Admissible set of robust PID controller values using Hurwitz Criterion for a Pure Integrating Process with dead time.

Thirunavukkarasu, I. ; George, V.I. ; Iyer, S. Narayana 等

Introduction

Most of the chemical and process industries are encountered with the unstable and higher order systems. We can model the higher order systems to FOPDT with dead, for the purpose of controller design. For example, in the study of the distillation column, the resultant FOPDT will have a very high time constant, which may lead the process to settle after a very long time[9]. In these cases some of the researchers mentioned that we can treat the model as a pure integrating process for the design of controller. Once the controller has been designed, the same can be implement for the higher order systems. In real time none of the system exists as a PIPDT.

Reflux boiler control in the distillation column tower , single storage tank with a constant displacement pump against a step input, satellite attitude control (Double integral process) are some of the real time examples of the integrating process. The study of distillation column is considered for the simulation. Model free PID controller design is more suitable for the unstable process with various uncertainties. The common problem with the ordinary conventional controller is that the PID values, which are tuned will operate only for one particular transfer function. The tuned controller value may lead to the poor time domain response or unstable output under some additional uncertainties added into the system. The Robust controller may find solution to these kinds of problems, the only disadvantages with the design of H-Infinity is that the order of the controller will be very high. In order to reduce the order of the controller and to ensure the robust performance, H-Infinity based PID controller was proposed in this paper for the PIPDT processes. Also the admissible set of PID controller was found, using the Hurwitz Criterion satisfying the robust performance condition [1]. The uncertainty was considered in the form of dead time in PIPDT process and the PID tuning values are found for uncertain system and plotted as a 3D plot.

Design Approach

Consider the mathematical model of the distillation column as G(s) = [0.0506e.sup.-6s] / S for the controller design [10]. The general structure of the PID controller was given by C(s) = [K.sub.i] + [K.sub.p] s + [K.sub.d] [s.sup.2] / s objective is to find the values [K.sub.p], [K.sub.i] and [K.sub.d] using the Hurwitz Criterion. The PID values are selected by satisfying the following condition.

1] [rho] (s, kp, kt, kd) ts Hurwitz. (1)

2] [psi] (s, kp, kt, kd, [theta], [phi]) ts Hurwitz (2)

3] [absolute value of [W.sub.1] ([infinity])] + [absolute value of [W.sub.2] ([infinity])T([infinity])] < 1. (3)

In this paper, the controller has been designed for the plant with uncertainty, the uncertainty was considered in the form of dead time. The perturbed plant with various dead times were given below (after the pade approximation), for which the controller needs to be designed.

Case-1: With dead time Td=6sec.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Case-2: With dead time Td=6sec.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Case-3: With dead time Td=6sec.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The root locus plot can help initially to get the rough values of [K.sub.p] for all the three cases.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The weighting functions [W.sub.1] (s) and [W.sub.2] (s) are quite sensitive and should be chosen carefully for all the three cases based on the frequency inputs of the plants. Let T(s) and S(s) be the complementary sensitivity function and the sensitivity function respectively.

T(s) = G(s)C(s) / 1+G(s)C(s) (7)

S(s) = 1 / 1+G(s)C(s) (8)

Also for the case 1, the weighting functions are selected as

[W.sub.1] (s) = 2 / s + 5 (9)

[W.sub.2] (s) = 5 s + 1 / s + 2 (10)

For G(s) in case 1, the T(s) and S(s) are found to be

T(s) = -0.0506 kd [s.sup.3] + (0.0168 kd - 0.0506 kp)[s.sup.2] +(0.0168 kp - 0.0506 kt)s + 0.0168 kt / (1 - 0.0506 kd)[s.sup.3] + (0.333 + 0.0168 kd - 0.0506 kp)[s.sup.2] + (0.0168 kp - 0.0506 kt)s + 0.0168 kt

S(s) = [s.sup.3] + 0.333 [s.sup.2] / (1 - 0.0506 kd)[s.sup.3] + (0.333 + 0.0168 kd - 0.0506 kp)[s.sup.2] + (0.0168 kp - 0.0506 kt)s + 0.0168 kt

Especially, we consider the problem of disturbance rejection for the plant with multiplicative uncertainty. This problem can be formulated as the following robust performance condition.

