Multivariable Control of a Turbogenerator Pilot Plant using the Polynomial Matrix Approach

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MULTIVARIABLE CONTROL OF A TURBOGENERATOR PILOT PLANT USING THE POLYNOMIAL MATRIX APPROACH U. Keuchel AlllfllII{[li( Cfllllru/ I. {[/}(J III 1111'\' , F{[/'Idly

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Abstrnct. A tur bogene rator se t , consisting of a turbine and a synchro nous generato r, is a typica l multi\'ar iab le electromechanica l system, which is determ ined by its highly nonlinear behaviou r and non negligible couplings between electrical and mechanica l con trol loops . Especia ll y if th is plant is not connected to th e rigid 50 Hz mains, th e cont rol task f o r v3rying load in different ope rat in g points is quite difficu lt. Decentr:llized conventional thr ee- term control does not give suffic ient control performa nce. In this paper a multivariable control strategy based on the "robust se r vo compensator" wi ll be inves ti ga ted. In con tra ry to th e st:mdard s tate-space approach, here all theory neceSSJ r y for modelting , ana lys is, design and impleme ntation will be formulated in fr eq uency domain, using th e polynomial matrix represen tat ion. Star tin g from measured input and output sign al s a reduced order left coprime po lynomial matrix fra cti on has been obtained using a leas t- sq ua res identification algorithm. The controller design comp ri ses solu tio n of two diophantine poly nomial ma trix equations a nd two spec tral factori za ti on problems for closed loop and observe r dynami cs, according to minimizat io n of a quadratic performance ind ex. Keywords. Frequency and vo lta ge control, polynomial mat r ix fractio n desc ri ptions, po le placement, linear quadratic op timal cont ro l, robust servo compensator, diophantine eq uati o ns, spectral factorization .

INTRODUCTION

THE OBSERVER/ CONTROLLER COMPENSATOR

The desig n of several types of controllers using the state >pace approach has meanwhile become a standard task for the control engineer. The availibility of validated numerical ~!go­ rithms for basic linear algebra and procedures for analysis and synthesis did lead to a wide spreading of this design methodology . This paper demonstrates how design algorithms for proportional state feedback, PI state feedback and the well known servo compensator can be carried over to algebraic design methods working with a sys tem represe ntation in form of polynomial matrix fractions descriptions (MFD).

This sectio n shall review the design of state feedback controller and state observer in frequency doma in , and will widely pafallel the approach of (Wolovich, 1977). But neither the ge neralized compensator, nor the design methods introduced there will be utilized . This paper does not deal with state feedback, but the observer/ controller compensator is an integral part of the servo compensator derived in the next section; indeed the inner control loop will be of state feedback type . A linea r time-invariant disc rete system in state space representation is desc ribed by (la) ~(k+l) = A ~(k) + 1l g(k)

The synthesis of control systems usi ng MFD's has some advantages compared to design algorithms formulated in sta te space representation:

= h ~(k).

y(k)

(Ib)

Due to non measurable sta te values - here an input-output desc ripti on is used - a state observer has to be introduced, which is represe nted by

Many approaches, which are known for the design of single-input si ngle-output control systems, ma y be formulated for multi-input multi-output sys tems in an analogous way .

i'
The canonical MFD's ma y be directly identifi ed from input and output measurements of the plant usi ng parameter estimation methods. For controller design no transformation of the system represe ntation is necessary. Thus it is easy to carry out controller desig n online.

y(k)

= =

(A - !! h) i'
(2a)

h i'
(2b)

Using the state feedback matrix E, poles and eigensystem of the closed loop system can be specified. For zero setpoint the feedback law is given by g(k) = f(k) = E ~(k) .

