Convergence and Robustness Properties of a Generic Regulator Refinement Scheme

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

CONVERGENCE AND ROBUSTNESS PROPERTIES OF A GENERIC REGULATOR REFINEMENT SCHEME Cs. Banyasz and L. Keviczky Computer and Automation Institute, Hungarian Academy of Sciences H-l/ l/ Budapest, Kende u 13-/7, HUNGARY Phone: +361-1665-435; Fax: +361-1667-503 e-mail: [email protected];[email protected]

Abstract: This paper investigates a generic optimal controller structure and regulator design method, where the identification and control errors are identical. Several special properties of this combined scheme are discussed. The error relationships, controller sensitivity behaviors and convergence properties are also treated. It is shown that this iterative scheme maximizes a robustness measure, too. This last property is demonstrated by a simulation example and simple corollaries for a first order disturbance rejection model. Keywords: combined identification and control, iterative controller refinement, robust control

closed-loop, where discrete-time representations are considered for computer controlled systems, i.e., the argument k of variables means the integer value discrete time (integer multiple of the sampling period) and Z-I means the backward shift operator

1. INTRODUCTION There are many variants of the two degree of freedom feedback control systems published since the classical works as, e.g., Horowitz (1963). Several of them were connected to pole placement design and partly to self-tuning and other type of adaptive regulators (see, e.g., Astrom and Wittenmark, 1984). Some iterative controller refinement techniques are also based on this basic principle. In our recent works a generic optimal controller structure and regulator design method (Keviczky and Banyasz, 1994, Keviczky, 1995, Keviczky and Banyasz, 1995b) have been discussed, where the controller polynomials can be obtained without the solution of a Diophantine equation and in the iterative scheme suggested for this structure the identification and control errors are identical (see later).

(z-IY(k) =y(k

-1)).

S, M,

Pr

and R are the process,

its model, a desired tracking (servo) reference model and the regulator transfer functions, respectively. Requiring a regulating (or disturbance rejection) behavior by Pw the optimal regulator is

R = CK = w

Pw G M-I I-PW GW M- W +

(I)

where

provides the

Fig. I. Generic scheme for all cases theoretical reachable optimal overall transfer functions in case of the ideal exact model matching M = S, assuming

The generic scheme can be summarized for all cases (e.g., for inverse stable or unstable plants) according to Fig. 1. Here Yr' u, Y and ware the reference, process input, output and disturbance signals in the

and

455

(4)

are stable. Here S+ and M+ mean the inverse stable (IS) , S_ and M_ the inverse unstable (IV) factors,

that Y~ does not necessarily change by iteration.

respectively. It can be well seen that the desired tracking Yt(k) = Pr Yr(k) and regulating

2. Using u;={u;(k);k=l, ... , N} identify a model between u; and Y;_I ={Y;_I(k) ; k=l, ... , N} based on the modeling step

Yd(k)=(I-Pw )w(k) behaviors cannot be reached exactly for IV processes because of the invariant factor S_. In some cases the influence of S_ can be attenuated only for given excitations by the extra serial compensators Gr = Gr(yp S_) and

M; =arg min Qic(.At',Au;, R;_I'Pw )= Me.L

(6)

= M; (.At', M;_I' R;_I' YpS, Pr, pw )

Gw = Gw (w, S_). Anyway the above method gives the unique explicit computation of the minimal order controller polynomials formulated in the general infinite number solutions of the optimal closed-loop model matching paradigm of Kucera (1981) , Wang and Chen (1988).

where Qic is a closed-loop identification criterion. 3. Calculate the optimal regulator using (I) then compute the process input u; as

k=I, ... ,N

2. ITERATIVE CONTROLLER REFINEMENT

(7)

and apply to the process in closed-loop.

