Cross-Optimization Aspects Concerning the Extended Symmetrical Optimum Method

Copyright @ IFAC Digital Control: Past, Present and Future of PID Control, Terrassa, Spain, 2000

CROSS-OPTIMIZATION ASPECTS CONCERNING THE EXTENDED SYMMETRICAL OPTIMUM METHOD

Steran Preitl and Radu-Emil Precup

"Politehnica" University ofTimisoara Department of Automation RO-1900 Timisoara, Romania, Bd, V. Parvan 2 Phone: +40-56-204333 ext. 688, 689, 681, 683, Fax: +40-56-192049 E-mail: [email protected]@[email protected]

Abstract: The paper presents cross-optimization aspects in controller parameter tuning in the case of the Extended Symmetrical Optimwn (briefly, ESO) method - introduced by the authors (Preitl and Precup, 1996, 1999a) - for a class of systems characterized by an openloop transfer function Ho(s): ko(l+sTc) Ho(s)=---s2(l+sTI:) The following optimization criteria were taken into consideration: - lSE-he 00

he = Io e2(t)dt , - generalized ISE - hg : 00

hg = Io [e2(t) +

re2(t)]dt,

- cross-optimization criterion (index) - he 00

he = Io [e2(t) + PV(t)]dt. Generally speaking, it is relatively difficult to express these three indices and the obtained results are not conclusive. It is proved in the paper that the generalization of the Symmetrical Optirnwn (SO) method to the form ofESO method determines that the use of the indices he, hg and he becomes no more necessary. Copyright © 2000 IFAC. Keywords: Cross-optimization, symmetrical optimum, PI and PlO controller, tuning parameters, control system performance.

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1. PROBLEM STATEMENT

"optimization" solutions, for the computation of the two parameters kc and Te the following specifications are considered to be significant:

The particular structure of control system containing a controller with homogenous information processing with respect to the two input channels and having as openloop transfer function (t.£) Ho(s): ko(l+sTc) Ho(s)=---s2(1+sTt)

(a) The optimization in terms ofthy SO Method (Kessler, 1958) is based on the relations (Astrom and Hligglund, 1995):

(I)

(4)

and it ensures: with: - the possibility of the univocal computation of the tuning parameters (this is an advantage):

ko =kckp, Te - controller time constant, Tt - small time constant,

I kc=-8Tlkp

is specific to electrical driving systems.

and

Te=4Tt;

(5)

- performance that are far away from being acceptable: the overshoot 01 l':: 43%, the settling time ~16.3TL, the first settling time tll'::3.7Tt, and the phase margin (reserve) (j>rl>:036° (this represents a shortcoming).

The double pole in origin is due to: - the controlled plant, and to - the I component brought by the conventional controller of PI or PlO (from one case to another according to (Preitl and Precup, 2000)).

(b) The optimization in terms of the Integral of Squared Error (ISE) criterion method (see, for example, (Graham and Lathrop, 1953)):

Corresponding to (1), the closed-loop t.f. obtains the expression (2):

ao

Hw(s)

2 he = Io e (t)dt ,

(2)

(6)

that can be evaluated on the basis ofParseval's relations: The basic two situations are those presented in (Preitl and Precup, 1996, 2000) and deal with the use of PI and PID controllers with homogenous information processing with respect to the two input channels having the following t.f.s: kc Hc(s) = - - (1+ sTc) s

2 ~2aoal + (b12 - 2bob2)aoa3 + bo a2a3 he = - - - - - - - - - - - - -

(7)

The computation of the parameters kc and Te based on minimizing the expressions (8) and (9): and

a12e

(3)

kc He(s) =--(1+sTe)(1+sTe'),

akc

s

~=o

with Ro = [kco Tco] T ,

(8)

aI 2e

aTe

where Te' compensates a large time constant of controlled plant (for example, T I).

=0,

(9)

Ro

does not ensure the univocal determination of the optimum values of the parameters kc and Te; this makes necessary the choice of the value of one parameter and then, on this base, the computation of the other parameter (Dragomir and Preitl, 1979).

The parameter tuning involved the computation of those values of kc and Te for which the system should fulfil certain performance requirements. Concerning the frequently used in practice two classical

224

(c) The optumzation in tenns of the ESO method introduced and developed afterwards by the authors in (precup and Preitl, 1996, 1999a, 2000) is based on the relations (4) generalized to the following expression:

1 0/0 G11

70

I"~ ;

60

(10)

SO where the parameter ensures:

~

is available to the designer and it

40

30 - the 1.[ Ho(s) to be dependent only on the value of the parameter ~ (for a given Tt):

20

;

/ ' ts

-l}l

(;f

11'

lA

Ho(s)=-----~ ~l~l s2(I+Tt s)

o

(I I-a)

1\

~\

/, ~

7 30 6

/.~

75

.....