The design of compensators for different types of disturbance and reference signalS may be unified usin g the "i nternal model principle" ,

(2c)

Desc ription of the co ntrol syste m using MFD's In an input-output view, f(k) may be interpreted as o utput of a d ynamic system having inputs g(k) and y(k). After ztransfo rm atio n Eq, (2) may be written as

Degrees of freedom may be eas il y recognized usin g the general so lution of the design equations, Thus, several new facilities as for example limitation of the actuating sig nal response for disturbances, while the se tpoi nt transfer functi on remains invariant , may be introduced.

f(z)

=

E (z 1- A +!! hrl 1l g(z) +

(3)

+ E (z 1- A +!! h)-I!! y(z),

The first section of this paper serves for a short re view on the design of of the state feedback controller with observer in frequency domain. The des ign problem is reduced to the so lution of a single polynomial mat rix equation . In the next sec tion th e control loop is augmented for asymptotic regulation of refe rence and dist urbance signals ha ving polyno mial type. Therefore a second polynomial desig n eq uation of the bilateral type has to be introduced . In the third section a quadratic opt imal control strateg y for the se rvo co mpensator sys tem will be illustrated. The optimal characteristic polynomial matri x will be evaluated usi ng a spec tral factor iza ti on procedure. The las t sect io n se rves for discussion of experimental results of the control strategy for voltage and frequency co ntrol of a turbo generator pilot plant.

which may be interpreted as filtering input and output signals by two transfe r functions represe nted by left coprime MFDs with common denominator matrix !::l,(z). So Eq. (3) may be rewritten as f(z) = ~-I(z) E(z) g(z) + ~-I(z) Q(z) y(z).

(4)

The se tpoi nt transfe r function matrix ~yv(z) of the closed loo p system is given by

~yv(z)

=

~( z) [(~(z) + ~(z)) f:J..(z) + Q(z) ~(z)r I ~(z).

(5)

The design of the observer/ controller sys tem con sists in determining pol ynomial matrices E(z) and Q(z), such that the closed loop sys tem yields the desired setpoint beh aviour

I:!I

2) The transfer zeros of the plant, in our case the invariant zeros of the numerator polynomial matrix .!l.(z) , do not interfer with the poles of the z-transformed reference signal.

(6)

which is reached, if E(z) and Q(z) are a solution of the unilateral diophantinc equation (!::!(z)+E(z» ~(z) + Q(z) !!(z) = !::!(z) ~(z)

(7)

3) The number of manipulating inputs is greater or equal

to the number of controlled outputs, r

with

~(z)~ -I (z):

plant model as minimal right coprime MFD,

!::!(z)

desired characteristic polynomial matrix for the observer dynamics and

4)

From classical control theory it is a well known fact, that the control loop must contain integral action in the forward path, such that step disturbances and setpoints can be regulated without offset. For ramp signals the controller must include double integral action, for sinusoidal signals with frequency Wo the transfer function of the forward path must be constrained to have poles at ±jwO' In this case the gain of the forward path approaches infinity for this special frequency , such that the control deviation vanishes. In the forward path modes will be excited, which compensate the disturbance or the setpoint.

desired characteristic polynomial matrix for the closed loop system. Realization of the controller The polynomial matrix £(z) in Eq. (4) has the same row degrees as the observer dynamics matrix !,{(z), which is constrained to be row reduced (Keuchel, 1988). Thus the controller transfer function matrix !,{-I (z) £(z) is proper, but not strictly proper. To remove the algebraic loop in the control law, Eq. (4), the controller has to be implemented using the modified control law

In the following it will be shown, which demands a multivariable system in MFD must fulfill for asymptotical regulation of output disturbances. The z-transform of the output disturbance is represented by the left MFD

(8)

,,/z) = ~-I(z) !!(z).

which is advantageous, because the linear combination (!::!(z) + E(z» is directly obtained as solution of Eq. (7).

K = gyv-I(Z=I).

~(z) = ~-I(z) ~(z).