Figure 2 using the generic controller structure shows an obvious way how to perform the identification step in a combined identification and control scheme (Keviczky, 1995), i.e., a regular identification algorithm should be used between the auxiliary variable and the measured controlled variable y. Note that uand Y must be obtained from the closed-loop operated by the generic optimal controller structure with the optimal regulator R and compensator K r .

4. Once M; and R; are found the closed-loop bandwidth can be increased by repeating the procedure. The iterative process is continued from step 1, while a stop condition is not fulfilled (until the ultimate control objective is achieved or it is terminated by reaching some vital constraints).

u

3. GENERIC SCHEME ERROR PROPERTIES The model based version of the generic scheme shown in Fig. 2 is the long searched ideal one for the combined identification and control problem, because in this case the control and modeline errors are identical. It is interesting to show that the common modeling and control error is A

£=y-y=

PrGrM_ 0 1 -tY +--w= I+RS r I+RS

p.G

1

(8)

= 1~ ;S £+Yr + 1+ RS w =-e

Fig. 2. The generic controller scheme for iterative controller refinement

where

u

Because depends on the model M, only an iterative control refinement procedure can be performed. It's simplest - so-called relaxation type iteration can be built in the following way for an offline case using N samples (i -th iteration is shown):

£=~= S-M M M

and

(9)

u;

1. Calculate the auxiliary variable based on the available model M;_I' the reference model Pr and the applied reference signal series

Y~

o

= {y~(k);k =1, ... , N}

u;(k) = Pr k

Gr(M~-I)[M~-lr Yr(k)

(5)

Fig. 3. Uncertainty model of the generic scheme

=1, . . ., N

Figure 3 represents the uncertainty model corresponding to error equation (8) . It was shown (Keviczky , 1995) that the frequency domain

Here i denotes the index of the iteration and note

456

weighting properties of this error is superior to the open-loop method and to others (parallel-closedloop schemes) usually applied in combined iterative refinement methods.

C

w(k)=-~(k)

(19)

D

where ~ is white noise, then the well-known d -step ahead LS predictor can be obtained in the form of

Straightforward calculations give the following equivalent form of the common error E

L

w(k+d)= Dw(k)+F~(k)=P~w(k)+F~(k)

(20)

Here D= LC+z-dF Here Er and Ew are the contributions of Yr and W to the error E, furthermore Eo is the remaining uncompensable error (in case of ideal process model matching: R= 0). Because generally Pw is also not

(22)

the ideal, it is an estimate of the true (or optimal) P~ predictor, the above .formula can be further rewritten

The above iterative procedure is based on the so-called relaxation type method, which does not use the derivative of a criterion instead finds the solution of an nonlinear equation x = f(x) by the simplest

as E

= GrM_R p.

l+RS rYr

_ GwMj P. w+ l+RS w

'-v---'

'-v---'

iteration Xi+1 (11)

E~w

'

.'

E~

Here Et.Pw occurred because of the not ideal Pw predictor and E~ is the theoretical (when both R= 0 and Rw = 0) residual error, where

(12)

de

= (Xi + f(x;))I2).

M(1+CM- 1S)Y= 1

1

I

=- M 1 + C(1 + R) Y ( =0 =: -

(14)

(23)

1 MY

Because the true derivative depends on the real closedloop, so on the true system S, this approximation is good assuming M = S (R = 0). Even if this formal approximation is valid, however, it is always difficult to filter by If M and not only for IU systems, but for delay time, too. Therefore it is necessary to find other natural approximation of this function. Observe that

(15)

are introduced, furthermore (16) ~

(or Xi+1

1

aM =

(l3)

are predicted variables based on the reference models. Finally

= f(Xi)

At the same time, of course, it is possible to formulate this iterative refinement procedure as the minimization of a quadratic criterion Q =E2(k)l2, forming either an on-line or off-line method. Both approaches require the gradient of E(k), because aQ/aM = E(k) de/aM . Let us calculate this gradient for the simplest case when the process is IS: so Gr = Gw = 1 and M_ =1 The form of this gradient is very instructive

