A Simplified Approach to Geometric Control

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A SIMPLIFIED APPROACH TO GEOMETRIC CONTROL M. Kiimmel l *, L. Foldager* and P. C. Hansen** *(.'hrm/cal ElIgillrprill~ Drparl11lnlt. TfclOl/cal l. ·1I1t'f'r.\/I." of [)nwwrk, 28()() I..Plgh."- IJ/'fjuulrk **ht\lilutf Jilt XUlIlt'r;cal A.lwlyJL\, Tfrlll/iml C,lil'pn;t)' uj IJnw/(Jrk, 28(H) I.ynghy. 1Jf'l11II(lI'k

Abstract. A s hort cut method, based upon Wo nham's geometri c approach for multivariable control, is presented and applied to control of a distillation co lumn. Simulation results for geometric control of the distillation column a re presented and com pared with the well-known PI and optimal controllers. Keywords . Geometric control; distillation co lumn control; di s turb ance r e jection; multivariable constrol systems; control computer applications; control sys t ems synt he sis; chemical industry; industrial control; process control . INTRODUCTION

Here the geomet ri c co ntr ol i s appli ed to disturbance rejection. Takamatsu et.a !. (1979 ) ha ve deve loped the theoretical results for disturbance rejection on a distillation co l umn base d on ~Jon ­ ham' s results, and they have evaluated the result s through si mulation of the basic geometric control . Also Shah, Fisher and Se borg ( 198 1 ) have consIde red this problem . A graph theore ti c approach to the sa me problem i s dev e lope d by Schizas a nd Evans (1981) .

In the early 1970's a new so called "geometric approach " to multivariable control was introduced by Wonham and s ubsequentl y developed by Wonham and his coworkers ( Wonham et.al. 1970a,b, 1974a, b,c, 1975 and 1979 ) . In this approach it is recognized that the properties of a linear system depend on the structure of the linear subspace generate d by the given system in the state space . The design of a control system is regarded as a synthes i s process in which the resulting system, i ncludi ng the control system, will generate a l inear s ubspace which has the desirab l e structure in the state space so as to satisfy t he design specification . By describing the design specifi cations of the feedback controller as the structure of t he feedback subspace generated by the controller system, the de s ign is treated more intuitive l y, and better insight is obtained concer ning conditions for the existence of a solu tion and other related problems such as decoupl ing control and disturbance localization (Shah, Seborg and Fisher, 1976, 1977 and 1981 ) .

The basic geometri c controller s upplemented with pole placement or with feedfor wa rd is further investigated. The re s ult s are co mpared to optimal control and to PI control with feedfor ward. Th e conclusion is that geometric cont r ol supplemented wi th feedfor ward i s s uperi or in pe rforman ce to the other controllers investigated. THE GEOMETRIC APPROACH The following development is a s ummar y of Takamats u ' s results with a s lightly different notation. The multivariable system is de sc ribed by

Geometric control has been implemented on some processes, e.g. evaporators, (Shah , Fisher and Seborg, 19B1 ) a nd ( Kummel and Ohrt, 1981 ) .

x ( t)

(1 )

C2'. ( t) ,

y et)

Even with the contribution of Shah, Fisher and Seborg the application of geometric control is not widespread, possibly due to the abstra c t nature of the underlying theor y . Assumi ng this to be t he case there is a need for a simp l ified design method. In this paper such a method based on a simpli f ication of Wonham's approach is presented .

where the state vector x( t ) has d i mension n, the c ontro l vector u ( t ) dimensi on m, the output vector y et ) dimension p, and the di s turbance vector g (t) dimen s ion d ( where p < m) . Th e matrices A( nxn ) , B(nxm ) , C(pxn ) , and D (nxd ) are time independent. The problem is now to find t he feedback matri x F (mxn ) in u

The design method is applied to multivariable control of a distillation column . The objective is to keep two output variables, the composition of the distillate and of the bottom product at certain desired va l ues by manipulating two con trol variables, the reflu x flow rate and the boi l- up rate. A mathematical model of a disti l la tion co l umn together with a geometric controller has bee n simulated in order to evaluate the effectiveness o f the control methods.