(9)

~(z) = - (~(z) + ~(z»-I ~(z) "y
The disturbance behaviour of the system will be influenced by the observer dynamics, which is represented by the invariant zeros and the structure of !'{(z). !'{( z) is of determinantal degree strictly less than A(z), if a reduced order observer will be designed. Choosing row degrees ~(z)

= max (degij

lim

~(z)

(14)

z~1

z~1

(9)

~(z)

The system ti-IQ is strictly proper, which means that the controller can cope with the disturbance after a delay. To add a feed through to thi~ system, the general solution of the diophantine equation (Q ,E) (Kucera, 1979) may be used. Starting from

=

~'O(z) ~(z)

,

(IS)

assures existence of the limit value in Eq. (14) and the control deviation vanishes for k ~ ~. Thus lim ~(k) = - lim (z-I)(~(z) + ~(z))-I ~'O(z) !!(z) = Q, (16) t .... oo z-+l and the controller must include the denominator polynomial matrix of the disturbance model

(10)

where I is a real valued matrix and EI-I(z)QI(z) a left coprime factorization of the plant model, leads to

gr (z) = ~-I (z) g'r
(17)

such that

(10)

~(z) = ~(z) ~-I(z) g'r(z)

so that J!(z) = - ~(z)~-I(z)!::!-I(z) (Q(z) - I EI(z» ,,/z) .

= lim (z-I)

(13)

may be calculated as

For existence of the limit value in Eq. (14), the z-transformed transfer function of the control deviation system must be stable. The characteristic polynomial matrix of the closed loop system (~+~) represents a stable system, ~(z) is usually marginally stable or unstable (otherwise disturbances would decay without control action). Choosing

The relation between actuating signal and output disturbance "y is given by

(!::!(z) + E'(z» = (!::!(z) + E(z» + I QI(z) ,

~(z)

= -lim (z-I) (~(z) + ~(z))-I ~(z) ~-I(z) !!
which correspond to a full order observer, gives additional degrees of freedom to shape the disturbance behaviour.

Q'(z) = Q(z) - I EI(z),

~(k)

k~~

~(z»,

J!(z) = - ~(z) ~-I(z) !::!-I(z) Q(z) ,,/z) .

(12)

Using the above definitions, the control deviation is given by

Modification of the observer

degri

(11)

A left coprime MFD of the open loop transfer function is

If a setpoint different from zero exists, a static prefilter K must be introduced, such that y(z) = gyv(z) K ~(z),

m.

The internal model principle

~(z)

J!(z) = (!::!(z) + E(z)f I !::!(z) (',C(z) - !::!-I(z) Q(z) y(z)),

~

The output y is physically measurable.

(18)

= ~-I(z) ~(z) g'(z) = ~-I(z) g'O(z).

(11 )

By equating leading coefficient matrices in Eq. (11) yields U P -I (11 ) D -I AR -zy =!'{R!.
I

The augmented plant with disturbance model The first step of the design procedure consists of forming the characteristic polynomial matrix of the disturbance model from a priori knowledge about reference and disturbance signals. This in almost all cases diagonal matrix will be connected in series with the right matri:< fraction model of the plant , to generate an internal model of the disturbance in the fowrward path.

THE SERVO COMPENSATOR

The aim of control is not only to give to the closed loop system a setpoint behaviour up to the choice of the designer, the controller should also make the system insensitive to parameter variations and input/output disturbances. In several cases the controller must be able to track a reference signal, for example steps and ramps, without steady state control deviation and it must be able to regulate as ymptotically deterministic disturbances. The control engineering formulation of the servo compensator problem leads to the 'internal model principle' (Bengtsson, 1977), which shall be shortly adressed here. (Davison and Goldenberg , 1975) show, that a solution of the servo compensator problem exists, if the following conditions are fulfilled:

For the augmented plant .!i(z) A-I(z) s,-I(z) a controller has to be designed, which stabilIZes the closed loop system. Two possiblities for feedback are offered: The plant input signal J! and the input of the disturbance model ',C. By cascaded proportional state feedback with observer, the inner loop may be given a characteristic polynomial matrix Q(z). To stabilize the poles introduced by the disturbance mOdel S,(z) in the closed loop system, a further degree of freedom is- necessary. The numerator H(z) of the disturbance model is still up to the designers choice-and serves for this purpose. The structure of the augmented control loop is depicted in Fig. I.

