EfficientlyAttentive Quantized Event-Triggered Systemslemmon/projects/NSF-05-1518/... · 2011-12-02 · LCCC Workshop - Lund Sweden - December 7, 2011 EfficientlyAttentive Quantized

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


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?


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)|


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


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.


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.

,


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


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.


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 ?


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


Lower Bound on Inter-sampling Interval[10,11,13]


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)|


Lower Bound on Inter-sampling Interval


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)

))


Upper Bound on Stabilizing Delay[11,13]


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))|

))


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


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|))


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]


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.


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


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/


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].


References1.) R.W. Brockett and D. Liberzon. Quantized feedback stabilization of linear systems. Automatic Control, IEEE Transactions on, 45(7):1279–1289, 2000.

2.) S. Tatikonda and S. Mitter. Control under communication constraints. Automatic Control, IEEE Transactions on, 49(7):1056–1068, 2004.

3.) D. Nesic and A.R. Teel. Input-output stability properties of networked control systems. IEEE Transactions on Automatic Control, 49(10):1650–1667, 2004.

4.) P. Tabuada. Event-triggered real-time scheduling of stabilizing control tasks. IEEE Transactions on Automatic Control, 52(9):1680–1685, September 2007.

7.) J.H. Sandee, P.M. Visser, and W.P.M.H. Heemels. Analysis and experimental validation of processor load for event-driven controllers. In IEEE Conference on Control and Applications (CCA), pages 1879–1884, Munich, Germany, 2006

6.) K.J. Astrom and B.M. Bernhardsson. Comparison of Riemann and Lebesgue sampling for first order stochastic systems. In Proceedings of the 41st IEEE Conference on Decision and Control, volume 2, pages 2011–2016, Las Vegas, Nevada, USA, December 10-13 2002.

5.) K-E Arzen. A simple event-based PID controller. In Procedingsof the 14th World Congress of the International Federation of Automatic Control (IFAC), Beijing, P.R. China, 1999.

8.) T. Henningsson, E. Johannesson, and A. Cervin. Sporadic event-based control of first-order linear stochastic systems. Automatica, 44(11):2890–2895, November 2008.

13.) X. Wang and M.D. Lemmon. Self-triggered feedback control systems with finite-gain l2 stability. IEEE Transactions on Automatic Control, 54(3):452–467, March 2009

15.) X. Wang and M.D. Lemmon (2011), Event-triggering in distributed networked control systems, IEEE Transactions on Automatic Control, volume 56, number 3, pages 586-601, March 2011.

14.) X. Wang and M.D. Lemmon (2010), Self-triggering under state-independent disturbances, IEEE Transactions of Automatic Control, vol 55, no. 6, pages 1494-1500, 2010.

11.) L. Li, M.D.Lemmon, and X. Wang (2012), Stabilizing Bit-Rates in Quantized Event Triggered Control Systems, submitted to Hybrid Systems: computation and control, Beijing, China, April 2012.

10.) X. Wang and M.D. Lemmon (2011), Attentively Efficient Controllers for Event-triggered Feedback Systems, IEEE Conference on Decision and Control, Orlando Florida, December 2011.

12.) L. Li, M.D.Lemmon, and X. Wang (2012), Stabilizing Bit-Rates in Perturbed Event Triggered Control Systems, submitted to IFAC Conference on Analysis and Design of Hybrid Systems, Eindhoven, Neitherlands, 2012.

16) D. Angeli, E.D. Sontag, and Y. Wang. A characterization of integral input-to-state stability. IEEE Transactions on Automatic Control, 45(6):1082–1097, 2000.

18) D. Liberzon and J.P. Hespanha. Stabilization of nonlinear systems with limited information feedback. Automatic Control, IEEE Transactions on, 50(6):910–915, 2005.

9.) M.D. Lemmon (2010), Event-triggered Feedback in Control, Estimation, and Optimization, in Networked Control Systems, editors A. Bemporad, M. Heemels, M. Johansson, Volume 405 Lecture Notes in Control and Information Sciences,pages 293-358, Springer-Verlag Berlin Heidelburg, 2010.

17) H. Deng, M. Krstic, and R. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Transactions on Automatic Control, vol. 46, no. 8, pp. 1237–1253, 2001.

19) Anta, A.; Tabuada, P.; , "Isochronous manifolds in self-triggered control," Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on , vol., no., pp.3194-3199, 15-18 Dec. 2009

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EfficientlyAttentive Quantized Event-Triggered Systemslemmon/projects/NSF-05-1518/... · 2011-12-02 · LCCC Workshop - Lund Sweden - December 7, 2011 EfficientlyAttentive Quantized