[[[absolute value of [W.sub.12] (s)S(s)] + [absolute value of [W.sub.2] (s)T(s)]].sub.[infinity]] < 1

This condition can be converted into a simultaneous polynomial stabilization by considering the following lemma.

Lemma 1: Let

[W.sub.1](s)S(s) = A(s) / B(s) = [a.sub.0] + [a.sub.1] s + ... + [a.sub.x][s.sup.x] / [b.sub.0] + [b.sub.1] s + ... + [b.sub.x] [s.sup.x]

A(s) / B(s)

= [s.sup.2] + 4[s.sup.3] + 4[s.sup.4] / 0.1265 lt + (0.1265 kp - 0.25047 kt)s + (2.5 + 0.1265 kd - 0.25047 kp - 0.00506 kt)[s..sup.2] + (5.05 - 0.25047 kd - 0.00506 kp)[s.sup.3] + (0.1 - 0.00506 kd)[s.sup.4]

And

[W.sub.2] (s)T(s) = C(s) / D(s) = [c.sub.0] + [c.sub.1] s + ... + [c.sub.x] [s.sup.x] [d.sub.0] + [d.sub.1] s + ... + [d.sub.x] [s.sup.x]

C(s) / D(s)

0.0253 kt + (0.0759 kt + 0.0253 kp)s + (0.0253 kd + 0.0759 kp - 0.253 kt)[s.sup.2] + (0.0759 kd - 0.253 kt) [s.sup.3] + (1 - 0.0506 kd)[s.sup.4]

Be stable and proper rational functions with [b.sub.x] and [d.sub.x] not equal to zero. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If and only if:

a) [absolute value of [a.sub.x] / [b.sub.x]] + [absolute value of [c.sub.x] / [d.sub.x]] [less than or equal to] 1

b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof: The necessity of condition (b) is established in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because B(s) and D(s) are Hurwitz, using Rouch's Theorem we conclude that, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is Hurwitz for all [] & [PHI] [??] [0,2[pi]].

Sufficiently proceeding by contradiction we assume that conditions (a) and (b) are true but,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a) Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is continuous function of [omega] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then there must exist at least one [[omega].sub.0] [??] R such that.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, it implies that there exists [] and [PHI] [??] [0,2[pi]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and it obviously contradicts condition (b).

The designing of PID controllers such that the robust performance condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) holds. Based on lemma 1, The problem of synthesizing PID controllers for robust performance can be converted into the problem of determining values of (kp, ki, kd) for which the following conditions hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For condition 3 to be hold [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] kd must lies in within the range (-4.9407, 3.2938).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

After the range of kd was fixed, the next step is to fix the range of kp such that the robust performance condition hold and also [rho] and [phi] must be Hurwitz within this range.

Let

N(s)=Ne([s.sup.2])+s No([s.sup.2]) (11)

D(s)=De([s.sup.2])+s Do([s.sup.2]) (12)

Using this decomposition of N(s) and D(s) into even and odd part will simplify the calculation that should be used for fixing the range of Kp. Using (11) and (2), M*(s) can be redefine as M* (s)=N(-s)= Ne([s.sup.2])-s No([s.sup.2])

The closed loop characteristics polynomial is

[delta](s, kp, ki, kd) = s D(s) + (ki + kd [s.sup.2])N(s) + kp s N(s)

Let n, m be the degree of [delta](s, kp, ki, kd) and M*(s) respectively. Multiplying [delta](s, kp, ki, kd) by M*(s) and examining the resulting polynomial it gives : o(s, kp, ki, kd) by M*(s) and examining the resulting polynomial it gives :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of degree n is Hurwitz if and only if

l([delta](s, kp, ki, kd)) = n

And

r ([delta](s, kp, ki, kd)) = o

Now, we consider this theorem

Let [??] (s) be a given real polynomial of degree n. then

l([delta) - r([delta]) = [[sigma].sub.l]([delta])

l([delta) - r([delta]) = [[sigma].sub.r]([delta])

Moreover, (s) is Hurwitz if and only if and only if [[sigma].sub.l]([delta]) = [[sigma].sub.r]([delta]) = n.

