EfficientlyAttentive Quantized Event-Triggered Systemslemmon/projects/NSF-05-1518/... ·...
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LCCC Workshop - Lund Sweden - December 7, 2011
EfficientlyAttentive Quantized Event-Triggered Systems
Lichun Li, Xiaofeng Wang, M.D. LemmonDept. of Electrical EngineeringUniversity of Notre DameNotre Dame, IN, USA
LCCC Workshop - Lund Sweden - December 7, 2011
Sampled Data Control System
Channel - Lossy with finite bit-rateDecoder - zero-order holdPlant/Controller - stable under perfect state feedback
Encoder - quantizes and samples system state
Plant andController
Channel
EncoderDecoder
x(t)
xkxk
x(t)
• sk∞k=0 = sampling instants
• Plant/Controller state, x(t) satisfies
x(t) = f(x(t), u(t), w(t)); x(0) = x0
• Sampled and Quantized state, xk =Q(x(sk))
• Reconstructed state x(t) = xk for t ∈[sk, sk+1).
Minimum Information for Stabilization?
LCCC Workshop - Lund Sweden - December 7, 2011
Dynamically Quantized Feedback
Necessary and sufficient bit rate for asymptotic stability
Discrete-time Linear System with one sample delay- state is periodically sampled with quantization[1,2]
Uncertainty setsat time instant k
Uncertainty setsat time instant k+1
R ≥ R =
n∑
i=1
max(0, log2 |λi|)
LCCC Workshop - Lund Sweden - M.D. Lemmon - December 7, 2011
Emulation Method
Select Sampling Instants to guarantee that thesporadically sampled system is also ISS.
“Emulation Method” for Sampled System Design- design controller, K, that leaves “continuously” sampled system input-to-state stable (ISS)[3].
There exist functions β ∈ KL and γ ∈ K such that
|z(t)| < maxβ(|z0|, t), γ(|e|L∞)
e ∈ L∞ and z(·) : R+ → Rn satisfies
z(t) = f(z(t),K(z(t) + e(t))
β(|z0|,t)
γ(|w|∞)
|z(t)|
LCCC Workshop - Lund Sweden - December 7, 2011
Event-Triggered Feedback[4]
State is sporadically sampled with no quantization
ξ(|x(t)|)
|e(t)|
s0 s1 s2 s3
Since the ”continuously” sampled system is ISS,there exists a positive definite function V (·) :Rn → R+ and functions α, γ ∈ K such that
∂V
∂xf(x,K(x+ e)) ≤ −α(|x|) + γ(|e|)
Select sampling instants, sk∞k=0
so that for all time, t,
|e(t)| ≤ γ−1 ((1− σ)α(|x(t)|)) ≡ ξ(|x(t)|)
where 0 < σ < 1
LCCC Workshop - Lund Sweden - December 7, 2011
Resource Utilization in Event-Triggered Systems
Prior work has suggested that event-triggered systemshave lower resource utilization than comparableperiodically sampled systems[5,6,7,8].
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
10
87654321
9
V R / V
L
Mean Sample Period (T)
a=1
a=0
a=-1
[Astrom 02]
0 2 4 6 8 10−5
0
5x 10−3
Erro
r [ra
d/s]
0 2 4 6 8 10−5
0
5x 10−3
Erro
r [ra
d/s]
0 1 2 3 4 5 6 7 8 9 100
5e3
1e4
Time [s]
# sa
mpl
es
Time−driven PID
event−driven PID
Time−driven PIDEvent−driven PID
[Sandee 2006]
But this may not always be the case.