+RwGwM_Pww +(1- P~GwM_)w ~

(21)

and the true (optimal ) predictor is

and Pw

1

-u=:--y M

(24)

is a possible good approximation. Here u is the input of the process. Thus computing the gradient the calculation of E is based on Y and U, however aE/aM should be computed using u instead of it if we want an approximate gradient in our identification. Let us observe that there is another possible approximation to properly calculate the gradient. If the Yr = ~Yr is substituted by Y then the auxiliary input u becomes a good approximation, so the original method identifying the model between it

(17)

is obtained, where (18)

One can recall that if the additive output noise is driven by 457

and Y becomes a first order gradient based one. In this case e and ae/aM are also calculated using U. This method should start by using Yr as the first excitation then Y:: Yr must be used in all sequential steps.

5. SOME CONVERGENCE PROPERTIES OF THE GENERIC SCHEME Equation (17) shows the different convergence investigation possibilities. If the iterative controller refinement procedure provides that both € = 0 and €w = 0 are fulfilled then the generic controller scheme gives the theoretical reachable residual error

The analogy with the derivative performing method of Gevers (1996) is obvious, however, the two techniques are different.

e~ = (1- P;GwM_)w. This error is the theoretical reachable optimal one for IS processes, when it

4. SENSITIVITY OF THE OPTIMAL REGULATOR

depends only on the optimal predictor P~ of w. For IV processes it depends on the invariant factor, too, so it can be optimized only for a given class of w and M_ by Gw = Gw (w, M_). The special structure of the generic scheme provides excellent ways to investigate optirnality and convergence separately.

Investigate the sensitivity of the optimal regulator when the process is IS: so Gr = Gw = 1 and M_ = 1 Using (1) it is easy to obtain that o M Ro-R .eR = - = - - = - € Ro Ro

for "tro

(25)

Assume, e.g., a disturbance free case when w = 0 . The I~. norm of the common error is now

t.

This means that any identification method minimizing II€IL will provide the most robust regulator

Ilell.. =

~ IIGrM-P.-YrILI11 +lRslLll€lloo = Cl C2II.eJL

(26)

Ll

for "tro

the initial model Mo = Mo{€o} gives a stable system then the iterative regulator refinement scheme (5)-(7) providing a monotonous decreasing relative uncertainty II€ results in

(27)

ilL

The last equation has much less importance, because S is not known, so any norm of can not be

!lLl/SII minimized easily, in spi te of IIM/RI!.. = IIN51!... The

(32)

relationship between the two regulator relative uncertainties is

In this section the properties of robustness measures for the generic scheme are also discussed for IS processes, when similarly to the process M+ = M, M_ = 1 and Gr = Gw = 1. (The results are very similar for IV processes, however, the necessary formulas are much more complicated.)

These equations calculate only the uncertainty coming from the process model. To calculate the influence of the uncertainty in Pw the whole first order difference of M should be formed as

M:: aR M1 + CJR M'. = aM Mw w

Straightforward calculations give that

Pm(R)=~nll+R51= Ill-~w

(33) 11

1+ Pw €

00

(29)

= -RM-1M1 + R p;l M'. 1-Pw w

and the upper limits can be obtained as (34)

Thus the relative uncertainty can be obtained by M M1 1 Mw €R = - : : - - - + - - - R M 1-Pw Pw

(31)

For bounded excitation Yr and for obviously stable Pr it is easy to see that Cl is bounded. For robust stable closed-loop system c2 is also bounded. Thus if

If the uncertainty in R is related to R instead of Ro as above the following formula is obtained

M € €R = - = - - - = - R 1+€ S

II~~; PrYrl!.. ~

where (30)

