On the Applicability of Certain Optimal Control Methods

Copyright© IFAC Control Science and Technology (8th Triennial World Congress) Kyoto. Japan. 1981

ON THE APPLICABILITY OF CERTAIN OPTIMAL CONTROL METHODS L. Keviczky* and K. S. P. Kumar** *Department of Automation, Technical University of Budapest, Budapest, Hungary **Center for Control Scz"ences, University of Minnesota, Minneapohs, USA

Abstract. It is shown that using optimal control strategies based on zero cancellation the requirement of the inverse stability of a discrete model is fulfilled only for a very narrow class of the sampled continuous processes. Keywords. Optimal control; zero cancellation; zeros of a discrete model; sampling interval.

INTRODUCTION The sampled data control systems have a very important role in the advanced control methods, mostly because a discrete-time technique can easily he realized by digital computers or microprocessors. The discrete-time approach is very popular between the scientists, too, because it is easy to get computer simulation results using the generally recursive formulas. However, it is easy to forget that in the practice the discrete control algorithms work together with continuous processes and a discrete model can be considered equivalent to the continuous one only at the sampling instants. The parameters of a discrete model, therefore the parameters of an optimal regulator and, of course, the intersampling properties of the transient processes strongly depend on the selection of the sampling interval, h. It is much easier to get nice digital simulation results using a carefully selected discrete model, than to investigate, what is the chance to obtain that model by using a reasonable h at a given continuous process. The appropriate choice of h is especially very important in the design of optimal control system, where sometimes a bad selection may eliminate the improvement caused by the optimality. In this paper optimal control algorithms based on d-step ahead prediction will be considered, but the results can also be useful at the other methods. Control Problem A Let z denote the backward shift operator and consider the process given by y(t)=A

(z)B(z)u(t-d)+A

-1

(z)C(z)e(t),

B(z)=bo+blz C(z)=l+clz

-1

-1

+ ... +bmz

+ ... +cnz

-m

-n

,

)

(2)

where znC(z) and zmB(z) have their zeros strictly inside the unit disc. Introduce the identity C(z)=A(z)F(z)+z-dG(z) ,

(3)

where F(z)=l+flz

-1

+ ... +fd_lz

-(d-l)

+gn_l z

-(n-l)

(4) (5)

are unique. The ~-step ahead LS predictor of y(t+d) having independent prediction error can be given by 1 y(t+d I t)-C(z) [B(z)F(z)u (t)+G(z)y(t)] (6) usingtheEq. (3). The minimum variance (MV) control strategy, minimizing V=E{[y(t+d)-y ]2 It} r

(7)

is given by the equation y (t+d It) =y

r

,

(8)

where y is a desired reference value. The solution rof (8), the optimal uO(t) can be obtained in the form (~strom, 1970) o A(z) G(z) u (t)-B(z)Yr+ B(z)F(z)[y(t)-Yr]'

-1

-1

prime polynomials A, B and A, C are given by -1 -n A(z)=l+alz + ... +anz

(9)

This means, that the optimal closed loop system is given by the feedforward A(z)/B(z) and the regulator G(z)/[B(z)F(z)].

(1)

where d is a time delay, yet) the output, u(t) the input and e(t) a sequence of independent, identically distributed r2ndom numbers with zero mean and variance A . The co-

475

L. Keviczky and K. S. P. Kumar

476 Control Problem B

(18)

Consider the noisefree form y(t)=A-l(z)B(z)u(t-d)

(10)

of the Eq. (1). Introduce the identity -d l=A(z)F' (z)+z G' (z) , where

(11)

F' (z)=l+f'z-l+ +f' z-(d-l) 1 .. . d-l , -1 ,-(n-l) G (z)=g~+giz + .. . +gn-lz

(12) (13)

are unique but different from F(z) and G(z). Using the Eq. (11) it is possible to rewrite the process equation (10) into the form y(t+d)=B(z)F' (z)u(t)+G' (z)y(t)=y(t+d It), (14) which can be considered an error less d-step ahead predictor. It is desired to find a regulator such that the overall transfer function of the closedloop system is given by the reference model N (z)