Lemma 2:

[[sigma].sub.l]([delta](s, kp, ki, kd) M*(s)) = n - (l(N(s)) - r(N(s)))

The task now is to find the sets of (kp,ki,kd) for which the formula in lemma 2 holds. By considering the even and odd parts of N(s) and using them to find M*(s), then substitute the result to find [delta](s, kp, ki, kd) M*(s) this product can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting (s=j[omega]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

After computing the equations above, the next step to determine the admissible set of (kp,ki,kd) is to fix the range kp. This can be done using root locus ideas as following.

Let [eta]([omega], kp) = [omega][U([omega]) + kp V([omega])]

Define V([omega])/U([sigma]) as a function of ([omega]). then find the derivative of V([omega])/U([sigma]) with respect to ([omega]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by setting d/d[omega] (V([omega])/U([sigma])), then from the real zeros of the equation a corresponding Kp

Values that are distinct and finite producing either real breakaway points or a root at the origin can be found from U([omega]) + kp V([omega]) = 0.

The next step is to find the real root distributions of U([omega]) + kp V([omega]) = 0 with respect to the origin.

The necessary condition for fixing the range of kp is that q([omea],kp) has at least

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

real, nonnegative, distinct roots of odd multiplicity. These ranges of kp satisfying this condition are called allowable.

Fixing a value for kp in within the range and evaluate q([omega],kp) at this value. The real, non-negative values, distinct finite zeros of qf([omega],kp) are the zeros of q([omega],kp). Let these zeros be

0= [[omega].sub.0]< [[omega].sub.1]< [[omega].sub.2]< ... ... < [[omega].sub.1-2]< [[omega].sub.1-1]

Then we have to construct a sequence of numbers [i.sub.0], [i.sub.1], [i.sub.2], ..., [i.sub.l], as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

With [i.sub.0], [i.sub.1], [i.sub.2], ..., [i.sub.l] defined in this way, we define the string I: N [right arrow] R as the following sequence of numbers: I := [[i.sub.0], [i.sub.1], [i.sub.2], ..., [i.sub.l]]. This admissible string must satisfy the following condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

For an admissible stringI, the set of (ki,kd) values can be determined from the following string of linear inequalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For which [M.sup.*] (j[[omega].sub.t])[not equal to] 0.

This procedure have to be repeated for all admissible strings to obtain the corresponding admissible (Ki,Kd) sets. The set of all stabilizing (Ki,Kd) values corresponding to the fixed kp is then given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we are going to apply this procedure to determine the admissible set of (Kp,Ki,Kd) values. First, let's consider the transfer function in case 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

N(s) and D(s) can be decomposed into even odd parts as following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evaluating the above equation at s=jw:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[P.sub.1]([omega]), [P.sub.2]([omega]), [q.sub.1]([omega]) and [q.sub.2]([omega]) are found to be:

[P.sub.i](w) = - 0.0055944 [w.sup.2] + 0.0506 [w.sup.4]

[p.sub.2] (w) = 0.00028224 + 0.00256036 [w.sup.2])

[q.sub.1] (w) = -0.0336498 [w.sup.3]

[q.sub.2] (w) = 0.00028224 w + 0.00256036 [w.sup.3]

q([omega],kp) can be written in the q([omega],kp)= [omega] [ U([omega]) + kp V([omega]) ] form as:

q (w, kp) = w[ -0. 0336498 [w.sup.2] + kp (0.00028224 + 0.00256036 [w.sup.2])]

Where

U(w) = - 0.0336498 [w.sup.2]

V(w) = 0.00028224 + 0.00256036 [w.sup.2]

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The only real zero of (5) is [omega]=0. Substituting this value in [U([omega]) + kp V([omega]) = 0], the corresponding value w is kp=0. This gives that the range of kp can be:

(-[infinity],0] or [0,[infinity]). Examining the distribution of the roots for kp belongs to each range will give a more accurate range for kp.

Since n+m=4 (even), there must be at least [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] real, non-negative distinct finite zeros in the range of kp. In this case there should be at least [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By chosen any arbitrary value for kp in each range, the exact range of kp can be determined. For this case in our hand the range of kp is [0, 13.115]. Taken any arbitrary value for kp in within the specified range, for instance let kp =0.65, and substituting the value in q(w,kp) gives:

q (w, 6.5 ) = 0.017007460 [w.sup.3] + 0.001834560 w

The real zeros of q([omega], 5) with odd multiplicity are [[omega].sub.0]=0 and [[omega].sub.1]=0.32843. Also define [[omega].sub.2]=[infinity].

