Optimal Fluctuations for Satisfactory Performance under Parameter Uncertainty

HJ Kadim LJMU, England, UK

enghkadi@livjm.ac.uk

Abstract

Maintaining constant performance in the presence of a set of changes in parameters and unwarranted events has become an essential aspect of present system designs. Knowing a predefined upper limit, for which a drop in performance is said to be satisfactory, enables autonomous systems to perform a control action to mitigate changes that violate such a predefined limit. This paper introduces an analytical model for optimisation of the maximum possible parameter fluctuations that permit robust operation.

1. Introduction A key challenge of multi-tasking [1], uncertainty [2] and cross-coupling effects of parameters [3], is the ability of a system to maintain consistent and satisfactory performance, under conflicting functional requirements as well as unwarranted events. Such requirements and events could be dictated by internal or/and external processes. Therefore, the design goal is to the development of systems that are able to autonomously adapt in response to an internal/external change. Adaptive systems employed in applications such as medical [4], defence and security [5][6], require the constant mode of operation and, therefore, special effort has been placed to develop fault tolerant systems. The emergence of wireless sensor networks [7][8] and desperate applications (e.g. control and condition monitoring of jet engines, structural monitoring of test prototype) will further increase the importance of fault tolerance. The main purpose of tolerant systems is to maintain satisfactory performance under the effect of uncertainty. It is important to note that some uncertainty is always present in the behaviour of a system – for instance, systems are not perfect and may exhibit unpredictable behaviour - as well as the

environment of the system – i.e. the impossibility of knowing in advance the exact disturbance a system may experience. There are two aspects associated with tolerant systems [9]: (i) Performance – i.e. how well a system copes with

unexpected disturbance. (ii) Robustness – i.e. how resilient a system is in the

face of internal changes in behaviour. The performance of a system could be viewed as

a function of both internal and external parameters that influence its behaviour. Depending on the nature and the degree of effect of parameters, a system may or may not exhibit a noticeable change in performance. Therefore, it is sensible to consider a reference-range (i.e. lower end of the range – upper end of the range), within which a system is considered to operate satisfactorily under uncertainty – aspect (ii) above. A system’s ability to satisfy such a requirement can also be considered an important part of its performance. In order to determine a system’s robustness to parameter fluctuations, internal and external, this paper presents an analytical model to determine the optimal upper limits necessary for satisfactory operation. One way of achieving that is by synthesising the degree of effect of parameter fluctuations on the performance of a system. The model is characterised by two kinds of equations: (i) Equations express the relationship between internal/external parameters; (ii) Equations express the constraints on the operation. This paper is structured as follows: Section 2 introduces analytical modelling of parameters. Simulation results are presented in Section 3. Conclusions are given in Section 4. 2. Analytical modellling A system may be considered as a function of a set of parameters that may affect its behaviour, with a varying degree. It is possible to model parameter fluctuations as infinitesimal transfer functions within

978-1-4577-1958-5/11/$26.00 ©2011 IEEE

the main transfer function of the system. If all the infinitesimal transfer functions are represented by p and are related to the main transfer function T as shown in Fig.1, then the transfer function ‘T’ of a system, in the presence of parameter fluctuations, can be defined as [10]:

pnTT (1)

With

0

2

1

с

00

00

00

Д

nn

(2)

where T and Tn represent the nominal transfer function and the true transfer function, respectively.

Figure 1 Modelling of parameter fluctuations The relationship ‘’ in (3)

)P€,E()P€,E( ooii

(3)

is given as follows: From Fig.2:

iс,1-in,

jj

1-jij

i

jo

,

1,1

Д)Tw)w(с

с

njni

ji

(2)

Considering changes in the biological layer:

inj

ii

njni

ji

TE ,,ji

ji

io

,

1,1

)сw(E

(3)

with

jo

jo

ji

jj

jiinj

o

ji

njni

ji

wwTI

:

1,

,

1,1

)( (4)

and

ioiiioin EEET

injjij

iii

ionjni

ji

TwweE

E

,,,, :

,21

11,

,

,,

1,1

(5)

where I is a unity matrix. It is assumed that in the absence of parameter fluctuations the designed system is stable. In the presence of parameter variations a system could maintain a stable behavioural operation as long as:

€ (6)

where represents a maximum bound that ensures stability and desired performance. For an optimal change in parameters that retain satisfactory performance, consider:

~nn TTT (7)

with

1

~

{ , ,..., }; 0, 0i i i nn n f i n

T T

(8)

where ~nT represents the nominal transfer function

under parameter fluctuations. Rewriting (7)

~nn TTT (9)

Restricting the changes in сД to k , and from (9)

nn TTT (10)

with

nn TTT (11)

and € (12)

From (10) and (11), and taking into account (12)

nn TTT (13)

with

1

~

{ , ,..., }; 0, 0

( )

( ) i i i n

n

g f i n

T sg

s

(14)

Two alternatives: (i) Manipulating (8) and (14) ~ ~ 2

nT g (15)

From (7) and (15) ~ ~ 2T g g (16)

(ii) Applying (14) to (7) 2

( , )n

nn

T f T

TT T

g

(17)

( , )

1n

n

n T f T

TT

T g

(18)

Choosing (ii), T/Tn is plotted as a function of g in Fig.2

io )( s

iE)( tnT

oE

jwiw

For an optimal fluctuation in parameters that maintains satisfactory performance, and from Fig.2:

)(;0,0~

1

fggnT

T (19)

To satisfy (19) – i.e. changes in parameters have negligible effect on the functional behaviour of a system - g has to take a small value. Therefore, changes in parameters have to be restricted to small changes.