= F 2'.

(2;

so that any dlsturban ce g ha s no influence on th e control l ed output y. Note that through this defi nition, C may be diffe rent from the usual output matrices. Inserting eq. ( 2 ) into eq. ( 1 ) yie ld s

lA l l correspondence to: M. Kummel, Chem . Eng. Dept., DTH, 2800 Lyngb y , Phone 02 . 88.32.88 - Telex : 37529 DT HDIA DK

(A+BF )2'. + Dg ,

(3 a )

C

( 3b ~

x •

With the initial conditi on x = 0, the output vec tor y is obtained from eqs. (3a ,b ) as 1701

M. KUmmel, L. Foldager and P. C. Hansen

1702 n-l L: k=O

y et )

(4 )

where B k ( t ) is a scalar function ( which can be determined through Cayley-Hamilton's theorem ) . For the output variable y to be zero, one must require

IQI ,

(5)

k=0,1,2, . .. ,n-l

The feed forward eq. (15 ) .

matrix

obtained

is

from

Pole shift. Regardless of whether feed forward control is applied or not, the dynamics of the system may need improvement, e.g . by pole shifting if some eigenvalues of A+BF lie too close to the positive half plane. In the feedback matrix F there will be, among the mxn elements, some which are not fixed through eq . ( 12 ) . These elements are potentially available for pole shifting.

where 0 i s the range of matrix D (0 is spanned by the columns of D) . Eq. ( 5) is equivalent to SIMPLIFIED GEOMETRIC DESIGN

( A+BF )k O~ N( C) ,

(6 )

k=0,1,2, . . . ,n-l

where N( C) , the null-space of C, is defined as

QI.

( 7)

Eq. ( 6 ) is s a tisfied i f

o ~N ( C )

(Ba )

( A+BF ) 0

~

(Bb )

0

and it can be shown that there exists a matrix F whi ch satisfies ( Bb ) if, and only if

Os

(9 )

V*,

where V* is the s upremal ( A,B )-invariant subspace in N( C) ( see Wonham, 1979 ) . V* can be obtained by the s equence:

vo

The geometric design method as presented above may appear to be difficult due to the abstract nature of the underlying theory, especially the synthesis of the supremal ( A,B )-invariant subspace V*. We shall now present an alternative design procedure which is easier to handle and which also gives a bet ter insight into what is required concerning the system outputs to obtain total di s turbance rejection. The restri c t i on i s that the output y consists of a subset of p elements of the n state variables x, where p obvious ly mu s t not exceed m, the number of controls. The matrix C "picks out" the particular variables of x to be decoupled, and to simpli fy the proof i t -I S no restriction to assume that those are the first p elements of ~. Hence, C has the structure

(10a ) -1

N( C) () A ( Vk+B ) ,

k=0,1,2, ...

(10b )

C

Note that A is allowed to be singular, in which case ( lOb ) takes the form, (11)

where A+ is the pseudo-inverse of A ( see e.g. Lawson & Hanson ( 1974 )) , A is the range of A, and N( A) is the null-space of A. When eq. ( 9 ) is satisfied, one must choose F such that ( A+BF) V*

c::: V*

(12 )

i s satisfied. Feedfor ward control . I f the system does not satisfy ( 9 ) , complete disturbance rejection is unattainable solely through geometric feedback control. However, through addition of feed forward loop s , eq. ( 2 ) is changed to

When C doe s not have this form, it can always be diagonalized by a change of bases for x and y. This diagonalization is obtained by use of singular value decomposition ( SVD ) , ( see e.g . Moore, 1979 ) . For the analysis below, three submatrices A*, B* and D* are introduced. Definition. The submatrices A* ( pxn ) , B* ( pxm ) and D* ( pxd ) are defined to consist of those rows in the state matrix A, the input matrix B and the disturbance matri x D, respectively, which refer to the p decoupled states. With the above assumption we obtain, _n_ -m-

There exists a feed forward matrix KFf,' which guaranties the "closed loop disturbance', D+BK[F' to be decoupled from the output, if and only if FF

)

C

V*

n

A*

p

-----------

•

I

(14 )

range ( D+BK

(15 ) D

or equi valentl y ,

(17)

The rank of C is obviously p.