The next step is to find the admissible strings [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [M.sup.*](s) has no zero at the origin and [M.sup.*](j [[omega].sub.t]) [not equal to] for t=, [i.sub.0] = [??]. [i.sub.l] also will have the value of [??] because n+m is even number.

Since m+n=4, which is even, and [M.sup.*](s) has no j[omega]-axis roots, then the set A(5) becomes.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since n - (l(N(s)) - r(N(s)) = 3 - (0 - 1) = 4 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then every admissible string I := {[l.sub.0], [l.sub.1], [l.sub.2]} must satisfy the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The only string which satisfy the condition above is I := {1, -1,1}. It follows that the stabilizing set (Ki,Kd) corresponding to kp=5 must satisfy the string of inequalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting for [[omega].sub.0], [[omega].sub.1] and [[omega].sub.2] in the above expression, we obtain the set [S.sub.1](5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The graph of this set of (Ki,Kd) values is shown in figure 5. By sweeping over different kp values within the range [0, 13.115] and repeating the above procedure at each stage, the set of (Kp, Ki,Kd) values can be generated. The sets of (Kp, Ki,Kd) values shown in figure 6.

[FIGURE 4 OMITTED]

In order to find a such common set of ([K.sub.p], [K.sub.i],[K.sub.d]) values that hold both [rho] and [phi] to be Hurwitz and satisfy the robust performance condition, Routh table was used. The first column of routh tables of [rho] and [phi] should be positive. According to the fact that for any system to be stable the first column in Routh table must be positive. The ranges of ([K.sub.p], [K.sub.i],[K.sub.d]) values that makes it positive are shown in the table below.

But in order to find a common set of values that satisfied both [rho] and [phi] and also satisfied the robust performance condition, the values in the above table have to be modified slightly. One of the sets that can satisfy the three conditions is in the table below:

Then, values obtained were substituted in the original equations of [rho] and [phi] to ensure they will make the system stable. Figures 7 and 8 below shows the first column of [phi] and [rho] with the values of ([K.sub.p], [K.sub.i],[K.sub.d]) respectively.

From table 3 and 4 above that the system is completely stable for the set of ([K.sub.p], [K.sub.i],[K.sub.d]) values. All the zeros of [rho] and [phi] are in the left side of the plane.

By sweeping over different [K.sub.p] values within the interval [0, 13.115) and repeating the above procedure at each stage, the set of stabilizing ([K.sub.p], [K.sub.i],[K.sub.d]) values can be generated. Figure 5 shows three layers which was obtained for [K.sub.p] =5, 5.2 and 5.4.

[FIGURE 5 OMITTED]

The design of the Robust PID controller satisfying the robust performance for case1 was completed. The same procedure is to be applied for the other cases. The resulting stabilization set of ([K.sub.i],[K.sub.d]) and the 3D stabilizing set of ([K.sub.p], [K.sub.i],[K.sub.d]) are shown for the case 2 (Plant with uncertainty) transfer function.

For CASE 2 Transfer Function.

For the plant transfer function of case 2, [rho] and [phi] equations are as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Robust Performance Condition

For the robust performance condition to be hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For this condition to be hold, kd must lies in the range (-4.9407, 3.2938).

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Conclusion

In this paper, the admissible sets of PID controller values were found using the Hurwitz criterion for the Pure Integrating Process with dead time. The PID Controller values obtained satisfies the robust performance condition for the system. The Controller has been designed for the plant with and without uncertainty and admissible set of 3D plot has been plotted. An attempt can also be made by using genetic algorithm (GA) approach to tune the exact PID values which lead to the better time domain response as a future work. From the obtained admissible set of PID values, the exact values needs to be identified which are giving the good time domain specifications.

References

[1] Ming-Tzu Ho and Chia-Yi Lin, "PID Controller Design for Robust Performance", IEEE Transactions on Automatic Control, VOL. 48, NO. 8, AUGUST 2003.

[2] Eduardo N. Goncalves, "A novel approach for [H.sub.2] / [H.sup.[infinity]] robust PID synthesis for uncertain systems", Journal of Process Control, Elsiver, Vol 18, Jan2008.

[3] Tae-Hyoung Kim "Robust PID controller tuning based on the constrained particle swarm Optimization", Journal of Automatica , Springer Publication, August 2007.