I) The plant has no unstable hidden modes; i.e . it has to be stabilizable.

122

Ei&...l

Block diagram of the servo compensator with reference feedforward

!Lw = (~(z) + ,!!(z))R -I NR K

Ei&....l

Block diagram of the servo compensator

The following polynomial matrices will enter the design dure: Plant model in its representation Y,(z) fi - I (z): nimal right coprime polynomial fraction characteristic polynomial matrix ~(z) : disturbance model characteristic polynomial matrix ~(z) : observer dynamics characteristic polynomial matrix ~(z) : closed loop s ys te m.

K proceas mimatrix

for the of the

Following Eq. (4) (~(z) fi(z) + ~(z) fi(z) + Q(z) Y,(z» = ~(z) j2(z)

represents the design equation for the proportional observer/controller compensator of the inner loop. Using this identity, the inner loop is assigned the transfer function Q,y,v(z) = Y,(z) j2-I(z). So the dynamic behaviour between setpoint and output given by =

Y,(z) (~(z)Wz)+!;!,(z)y,(z»-1 Wz)

=

Y,(z) ~-I(z) Wz) .

(19)

To assign a chosen characteristic polynomial matrix .!;.(z) to the closed loop system, the diophantine equations ~(z)

j2(z) + !;!,(z) Y,(z) =

(~(z)

NR -I (~(z) + ~(z»R !Lw .

=

(24)

As for the proportional state feedback controller the amplitude of the first controller action following an output disturbance may be chosen be the design engineer, if a full order observer system is used. The underlying proportional state feedback controller is to be designed in the same way, also in case of the servo compensator approach.

of the

A design equation for the unknown controller matrices !:!(z) , E(z) und g(z), has to be specified, for assigning to the clOsed loop system of Fig. I a characteristic polynomial matrix ~(z). The set-point transfer matrix of the closed loop system will be considered first.

Q,yw(z)

(23)

leads to

~(z)

+ ~(z)) fi(z) + Q(z) Y,(z) = ~(z) j2(z)

(20) (21)

have to be solved. The bilateral diophantine equation (20), is used to evaluate the feed forward matrix !:!(z) for the disturbance model and the characteristic polynomial matrix Q(z) of the inner control loop. Subsequently the unilateral diophantine equation (21), for design of the controller/ observer compensator has to be solved. Due to numerical reasons it is important , to work with low polynomial degrees (Keuchel, 1988) during solution of the diophantine design equations. The synthesis procedure given above has the advantage, that the determinantal degree of all polynomial matrices is minimal and there is no necessity to form an augmented system representation. Modification of the control law by setpoint feedforward In the control structure of Fig. I the setpoint signal has to pass the system ~-I (z) !:!(z), before an actuating signal will be generated, this ~ill caUSe a delay due to the strictly proper disturbance model. rhls drawback can be compensated, if the setpoint is fed forward using a matrix K as shown in Fig. 2. The amplitude in the first sampling step after a step setpoint change ~ is given by

EVALUATION OF THE OPTIMAL REGULATOR

According to (Kwakernak & Sivan, 1973) the discrete canonical system in state space description is given by

[

~(k+l)l

unit step setpoint will be a prespecified value !Lw' Using

~

,(25)

~T ~y ~

Supposing stabilizability and detectability, the characteristic routs of this equation having moduli strictly less than I constitute the characteristic values of the steady state optimal regulator, which minimizes the performance criterion

., I = E ~T(k) Q ~(k) + l!T(k)

R

l!(k)

(26)

k=O

The characteristic polynomial matrix of the quadratic optimal clo.sed loop system j2(z), having all its invariant zeros in the unit .clfcle of the complex z-plane, is given by spectral factoflsatlOn of j2.(z) ~u j2(z) with j2.(z)

=

fi.(z) ~ufi(Z) + Y,.(z) ~yY,(z)

(27)

j2T(I / z).

