Reset Control Systems: Theory and Application · Clegg and FORE Exponential Stability Set-point...

Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems

Reset Control Systems: Theory and Application

Luca Zaccarian

LAAS-CNRS, MAC group


Outline

1 Clegg integrators and First Order Reset Elements (FORE)

2 Exponential stability of reset control systems

3 Set-point regulation of linear plants using adaptive FORE

4 A collection of results using hybrid dynamical systems


An analog integrator and its Clegg extension (1956)

Integrators: core components of dynamical control systems

∫ xcẋcv Bc

Ac

Example: PI controller

ẋc = Acxc + Bcv

∫ xcvkp

kiC

R vCv xc

• In an analog integrator, the stateinformation is stored in a capacitor


An analog integrator and its Clegg extension (1956)

Integrators: core components of dynamical control systems

∫ xcẋcv Bc

Ac

Example: PI controller

ẋc = Acxc + Bcv

∫ xcvkp

kiC

R

v

C

R vC1

vC2

Rd xc-

-

• Clegg’s integrator (1956):• feedback diodes: the positive part ofxc is all and only coming from theupper capacitor (and viceversa)• input diodes: when v ≤ 0 the uppercapacitor is reset and the lower oneintegrates (and viceversa) [Rd � 1]• As a consequence ⇒ v and xc neverhave opposite signs


Hybrid dynamics may flow or jump (or both)

Hybrid Clegg integrator:

ẋc (t, j) =1

RCv(t, j), xc (t, j)v(t, j) ≥ 0,

xc (t, j + 1) = 0, xc (t, j)v(t, j) ≤ 0,

• Flow set: set where xc may flow (1st eq’n)• Jump set: set where xc may jump (2nd eq’n)

DC

xc

v

C

• Flow and Jump sets: for our example• Flow set defined as C := {(xc , v) : xcv ≥ 0},• Jump set defined as D := {(xc , v) : xcv ≤ 0}.

• Hybrid time domain: domain of solution xc has two directions:• t: elapsed flow times (number of seconds flown by xc )• j : elapsed jump times (number of jumps performed by xc )


Hybrid dynamics may flow or jump (or both)

Hybrid Clegg integrator:

ẋc = (RC )−1v , xcv ≥ 0,

x+c = 0, xcv ≤ 0,

• Flow set C := {(xc , v) : xcv ≥ 0} is closed• Jump set D := {(xc , v) : xcv ≤ 0} is closed• Stability is robust! (Goebel 2006)

DC

xc

v

CPrevious models (Clegg ’56, Horowitz ’73, Hollot ’04):

ẋr = (RC )−1u, if u 6= 0,

x+r = 0, if u = 0,

• Imprecise: solutions ∃ s.t. xcv < 0, butClegg’s xc and v always have same sign!

• Unrobust: C is almost all R2(arbitrary small noise disastrous)

• Unsuitable: Adds extra solutions⇒ Lyapunov results too conservative!

CC

xc

v

CD

D


Stabilization using hybrid jumps to zero

First Order Reset Element:

ẋc = acxc + bcv , xcv ≥ 0,x+c = 0, xcv ≤ 0,

Theorem If P is linear, minimum phaseand relative degree one, then

FOREy

d

Puvxc

ac , bc or (ac , bc) large enough ⇒ global exponential stabilityTheorem In the planar case, γdy shrinks to zero as parameters grow

Simulationuses:

P = 1s

bc = 1

0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

Time

Pla

nt

ou

tpu

t

Linear (a

c=−1)

ac=−3

ac=−1

ac=1

ac=3

Interpretation: Resets remove overshoots, instability improves transient


Piecewise quadratic Lyapunov function construction

• Given N ≥ 2 (number of sectors)• Patching angles:−θ� = θ0 < θ1 < · · · < θN =

π

2+ θ�

• Patching hyperplanes (Cp = [0 · · · 0 1])Θi =

[01×(n−2) sin(θi ) cos(θi )

]T

• Sector matrices:S0 := Θ0Θ

TN + ΘNΘ

T0

Si := −(Θi ΘTi−1 + Θi−1ΘTi ), i = 1, . . . ,N,

S�1 :=

0(n−2)×(n−2) 0 00 0 sin(θ�)0 sin(θ�) −2 cos(θ�)