Yr m(t)=M(z)y r (t)= -D( z) z

-d

y r (t)

which means the application of a zero-order hold (or the step-wise approximation of the input signal at the discretization of the state-equations). The well-known transformation of 1 H(s)=K 1+sT (19) using (18) is given by b lZ -1 G(z)= -1 l+alz

(20)

where -x al=-e

(21) x

(22 ) b =K(1-e- ) . l Here we introduced the relative sampling rate x=

(23)

where h is the length of the sampling interval.

(15) LOCATION OF ZEROS OF DISCRETE TRANSFER FUNCTIONS

An optimal control strategy giving the requested M(z) can be obtained from the equality y(t+d It)=y

(t+d) , (16) rm which has the solution for the optimal input (Keviczky, 1979a) o G' (z) N(z) u (t)=B(z)F' (z) [G' (z)D(z)Yr(t)-y(t)].

h T

(17)

The optimal closed-loop system is given by the feedforward N(z)/[G'(z)D(z)] and the regulator G' (z)/[B(z)F' (z)]. In both control problems (and of course in many othey situations, too) the optimal regulator contains the numerator of the process transfer function. This cancellation step involves that transfer functions having zeros only inside the unit circle can be controlled by the above algorithms, otherwise the well-known infinite sensitivity problem would arise (Jury, 1958), the open loop process zeros get into the characteristic equation of the closed-loop system. Therefore it is very important to know the location of zeros of a discrete transfer function. The adaptive versions of these algorithms, based on the recursive least-squares estimation of the predictor parameters, are called selftuning (ST) regulators (Astrom and others,1973).

TRANSFER FUNCTIONS OF SAMPLED CONTINUOUS PROCESSES

Investigating the problem to get inverse unstable G(z), it is known that H(s) having nonminimumphase character results in zeros of G(z) outside the unit circle for short sampling interval. It is, however, not widely known that we can have inverse unstable G(z) for minimum-phase H(s), too, applying very large or very small h, depending on the structure and parameters of H(s). Because this is a very frequent possibility in the practice, we have to deal with it. Let G(z) be given by n n -i IT (z-z.) Z b.z r i=l r i=O (24) b G(z) 0 n n -i 1+ Z a.z IT (z-p i) r i=l where the transformation of poles can be computed by the well-known exponential transformation (25)

p =e-hPci . i=l 2, ... ,n . i

"

Here Pci denotes a continuous pole. (Note that the transformation of poles does not depend on the applied hold, while the transformation of zeros does.) Using an infinitely large sampling interval h=oo , the equivalence means only the equality of static gains n

We can talk about the equivalence of a discrete G(z) and a continuous transfer function H(s) only requesting the equivalence of the output responses for given input excitation or assuming a given order hold serially connected to the continuous process or applying a given order approximation for the intersampling input signal, which assumptions are equivalent (Keviczky, 1979b). The most commonly used step response equivalent (SRE) transformation is given by

H(O)=G(1)=b

o

IT (1-z.) i=l r n

n

=b

IT (l-p.) i=l r h=oo or H (0)

b

o

H(O) H(oo)

IT (1-z.)

o i=l

(26)

r

n

IT (1-z.) , i=l r

(27)

if b =H(oo)fO. The Eq. (27) has the unique solo

ution

477

Applicability of Certain Optimal Control Methods H(O) zl (h=oo)=zl (00)=1- H( oo) ; z2= ... =zn=O.

