on the design of wireless event-triggered control systems · on the design of wireless...

On the Design of Wireless Event-triggered Control

Systems

Daniel E. Quevedo

Department of Electrical Engineering (EIM-E)

Paderborn University, Germany

[email protected]

February 2017

Processing Power Mismatch

5

An Inherent Assumption

Pro

cess

ing

pow

er

Time

Provided by microprocessor

Required by control algorithm

•  The control designer designs a control algorithm assuming that a control input can be calculated at every time step.

•  The processor designer assumes that the worst case execution time the microprocessor can provide is good enough.

Controller

Sensor System Actuator

6

Challenging the Assumption

Pro

cess

ing

pow

er

Time



•  Either the processor availability or the computation required by the control algorithms is uncertain and time-varying.

•  The actuator still needs a control input at every time step.

Controller


The traditional assumption about the processor always being ableto execute the control algorithm during the available computationslot may break down.Overdimensioning the processor or increasing the slot is often notan option, see Airbus European Patent Application “Robustsystem control method with short execution deadlines”, 2013.

Daniel Quevedo ([email protected]) Control with limited resources 2013, Uni Melbourne 3 / 34

0 20 40 60 80 1000

5

10

15

20

25

30

Daniel Quevedo ([email protected]) Control with limited resources February 2017 1 / 43

More general situation:

Processing Power Mismatch in Networked and

Embedded Systems

Traditional

5


Pro

cess

ing

pow

er

Time





Controller


Networked/Embedded

6


Pro

cess

ing

pow

er

Time





Controller


The traditional assumption about the processor always being able

to execute the control algorithm during the available computation

slot may break down.


This talk

1

presents a modeling framework for closed-loop control, when

processing and communication resources are limited,

Controller Plant

Channel

2

describes an intuitive algorithm to synthesise such loops,

controls

delete buffer

take u(k)from buffer

calculate

3

illustrates how stability can be analysed using random-time

state-dependent drift conditions.

k

8

d

|x|

k

0

k

1

k

2

k

3

k

4

k

5

k

6

k

7


Outline

1

Event-triggered Control with Dropouts

System Model

Baseline Algorithm

Numerical Example

2

Anytime Control Algorithm

Method Description

Numerical Example - revisited

3

Stability Analysis

Assumptions

The Baseline Algorithm

The Anytime Algorithm

Comparison of Bounds

4

Conclusions


Event-triggered Control with Dropouts System Model

System Model

availability

Controller

Plant

x(k + 1) = f (x(k), u(k))

u(k) x(k)

Erasure Channel

|x(k)| < d ?

�(k)

processor

The quantity d is a design parameter which trades communication

channel utilization for control performance.

Transmission between sensor and controller node is through a

delay-free (wireless) link with dropouts:

�(k) =

8><

>:

0 if x(k) is received with errors (a dropout occurs),

1 if x(k) is received error-free,

2 if the sensor did not transmit at time k (i.e., |x(k)| < d).


Event-triggered Control with Dropouts Baseline Algorithm

We are interested in a situation, where a “good” state-feedback

control policy : Rn ! Rp

has been pre-designed.

Processing resources for control may, at times, be insufficient to

evaluate within the pre-allocated time-slot of length ⌧ .

A direct implementation of the control policy , yields the

Baseline Algorithm

u(k) =

8><

>:

(x(k)), if �(k) = 1 and (x(k)) was evaluated

between times kT and kT + ⌧ ,

0p

, otherwise,

where u(k) with k 2 N0

denotes the plant input which is

applied during the interval [kT + ⌧, (k + 1)T + ⌧).


Event-triggered Control with Dropouts Numerical Example

Numerical Example - Baseline Algorithm

Plant model and controller

Plant model with i.i.d. disturbance distributed as N (0, 1):

x(k + 1) = �x(k) + 0.1 sin(x(k)) + u(k) + w(k), x(0) = 20.