S�2 :=

0(n−2)×(n−2) 0 00 −2 cos(θ�) sin(θ�)0 sin(θ�) 0

��

��

��

��

��

��

��

��

��

��

��

xr axis

P1

PN

P2

PN−1

y axis

P0

Sǫ1

Sǫ2

θ1

θ2

θ0

θN−2θN−1 θN

S0

S2

S0

SN−1SN

S1

Hybrid closed-loop:

ẋ = AF x + Bdd , x ∈ Cx+ = AJx , x ∈ D


Piecewise quadratic Lyapunov theorem

Theorem: If the following LMIs in the green unknowns (whereZ = [In−2 0(n−2)×2]) are feasible:

(Flow)

ATF Pi + PiAF + τFiSi PiBd CT

? −γI 0? ? −γI

< 0, i = 1, . . . ,N,

(Jump) ATJ P1AJ − P0 + τJS0 ≤ 0(Cont ′ty) ΘTi⊥ (Pi − Pi+1) Θi⊥ = 0, i = 0, . . . ,N − 1,(Cont ′ty) ΘTN⊥(PN − P0)ΘN⊥ = 0(Overlap) ATJ P1AJ − P1 + τ�1S�1 ≤ 0(Overlap) ATJ P1AJ − PN + τ�2S�2 ≤ 0

(Origin)

Z (ATF P0 + P0AF )ZT ZP0Bd ZC

T

? −γI 0? ? −γI

< 0,

then global exponential stability + finite L2 gain γdy from d to y


Example 1: Clegg (ac = 0) connected to an integrator

• Block diagram:

1s

yd

xcClegg

ac = 0

• Output response (overcomeslinear systems limitations)

0 2 4 6 8 10−1

−0.5

0

0.5

1

Clegg (ac=0)

Linear (ac=0)

• Quadratic Lyapunov functionsare unsuitable

• Gain γdy estimates (N = # of sectors)N 2 4 8 50

gain γdy 2.834 1.377 0.914 0.87

• A lower bound:√

π8 ≈ 0.626

• Lyapunov func’n level sets for N = 4

−2 −1 0 1 2

−3

−2

−1

0

1

2

3

y

xr

−2 −1 0 1 2

−3

−2

−1

0

1

2

3

y

xr

• P1, . . . ,P4 cover 2nd/4th quadrants• P0 covers 1st/3rd quadrants


Example 2: FORE (any ac) and linear plant (Hollot et al.)

• Block diagram (P = s+1s(s+0.2) )

P yd

xcFORE

• ac = 1: level set with N = 50

−4 −2 0 2 4−30

−20

−10

0

10

20

30

y

xr

−4 −2 0 2 4−25

−20

−15

−10

−5

0

5

10

15

20

25

y

xr

• Gain γdy estimates

−5 0 50

1

2

3

4

5

6

7

8

ac

L2 g

ain

s

Linear CLS

Reset CLS (Thm 3, ACC 2005)

Reset CLS (this theorem)

• Time responses

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

Linear (a

c=−1)

ac=−3

ac=−1

ac=1

ac=3

ac=5


Stabilization using hybrid jumps to zero (recall)

First Order Reset Element:

ẋc = acxc + bcv , xcv ≥ 0,x+c = 0, xcv ≤ 0,

Theorem If P is linear, minimum phaseand relative degree one, then

FOREy

d

Puvxc

ac , bc or (ac , bc) large enough ⇒ global exponential stabilityTheorem In the planar case, γdy shrinks to zero as parameters grow

Simulationuses:

P = 1s

bc = 1

0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

Time

Pla

nt

ou

tpu

t

Linear (a

c=−1)

ac=−3

ac=−1

ac=1

ac=3

Interpretation: Resets remove overshoots, instability improves transient


Set point adaptive regulation using hybrid jumps to zero

• Parametric feedforward uff = Ψ(r)Tαẋc = acxc + bcv ,α̇ = 0,

if xcv ≥ 0,

x+c = 0,

α+ = α + λ Ψ(r)|Ψ(r)|2 xc ,if xcv ≤ 0,

FOREy

d

Pr ++uxcv

αuff

Theorem: If FORE stabilizes withr = 0, then for constant r , y → r

Lemma: Tuning of λ usingdiscrete-time rules (Ziegler-Nichols)