(28)

This is the case if the degree of the numerator of H(s) is equal to the degree of its denominator. If bo=H(OO)=O, then b has the same role as b l o and H(O) zl (h= oo)=zl (00)=1- b (h=oo) ; z2= ... =zn_l =0 1 (29) is obtained instead of (28). It is easy to check that bl(h= oo) =H(O), therefore zl(oo)=O. Let m and n be the degrees of the numerator and t5e denominator in H(s), respectively. If mc=n, G(z) has n zeros, which (n-l) tend to zero and one tends to -1 H(O) (30) \1- - H( oo) from, when h becomes infinitely large. If m < n, G(z) has (n-l) zeros, which tend to zero af infinite large h. Investigating the loca tions of zeros of G(z) when h tends to zero, it has to be taken into consideration that the discrete model will be equivalent to the high frequency approximation of the continuous process. Assuming the form m

c

IT (1+s'i) H(s)= .=.i_=",-l_ __

(31)

n

IT (1+sT.) 1. i=l this approximation means an (n-m ) order integc rator as m c IT 1. i=l 1---(32) H(s) I n-m I s=oo nIT s c Ti i=l The transfer function 1 1 (33) R(s)= n-m v S C s

,.

has the SRE transformation (Keviczky, 1979b) v - ( z) = -'--'::'!-v-'-'z=--=-"(-l)v(z-l) V Cl [ Z -a h 11 (34) G • _Cla z-e a=O The approximate G(z) can be found in Table 1 for v=1,2, ... ,8 , where the numerically comput-ed zeros are also given. When h tends to zero the poles of G(z) will go to the point 1, according to (25). Zeros of mc number must also tend to this point because of the equivalence requested. The remaining

1 n "\. v

I

2 ,

." -----

; '~-+~------------~

i '

0

!

2

o 2 ,

'-

- .:~.-

"/

,

3 ,..,.'.~

,

20~'_ 03 "~:~'­

,0/ ,

3 0

o ,

2

'" 2 2

3

,

'O _'~-_

Fig. 1. Zeros of SRE discrete transfer fun c tions in the function of sampling interval ( n $4) n-m (if m =n) or n-m -1 (if m 2 , the SRE discrete transfer function always has zeroes) outside the unit circle.Further considerations on the number of inverse unstable zeros can be obtained by investigating th e equivalence of the phase-frequency characteristics of the continuous and discrete frequen cy-responses at infinitely large frequency, when h tends to zero. Remember that the serially co nnected zero-order hold and the continuous process are equivalent to the SRE discrete transfer function (Keviczky, 1979b). The phase of the continuous frequen cy response at infinitely large frequency can be computed (see Eq. (33)) by ~ (oo)=-(n-m ) ~ = - ~2 c c 2

(35)

The discrete freqvegcy-response G(jw) can be obtained by the z=e Jw substitution, and it is reasonable to compute its limit for infinite w value by using the Shannon's limit frequency ws=7T/h, where (36) is invariant to the change of h (Jury, 1958). It makes the phase easy to compute, because G(j w) now has the form . h . h JW ) G(z=e =G(z=eJ ws) 7T =G(-l) I = w= w=w ~ h=O h=O h=Os h

I

i

z....os

zeros

0'

I

n -o . U ()6 , -l.HU, -l) . lO~

-l ot . Ji:- I . n u : _1 .... 1 : - O. IlH -o.~Hl,

",'

~

.7.14'.'.UU.~.UUh'.UUhl"lIhl.Hh.1 (1_11 '

-0.00'1 "

-1l' . ~11 -Il.,n, -).117' : _O.UI l -O.01iH: -O.OO(l H : -1. 0

:

IT (-l-z ) i=l i =b o n IT (-l-p.) i=l 1.