Control policy is taken as:

(x) = x � 0.1 sin(x) + ⇢|x |, ⇢ = 0.9.

Resources

Pr{�(k) = 1 | |x(k)| � d} = 0.5

Pr{processor is available |�(k) = 1} = 0.8

The closed loop is characterised by:

x(k + 1) =

8><

>:

⇢|x(k)|+ w(k), if processor is available

and �(k) = 1,

�x(k) + 0.1 sin(x(k)) + w(k), otherwise.


Event-triggered Control with Dropouts Numerical Example

Performance (averaged over 1000 realisations)

Empirical Cost

1

500

700X

k=201

x

2(k)

!.

0 2 4 6 8 105

10

15

20

25

30

d

Larger thresholds give a worse

control performance!

Channel Utilisation (%)

Percentage of time steps at

which the sensor transmits, i.e.,

where �(k) 2 {0, 1}.

0 2 4 6 8 100

20

40

60

80

100

d

Larger thresholds lead to less

communication uses!


Anytime Control Algorithm Method Description

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions



Event-triggered Anytime Control Algorithm

1

availability

Controller

Plant

x(k + 1) = f (x(k), u(k))

u(k) x(k)

Erasure Channel

|x(k)| < d ?

�(k)

processor

The purpose is to make efficient use of the communication and

processing resources available.

The algorithm calculates sequences of tentative future inputs.

These are stored in a local buffer.

Buffered values may be used when, at some future time steps, the

processor availability precludes any control calculations, or a

dropout occurs (�(k) = 0).

1

Q., Gupta, Ma and Y¨uksel, IEEE TAC, 2014.



�(k) = 0

calculate controls (if possible)

and write them into buffer

set u(k) 0p

;

delete buffer

�(k) = 2

�(k) = 1

the buffer

take u(k) from

If �(k) = 1, then x(k) is used to calculate N(k) 2 {0, 1, . . . ,⇤}tentative control values using :

u

0

(k) = (x(k))

u

1

(k) = �f (x(k), u

0

(k))�, etc.

This sequence is stored in a local buffer of size ⇤. Its contents may

be used when the processor is unavailable or when �(k + `) = 0.



Comments

The algorithm does not require prior knowledge of processor

availability. Therefore, the control task can be preempted.

The buffer state b(k) provides the current plant input,

u(k) = b

1

(k).

If N(k) > 1, then b(k) also contains suggested future inputs.

If the buffer runs out of tentative plant inputs, then u(k) = 0p

.

It is convenient to introduce the

effective buffer length

�(k) =

8><

>:

N(k) if N(k) � 1,

max{0,�(k � 1)� 1} if N(k) = 0 and �(k) 2 {0, 1},

0 if �(k) = 2.



Example

Suppose that ⇤ = 4 and that {N(0),N(1),N(2),N(3)} = {4, 0, 1, 2}.

The Anytime algorithm provides

{b(0), b(1), b(2), b(3)} =

8>><

>>:

2

664

u

0

(0)u

1

(0)u

2

(0)u

3

(0)

3

775,

2

664

u

1

(0)u

2

(0)u

3

(0)0

p

3

775,

2

664

u

0

(2)0

p

0p

0p

3

775,

2

664

u

0

(3)u

1

(3)0

p

0p

3

775

9>>=

>>;

{�(0),�(1),�(2),�(3)} = {4, 3, 1, 2}, and plant inputs

{u(0), . . . , u(3)} = {(x(0)),�f (x(0),(x(0)))

�,(x(2)),(x(3))}.

If the baseline-algorithm is used, then

{u(0), u(1), u(2), u(3)} = {(x(0)), 0p

,(x(2)),(x(3))}.


Anytime Control Algorithm Numerical Example - revisited

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions




Resources

Suppose that ⇤ = 4 and that

Pr{�(k) = 1 | |x(k)| � d} = 0.5

Pr{N(k) = j |�(k) = 1} = 0.2, j 2 {0, 1, 2, 3, 4}

200 300 400 500 600 7001.8

2

2.2

2.4

2.6

2.8

3

3.2

k

x

Anytime AlgorithmBaseline Algorithm

State trajectory

with d = 2.