A

res ulting in a closed loop equation

H

-p

( 13 )

F ~ + KFF g,

0

[ 1

0

in whi ch B is the range of B. This sequence always converges after at most k = dim ( N( C)) steps when Vk+l = Vk , and then V* = Vk ·

u =

..

n N( C)

I

Aa

•I

-dD*

tp

n

-----------

~

B

n

B*

p

-----------

•

Ba

(1B )

DO Dc V* + B.

(16 )

If eq. ( 16 ) holds the design of the geometric c ontroller can be completed, resulting in the desired disturbance rejection.

These three, easily obtained submatrices are very useful, s ince the follo wing holds:

A Simplified Approach to Geometric Control Theorem: If B* has full rank ( p ) then the supremal ( A,B ) invariant s ubspace V* i s equal to N( C) , the null-space of C. Proof:

First, VJe notic e that N( C) the columns of the matrix

is sp a nned by

_ -- n-p --_

o N

c

p

-----------

n

o

(l9 )

I

n- p

!

o Since the rank of B* i s p, ther e exist p linearly independent columns o f B* and therefore p linearly independent co lumns of B. Another consequ e nce of B* having full rank i s that we cannot wi th th e above p columns of B produce any linear c ombina tion with zeroes at th e first p entries . Hence, the columns of N are linearl y independent of these p columns o'f B. This means that the matrix [N : BJ ha s ( n-p )+p = n linearly independent co c lumns - it has full rank. Thus, N( C) + B

=

Rn .

1703

3)

Determine the e l ement s of F ,·,hi c h are nee de di to sat i sfy eq. ( Zl ) .

4)

If the rank of D* eq. ( 22 ) .

Example:

f

0, de vel op KFF from

Bi nary Distillation Column

In thi s example , we have us ed th e math emat ica l mod e l for a 9 plate distillation c olumn ~ ith con denser and reboiler, derived by Holland ( 197 5 ) and subs equently used by Takama t s u ( 1979 ) . De tail s of the model are given in Holland ( 1975 ) . The di s turbance is a change in composition a nd flo w o f the feed stream. The output s are top and bott om compo sitions and the manipulated variab l es ar e fl m·, rates of the reflux fl o", and the vapor s tre am. When this model is linearized ar ound th e stead} state cond iti on s , the model listed i n tabl e 1 ap pears. The design goal i s now to decoupl e the Input di s turbances from the t wo outputs . 1.

Let the decoupled states be Xl and XII. Nlth this choice of outputs, the s ubmatrl ces be come:

A*

( 20 ) -0.1740.1050000000

Next, we in s_'Yt eq . h20 ) into eq. (lOb ) , for k=O. Obvious l y, A Rn = R , which means that VI = V = N(C ) , and that V* = N( C) . If A is singular , ~he subspace inside in e q. ( 11 ) i s again Rn, and V* = N(C) , independent of th e rank of A.

o

[

o

0

0

0

l

0 0 0 0 0 0 0 0.115 - 0.17 1J

B*

If B* has ful l rank, then the follo wing hold s : 1)

2)

If the rank of D* i s null, i . e . D* holds only null elements, then total di s turbance rejection can be e s tabli s hed through geometri c feedback control u = F ~ s in ce D S V* = N( C) wil l be satis f ied.

f 0, then total di s turbance re j ection is poss ible through geometric feedback control combined VJi th feed forward u = F ~ + K F g · nThis is true, s ince D S V* + B = N( C) + = R ( cf. ( 20 » , whi c h means that eq. (l6 ) i s al wa ys sa ti s fied, when B* has full rank. If the rank of D*

a

The feedback matrix can be derived from eq . (l2 ) , which in thi s de s ign method simplifies to ( A* + B* F )N c

0,

( 21 )

where the multip li cation ",ith N simply "picks out" the n-p columns of ( A*+B*F ) w'hich do not cor respond to the dec oupled states . The feedfor ward matrix KFF c an, if ne e ded, be deri ve d from eq . ( 15 ) which in this design method simplifies to D* + B* KFF

=

O.