(37)

for infinitely large w values. This value is always real and its phase, considering its sign,

L. Keviczky and K. S. P. Kumar

478

can be evaluated using the following considerations. For stable, minimum-phase continuous system 1Pi l<1, i=1,2, ... ,n (38)

(53) ca~ be ensured if and only if one of the zeros z is -1 . Then

I

(z-z * )

(54 )

=[-1-(-1) )=-0 ,

iJ.F W

s

and (39)

zi
(40)

sign(-l-Pi)= sign( -l ) and

r

sign (-l ) , if (41)

1

As it was shown earlier inverse unstable zeros at h=O occ ured for H(s) with mc
and this factor corresponds to the -n/2 phase value. Anyway, the sign computation by (44) is the same, thus (53), similarly to (51), results in the equation (_l)k=(_l)mu+l (55) which gives the solution n-m - 2 m =k-l= ____c__ = v- 2 (56) u 2 2 Summarizing the above results, the number of inverse unstable zeros is given by

u

sign[G (-l)

n n-m -1 m )=(-1) (-1) u (1) u=

I

m u ,

h=O =(_l)mu+l= cos[-(m +l) n )

.

if v=n-m

n-m - 2 c -2--

,

i f v=n-m

(42)

u

can be obtained . Be cause G(-l) is real, this form ula c orresponds to the following limit value of the sampled phase-frequenc y response

I

[

n - m -1 c - 2 --

I

I

c c

is odd and v>2

(57)

is even.

Table 2 gives the inverse unstable and other zeros in the function of n and m and c ontains the co rr esponding v. q)c( ro) and c ~ (ro)- n/ 2 values, too. c

o ( w=ro) =:; (w ) =arc[G(-l) )= h=O s s h=O h=O s (43) = ar c cos[ - (mu+l) n )=-(mu+l) n I m +1 and (44 ) c os q) ( W ) I =(-1) u s s h=O The transfer function of a zero order hold is given by -hs l-e H (s)= - - (45 ) o

s

which has the phase - frequency response (Jury. 1958) wh (46) Qo( w)=-Z Its value at the Shannon ' s limit-frequency is (w ) = - ~ 2 ' (47) o s i.e . independent of h. The equivalence of the phase-frequency responses can be formulated by the equation to

yC w)

Iw=oo+ 0 0 (ws ) Ih=O =Qs ( Ws ) Ih=O

or -en-m ) ~ - ~ c 2 2

=

s

s

It can be seen on Fig. 1, that all zeros can be located inside the unit circle by an approp riate choice of the sampling interval. This fact unfortunately remains a theoretical advan tage only, as it will be shown next. For cases m =n G(z) can have a discrete zero C outside the unit circle if ~-l,i.e. when H( ro) <

h=O

-[ ( 2k+l)+l)n_-(k+l)n=-(m +l) n 2 u or which gives the solution n-m -1 m =k= ____c __ = v-I u 2 --2-

.

(49)

l I

2 10

.

"c (

-II 1-1-./1 !_lol Il "V" .. ,n'
: 2 1 - .,

,

I 'I' !' 1-,/2 ! \ 2 12 ; 0 1

0

t

(50)

-) ,, /2

:

I

o ~ h (! r

l e rO$

I 0

!

, 0

I

-2 ~

. 1

: 0 ;

- ..,/ 2

: 0 , ; 0 I

-'in/2

. .

1. 0 1. 0 ; 1. 0

-). 7 )21

-h /2

I

- 0 . 2650 - loO; 1. 0 1 0: 1 0

1.0: 1.0:1.0

,

- 0 . 10 1 :-1.0

1

I

0 0

I

: 0

(52 )

-1. 0

0

- r,!2

i

Iw=oo-hi>o (w s ) Ih=O =-[ (2k)+l)¥

I

zero s

1. 0

(ii) Be v=n-m even, i.e. v=2k, then using (35) c and (47) the Eq. (48) is

=-kn - - =q) (w ) 2 s s h=O

(58)

T"b l e 2 / a

ol.c "I'

(51)

n

H(O)

rr-r.--,-----,-,------~~,---------·- ---

(i) Be V=n-m odd, i.e. v=2k+l, then using (43) the Eq . c(49) becomes

q) (w)

"21

by choosing h large enough. This means that a sampling interval h small enough always results in inverse stable zeros of G(z) for this structure. Unfortunately this case, when there is a proportional channel between the input and out put of the process, cannot be considered a frequent one.