The anytime

control algorithm

outperforms the

baseline controller.



Performance (averaged over 1000 realisations)

Empirical Cost

1

500

700X

k=201

x

2(k)

!.

0 2 4 6 8 100

5

10

15

20

25

30

d



Percentage of time steps at

which the sensor transmits, i.e.,

where �(k) 2 {0, 1}.

0 2 4 6 8 100

20

40

60

80

100

d




Design trade-off – choice of threshold d

Empirical Cost versus Channel Utilisation

0 20 40 60 80 1000

5

10

15

20

25

30


Empi

rical

Cos

t


Performance Gains

For a given transmission rate, the proposed anytime control algorithm

reduces the empirical cost by approximately 40-50%.


Stability Analysis Assumptions

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions



Standing Assumptions

Globally Stabilising Nominal Controller

There exist functions V : Rn ! R�0

, '1

,'2

2 K1,

a

a constant

⇢ 2 [0, 1), and a control policy : Rn ! Rp

, such that

'1

(|x |) V (x) '2

(|x |),V (f (x ,(x))) ⇢V (x), 8x 2 Rn.

a

A function ' : R�0

! R�0

is of class-K1 (' 2 K1), if it is continuous,

zero at zero, strictly increasing, and unbounded.

Open-loop bound

With V as above, there exists ↵ 2 R�0

such that

a

V (f (x , 0p

)) ↵V (x), 8x 2 Rn.

a

For the numerical example described before, we have ↵ = 1.1.



Processor availability is i.i.d.

The process {N}N0

has conditional probability distribution

Pr{N(k) = j |�(k) = 1} = p

j

, j 2 {0, 1, 2, . . . ,⇤},

where p

j

2 [0, 1) are given.

For other realizations of �(k), no plant inputs are calculated:

Pr{N(k) = 0 |�(k) 2 {0, 2}} = 1.

Packet dropouts are i.i.d.

The transmissions are Bernoulli with packet transmission

success probability

Pr{�(k) = 1 | |x(k)| � d} = q.


Stability Analysis The Baseline Algorithm

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions




The closed loop is characterised by:

x(k + 1) =

(f (x(k),(x(k))), if N(k) � 1,

f (x(k), 0p

), if N(k) = 0.

We denote via p

0

the probability that the controller is unable to

calculate any control input, despite x(k) being available.

Stochastic Stability with the Baseline Algorithm

Suppose that E�'

2

(|x(0)|) < 1 and that

� , (1 � q)↵+ q

�p

0

↵+ (1 � p

0

)⇢�< 1.

Then there exist finite � and µ such that

E�'

1

(|x(k)|) ��k + µ, 8k 2 N

0

.

If d = 0, then µ = 0.



Sketch of proof

The process {x(k)}, k 2 N0

is Markovian.

Using the assumptions and writing x for x(0), we have:

E�

V (x(1))��x

=

2X

j=0

E�

V (x(1))��x ,�(0) = j

Pr{�(0) = j | x}

< �V (x) + (↵� �)'2

(d), 8x .

Thus, using the Markov property, we obtain

E�'

1

(|x(k)|) | x �k

V (x) +(↵� �)'

2

(d)

1 � �< 1, 8k 2 N

0

.

Taking expectation with respect to the distribution of the initial

state x(0) establishes the result.


Stability Analysis The Anytime Algorithm

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions



Preliminaries

x(k)

calculate controls (if possible)

and write them into buffer

set u(k) 0p

;

delete buffer

�(k) = 1

the buffer

take u(k) from

�(k) = 0

�(k) = 2

|x(k)| � d|x(k)| < d

�x(k),b(k)

�

Due to buffering, {x(k)}, k 2 N0

is not a Markov process.