~o ~

1

1 11 Pm =Pm ( €=O ) = 11 I-P w

=-€+--€ I-Pw w

458

' I~

00

=mm (!) I-Pw

(35)

runs an additive white noise was used as output disturbance with a standard deviation A = 0.01. Start the above iterative controller refinement by the initial model

so p~ is the theoretically best optimal (maximal) robustness measure obtained if Ilell.. ~ 0 (Le., for exact model matching case) during the iterative controller refinement. Since it is easy to construct lower limits, too, the following inequalities hold during the iteration

l-llpwll..IIedl.. ::;~nll + Pw eil ::; P~~Ri) ::;

(42)

providing convergence if lie i 11.. ~ 0 .

and perform the sequential steps, which are repeated off-line closed-loop simultaneous identification and controller design steps. Figure 4 shows the relative control and identification loss functions (variances) by iteration. It can be well seen that the iteration is quite fast reaching the optimal values after 10 steps. Figure 5 shows the maximum of the multiplicative model error ~edl.. during the iteration. Fig. 6 shows Pm by iteration, how it reaches its optimal

Corollary I For a first order disturbance rejection reference model

maximum value p~. The two upper and two lower limits (see above) are shown in Fig. 7 limiting the relative robustness measure Pm/P~ .

(36)

Pw eilL ::; 1+ IIpwll.. lIeil ..

::; 111 + which means that

(37)

=

p

0.2z-

1

(38)

1-0.8z

w

\:

\

one can easily compute that 1.5

p~ =0.9

:::::: Relative conlIOlloss [unction

8°f~ '

0.5 .. . .

40 .. • -:- . . -', ... : .. . . :. -. -:- .. - -:- .. . -:- _ . - ... .

.

.

,

20 \ -:. :.. . . .: . .

,

.

.

.

,

:

..

: : :

. :

:

"" :'
····l· \·····.- ···:····j····J····:· ····j· ····

(39)

60 . . -:" . . . ':- ... : ... . : . . . ":' ... ":- . . . :- ...

.

,

:

j

· ···~·· ·· ·\:·· ·· j ··· ·j · · ·· : ···· :· · · ·· i

00

::

:~

2

6

4

:

:

::

8 10 12 Number of ilellllions

14

16

18

. : .... : .. .. :.... . :.. .. : . . . . . . . .

O~~--------~~------------~ o 2 4 6 8 10 12 14 16 18

Fig. 5. Maximum of the multiplicative model error

Relative identification Joss function

4O~Ji " .::::lj 30 . \. -: ... ..: . ... ; .. .. ; . . . -: .. ... :- .. . .:- . .. ; ... .

20 .. \

10

. .. . .: . .. . : .. . . : .. .. :.... . :.• . • .: .•.• ; .. . .

"< .:....:.. .. :.... :.... :.....:.. .. .:.. ..:.. ..

00

2

4

6

8 10 12 Number of iterations

14

16

0.8

r

1 0.5zan d P w 1-0.5z- 1

_

. . 0:". ........ 0:". ........ . ....

.

.

.

~

.

.

.

..

.

.., .. .. ................. , .. ...... 0." ........ "." ....... "' .. ..

: Pm minp + R.S1 ........... , ••••••••• • ••.•••• • ••• •• ••••••• • . •• "' • • a

0

•

•

•

•

: P::' =Pm(e=O)=-1 I L =fu..:..!l=0.9 :

.:

.

..

..

.

2

4

6

8

I-P.,

2

:

.

..

14

16

0.2 ••. •• ••.•••••••••••••••••••••••••.•.• • ••••.••••••.•

(40)

_0._2..;,..z-_1"7" = 1-0.8z-1

..

=

0.4

.

.

10

12

Number of itellltions

Fig. 6. Minimum of robustness distance Pm(R) = minl1 + R51 by iteration

i.e., d = 1 here. Apply unity gain tracking and disturbance rejection reference models p =

.

.. , .. ....

0.6

Let the IS sampled process be given by

0.125Z- I (1 + 0.6z- l ) S = -;-----:+-;----':-;(1- 0.5z- 1)(1- 0.8z- l )

.