In conclusion, the design procedure for the simp lifi ed method is:

Z) IWC4-B*

Check that B* has fu ll rank ( p ) .

Since two states must be decoupl e d t he rank o f B* must be t wo. Thi s is, however , not th e c as e . Con sequently, a geometric c ontroller i s not f eas ible for decoupling of Xl and xl! and apparentl y it cannot be designed by thIS me thod . Howev er, through study of the matri ce s A, Band D it i s ob served that Xl is onl y a function o f Xl an d x ' 2 and for this reason Xl can be dec oupleB thr ough decoupling of x . 2 2 . Th e decoupled states are now x and xl!. 2 this case the s ubmatrice s become

A*

In

.... a 2 . 11 ] • ... a l 1.11

-0.943 0.469 0 0 0 0 0 0

o

( 22 )

Accordingly, a design procedure is developed where only the submatrices A*, 8* and D* are examined, followed by the solution of eq. ( 21 ) and possibly eq . ( 22 ) . At the same time thi s procedure gi ves a quick answer to what i s required to obtain total disturbance rejection on the s ystem .

1 ) Determine the submatrices A*, B* and D*.

D*

o

J

o0

0

0 0 0 0 0 .1 15 -0.00244l

B*

[b 2 . 1 b 2 . 2 b ll . l b l 1. 2

[0.00328 0.00032

D*

[d 2 . 1 d Z . 2 ] dILl d l 1.Z

[~ 0.0~032]

-0. 00042J '

Here the rank of B* = 2 and the rank of D* f 0 so it is possible, through combi nin g geometric feedback with feed f orwa rd control, to decouple top and bottom product compositions from the disturbanc es

M. KUmmel, L. Foldager and P. C. Hansen

1704

0 1 and 02' K is found by eq . (22) which, with tne relevant eFfements from table 1, yields KFF =

1 . 308 1. 758

[ 00

The feedback matrix Le. (A* +8*F )N

[a2~1+e

J.

(23)

F is derived from eq.

(21) ,

c

a 2 . 3+e

e

e

e

e

e

e

e

e

e

e

e

e

e

where the elements e in the column are derived from f 1. j + b 2 . 2

all~lD+e]

i ' th

e1.j

b2 . 1

e2. j

b ll . l f l.j + b ll . 2 f 2. j

row and

= 0 (24) j' th

f2. j 1,3,4 ... lD (25)

Note that since x and xl are to be decoupled, 2 the 2nd and 11th column of A*+8*F do not appear in eq. ( 24 ) . This l eads to 9 sets of 2 equations with 2 unknowns which, when solved , yie ld the following feedback matrix: Fl

=

- 367 . 36 f1.2 -330.06 [ -279.89 f 2 . 2 -251.47

o. ... 0

11] .

470. 17 f 1. 0 . ... 0632.04 f . 2 11

Th e e lements in columns 2 and 11 are available for pole shi fting. We st ress that the design procedure present ed here is a s hort cut method which does not necessar il y yield the general so lution . For example, application of the complete design process shows that the elements of the first column of F are not fixed . Also , there may exist a so lution even when B* is not of full rank. Pole s hift. In some situations, feed forward control is not applicable, either due to lack of necessary measurements or due to drift in the open loop system caused by inaccuracies in the applied mathematical model . The problem then arises whether F can be chosen to ensure that at least some of the outputs y are decoupled. In this example we note that the only violation of eq. (8a) is caused by d 1l . t O. Hen ce , Yl = x may still be decoupl2 2 ed If we require i=1,2, .. . ,10 i =ll