This complicates the analysis significantly.



Preliminaries

To study stability of this event-based anytime algorithm, we will

obtain a state-dependent random-time drift condition of the form:

2

E{V (x(ki+1

)) | x(ki

) = �} D + ⌦V (�), ⌦ < 1,

where {k

0

, k1

, k2

, . . . } are special random time instants.

If {x(ki

) : i 2 N0

} is Markovian, then the above would ensure

exponential boundedness at the instants k

i

:

E�

V (x(ki

)) | x(k0

) = � ⌦i

V (�) +D

1 � ⌦, 8i 2 N

0

.

Depending on

1

system behaviour in-between instants k

i

2

the distribution of {k

i+1

� k

i

}boundedness at all instants k 2 N

0

may follow.

2

Y¨uksel and Meyn, IEEE TAC, 2013.



The Randomly Sampled Process

We begin our analysis, by denoting the random time steps where

the buffer is empty via

K = {k

i

}, i 2 N0

,

where

k

i+1

= inf

�k 2 N : k > k

i

, �(k) = 0

, k

0

= 0.

We also describe the amount of time steps between consecutive

elements of K via

�i

, k

i+1

� k

i

, i 2 N0

The following property of the randomly sampled process x is key:

Lemma

The plant state sequence at the time steps k

i

2 K, namely

{x(ki

) : k

i

2 K}, is Markovian.



System Behaviour

�7

k

0

k

1

k

2

k

3

k

4

k

5

k

6

k

7

k

8

�3

�4

�6

�5

d

|x |

�0

�1

�2

In the present disturbance-free case, the plant inputs are simply

given by

u(ki

) = 0p

, 8k

i

2 Ku(k

i

+ `) = (x(ki

+ `)), 8` 2 {1, . . . ,�i

� 1}.



Evolution of V (x(ki

))

�7

k

0

k

1

k

2

k

3

k

4

k

5

k

6

k

7

k

8

�3

�4

�6

�5

d

|x |

�0

�1

�2

It is then easy to see that

E{V (x(ki+1

)) | x(ki

) = �,�i

= �} ↵⇢��1

V (�), 8� 2 Rn.

Unfortunately, due to the event-triggering mechanism, the

distribution of {�i

}i2N

0

depends on x(ki

) and is difficult to

characterise.

Hence establishing a suitable drift condition is not so easy!



Conditioning Events

�6

k

0

k

1

k

2

k

3

k

4

k

5

k

6

k

7

k

8

�7

�2

�1

�0

d

|x |

�3

�4

�5

It turns out to be convenient to distinguish between the two cases:

1

the buffer is emptied due to lack of resources (�(ki+1

) 2 {0, 1}), and

2

it is emptied triggered by the plant state being in the desired region.

Only the first case influences stability conditions.



Finding an Upper-bound on the Drift

�6

k

0

k

1

k

2

k

3

k

4

k

5

k

6

k

7

k

8

�7

�2

�1

�0

d

|x |

�3

�4

�5

Accordingly, one can condition on �(ki+1

) and use the law of total

expectation to show that

E{V (x(ki+1

)) | x(ki

)} '2

(d) + E{V (x(ki+1

)) | x(ki

),�(ki+1

) 6= 2}.

Now, we can condition on �i

to

1

establish exponential boundedness at the instants k

i

2 K, and

2

upper-bound E�P

k

i+1

�1

k=k

i

V (x(k))��x(k

i

),�(ki+1

) 6= 2

.

This leads to...



Main Result

Stochastic Stability with the proposed Algorithm

Suppose that E�'

2

(|x(0)|) < 1 and that

⌦ ,X

�2N↵⇢��1Pr{�

i

= � |�(ki+1

) 6= 2} < 1.

Then there exist finite �, µ such that

max

k2{k

i

,ki

+1,...,ki+1

�1}E�'

1

(|x(k)|) �⌦i + µ, 8i 2 N.

If d = 0, then µ = 0.