[4] Rong-Maw Jan, "Robust PID control design for permanent magnet synchronous motor: A genetic approach", Journal of Electrical Power system, Jan 2007.

[5] Masami Saeki, "Fixed structure PID controller design for standard [H.sup.[infinity]] control Problem" Journal of Automatica, Springer Publication, OCT 2005.

[6] Guillermo J. Silva, "PID Controllers for time-delay Systems", Birkhauder Publication, ISBN 0-8176-4266-8, 2004.

[7] Weidong Zhang, "Design PID controllers for desired time-domain or frequency-domain response", ISA Transactions, vol. no: 41, 2002. Pages 511-520.

[8] Weidong Zhang, "[H.sup.[infinity]] PID controller design for runaway processes with time delay", ISA Transactions, Vol. No: 41, 2002. Pages 317-322.

[9] Wayne Bequete, "Process Control: Modeling, Design and Simulation", Prentice Hall, ISBN-0133536408.

[10] Tuyres, Luyben, "Tuning PI Controller for Integrating/Dead Time Processes", Ind Eng. Chemical Res, 1992, 31, 2625-2628.

I. Thirunavukkarasu (1), V.I. George (2), S. Narayana Iyer (3) G. Saravana Kumar (4) and Yousuf Al-Abrii (5)

(1,4) Faculty, (2) Prof & Head, (3) Dean-R&D, (5) Student, Sultan Qaboos University Dept of Instrumentation & Control Engg, M.LT, Manipal University, Karnataka, India (3) Nanmit, NITTE, Karnataka, India

Table 1: the ranges of ([K.sub.p], [K.sub.i], [K.sub.d]) values for
G(s) of case 1.

Case 1      [K.sub.d]            [K.sub.p]

  P      [K.sub.d] <19.76    For [K.sub.d] =2
                              [K.sub.p] <7.24

[PSI]      [K.sub.d] <      For [K.sub.d] = 1.5
              3.878          [K.sub.p] <26.96

Case 1               [K.sub.i]

  P       For [K.sub.d] =2, [K.sub.p] =5
               0 < [K.sub.i] < 0.458

[PSI]    For [K.sub.d] =1.5, [K.sub.p] =5
              0 < [K.sub.i] < 77.433

Table 2: checked set of ([K.sub.p], [K.sub.i],[K.sub.d]) values.

         K.sub.d]   [K.sub.p]   [K.sub.i]

Case 1     0.5          5          0.2

Note: These are the common values of Kp,Ki and Kd which are satisfying
the condition [rho] and [phi].

Table 3: Routh Table for the condition 2 ([psi])

Routh-Hurwitz: 1st    TF Zeros are:
column is:

[s.sup.5]: 0.87108    -6.4886
[s.sup.4]: 7.0089     -1.2726
[s.sup.3]: 8.8684     -0.11902-0.35566i
[s.sup.2]: 2.1875     -0.11902+0.35566i
[s.sup.1]: 0.93858    -0.046609
[s.sup.0]: 0.047362

Table 4: Routh Table for the condilion 1 ([rho])

Routh-Nurwitz: 1st    TF Zeros are:
column is:

[s.sup.3]: 0.9747     -0.021974-0.270671
[s.sup.2]: 0.0884     -0.021974+0.270671
[s.sup.1]: 0.036833   -0.046746
[s.sup.0]: 0.00336

Table 5: The ranges of ([K.sub.p], [K.sub.i], [K.sub.d]) values for
G(s) of case 2.

Case 2       [K.sub.d]            [K.sub.p]

  P       [K.sub.d] <19.76     For [K.sub.d] =2
                               [K.sub.p] <10.88

[PSI]     [K.sub.d] <3.878     For [K.sub.d] =2
                                  [K.sub.p]
                                   <25.334

Case 2              [K.sub.i]

  P       For [K.sub.d] =2, [K.sub.p] =6
               0< [K.sub.i] <1.062

[PSI]     For [K.sub.d] =2, [K.sub.p] =6
               0< [K.sub.i] <69.07

Table 6: Checked set of ([K.sub.p], [K.sub.i],[K.sub.d]) values.

         [K.sub.d]   [K.sub.p]   [K.sub.i]
Case 1      0.5          6          0.5