(26)

where Vc is the s ub space ( 27 )

Eq. ( 26 ) leads to the equation (2 8 ) which imposes the following relation between elements of the last column of F: (29 )

With this r estriction , we have chosen F2 = -369.36 - 400 - 330 . 06 0 0 0 0 0 0 470.17 15001 [ - 279.89 - 200 - 251 . 47 000 0 0 0 632 . 04 2000J '

Simulation Th e nine plate di sti llation column with reboiler and condenser under control of the derived geometric control l er was simulated with a 15% step disturbance in either feed composition or feed rate . The contro ll ers were al l de signed based on a linearized math ematical model of the co lumn. Ho wever, in the simulation of the column the non - linear model was applied. The re sulti ng ste p responses are presented in figs. 1 and 2 wit h only 7 of the 11 states , i.e . the condenser: xl' x 3 ' x 5 ' the feed plate: x ' x ' x9 and the reboIler: x ll . Num6 7 bers on t he fIgures are absolute and ( in parenthesi s) relative deviations from steady state at t =17 min, and the similar for all the s imulat ed controllers . Fig. 1 shows the r es ult of the geometric control, i.e. controller F] wit h zero e l ements in the 2nd and llth column, fo a step disturbance 6 01 and in fig. 2 of lfl . Th ese c ur ves are sImIla r fo Taka mat s u' s result. He has derived a controller ac cording to Wonham ' s method which is identical to the derived simplified controller, but ,"ith zero elements in the fir s t column of the F matrix. The result is a s expected that 6 0 is comp l etely de coupled from the output xl anb xlI' however, 6 ° 2 is only decoupl ed from xl ou t not xlI' FIgS . 1 ana 2 further show geometric control WIE h pole shift, i.e. controller F . 601 disturbance in fig . 1 2 gives almost the same result with a nd WIthout pole shift. This i s so because the contro ller F 1 i . e . the elements in the 1st, 3rd and 10th column decouple the states x~ and x ' which are the states ll to be used for po-.te placement. He nce the pole placement does not contribute to the closed-loop gain, and therefore the transients in the column are unaffected. For the same r eason the response in fig. 2 shows an improvement in the transient and considerable r ed uction approximately 65% in offs et in the case of 602 disturbance. This is because the bottom product cannot be decoupled through geometric f ee dback control only . Fig . 2 further s hows geometric control with feed forward eq . (23 ) to a step disturbance in 6 02' The responses show that the feed forward controller de couples the disturbance from the column . Accord i ngly, neither xl nor x is disturbed , nor are ll any of the states x ... x in the column. This IO 2 could be expected co ns ider In g the McCabe - Thi e l e diagram. Onl y the f eed flow rate i s change d during this s imulation, a nd when the top and bottom product are held consta nt, there is on l y one solution f or the composition profile, i . e . the original stationary profile. For comparison fig s . 3 and 4 show PI control derived in appendix wit h dynamic compensation and with feedfor ward. Even though the con troll er does not decouple xl and x from 6 02 it does so for ll 6° , and all together this co ntroll er performs 1 s ur prisingly we ll. I n figs. 3 and 4 the performance of the optimal controlle r eq. ( A3) is further shm,n . Again for 6° , x and xlI are ef f ectively decoupled, how1 ever, ~or 602 xl and xl] show a minor offset about the same magnitude as (he PI contro ller. In figs. 1 and 3 there is almost no difference bet ween geo metric and optimal contro l because for comparison in the optimal controller xl and x are heavily ll weighted, and further the optimal control pays very little attention to the other states. During all the performed simulations the maximum amplitude of the control signal has been observed. Th e general result i s that the geometric control lers require the sma llest magnitudes, the optimal

