N/A
N/A
Protected

Copied!
6
0
0

Full text

(1)

f

g

1

Fq Z]

Z

Fq

## 3System Description

### The only formal description of the controller available to use was the actual implemented 1200 line Pascal code.

1This work was supported by the Swedish Research Council for Engineering Sciences (TFR) and the Swedish National Board for Industrial and Technical Development (NUTEK), which is gratefully acknowledged.

(2)

### Figure 1: The fighter JAS 39 Gripen.

Landing Gear

Controller Landing Gear

Pilot

Other system units

p a

s

m

## 4 Modeling

f0:::15g

### Timer variables and time conditions in the code have been replaced by binary state variables (flip flops) and corresponding input signals. A time condition becoming true in the original code corre- sponds to the timer input signal triggering the state variable. Once triggered, the state variable will be true until there is an explicit timer reset.

Comment Syntax Domains

B

I

I2!I

I2!I

I2!B

I2!B

I2!B

B !B

B2 !B

B2 !B

:::

I

B

I

B

I

B

### The landing gear controller code is one part of the software loop in the aircraft system. This means that the state of the code is stored until next iteration of the code. If we want to write the system as

x +

=f(xu) y=g(xu)

x

x+

u

y

uy

x

Inputs\Outputs

(3)

u )

x )

) y

) x

+

Inputs )

) Outputs

M(zz+)

z

z+

2

3

### follows the control flow graph of the program. The value of each program expression is determined by the current values of symbols and the actual program expression, i.e. the compilation function is of the form:

:PascalState!State

=fv

1 7!e

1

:::v

n 7!e

n g

vi

ei

?



?

pe = 0

B

B

B

B

B

B

B

@

### y2 := e END;

1

C

C

C

C

C

C

C

A

2Input, state and output variables.

3Boolean expressions are essentially polynomials over the fieldF2.

### with the initial symbol table

=fq7!q c7!c d7!d e7!e 

y17!y1 y27!y2 g

### we will get

 +

= (pe)=fq7!q c7!c d7!d 

e7!e y17!(q ^c )_(:q ^d )

y27!(q ^y2 )_(:q ^e )

### The final Boolean relation is computed from the final symbol table



nal

=fx +

7!f(xu)y7!g(xu)g

M(zz +

)=x +

\$f(xu)^y\$g(xu)

z= xyu]

M(zz+)

## 5Analysis – Verification

M(zz +

)

M(zz +

)^Q

1 (z)^Q

2 (z

+

)=0

M(zz+)

Q1(z)

Q2(z+)

z

z+

(4)

1 0

1

0

0 1

0

0

M(xx +

)^Q1(x)^Q2(x +

)=(x +

1

+1+x1+u1)^x1^x +

1

=1+u

1 :

1+u1 = 0

u1 = 1

0

0

u1 = 1

n

Fp

pn

2n

NP 6=P

n

n

M(zz+)

k

I(z)=0

Rk(z)

R

0

(z):=I(z)

R

k +1

(z):=R

k

(z)_(9~z(R

k

(~z)^M(~zz)))

Rd+1(z)=Rd(z)

d



d

d

n

z = z1:::zn]

2n

n

n

n



Rk(z) = Rk +1(z)





10 000

226

0 1 2

x1

x2

x

1 x

2 State

(5)

### Using this encoding we get the polynomial model

M(xx +

)=((:x1^:x2)^(:x +

1

^x +

2 ))_

((:x

1

^x

2 )^(x

+

1

^:x +

2 ))

0

### as

R

0

(x):=I(x)=:x

1

^:x

2

R

1

(x):=((:x

1 )^(:x

2

))_((:x

1 )^x

2 )

R2(x):=((:x1)^(:x2))_((:x1)^x2)_(x1^(:x2))

R3(x):=((:x1)^(:x2))_((:x1)^x2)_(x1^(:x2))

k = 2

2

M(xx+)

2

Q(z)

Q(z) = 0

S(z)

M(zz+)

2

x1^:x2]:

Verify(M(xx +

)

x1^:x2])=

=9x +

M(xx +

)^(x +

1

^:x +

2 )

=(:x

1 )^x

2 :

1

2

### in one step.

Temporal Algebra Natural Language

Q(z)

Q(z)

Q(z)] Q(z)

### EU

Q1(z)Q2(z)] Q1(z)

Q2(z)

Q(z)] Q(z)

Q(z)] Q(z)

Q(z)] Q(z)

### AU

Q1(z)Q2(z)] Q1(z)

Q2(z)

Q(z)] Q(z)

Q(z)] Q(z)

2

0

f012g



P(z)

P(z)]

P(z)]





A

B

QA(z)

A

QB(z)

(6)

B

QA(z)]!

:QB(z)]:

M(zz+)

R (x)

226 108

^

M(zz +

)=R (x)^M(zz +

)^R (x +

)

M^(zz+)

P(u)

P(u)

M(zz+)

S(z)

M(zz +

)

S(z)

## References

### 1863–1864, 1995.

References

Related documents

geom point Scatterplots geom line Lineplots geom boxplot Boxplot geom histogram Histograms geom bar Barchart.. Bartoszek (STIMA LiU)

The number of such polynomials is given by formulae presented in , but here we will be content with presenting a polynomial that exhibits properties regarding irreducible

Light absorption in folded solar cells was modelled, and combinations of different active layer thicknesses, folding angles and materials were studied.. A beneficial light

Surface and Semiconductor Physics Division Department of Physics, Chemistry and Biology Link¨ oping University, S-581 83 Link¨ oping, Sweden. Link¨ oping 2010 Johan Eriksson A tomic

Genom att relatera forskningen kring innovation i offentlig sektor till konkreta exempel tydliggörs varför det är viktigt att översätta begreppet innovation. I bilagan presenteras ett

In this picture MD i denotes a modeling domain, ID j denotes an implementation domain and PDS denotes polynomial dy- namical systems (over finite fields).... In figure 2 we receive

From a control theory users point of view, tempo- ral algebra offers a straight forward way of translat- ing informal verbal specification into a formal alge- braic specification

A ven om vi vill anvanda en annan parametrisering kan detta darfor vara ett bra satt att initialisera modellen, da lampliga begynnelsevarden for t ex OE och hinging hyperplanes

Department of Computer and Information Science Link¨ oping University. SE-581 83 Link¨

Our team (”Link¨oping Humanoids”) represents the student association FIA Robotics from Link¨oping University (LiU) and the Division for Artificial Intelligence and Integrated

Idén med att göra en simuleringsmodell för SVM är att dels se vad för svårigheter som kan uppstå, dels att göra modellen syntetiserbar för att undersöka om tidskraven uppfylls

Department of Management and Engineering Link ¨oping University, SE-581 83, Link ¨oping,