Codesign Strategy based upon Takagi Sugeno Control

Design and Real-Time Computing Integration

Héctor Benítez-Pérez

Departamento de Ingeniería de Sistemas Computacionales y Automatización

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas

Universidad Nacional Autónoma de Mexico

Apdo. Postal 20-726, Del. A. Obregón, México D. F., CP. 01000, Mexico

Abstract. Nowadays the idea of network control systems design considering

the restriction results from schedulling analysis becomes a challenge based

upon the persepctive of codesign point view since both analytic tools are

pursued. A clear strategy is to define in cascade mode the scheduling analysis

and afterwards the stability analysis of the respective control strategy. However,

any modification in both structures has an integrated impact which is necessary

to measure. In that respect the use of time delay impact is a suitable strategy to

be followed and is explored in this paper. The use of codesign is to pursuit as a

two objective strategy the definition of a valid metric that represents the effects

in both, following the idea that stability analysis is affected according to the

schedullability analysis. In both analysis a relaxation at the local conditions is

feasible but it will have a global cost giving a non valuable approximation.

1 Introduction

Nowadays, the use of multiple tools for complex systems design like Real-Time

distributed systems need any background in terms of design, for instance, the relation

between analysis constrains expresed in different metrics like reliability,

schedullability, safety, stability and so on. The need to relate these measures can be

pursued in terms of a hollistic design or codesign sctrategy [4]. In order to define this

kind of strategy it is necessary to determine the effects of each technique over the rest.

Several approaches can be persued like decision trees, common metrics definitions,

stochastic processes and others. However, this problem remains open in terms of a

standard approximation amongst the complexity of the goal. One interesting

approximation is based upon the codesign way of thinking, by choosing one specific

aspect from each technique. Therefore, the individual achievemnt of every technique

considering its effects over the rest should be pursuit. The objective of this paper is to

review this approximation over a Real-Time Distributed System considering its

effects over a specific application such as dynamic system. Specifically, this is

studied by the use of schedulling and control design, where schedulability and

strability analysis are reviewed to guarantee the feasability of this strategy.

Benítez-Pérez H. (2008).

Codesign Strategy based upon Takagi Sugeno Control Design and Real-Time Computing Integration.

In Proceedings of the 4th International Workshop on Artiﬁcial Neural Networks and Intelligent Information Processing, pages 80-88

DOI: 10.5220/0001502800800088

 SciTePress

Although this is dependant on the specific strategy to be followed, global

characteristics such as the respective analysis can be pursued.

Further on, the union of different techniques allows a holistic view of the problem,

although, the result can be restricted to specific algorithms and the inherent restriction

of the case study.

Section 2 presents a review of structural codesign based upon the schedulling

approximation. Section 3 presents control codesign where fuzzy logic control law is

used in order to incorporate scheduler information onto stability analysis. Section 4

presents some concluding remarks of thi approximation.

2 Structural Codesign

The codesign proposal is based upon the iteration between schedulability and stability

analysis following online approximation as shown in Fig. 1.

SCHEDULLING

EVALUATION

Stability

Test

Time Delays

Evaluation

Valid Scheduler

Reconfiguration

Yes

Scheduler Proposal

External

Event

Fig. 1. Dynamic Reconfiguration in terms of Codesign strategy.

Any classical scheduller [1] bounds the time behaviour of the tasks in terms of their

own priority where certain modification amongst them produces important

differences. For instance, consumption time from tasks named as sensors (

t ),

actuators (

t ) and controllers (

t ) can be seen as follows:

cjcjcj

ajajaj

sjsjsj

ttt

(1)

Where the total time spent by the tasks is equal to

∑∑∑

111

tttt

===

++=

(2)

Index i is the number of tasks involved per structural elements like sensors (M),

actuators (P) and controllers (N). Index j is the current scenario defined by the

scheduller. These time delyas that are the result of priority modification on the

peripheral elements as individual manner should change the design parameters at the

control law. At least these time delays provide enough information to perform an

adecuate control law design. Fig. 2 shows how

t should be bounded to Control

period of time.

sensors

controllers

actuators

Total time Consumed by system

Time

Fig. 2. Bounded Time Inherent to Control Period of Time.

Where T is a long enough time window where

t should take place.

tΔ

is the

variation presented per element [12] during element actuations according to the

pursued scheduler algorithm such as EDF, RM or FTT [1] [2] [3] [15].