A Simplified Approach to GeGmetric Control controller slightly larger magnitude, and the PI controller almost twice the magnitude required by geometric control. For all th e responses the largest magnitude in the control signa l is observed for feed flow rate disturbance. Discussion of the Simulation Results The result of these simulations is that disturbances in feed composition can be rejected from the top and bottom plate through basic geometric control. Disturbances in feed rate can be simi larl y decoupled from the top plate, but not from the bottom plate. If the geometric feedback contro ller is supplemented "Jith feedforward contro l the entire co lumn can be decoupled from feed rate disturbances. The basic geometric controller is supplemented wit h pole placement. The result is most important for feed rate disturbances since the offset in the bottom plate is reduced with 65% compared to basic geometric control. Also th e improvement of the transient responses must be appreciated. The result for the optimal controller shows that it performs just as well on disturbances In feed compo sition as the basic geome trIc contro ll er. However, disturbances in feed r a te influence both the top and the bottom products. Furth er, the simulations show that PI control with dynamic compensa t ion and feed forward pe rforms excellently. Ho wever, it does not control the column as well as the geometric controller for disturbances in feed composition or in feed rate. In particular, the disturbance in feed rate is handled better by the geometric controller than by the optimal controller or by the PI controller, since with feed forward added the entire column is decoupled from the disturbances. Both geometric, feed forward and optimal control do, ho wever, rely on an accurate mathematical model of the column and are vulnerable to fluctuatIons in plant parameters. This uncertainty must be kept in mind since the potential discrepancy bet ween the physical plant and the model is not exposed through the simulations. This is particularly important with feedforward. Finally, the magnitude of the control s ignal has been examined with the result that among the investigated control methods, the geometric controller requires the smallest magnitudes of the signals.

1705

Lawson, C.L. and J.R. Hanson ( 1974 ) . Solving least squares problems . Prentice Hall Series in Automatic Computation, Englewood Cliffs, Ne w Jersey. Moore, B.C. (1979 ) . Singular value analysis of linear systems. Proc. 17th IEEE Conf. Decision Contr., Jan. 1979, pp. 66. Morse, A.S. and W.M . Wonham ( 1970 ) . Decoupling and poleassignment by dynamic compensation. SIAM J. Control, 8, 317. Shah, S.L., D.e Seborg and D.G. Fisher (1977 ) . Disturbance localization In linear systems by eigenvector assignment. Int. J. Control, 26, 853. Shah, S.L., D.E. Seborg and D.G. Fisher (1981). Design and experImental evaluation of controllers for process undisturbabillty. AIChE Journal 27, 13l. Schizas ,-C. and F.J. Evans ( 1981 ) . A graph theoretic approach to multivariable contro l systems design. Automatica 17 no. 2, 371. Takahashi, Y., M.J. Robin s and D.M. Auslander (1972 ) . Control and Dynamic Systems. AddisonWesley Publ. Comp. Takamatsu, T., I. Hashimoto and Y. Nakai ( 1979 ) . A geometric approach to multivariable control system design of a distillation column. Automatica, 15, 387. Wonham, W.M.-C1979 ) . Linear multivariable control: A geometric approach, 2. ed. , Springer-Verlag, Ne w York. APPENDIX Traditional Controllers PI Control Wi th P I control the reflux Vi as controlled using measurements of the composi tion of the top product. Furthermore, a feedfor ward loop where the heat input to the reboiler is controlled from the disturbances in fe ed composition and flo w rate is used. Th e design of the PI controller was based on a simplified model, 2nd order and dead time, of the distillation column and traditional design, resulting in G( s )P[ K (1+1/1 [s ) ; ( K, 1[ ) = (113, 3.1 min ) . The dynamfcs was improv,::~ through a lead-lag term D(s ) = (s + 0.0244 )( s+1 ) The feed forward controller ," as designed through linearization of the r elevant equations to, 6U = K 60 +K 6Q = 5.33601 +1.76602 ( AI ) FF2 2 FFl 1 2