λ = 0

λ = 0.32

λ = 0.64

UnitCircle

Example: EGR Experiment (next slide)

0 0.1 0.2 0.3 0.4 0.5 0.6

20

40

60

80

Po

sitio

n [

%]

0 0.1 0.2 0.3 0.4 0.5 0.6−1

0

1

xc

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.1

0.2

0.3

0.4

uff

Time [s]

Reference

λ=0.04

λ=0.32

λ=0.58


Fast regulation of EGR valve position in Diesel engines

• EGR: Recirculates Exhaust Gasin Diesel engines

• Subject to strong disturbances⇒ need aggressive controllers(recall exp. unstable transients)

34.6 34.7 34.8 34.9 35 35.1 35.2 35.3 35.4

0.2

0.25

0.3

0.35

0.4

0.45

Va

lve

Po

sitio

n [

V]

Experimental

Simulated

34.6 34.7 34.8 34.9 35 35.1 35.2 35.3 35.4−0.5

−0.4

−0.3

−0.2

−0.1

0

Du

ty C

ycle

Time [s]

• Identified valve transfer function:EGRValve

voltage position

P(s) =2200

(s + 164.4)(s + 10.69).


Laboratory experiments close to time-optimal

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

20

30

40

50

60

70

Positio

n y

[%

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

−1

−0.5

0

0.5

1

Duty

Cycle

u

Time [s]

Reference

Adaptive FORE

Time Optimal

PI

• Time-optimal:unrobust, obtained viatrial and error

• PI:Tuned using standardMATLAB tools

• Adaptive FORE:Response afterα→ α∗ =(0.128, 0.087, 0.115)

• Note the exponentially diverging voltage:aggressive action for disturbance rejection on the real engine


Experiments on Diesel engine testbench (JKU)

• Specs: 2 liter, 4 cylinder passengercar turbocharged Diesel engine

• Compared: to factory EGR valvecontroller coded in ECU(gain scheduled PI with feedforward)

• Test cycle: Urban part ofNew European Driving Cycle

• Relevance: Faster EGR positioning⇒ Reduced NOx emissions


Adaptive FORE provides substantial performance increase

115 116 117 118 119 120 121 122 123 124 125 126

0

50

100

Po

sitio

n y

[%

]

ECU

Position

Reference

115 116 117 118 119 120 121 122 123 124 125 126

0

50

100

Po

sitio

n y

[%

]

FORE

Position

Reference

115 116 117 118 119 120 121 122 123 124 125 1260

0.5

1

Time [s]

NO

x (

no

rma

lize

d)

FORE

ECU

• Mean squared error: ECU = 6.68 (100%), FORE = 1.53 (23 %)• Improvement most important with EGR almost closed (t ≈ 117, 124)


From discrete + continuous to hybrid dynamical systems

[Branicky, Collins, Henzinger, Liberzon, Lygeros, Sastry, Tavernini, ...]

Continuous dynamical system Discrete dynamical systemdx(t)

dt= f (x(t)), ∀t ∈ R≥0 x(k + 1) = g(x(k)), ∀k ∈ Z≥0

(possible discrete variables)︸︷︷︸⇓Hybrid dynamical system

dx(t, k)

dt= f (x(t, k)), x(t, k) ∈ C ⊂ Rn (a.e.)

x(t, k + 1) = g(x(t, k)), x(t, k) ∈ D ⊂ Rn

• Continuous time domain t ∈ R≥0 and Discrete time domaink ∈ Z≥0 merged into Hybrid time domain (t, k) ∈ R≥0 × Z≥0• Solution x can “flow” if ∈ C, can “jump” if ∈ D• Fundamental stability results now available (Converse Lyapunovtheorems, ISS, invariance principle, Lp stability) [Teel ’04 → ’12]


Hybrid resets induce passivity in nonlinear controllers

• Inspired by [Carrasco, Banos, Van der Schaft, 2009]• Given any (nonlinear) controller K [under mild growth conditions]