The conditional distribution

Pr{�i

= � |�(ki+1

) 6= 2} = Pr��

i

= �� |x(k

i+1

)| � d

.

depends only on p

j

and q and can be easily characterised using

finite Markov Chain methods (“first return times to � = 0”).



Comments

Our result establishes a condition on

Iplant,

Icontrol law,

Ichannel, and

Iprocessor availability

which ensures stochastic stability of the event-based anytime

control loop.

The analysis can be extended to situations where processor and

communication resources are not i.i.d., but correlated.

Under continuity assumptions, related stability conditions can be

derived for plant models with disturbances.

a

a

Q., Ma and Gupta, IEEE TAC, 2015


Stability Analysis Comparison of Bounds

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions


Stability Analysis Comparison of Bounds

Resource Limitation Model

Suppose that the buffer size is set to ⇤ = 4 and that

Pr{�(k) = 1 | |x(k)| � d} = 0.5

Pr{N(k) = j |�(k) = 1} = 0.2, j 2 {0, 1, 2, 3, 4}

�

�

1 1.1 1.2 1.3 1.4 1.5 1.60

0.2

0.4

0.6

0.8

1Baseline AlgorithmAnytime Algorithm

The stability regions in the ↵-⇢ plane established for the anytime

control algorithm are larger than those for the baseline controller.


Conclusions

Outline

1


System Model

Baseline Algorithm

Numerical Example

2


Method Description


3

Stability Analysis

Assumptions




4

Conclusions


Conclusions

Conclusions

We have presented a framework for the study of control when

processor availability and communication resources are random.

IThe sensor node is event-triggered

and transmits data using a

communication link prone to dropouts.

Controller Plant

Channel

IThe control algorithm is executed with a processor that can provide

only time-varying and a priori unknown resources.

To better utilise the processor, the plant

inputs are calculated by an algorithm

that provides sequences.

controls

delete buffer

take u(k)from buffer

calculate

For general non-linear systems, we

used stochastic Lyapunov methods to

obtain sufficient conditions for stability.

k

8

d

|x|

k

0

k

1

k

2

k

3

k

4

k

5

k

6

k

7


Conclusions

Future Work

Many research problems remain open, e.g.,

characterising closed loop performance,

considering more than one control policy (e.g., numerics)

developing processor scheduling and cooperation strategies.

Further Reading

1

Quevedo, Gupta, Ma and Y¨uksel, “Stochastic Stability of

Event-Triggered Anytime Control,” IEEE TAC, Dec. 2014

2

Quevedo, Ma and Gupta, “Anytime Control using Input Sequences

with Markovian Processor Availability,” IEEE TAC, Feb. 2015

3

Huang, Dang, Ling and Quevedo, “Event-triggered Anytime

Control with Two Controllers,” Proc. IEEE CDC, 2015.

4

Ma, Gupta and Quevedo, “Collaborative Processing in Distributed

Control for Resource Constrained Systems,” IET Control Theory &

Applications, to be published.


Conclusions

Acknowledgements

A/Prof. Vijay Gupta

Department of Electrical Engineering

University of Notre Dame, USA

Dr. Wann-Jiun Ma

Department of Mechanical Engineering and

Materials Science Engineering

Duke University, USA

A/Prof. Serdar Y¨uksel

Department of Mathematics and Statistics

Queen’s University, Ontario, Canada

Processing Power Mismatch

5


Pro

cess

ing

pow

er

Time





Controller


6


Pro

cess

ing

pow

er

Time





Controller


The traditional assumption about the processor always being ableto execute the control algorithm during the available computationslot may break down.Overdimensioning the processor or increasing the slot is often notan option, see Airbus European Patent Application “Robustsystem control method with short execution deadlines”, 2013.

Daniel Quevedo ([email protected]) Control with limited resources 2013, Uni Melbourne 3 / 34

Thank You!


on the design of wireless event-triggered control systems · on the design of wireless...

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