooomoooon

P

` k0ptqk1ptq 9eptqlooooooomooooooon

D

` uF ptqloomoon

e.g. feed forward

(FC‹2)

‚ k0ptq “1

ψ0ptq´|eptq|and k1ptq “

1ψ1ptq´| 9eptq|

‚ uF p¨q P L8pRě0;Rq‚ @ i P t0, 1u Dci ą 0: ςip¨q PW1,8

pRě0; rci,8qqC. Hackl | 24.06.2016 | Non-identifier based adaptive control in mechatronics 14/20

Funnel control with derivative feedbackFunnel controller (variant 2, see [1, Theorem 4.13])

ep¨q

ep0q ψ0p¨q

ψ0p0q

´ψ0p0q

eptq

ψ0ptq

´λ0 9ep¨q9ep0q

ψ1p¨q

ψ1p0q

´ψ1p0q

9eptq

ψ1ptq

λ1

tt TIME t rss

funnel

uptq “ k0ptq2eptq

loooomoooon

P


D

` uF ptqloomoon

e.g. feed forward

(FC2)

‚ k0ptq “ς0ptq

ψ0ptq´|eptq|and k1ptq “

ς1ptq

ψ1ptq´| 9eptq|

‚ uF p¨q P L8pRě0;Rq‚ @ i P t0, 1u Dci ą 0: ςip¨q PW1,8

pRě0; rci,8qqC. Hackl | 24.06.2016 | Non-identifier based adaptive control in mechatronics 15/20

Funnel control with derivative feedbackImplementation at laboratory setup

test benchimplementation

e

9e

nm

9nm

φ

ω

y “ φ ` nm

9y “ ω ` 9nm

yref “ φref

9yref “ ωrefu “ mM

´

´

P

D

mLdrive load

uptq “ k0ptq2 eptq

loooomoooon

P


D

(FC‹2)

uptq “ k0ptq2 eptq

loooomoooon

P


D

where k1ptq ě2?γ0

(FC2)


Measurement results (1MS): (FC‹2), (FC2)

Position

(Win

kel)

φ+n

m[π

rad]

0

1

2

3

4

5

φref (·) φref (·) ±ψ0(·)

Ges

chw

indig

keit

ω+n

m[π

rad/s]

−9

−6

−3

0

3

6

9

ωref (·) ωref(·) ±ψ1(·)

Zeit t [s]

Moto

rmom

ent

mM

[Nm

]

0 5 10 15 20 25 30 35 40 45 50

−5

0

5

10

15

20

−mL(·)


Measurement results (1MS): (FC‹2), (FC2)

Positionsfeh

ler

e[π

rad]

−0.1

−0.05

0

0.05

0.1

±ψ0(·)

P-V

erst

ark

ung

k2 0

[Nm

/ra

d]

0

20

40

60

80

Ges

chw

.-Feh

ler

e[π

rad/s]

−3−2−1

0123

±ψ1(·)

Zeit t [s]

D-V

erst

ark

ung

k1,k0k1

[Nm

s/ra

d]

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20


Theorem (Funnel position control of unsaturated 1MS)Consider the mechatronic system (1MSφ) with (1MSφ-Data) and uA “ 8 (unsaturated actuator).Then, for any funnel boundary pψ0p¨q, ψ1p¨qq P B2, gain scaling function ς0p¨q, ς1p¨q P B1,reference signal yrefp¨q PW2,8

pRě0;Rq and initial value pφp0q, ωp0qq “ pφ0, ω0q P R2,satisfying

|yrefp0q ´ φp0q| ă ψ0p0q and | 9yrefp0q ´ ωp0q| ă ψ1p0q, (FC2-init)

the funnel controller

uptq “ k0ptq2eptq ` k0ptqk1ptq 9eptq where eptq “ yrefptq ´ yptq,

k0ptq “ς0ptq

ψ0ptq ´ |eptq|and k1ptq “

ς1ptq

ψ1ptq ´ | 9eptq|

,

.

-

(FC2)

applied to (1MSφ) yields a closed-loop initial-value problem with the properties:

(i) there exists a unique solution pφ, ωq : r0, T q Ñ R with maximal T P p0,8s;

(ii) the solution pφp¨q, ωp¨qq does not have finite escape time, i.e. T “ 8;

(iii) tracking error and its derivative are uniformly bounded away from the funnel boundary, i.e.

D ε0, ε1 ą 0 @ t ě 0 : ψ0ptq ´ |eptq| ě ε0 and ψ1ptq ´ | 9eptq| ě ε1;

(iv) gains and control action are uniformly bounded, i.e. k0p¨q, k1p¨q, up¨q P L8pRě0;Rq.


References I

[1] C. M. Hackl, Contributions to High-gain Adaptive Control in Mechatronics. PhD thesis, Lehrstuhl fürElektrische Antriebssysteme und Leistungselektronik, Technische Universität München (TUM), Germany,2012.

[2] C. M. Hackl, N. Hopfe, A. Ilchmann, M. Mueller, and S. Trenn, “Funnel control for systems with relativedegree two,” SIAM Journal on Control and Optimization, vol. 51, no. 2, pp. 965–995, 2013.


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