Ku y K̂Hû ŷContinuous-time Hybrid

ControllerController

a hybrid modification K̂H is given such that• the state x̂c of K̂H is sometimes reset to zero• whenever K̂H flows, it mimicks K• (continuous-time) output strict passivity of K̂H holds:∫ ∞

0ŷ(t)T û(t)dt ≥ ε2‖ŷ(·)‖22

• Promising for HMI, to avoid man-induced instabilities (PIO, etc)


Example: Reset passivation of a robot controller

Two-link robotpassivated by aninner loop

q1

q2

m1, I1

m2, I2

g

l2

l1

a2

a1

z

x

H(q)−∂V (q,r)∂q

u û

q

q̇

Robot−

+

K̂H

P

kH

up

θv

r

stable, no resets (dash-dot); stable with resets (dash); unstable (solid)

0 0.5 1 1.5 2 2.5

−10

0

10

20

Jo

int

sp

ee

ds d

q

0 0.5 1 1.5 2 2.5

0

10

20

30

0 0.5 1 1.5 2 2.5

0

5

10

Jo

int

po

sitio

ns q

0 0.5 1 1.5 2 2.5

0

2

4

6

0 0.5 1 1.5 2 2.5

0

5

10

15

Re

se

t co

ntr

ol in

pu

ts u

0 0.5 1 1.5 2 2.5

0

2

4

6

0 0.5 1 1.5 2 2.570

80

90

100

110

To

rqu

e in

pu

ts u

p

0 0.5 1 1.5 2 2.50

5

10

15

20


Hybrid systems tools utilized in diverse scenarios

P

K

w z

yc

u y

continuousZOH

discrete

sampler

Sampled-data control design with saturation• uniform sampling time assumed• jumps correspond to sampling actions

Lazy sensors for reduced transmission rate• reduce sample transmission over networks• used in cyber-physical systems• jumps occur at samples transmissions

K PP

u y

NScheduling

policy

Extrainformation

ξMeasurementdevices

xp

xcNonlinear stabilization via hybrid loops• enforces jumps of the controller state in some sets• so as to guarantee decrease of suitable functions• state jumps to the minimum of a Lyapunov function• useful, e.g., for overshoot reduction with SISO plants


Impacting systems: billiard ball x tracks billiard ball z

Lyapunov functions Vi(x, z) along hybrid trajectory

Idea: Follow the “closest” ballamong all the target balls z mir-rored by the walls

Linear FeedbackHybrid Feedback

where

mk(z) = Mkz + ck, k = 1, . . . , 8m0(z) = z (real ball)

(mirrored balls)

Use Hybrid Lyapunov function

V (x, z) = mini∈{0,...,9}

|x−mi(z)|P︸︷︷︸Vi(x,z)

classic.aviMedia File (video/avi)

hybrid.aviMedia File (video/avi)


Example: Clegg response to a sine input

0 24 6

8 10

0

1

2

3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

TimeJumps

Inte

gra

tor

valu

e

• Solution (bold black) as a function ofthe hybrid time domain (red)• Input selected as v(t, j) = sin(t)(dashed line below)• State xc is reset upon entering the2nd and 4th quadrants(in this case ≡ at the zero crossing)

• Solid: projection of xc onthe ordinary time axis t• Dash: projection of u onthe ordinary time axis t

0 1 2 3 4 5 6 7 8 9 10

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time


Feedforward: α converges to suitable parametrization

0 20 40 60 80 100−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Valve Position [%]

Va

lve

In

pu

t [V

]

Bare EGR valve

EGR valve with elastic band

Measured steady−state I/O pairs • ?: steady-state input/outputpairs (stiction!!)

• Red Solid: uff = ΨT (r)α∗, withα∗ steady-state for α

• Black dashed: uff = ΨT (r)ᾱ∗when pulling the valve with anelastic band

Clegg integrators and First Order Reset Elements (FORE)

Exponential stability of reset control systems

Set-point regulation of linear plants using adaptive FORE

A collection of results using hybrid dynamical systems

Reset Control Systems: Theory and Application · Clegg and FORE Exponential Stability Set-point...

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