To guarantee schedulability is necessary an effective performance from the control

law [4]. This can only be pursued if only if the time delays exist within the bounded

time delays used to design a suitable control law as a classical gain schedulling

strategy. When task schedulling is performed, it implies a variation

tΔ

giving a

modification to the control law. Therefore the classical scehdulability analysis [1] can

be modified in order to incorporate this kind of uncertainty giving the following result

≤

∑

U (3)

Where c

represents the consumption time of each task,

is the related uncertainty,

is the related period, N is the number of tasks and U is the total relation between

cosumptions and periods. This last value should be less than one in order to guarantee

schedulability. It is important to deploy that any classical schedulling algorithm

should fit into this condition as longs as the tasks are periodicals (which is the case

herein) and the inherent uncertaintes should be fit into the same condition.

In fact, these time delays can be seen like a phase modification within the

communication period from the involved processes. This scenario presents a complete

phase modification at the entire system.

The communication network plays a key role in order to define the behaviour of the

dynamic system in terms of time variance giving a nonlinear behaviour. In order to

understand such a nonlinear behaviour, time delays are incorporated by the use of

real-time system theory that allows time delays to be bounded even in the case of

causal modifications due to external effects. In order to model this behaviour a

reconfigurable real time schedulling algorithm is proposed, named Structural

reconfiguration algorithm (SRA).

This algorithm bounds Time delays through a real-time scheduling algorithm within

communication network. According to Figure 1, structural reconfiguration takes place

as a result of Earliest Deadline First (EDF) Schedulling algorithm and the associated

user request. This reconfiguration causes a control law modification [1] which is the

actual control law reconfiguration.

Schedulling approach pontentially modifies frequency execution and communication

of tasks [5] in order to give certain priority to some of them during a bounded time as

shown in Fig. 3. Furthermore, in this kind of strategy Tasks modifies their priority, it

does not imply that neither the period nor the consumtion times are modified.

Therefore the tasks would have a bounded delay within the sampling time wich is

reflected as changing on the phase.

Fig. 3. Task Frequency modification as result of Scheduller.

Potential modifications onto schedulling approach deploy change in the priorities that

affects time delays and the respective control law. The delays are measured as

tΔ

[14] and bounded into the inherent control period of time [6] [7].

Now by taking partial results from schedulling algorithm like

t and the related

tΔ

the actual time delays are used at the control law for parameters design as shown in

following section. The involved time delays are depicted as

and come from this

scheduling design. Other delays like actuators and control delays are not used in the

design of the control law, although play an important role.

Therefore schedulling and control analysis merge together when time delays are

complete bounded even in the case of time variance. The main restriction is in terms

of predictable time delays.

The approach followed at the control reconfiguration does not take into account

scheuller decision in a direct manner. It takes the time delyas as bounded values

already defined and used to design a suitabe control law. Therefore, acording to

current state plant values, the related fuzzy rule is selected.

3 Control Reconfiguration Approach

The control law is defined as a group of Fuzzy TKS [8] [9] [10] control law related to

each local linear system. At the beginning the general structure of each fuzzy rule is:

r if

x is

A and

x is

A and …

x is

A then

(

)()

txQkf

(4)

where

{}

N,....,1i =

, N is the number of fuzzy rules,

{

}

x,....,x

are current states of

the plant,

A are the gaussians membership functions like:

()

exp

iji

(5)

where:

and

are constants to be tuned.

Similar to fuzzy system plant [9], fuzzy control representation is integrated as:

()

∏

iji

xAw

(6)

And

()

∑

txQw

(7)

Wher

Q is the related i’

control gain. The configuration of the fuzzy logic control

(FLC) integrated to the plant, epxressed as well in terms of Fuzzy Takagi Sugeno

approach is represented in Fig. 4 [11]. The closed loop system is pursued in terms of

local plant and related control gain per rule. In order to pursue this strategy, plant

model is shown in terms of its state space representation.

Fig. 4. Plant configuration using FLC control.

Using the proposed dynamic plant based on state space representation, see [11]:

( ) () ()

()

kxcy

kuBkxAkx

+=+1

(8)

Specially

is stated as

()

∑∑

∫

τ-tA

dτeBρB

(9)

where

=ρ

and

∑

=ρ

taking into account that N are the total number of possible

faults and M are the involved time delays from each fault. Current communication

time delays are expressed as

1j−

and

remember that

∑

≤τ

(total period of

time inherent from control law design) and

in general terms is integrated as

→

⎥

⎦

⎤

⎢

⎣

⎡

, i fault element

where

bb →

are the elements conformed at the input of the plant (such as

actuators) and 0

is the lost element due to local sensor fault where

represents only

one scenario. Remember that the only considered faults are sensor faults. Therefore

one input signal is measured. This can lose its confidence but not current value [10].