(48)

- (v+l) ~
SOME ASPECTS OF THE CHOICE OF SAMPLING INTERVAL

I ,

1

-1.11 2 1

- 0 .2680: La -1. 0 ; 1. 0 : 1.0

loO: 1. 0 : 1.:1 1 . 0;1 . 0:1.0;l.0

! :~ l~;~' i

- O. 0 411 : -0.4)0c.

I

-0 . 2610;1.0:1.0

Sl2 , ~ l -h/2 ') 1" -, S

4 11: - .,/2

') !s

'0

~

0

- 0. 101;-1.0;1.0 -1 . 0; 1 .0:1.0 : 1 . 0 1.0;1 . 0Jl.0:1 .0 1.0: 1.0: 1 . 0; 1 . 0; 1.0

-0.OUS2: -0. 2202: - 1 .0 -0.0431:-0.4 )06; 1 . 0

- 0 .101 :-1.0:1.0:1.0

- 0 .2680; 1. 0 : 1.0 : 1.0

(53 )

Because G(-l) is real the phase value -n/2 in

-lo O; 1 .0 1 1 . 0; 1.0: 1.0

1.0;1 . 0;1.0:1.011.0

1 . 0 ; 1 .0: 1.0; 1.011.0: 1.0

Applicability of Certain Optimal Control Methods

I;

- IO'.ll:-'.U",-L 'Ul

1

I

-<:I.l1U,~ . Uu,-o . ootln -o.O I.~l,

479

-0.1 101 , -1 .0, 1 .0

-o.Otll:-o. ,~ ,I.O:l.O

-0.101:-1 .0: 1.0,1.0,1.0

-O.Uk), I .O,I. O,I.O,I.Q - 1.0,1.0,1 . 0,1.0,1.0,1.0

10

1.0:1.0,1.0,1.0,1. 0 ,1.0

15

1.0,1 . 0,\.0,1.0:1.0,1.0,1 . 0

, e ;;:

_:lI.~.;_;l.'·":·I

I

.• Hl

_: .""

-O.1 1l 1,-Q.Q1I H ,-O.OCM 1H ,-I.Q

j I

-O.llh,-o.H~l;·o.OOtl n , I .O -o.OIHl,-o.H01:-I.O:\.O,i.O

I:: .

-:~:~:;:~:::~~~ ;~~:;~~: ;~O

I:: ;

:~:::O~:;O;:;O;:;O;:;O;:;OO

-1.0

~~'~' __~~'~____________L-~:~::~:::~::_::~::~::~::_ :::_::_::_::_::_,,_,

For case m 2. The number of inverse unstable zeros is ~iven by (57). This means that a sampling interval large enough always results in inverse stable zeros of G(z). Unfortunately in most cases h must be so large that a considerable part of the pr ocess dynamics is practically negle c t ed.

-1

Example Consider a third order continuous process, given by the transfer function 1

H(s)=

(l+ST ) (1+sT ) (1+ST ) l 2 3

(59)

for the control problem A and (60) Tl=2 sec; T =5 sec; T3=10 sec 2 The pro c ess has a n output disturbance generated from an independent, Gaussian, zero mean white-noise with variance one by a highpass filter 1+sT 4 (61) H (s)= - 1 T + ' n s 5 where (62) Now v=n-m =3-0=3>2 is even for H(s) and the c number of zeros outside the unit circle is m =1 at h=O according to (57). This means that d~ c r eas ing h one of the zeros moves out of the unit disc, as Fig. 1 shows. Using Eq. (18) the SRE transformation of (59) can be obtained in the form (z-zl)(z-z2) G(z)=b , (63) 1 (z- Pl)(z- P2)(z- P3) where -0.5h (64) Pl=e and the zeros are the roots of the second order equation 2

z +q z+q2=0 l with the coefficients

ql =

ge -o.lh_ l6e-o.2h+ 5e -o.3h_ 5e-o.Sh+l 6e -o.6h_ ge -0. 7h(66) 6-e -o.Sh+ -o·2h_ -o . lh 1Oe l5e e-o·3h_10e-o·6h+15e-o·7h_6e-o.8h

q2

(65)