LCCC Workshop - Lund Sweden - December 7, 2011
Zeno Sampling and Efficient Attentiveness[9,10]
• Consider the following system
x(t) = f(x(t)) + u(t) with u(t) = −2f(xk)
wheresublinear superlinear
f(x) = sgn(x)√
|x| f(x) = x3
• Given sampling instant sk, the next sam-pling instant sk+1 is the first time when thestate leaves
Ωk =x ∈ Rn : (xk − x)2 ≤ |x|2 ≡ θ(|x|)
0 1 2 3 4 5 6 7 8 9 1010−15
10−5
105
1 2 4 7 100
0.20.40.60.8
1
Intersample Period
θ(x(t))
e2(t)
time
f (x ) = sgn( x ) |x |sublinear
Gap and Threshold History
985 63
0
10−10
4 5 6 7 8 9 10
4 7 10
Intersample Period
θ(x(t))
time
f (x ) = x 3superlinear
Gap and Threhold Trajectory0 1 2 3
0 1 2 3 5 6 8 9
−20
−10
10
0.50.40.30.20.10.0
0
e2(t)
Zeno-sampling : infinite samples over a finite intervalEfficiently Attentive : Intersampling Intervals get larger assystem state approach origin.
,
LCCC Workshop - Lund Sweden - December 7, 2011
Dynamic Quantization versus Event-Triggering
Similarities- both attempt to reduce “information” over channel.- quantization in “time” versus “space.- both discretize a continuous-time control system
Differences - Single sample delay (Q) versus small delay (ET)- Utilization measured by bit rate versus inter-sampling time- Dynamic quantizers achieve minimum stabilizing bit rate, but lower resource utilization not guaranteed for event-triggered systems
Objectives:- Unification quantization and event-triggering- Design Event-triggers to control resource utilization by ensuring efficient attentiveness
LCCC Workshop - Lund Sweden - December 7, 2011
Quantized Event-Triggered Networked System[11,12]
• Sampling Instants, sk∞k=0, and Arrival Instants ak∞k=0.
• State Equation is x(t) = f(x(t), k(xk))where xk is the quantized state and t ∈ [ak, ak+1).
• Quantized state, xk, satisfies |x(sk)− xk| < E(|x(sk)|)where E ∈ K is the Quantization Error.
• Controller, K, ensures ”continuously” sampled system is ISSwith respect to the gap ek(t) = x(t)− xk.
LCCC Workshop - Lund Sweden - December 7, 2011
Modeling of Channel Delay and Quantization[13,14,15]
Quantization Error, E(|xk|) Delay = Dk = ak-skInter-sampling Interval = Tk = sk+1-skAdmissible Sampling: sk < ak < sk+1
sk ak sk+1 ak+1sk-1 ak-1
Dk
Tk
ek-1(t)ek(t) ek+1(t)
gap
func
tion,
ej
E(|xk|) = quantization error
What are dynamics of gap function, ek(t) = x(t)-xk ?
LCCC Workshop - Lund Sweden - December 7, 2011
Dynamics of the Gap Function[11,12]
Since x(t) ∈ Ωk, the gap, ek(t), satisfies thefollowing differential inequality for t ∈ [sk , ak+1 )d|ek|dt
≤ |ek(t)| ≤ |f(xk,K(xk))|+ Lk|ek(t)|, ek(sk) < E(|xk|)
This is a linear differential inequality and we canuse the comparison principle to bound the gap |ek(t) |
xk
x : |x-xk| < ξ(|xk|)
ξ(|xk|)
Ωk
Lipschitz on Compacts: Lk > 0 such that
|f(x(t),K(xk))| ≤ |f(xk,K(xk))|+ Lk|ek(t)|
for all x(t) = xk + ek(t) ∈ Ωk where
Ωk =x ∈ Rn : |x| ≤ |xk|+ ξ(|xk|)
and ξ(s) = sup r : ξ(s− r), s < r
LCCC Workshop - Lund Sweden - December 7, 2011
Lower Bound on Inter-sampling Interval[10,11,13]
sk ak sk+1 ak+1sk-1 ak-1
Dk
Tk
ek(sk)
gap
func
tion,
ej
E(|xk|) = quantization error
ek(sk+1)ek(ak)
• kth Gap at kth transmission time is |ek(sk)| < E(|xk|)
• kth Gap at k + 1st transmission time is
|ek(sk+1)| ≤ E(|xk|)eLkTk +Ψ(xk, xk−1)
Lk
(eLkTk − 1
)
where Ψ(x,xk−1) = |f(xk,K(xk))|+ 2|f(xk,K(xk−1)|
LCCC Workshop - Lund Sweden - December 7, 2011
Lower Bound on Inter-sampling Interval
sk ak sk+1 ak+1sk-1 ak-1
Dk
Tk
ek(sk)gap
func
tion,
ej
ek(sk+1)θ(|xk|)event-trigger
• sk+1, is the first time when |ek(t)| = θ(|xk|). We refer toθ(·) : R → R as the event-triggering function.