Since this approximation current

considers local sensor faults and related time

delays of

()

∑

∫

τ-

deBB

(10)

Remember that the related time delays are the result of structural reconfiguration

(SRA) explained before are calculated according to eqns. 4 and 5.

Back to the controller definition where N is the number of possible scenarios,

therefore, number of rules.

()

[]

∑

jjz

txhQQ

(11)

()

∑

txh

(12)

as for the plant integrated to the controller in closed loop, this is expressed as:

xQBxAuBxA

-⇒+ (13)

()

[]

()()

[]

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

∑∑

∫

-⇒

txhQdeBxxA

()

()()

[]

()

txtxhQdeBhA

⎥

⎦

⎤

⎢

⎣

⎡

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

===

∑∑∑

∫

111

-⇒

()(){}

∑

txhAA

Then the proposed Lyapunov function is:

()

() ()

txPtxtxv

(14)

And its derivative is expressed in eqn 14 as a necessary condition for stability

()

() () () ()

0≤txPtxtxPtxtxv



(15)

In terms of the plant integrated to the control law this is expressed as follows:

() ()

xPxxQBAPxxPQBAxv



++= --

(16)

remember that

()

∑∑

∫

τ-

hdeBB

where M is the number of time delays per scenario within the control law inherent

period.

()()

{}

∑

jjz

txhQQ

Therefore the core of lyapunov function is given as :

()

xQBAx



(17)

Therefore the derivative of the energy as expressed in 15 can be expressed as:

()

(

)

{

}

xPQBAPPQBAx



++= -- (18)

()

∑

iiz

PtxgP

()

∑

iiz

PtxgP



and

QBAg -= (19)

Now by expressing the same energy function in terms of an inequality relation in a

relaxed manner, considering all the possible P

matrices equals in terms of the same

matrix (

QBAg -=

) for any condition considered along the N fuzzy rules, energy

function can be expressed as:

()

()()

()()()

() ()

∑∑

∑

--≤

iji

izz

QBAQBA

xxhxh

xQBAPPQBAxxhv



(20)

Therefore, the controller design can be expressed as:

0-- >++

izz

PQBBQPPAAP

(21)

Remember that i has a value between 1 to N. Therefore for every given plant and the

respective controller by decomposing this equation, the P

matrix should be bounded

as:

--- - 0

TT T T

Pp PP TP P TP P

ii ji i i j j

zzzzzj jzzi iz

PA A P PA A P PQ B B QP PQ B B QP

++ +> (22)

Remember that that also j has a value between 1 to N related to the number of rules.

Therefore in terms of Linear Matrix Inequality [9] is given by

zzjizi

izz

PPQBPA

BPQAPP

(23)

This condition is given for every single time delay and local fault appearance.

Furthermore the stability and the convergence of states should be assured by the

adequate selection of matrices P

and the related parameters from both fuzzy systems.

In this case a recommendable procedure to follow is multi-objective optimization in

order to define those suitable values [12].

The whole system considering this codesign strategy, has been implemented in

several environments such as simulation based [12] using True-Time [16] and real-

life using CANBUS [13]. Although this approximation is out of the scope of the

paper, these implementations have given enough information in terms of the practical

experience for current approach. Moreover, related strategies for codesign control

theory have been reviewed with preliminar succesful results in [6].

4 Conclusions

The use of codesign as a suitable strategy for networked control and scheduling

analysis is a real possibility as explored in this paper. Although it is restricted to the

fesability of both techniques, this can be approximated as an iteractive procedure

where both techniques need to achieve an aggrement.

In this case time delays are approximated and bounded through a suitable scheulling

policy which affects the results of current selected controller. The exploration

followed in here is based upon classical schedulling algorithms and fuzzy takagi

sugeno approach. The key characteristic of last approach is design of local control law

considering bounded time delays per valid scenario from schedulling results.

Several results need to be highlighted such as the convergence of variable time

delays due to the use of schedulling approximation and the restricted and known

modification onto control law design. Furthermore, bounded time delays as long as

they are from the same source, like sensor delays, they modifiy similar control

paramters, therefore, control structure does not need to be modified on a large scale.

Future work need to be focus onto strucutral modification from the control law, as

well as a deeper study from time delays source. For instace, the complex computing

relationship stablished through the operating system, middleware transactions,

interprocees communications, communication network protocols, and others.

Acknowledgements

The author would like to thank the financial support of DISCA-IIMAS-UNAM, and

UNAM-PAPIIT (IN105303 and IN101307) Mexico in connection with this work. The

present paper is part of an ongoing effort of the High-Performance Computing

project, within the `MacroproyectoTecnologías para la Universidad de la Informacíon

y la Computacion'' of the National Autonomous University of Mexico (UNAM).

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