Figure 2 Optimal changes in g that maintain satisfactory behavioural performance

3. Simulation results A given system could be viewed as a process with its performance being a function of parameter fluctuations. Fig.4 and Fig.5 show the degree of effect of parameter uncertainty on the nominal response, for varying κ. Depending on the applications, and form Figures 4 and 5, upper and lower limits (< ) for

parameter fluctuations could be identified that determine acceptable performance under parameter uncertainty. Simulation I: Consider an optical biosensor, as shown in Fig.3, with the following parameters:

i and j are parameters representing stimulation

and absorption processes, respectively -

),( tzfi , ),( tzfj . ~,zn is a parameter that

is a function of distance ‘z’ and ; )(tf - e.g.

temperature. In general,

n

ii

m

iin psszsKsT

1

2

1

)()(),( (20)

Taking into consideration (6), and by applying (19) to Tn in (20):

)(1

^

^ fp

n

jj

j

ps

(21)

From (21), a set of transient responses can be generated, as shown in Fig4.

Figure 3 (a) Optical waveguide; (b) Block diagram representation of the functional behaviour of the waveguide in (a).

Figure 4 Behavioural responses under varying k.

0

nTT

T/T

n→

g→

- z

l

~, zn

z

ya

(a)

)(1

^

^ fp

n

jj

j

ps

tconsR

Rtan:;116.0:

tconsR

Rtan:;116.0:

Time (sec)

Am

plitu

de (

V)

(b)

-I

ei(t)

∑ ~,

~

zn

n

eo(t)

∑ -I

~,

~

zn

n

i

j

zn

n

~,

~

i

j

eo(t)

∑

ei(t)

∑

I

Simulation II: Consider an aircraft with rolling about wind-axes/inertia-axes motion, as shown in Fig.5, with the following parameters: r~ := rτ; where r: rate of yaw, τ: magnitude of time unit. nr=Nr/Ix, where Nr: yawing moment derivative due to rate of yaw. nv= μNv/ix, where μ: propeller efficiency, Nv: yawing moment derivative due to sideslip velocity. yv: side-force derivative due to sideslip velocity

Figure 5 (a) Block diagram representation of motion in directional oscillation with lateral freedom; (b) Phase responses under parameter fluctuations. 4. Conclusion Predicting the performance of a system, under parameter uncertainty, is desirable for determining the robustness of the system as well as for testing and verification of its functional behaviour. Under normal operation, responses that are not confined to a predefined set of responses – governed by an upper limit of uncertainty- indicate abnormal behaviour. Such a set of responses could be employed, by an autonomous system, as a reference, to perform e.g. self-test, to verify a functional property, or to perform a control action to counterbalance internal/external changes that impair its function. 5. References

[1] H.J. Kadim, “Analytical Modelling for Adaptive

Multi-Purpose On-Chip Optical Interconnect”, NASA/ESA Int. Conf. on Adaptive Hardware and Systems, Edinburgh, UK, 5-8 August 2007, pp. 209-213.

[2] J.M. Maciejowski, Multivariable feedback design, (Addison-Wesley Publishers 1994).

[3] H.J. Kadim, “State-Space Modelling of Anticipatory Behaviour for Self-Adaptability with Applications to Biosensors”, NASA/ESA Int. Conf. on Adaptive Hardware and Systems, ESTEC, ESA, Noordwijk, the Netherlands, June 2008, pp 467-471.

[4] P. Si, A. Hu, S. Malpas and D. Budgett, “A Frequency Control Method for Regulating Wireless Power to Implantable Devices”, IEEE Trans. On BioMedical Circuits and Systems, Vol. 2, No. 1, March 2008, pp. 22-29.

[5] H.J. Kadim, “Analytical Modelling: An Investigation into the use of Smart biosensors as Stealth Countermeasures”, IEEE/ECSIS Symposium on Bio-inspired, Learning, and Intelligent Systems for Security, Edinburgh, UK, August 2008, pp.97-100.

[6] N. Bartzoudis and K. McDonald-Maier, “An Adaptive Processing Node Architecture for Validating Sensors Reliability in a Wind Farm”, Bio-Inspired Learning and Intelligent Systems for Security, BLISS 2007, pp. 83-86.

[7] E. Yang, N. Haridas, A. EL-Rayis, a.t. Erdogan, T. Arslan and N. Barton, “Multiobjective Optimal Design of MEMS-based Reconfigurable and Evolvable Sensor Networks for Space Applications,” NASA/ESA Int. Conf. on Adaptive Hardware and Systems, Edinburgh, UK, 5-8 August 2007, pp. 27-34.

(a)

Frequency (rad/sec)

Pha

se (

degr

ees)

y

x

N

Nr

r

YNvβ

β

y

x

x

…tconsyrv

v

nntan:

~ 10.0

ooo tconsyvr

v

nntan:

~ 10.0

Nominal response, unity parameters

+++ tconsyvr

v

nntan:

~ 50.0 tconsyrv

v

nntan:

~ 40.0

(b)

 

0..0

..::

0..0

 

-I

ei(t)

∑

-I

vr yn ~

eo(t)

ei(t)

∑

I

vvr nyn ∑

[8] T. Vladimirova, et al, “Characterising Wireless Sensor Motes for Space Applications,” NASA/ESA Int. Conf. on Adaptive Hardware and Systems, Edinburgh, UK, 5-8 August 2007, pp. 43-50.

[9] R.C. Dorf and R.H. Bishop, Modern control systems, (Addison-Wesley Publishing Co. 1995).

[10] H.J. Kadim, “Mathematical Modelling of Parameter Fluctuations with Applications to Fault Detection in Analogue VLSI Circuits”, Radioelectronics and Informatics Journal, Issue No. 1, 2004, pp 103-108.