V

15 3

f'..... I-- (5,1%)

-

,e

jo.-.

1 2

4

6

10 12 14 16

8

A

0".

p

20 4

"- ~ k(

Fig.I. Performance functions of~.

this expression ensures for each value of ~ the "symmetry" of the open-loop Bode plots with respect to the crossover frequency Wc and maximum value for the . phase margin
3'i

/ "" t;

1\

10 l+~Tt S

,

1,

_~ 4~ 8

-s 1

18 20 ~ A

[%], ts (= tsTt) and tl (= t.Tt) as

2. CONNECTIONS BETWEEN OPTIMIZATION IN TERMS OF ESO METHOD AND OPTIMIZATION IN TERMS OF INTEGRAL INDICES he, h g AND he The quadratic integral indices frequently used in the computation of tuning parameters are the followings (Graham and Lathrop, 1953):

Hw(s) = - - - - - - - - - ~ ~l~l S3+ ~ ~II2Tt2l+ ~Tt s+1 (I I-b)

- the ISE index:

- the univocal determination of the tuning parameters (this is an advantage) in the form:

(13)

kc-----

(12)

~ ~II2Tt2kp

(the particular case for

~

- the generalized ISE index: = 4 results in the relation (5));


hg = he + r fo e2(t)dt ,

- performance according to the proposed goal, by the possibility to vary the value of ~ (usually, between 4 and 16), Fig. I.

(14)

- the quadratic cross-optimization index: 00

he = he + p2fo u2(t)dt ,

The fact that all coefficients of Ho(s) and Hw(s) depend on the parameter ~ and that the whole controller development is reduced to the choice of a single parameter determines that, with respect to this parameter, the optimization based on integral quadratic indices will have a unique solution.

(15)

where t and p are weighting parameters available to user's choice. In the conditions (10) and (11) the computation of the expressions he, h g and he results in the fonns presented in (16), (17) and (18), respectively:

The paper perfonns an analysis of the connections between the three integral quadratic indices he, hg and he' and the value of the parameter ~.

Tt

13(~.12+1)

~ Tt

he=-_x_-2

225

[}-1

(16) 2(~112 -1)

Tlx S+r2(x3+x2-x+ 1)

r+~Tl:2

hcmin =- - - - - - - -

h8 = - - - 2Trl~112-1)

with x - the real solution (larger than 4) of the following sixth order equation:

for t = 0 it is obtained that 128 = I2e ; (1 +~II2)[

P~112+ r2 J>- r ~112 + r2]

~2 ~11brl:2+r2

he= 2Tl:

(23)

x ~ ~112 (~-1)

with r = p/kp .

and the graphics he = t{p,r), detailed for the values r = 0, 0.25, 0.5, 1 (continuous line for r = 0, dotted line for r = 0.25, dashed line for r = 0.5 and dashed and doted line for r = 1), according to Fig.4, and r = 0, 0.5, 1, 2.5 (continuous line for r = 0, dotted line for r = 0.5, dashed line for r = 1 and dashed and doted line for r = 2.5), according to Fig.5.

(18)

For p = 0 it is obtained that he = I2e . All these indices have the only parameter ~. The minimization of the expressions of the integral indices (16), (17) and (18) on the basis of the condition

By having determined the values of ~, the values of the parameters lee and Tc are univocal determined by means ofthe relation (12).

(19):

= 0,

These results point out the fact that all optimization procedures based on integral quadratic indices can be reduced to the proper choice of the value of the parameter ~.

(19)

leads to the following expressions:

hemin = 2Tl:

(22)

2Tu3(x-I)

(17)

~

for

=4 ,

(20) 3. INTERPRETATIONS ANC CONNECTIONS

with the graphics b

= t{~)

according to Fig.2.

hgmin = Td1+[I+(t/Td]ll2} ,

The minimum value of the index b is obtained for ~ = 4, and this fact was expected. Any solution

(21)

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,,-~

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bott.

2lJ

belt.

Fig.2. he versus ~.

Fig.3. hg versus ~ for t

with the graphics hg = t{~,t), detailed for the values t 0, 0.25, 0.5, 1 (continuous line for t = 0, dotted line for t = 0.25, dashed line for t = 0.5 and dashed and doted line for t = 1), according to Fig.3.

