reset control systems: theory and application · clegg and fore exponential stability set-point...
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
Reset Control Systems: Theory and Application
Luca Zaccarian
LAAS-CNRS, MAC group
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation 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
-
Clegg and FORE Exponential Stability Set-point Regulation 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
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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 )
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
-
Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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).
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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)
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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]
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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)
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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)
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Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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
-
Clegg and FORE Exponential Stability Set-point Regulation Hybrid Dynamical Systems
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