Conclusion on the Design Methods The simpli fied algebraic design method offers an attractive alternative to Wonham ' s orIginal method and it does not to the same extent require a deep insight into the abstract nature of the underlying theor y. The method might therefore be more sui table for industrial applications and contribute to a wider appreciation of the potential of geometric control.

where 0] is the feed compositIon and 02 the feed rate. Tile t wo constants 1<,[1 and KFF2 are determined from simu lation resul s to be the exact con trol required to cancel a 15% step change In either 01 or 02' [n this way It Vi as attempted to desIgn as effective a PI controller as possIble in order to make it a reasonable competitor to a geo metric controller. Optimal Control

REFERENCES Fabian, E. and H.M. \'!onham (1974 ) . Decoupling and disturbance rejection. IEEE lrans. Autom. Control, AC-13, 399. Fabian, E.--a;::;(j \·J.M. Uonham (1974 ) . Genenc solv ability of the decoupling problem. SIAM J. Control, 11, 688. Fabian, E. and N.M. Uonham ( 1975 ) . Decoupling and data sensitiVIty . IEEE lrans. Autom . Control AC-20, 338. Holland, C.D. ( 1975 ) . Fundamentals and modeling of separation processes. Prentice-Hall. Kummel, M. and E. Ohrt (1981). Practical evaluation of geometric contro l. Proc. of 8th IFAC Congress, Kyoto, Japan, pp. 2679.

An optimal controller was designed in compare a more advanced controller than troller with the geometnc controller. theory is we ll kn own, it is not repeated can be found in references, such as ( 1972 ) . The performance functIon ~as,

order to a P[ con Since the here, but lakahashi

1/2 rT ( x T F X + uT F u )dt Jo xuSince the optimal controller's primary task ~ as to control top and bottom composition without consid ering the amp litud e of the control signal, the weIg ht matrix is F = I. For numerIcal reasons the weigh t matrix F w~s chosen to be I except f or the t wo end -element~, which refer to the top and bottom compositions. J

M. KUmmel, L. Foldager and P . C. Hans en

1706

8 Th ese t wo el ements were chosen to be f x( l,l ) = fx (l l,ll ) = 1.10 . The r es ulting fee db ack matri x G was, G

\82 38.06, 616 . 42, 163. 43, 40.57, 9 . 01, 0.11, - 6. 16, - 21. 07, - 55 . 50 , -86. 50, 2535.83\ 1492.70, 252.21, 98.08, 31. 37, 8.65, 0.15, -6 .07, -1 9 .47, - 59 . 39, -220. 51, - 9250 .48

All the deve lope d con tr oll ers we r e based on a sampling time of 0.20 min. L1ne!rl1~

TAa.E 1:

- 0.114 0.522

Hil t helllatlcal

0. 10) ..0.94' 0.522

~l

or Binary Oht1l1 a Uon Coltftn

0. 069 -0.991

0.519

0.512

- L051

O.~9

0.511

- 1.118 0.521

0. 66;-1.~

0,718

a.m

ro

·}.6J,00

0.799

0.912

-I. ~11 0.922

O.O:n64 0.00400 O.OOH6

o

C.OOXl6

C

O.OO)}6

O.OL'1l6 0.00786

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rU£l1~

- 0.171

1 000000000 J [ O()OOOOOOOI

C ..

o.'XiJOO

o . :x;}OO

O.OOJJ1

0.(0))2

U. .... U.... llHJfl

!

0.0000)

(). )·1\;)1

n

(1)

1.101

0.115

O.rmue

0.00108

q

1.0'11 -1.94J

o o o

O.(X)J78

l

0.901 -1.82J 0.912

(0)

x,~----------------~~~----~------~ q

n.0052'::1

(11.72)

or

n.nn~, IF.

(0.711

q

~

O.",n., , S 0<> xs~~ ___ ~ ______~==~="I~,:,,:.o::~n:'"~::(=~:~O='=I~~~

X)I

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