18

Fig. 4. Loss functions in the iterative refinement Example

~

........ , .. ........ ,.. .. .. .. ~ ......... , .. ........ ~ ........

Cl)

(41)

6. CONCLUSIONS

for the optimal controller design procedure (Gr=Gw=l; M+=M and M+=M). A unity amplitude square wave input signal with periodic time 40 samples was applied and the off-line procedure used N = 100 samples. In the simulation

This paper demonstrates the necessity of iterative controller refinement methods in control engineering practice. A generic optimal controller structure and regulator design method, where the identification and control errors are identical, are used in the 459

,,~ : :: ...... ~ ••• : •• • ••: ••••• :. . . . . , .... •.

2.5

I

Keviczky, L. and Cs. Banyasz (1995b). A new poleplacement regulator design method. 34th IEEE Conf. Decision and Control, New Orleans, USA, 3408-3413. Kucera, V. (1981). Exact model matching, polynomial equation approach. lnt. J. Systems Sci., 12, 1477~ 1484 . Wang. S. and B. Chen (1988). Optimal model matching control for time-delay systems. Int. J. Control, 47, 3, 883-894 .

+lpwLlt1.. :._ h

.. +

p.

AI w' l.. ...•

2 •••• : . •\ : ••• ••: •••• •:••••••••• _ minjI+P.,1j : . .

:

p

1.5

....

:

:

. . . . ~ ...I:-.I:':I:-.~. • . :::-.

• ••••1-.. . : .., .. ,,~.:. ••••• :.....

~~

'--Pm__-.;_:-.....~-.;..:7"';:'-""~.... -::.:~..::.:=..==..f;II...•••• ' . r~. •••• ~.•••• ~.---

.

r,

..

~ ~;~----~0.5 . ~:::.'-::r';,;,,~. o:" ... o:" . ... ~ .... ~ ....

.

':0 ... 0: ... . o ····;. · ·/~· , . .....:.· .. ·:·. .... .; .. .. ~.......:..... .; ... .

.{I.51···)/ · ·~···· .~ _}_

o

....+....;.....~ .... +.... ~.....

t'

,

,

,

,

.

2

4

6

8

10

12

14

16

18

Number of iterations

Fig. 7. Combined relative robustness limits investigation. Several special properties of this combined scheme are discussed. The error relationships, controller sensitivity behaviors and convergence properties are also treated. It is shown that this iterative scheme maximizes a robustness measure, too. This last property is demonstrated by a simulation example and simple corollaries for a first order disturbance rejection model.

Acknowledgements - This work was supported in part by US ARO Grant No. DAAH04-96-1-0068 and the Hungarian NSF (OTKA), which is gratefully acknowledged by the authors. 7. REFERENCES Astrom, K.J. and B. Wittenmark (1984). Computer Controlled Systems-Theory and Design. Prentice Hall, Englewood Cliffs, N.J. Astrom, K.J. (1993). Matching criteria for control and identification. ECC'93. Groningen , Netherlands, 248-251. Gevers, M. (1991). Connecting Identification and Robust Control : a New Challenge, 9th IFACIIFORS Symp. SYSID'91, Budapest, Hungary, 1-10. Gevers, M. (1996). Learning from identification to control. Joint COSY Workshop, European Science Foundation, Valencia. Horowitz, I.M. (1963). Synthesis of Feedback Systems. Academic Press, N.Y. and London. Keviczky, L. and Cs. Banyasz (1994) A New Structure to Design Optimal Control Systems, IFAC Workshop on New Trends in Design of Control Systems, Smolenice, Slovak Republic, 102-105. Keviczky, L. (1995). Combined Identification and Control: Another way, (Invited plenary paper.) 5th IFAC Symp. ACASP'95, Budapest, Hungary, 13-30. Keviczky, L. and Cs. Banyasz (1995a). A new optimal regulator design method. European Control Conference, Rome, Italy, 3359-3364.

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