• With |ek(sk+1)| = θ(|xk|), our earlier constraint becomes
Tk >1
Lk
(ln
(1 +
Lkθ(|xk|)Ψ(xk, xk−1)
)− ln
(1 +
LkE(|xk|)Ψ(xk, xk−1)
))
LCCC Workshop - Lund Sweden - December 7, 2011
Upper Bound on Stabilizing Delay[11,13]
sk ak sk+1 ak+1sk-1 ak-1
Dk+1
gap
func
tion,
ej
ek(ak+1)θ(|xk|)ξ(|xk|)
event-trigger
stability-threshold
• Using similar techniques, we find
|ek(ak+1)| ≤ θ(|xk|)eLkDk+1 +|f(xk,K(xk))|
Lk
(eLkDk+1 − 1
)
• Asymptotic stability requires |ek(ak+1)| ≤ ξ(|xk|), so the sta-bilizing delay satisfies
Dk+1 ≤ 1
Lk
(ln
(1 + Lk
ξ(|xk|)|f(xk,K(xk))|
)− ln
(1 + Lk
θ(|xk|)|f(xk,K(xk))|
))
LCCC Workshop - Lund Sweden - December 7, 2011
Asymptotic Stability[11,12]
Assume that the event-triggering threshold θ satisfies
E(s) ≤ θ(s) ≤ ξ(s)
for any s ∈ R+ and assume the delay Dk < minT k, Dk where
T k =1
Lk
(ln
(1 +
Lkθ(|xk|)Ψ(xk, xk−1)
)− ln
(1 +
LkE(|xk|)Ψ(xk, xk−1)
))
Dk =1
Lk−1
(ln
(1 + Lk−1
ξ(|xk−1|)|f(xk−1,K(xk−1))|
)
− ln
(1 + Lk−1
θ(|xk−1|)|f(xk−1,K(xk−1))|
))
Then the closed-loop quantized event-triggered system is asymp-totically stable with an inter-sampling interval Tk which is alwaysbounded below by T k > 0.
Quantization Error Stability Threshold
LCCC Workshop - Lund Sweden - December 7, 2011
Stabilizing Bit Rate[11,12]
state trajectory
xk
x : |x− xk| = θ(|xk|)
E(|xk|)quantization
error
Nk=Number of Bits used to represent sampled state xk
Assuming |x| is the sup-norm, then
Nk = log2 2n+ (n− 1)
⌈log2
⌈θ(|xk−1|)E(|xk|)
⌉⌉bits
To guarantee the asymptotic stability of the closed-loop system,these bits must be delivered within the delay
Dk ≤ Dk ≡ 1
Lk−1
(ln
(1 + Lk−1
ξ(|xk−1|)|f(xk−1,K(xk−1))|
)− ln
(1 + Lk−1
θ(|xk−1||f(xk−1,K(xk−1))|
))
A bit rate stabilizing this system must therefore be
Rk =Nk
Dk> Rk ≡ Lk
ln 2(A(|xk|)(n− 1) +B(|xk|))
LCCC Workshop - Lund Sweden - December 7, 2011
Conditions for Zero Bit Rate[11]
In some cases, we can show that the stabilizing bit rate goesto zero as the system approaches its equilibrium point.