=

0, 0.25, 0.5, 1.

concerning larger values of ~ becomes " sub-optimal" from this point of view (b (~ > 4) > I2emin ) even if the empirical performance indices cr., ts, tl and
226

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, 2.5

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. . . .: . . . p P . . . .' . . . . .

:

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12

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: :

16

18

obtain their mmunum values. These connections are presented in Fig.6 (with continuous line for t and dotted line for p), for the domain of values ~ E [4, 16], for kp = 1 and TI: = 1. The following question can arise related to whether for the class of systems with Ho(s) of the fonn (1) the use of the indices he, h g and he is useful or not; the authors' opinion is no. The explanation consists of the generalization of the relations (4) specific to the SO method to the fonn (10) (the ESO method) when the possibility to univocal detennine the controller parameters kc and Tc is ensured for a large range of achievable perfonnance, Fig.l, by the simple variation of the parameter ~.

20

bett.

FigA. he versus

~

Finally, it has to be outlined that although the presentation deals with the "continuous time" case, the case of the digital controller is immediately obtained by digitization by means of any known method.

for r = 0, 0.25, 0.5, 1.

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I

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I

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I

,

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: : : \; : : :

f·· ......:····

:

: : : : : : :

:

\:

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4. CONCLUSIONS The paper points out the fact that for a class of control systems with the open-loop t.f. (1), some direct connections can be obtained between the integral indices he, h g and he (that are frequently used in practice) and the value of the parameter ~ by the generalization of the SO method to the ESO method statement (Preitl and Precup, 1999b).

(j B .... \ •• ~ ••••• ~ ...... _~-----~-----: ••••• :-•••• : ••••• : •••• •

,

,

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,

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1-_

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:

:

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2

4

:

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:

12

14

16

18

' '- ~

2 ,- ••

la

20

belt.

Fig.5. he versus

~

for r = 0, 0.5, 1,2.5. i ,.

It results that in this case the minimization of the index he does not lead to spectacular/acceptable results (see, for example, the analysis from (Graham and Lathrop, 1953) concerning the second order system).

fill" ...... 1

I

i

I

.-

. p. ... j, . . " ) ...

'.:j'/

II

,, .

",.

,

.1

r

·4··· .

The favorable effects of using the indices hg and 120, obtained by increasing the weighting coefficients t and p, respectively, lead to: - the increase of the value of the indices with respect to the value heroin (this is obvious); _ the increase of the value of the parameter ~, accompanied by control system perfonnance enhancement, defined by means of empirical indices (see also Fig.!) cri, ts, tl and fPr •

Fig.6. t and p versus the value of ~ corresponding to the minimum ofI2g and he.

From the user's point of view it is useful to know the connections between the values of~ and those values oft and p, for which the indices I2g and 120, respectively,

These connections detennine that when the algorithmic design of the PI and PID controllers (for the class of systems (1)) is perfonned, it is not justified to use the

A

.1

os .. / .

J/ 4

10

12

14

16

1lc1Ia(pJ

227

indices he, h g and he; their place can be taken, with very good results, by the ESO method, by the simple variation of the value of the parameter (3.

Preitl, S. and R.-E. Precup (1996). On the Algorithmic Design of a Class of Systems Based on Providing the Symmetry of Open-loop Bode Plots. Buletinul Stiintific al u.P. Timisoara. Transactions on Automatic Control and Computer Science, 41(55), 47-55. Preitl, S. and R.-E. Precup (1999a). An Extension of Tuning Relations after Symmetrical Optimum Method for PI and PlO Controllers. Automatica, 35, 10, 1731-1736. Preitl, S. and R.-E. Precup (1999b). Cross-Optimization Aspects Concerning the Extended Symmetrical Optimum Method. Buletinul Stiintific al u.P. Timisoara. Tranactions on Automatic Control and Computer Science, 44(58) (to appear). Preitl, S. and R.-E. Precup (2000). Extended Symmetrical Optimum (ESO) Method: a New Tuning Strategy for PI/PlO Controllers. PID'OO IFAC Workshop on Digital Control, Terassa, Spain.

REFERENCES

Astrom, KJ. and T. Hligglund (1995). PlD Controllers Theory: Design and Tuning. Instrument Society of America, Research Triangle Park. Dragomir, T.-L. and S. Preitl (1979). System Theory and Control Engineering. IPTVT Publishers, Timisoara, 2 (in Romanian). Graham, D. and R.C. Lathrop (1953). The Synthesis of "Optimum" Transient Response: Criteria and Standard Forms. Transactions of AIEE. Part II: Applications and Industry, 72, 273-288. Kessler, C. (1958). Das symmetrische Optimum. Regelungstechnik, 6, 395-400 and 432-436.

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