= 6-e -o.Sh+ 1Oe-o.2h_ 15e-o.lh

(67)

The zeros are plotted in the function of h on Fig. 2. When h approaches zero one of the zeros tends to -2+13 and the other one to -2-/3

·1-(l

Fig. 2. Zeros of a third order SRE discre te transf er functi on in the function of sampling i nterval 2 which are the roots of the polynomial z +4z+l given in Table 1. The latter zero is predicted by the formula (57). The sampling interval h must be greater than h ~7 .8 se c to get both zeros inside the unit r Cci rcle. Figs. 3 and 4 show the input and output re co rds at the application of the self-tuning (ST) regulator (Astrom and others, 1973) for h=l sec and h=lO sec, respectivel y . On Fi g. 3 it can be seen that only the limits applied for the control input u (t) keeps the variables bounded. The instability of the closed loop system oc cured because of the zero (-3.07 from Fig. 2) outside the unit c ircle. Fig. 4 shows a normal work of the ST regulator: af ter a learning period u(t) does not violate the applied input restrictions. Both zeros lie inside the unit disc, as Fig. 2 shows, however, applying this large sampling interval h=lO se c neglects the two smaller time constants practically.

CONCLU S IONS It is shown that using optimal control strategies based on zero cancellation the requirement of the inverse stability of a dis crete model is fulfilled only for a very narrow class of the sampled continuous processes. If the degree of the denominator of the continuous transfer function is higher by three than the degree of the numerator, then the discrete model always has zeroes) outside the unit circle for practically acceptable small sampling rates. The number and the values of inverse unstable zeros are given for zero and infinite large sampling intervals. The relationship between the location of the zeros and the applied sampling

480

L. Keviczky and K. S. P. Kumar

l,t t'I, IQ

st'c

~~~~,~~~~Ut~~ ' 1.'1 l ul!1

Fig. 3. Input and output records at the application of ST regulator for a third order continuous plant interval is also given . A simple example is presented to demonstrate a theoretical way to avoid this problem for minimum-phase, stable processes using very large sampling intervals. It is shown that this solution is unfortunately not acceptable at practical applications . It can finally be concluded th at only suboptimal control strategies can be considered as well applicable algorithms which are not based on zero-cancellation. The following approaches are suggested as possible solutions of the discussed problems. a)

b)

c)

d)

e)

Increased sampling interval, which is applicable only in very few cases, mostly not acceptable. Introducing an artificial proportional channel between the input and output. It was shown that the presence of b makes o the large h values only to produce a zero outside the unit circle. Suboptimal regul ator using lower order denominator neccessary to the optimality, eliminating the possibility of poles for cancellation. Increasing d, i.e. the prediction horizont, considering the inverse unstable zeros as the zeros of Pade-approximation of additional delays. Application of generalized cri teria, penalizing the control action.

Future investigations should be devoted to analyse these solutions, as possible alternatives, to make the MV and ST regulators more widely acceptable for practical industrial control problems.

Fig. 4. Input and output records at the application of ST regulator for a third order continuous plant REFERENCES Astrom, K.J. (1970). Introduction to stochastic control theory. Academic Press. New York. Astrom, K. J. and B. Wittenmark (1973). On self-tuning regulators. Automatica, ~, 185-199. Jury, E. 1. (1958). Sampled data control sys tems. John Wiley. New York. Keviczky, L. (1979a). A multivariable self tuning regulator for deterministic design. Cent er for Control Sciences, University of Minnesota. Keviczky, L. (1979b). On the transfer functions of sampled continuous systems . Center for Control Sciences, University of Minnesota.