System Equations x1 = x31 + 2x3
2 + u
x2 = −x31 − x3
2
Switching Condition: |ek(t)| = θ(|xk|) = 0.015|xk|u = −2x3
1 − x32
Feedback Control
0 20 40 60 80 100−2
−1
0
1
2
3
time:s
stat
e
x1 using event triggered quantizationx2 using event triggered quantizationx1 using periodic quantizationx2 using periodic quantization
0 50 100 150 200 250 3000
1
2
3
4
5
time:s
log 10
(bit−
rate
): bi
t/s
stabilizing bit−rate using event triggered quantizationstabilizing bit−rate using peiodic quantizationlower bound on stabilizing bit−rate using peiodic quantization in [8]
LCCC Workshop - Lund Sweden - December 7, 2011
Efficiently Attentive Stabilizing Bit Rates[10,12]
A bit rate is efficiently attentive if it is an increasing function of state
It can be very difficult to determine the minimal stabilizing bit rate. In such cases, a reasonable option is to require thebit rate to be efficiently attentive
Assume that the delay Dk < Dk, and the event-triggering satisfies
E(s) ≤ θ(s) ≤ ξ(s)
Let φc, φu ∈ K such that |f(x,K(x))| ≤ φc(|x|) and |K(x)| ≤ φu(|x|).If we also know that
lims→0
θ(s)
E(s)< ∞, lim
s→0
φc(s)
θ(s)< ∞, lim
s→0
φu(s)
θ(s)< ∞
then there exists a continuous, positive definite, increasing function R(|xk|) suchthat if the actual bit rate is greater than R, then the system is asymptotically stable.
LCCC Workshop - Lund Sweden - December 7, 2011
Simulation Example[12,18]
System EquationsThis example extends prior work to essentially bounded disturbances
Controllerx1 = x3
1 + 2x32 + u1 + w1
x2 = −x31 − x3
2 + u2 + w2u1 = −3x3
1, u2 = −3x32
Event Trigger and Quantization Mapθ1(s, w) = 0.075s1.5 + 0.05w
E1(s, w) = 0.025s1.5 + 0.017w
0 50 100 150 200 250 3000
1
2
3
4
time:s
log 10
(bit−
rate
): bi
t/s
r 1
r 1r 2
r 2
predicted bit-rate bound
actual stabilizing bit-rate
Predicted bit-ratesare conservativebound on actualbit-rates
LCCC Workshop - Lund Sweden - December 7, 2011
Wireless Networked Control Systems[15]
Event-triggering gives rise to sporadic message streamsin wireless networked control systems.
u1 u2 u3
y1 y2 y3y1 y3
y2
After an impulsive disturbance is applied to middle cart,one can bound the future inter-sampling times and bit-raterequirements of all controllers.Can we use this information to reschedule controller transmissionto maintain both overall physical system performance whilestaying within communication network’s capacity limits?
LCCC Workshop - Lund Sweden - M.D. Lemmon - December 7, 2011
Safety-Critical Systems
One concern with event-triggered systems is that they are illsuited for safety-critical systems.In the presence of disturbances, however, event-triggered solutions must be implemented with a minimum samplingfrequency whose size is determined by the “disturbance”. With efficiently attentive systems, event frequency increasesin the presence of impulsive disturbances.
http://www.nd.edu/~lemmon/projects/NSF-05-1518/heli-movie/
LCCC Workshop - Lund Sweden - December 7, 2011
Future Directions
Event-trigger design is based on existence of ISS controller (input disturbances). This may not always be possible. One solution may be to extend framework to iISS controllers[16].These results provide some guidance on the selection ofcontrollers, quantizers, and event-triggers. We still need toformalize this guidance into a design procedure.
Similarities between dynamic quantization and the quantized event-triggers. When do we achieve the known necessary andsufficient stabilizing bit rates for linear systems?
Efficiently attentive systems provide a basis for co-design of communication/controller in a deterministic setting. Is it possible toextend these ideas to a stochastic setting? One possible approachwould involve the use of stochastic ISS concepts[17].
Relax conservativeness of bounds[19].
LCCC Workshop - Lund Sweden - December 7, 2011
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