The invariant zeros of polynomial matrices on the left hand Side .of GI. (27) exhibit symmetry to the unit circle. The polynomial ~atflces on the right hand side may be factorized using a numerical method (Youla &. Kazanian, 1978), such that the solutIOn j2(z) contains all stable roots.

4 fu....1

~-1

- .1

B-

~

Augmented system of the servo compensator

According to Fig. 3 one can interpret the actuating signal u and the "internal state" 1(s) in combination with the output vector .v(s) as output. signals of an augmented plant. The input vector .v (s) of the disturbance model is the actuating value of the augmented open loop system. From a coprime factorization of the augmented system [.v i l!]T = y,'fi,-I.v = [Y,

I fi] T (~ fi) - I .v

(28)

one obtains in combination with Eq. (5.4.1.14,20) ~. ~v ~ = (fi ~). ~v (~ fi) +

(22)

K has to calculated, such that the manipulating signal for a

= [

JI!(k)

+

m.

!~.] diag [ ~y ~i ~u ] [~ ! fi]T,

(29)

which can be written after performing the multiplication on the right hand side as

Difficult to handle are operating points near the point of inflection of the generator voltage characteristics. In this paper experiments using operating point AP2 are illustrated to demonstrate insensitivity against parameter variations. All controllers have been designed for this operating point and a sampling time of 100 ms and also parametric identification was carried out under these experimental conditions.

~. ~v ~ = ~·~·~v ~ ~ + ~. ~u ~ + ~.~y~ + ~i (30). The role of matrices ~u and ~ is evidently clear. The choice of ~v may be discussed in the design of a controller with integral control law: If the disturbance model is a step function, then ~v will weight the derivatives of the actuating value. A weighti.ng matrix ~i > 0 causes a shift of the closed loop poles, whIch are zeros of ~(s), to the origin of the unit circle.

Identification of a parametric model The open loop system was excited using a zero-mean discrete white noise signal having a pulse width of 500 ms . To work under almost linear conditions the maximal amplitude of the input signals were constrained to 0.25 in magnitude. The estimation algorithm used was similar to the method of (Guidorzi, 1975). The structure indices VI = v2 = 2 which are equal to the row degrees of the denominator polynomial matrix had been chosen, such that a 4th order model had been obtained. The parameters of the left coprime MFD in combination with the invariant zeros of the polynomial matrices A(z) and B(z) = = are shown in Table I.

CONTROL OF A TURBOGENERATOR SET

To show the practical applicability of the proposed control strategies, a turbogenerator pilot plant has been chosen for experimental studies. The parametric descriptions, on which the synthesis was based, have been obtained from an identification procedure. The first section states the description of the plant. By measured static graphs and step responses the nonlinear characteristic of the control system will be illustrated. Experimental results will be depicted and discussed in detail.

a ll ( z) a l2 ( z) a 21 (z) a 22 (z} b ll ( z} b l2 ( z) b 21 ( z} b 22 ( Z}

The pilot plant turbogenerator Fig. 4 shows the experimental setup of the turbogenerator set. A compressed air turbine, which is fed by a central pressure pump, drives a synchronous generator working in insulated operation not connected to the rigid mains. As load serves a symmetric resistance, which may be switched between 1.8 0 and 2.8 o. Controlled signals are: frequency n, measured by a tachogenerator, and the mean value U A of generator Voltage. The opening V of the valve which steers the f1owthrough of the compressed air and the excitation current i of the generator are actuating signals. e

= 1. 8976.10- 1 = -8.3982.10- 3 = 6.2961.10- 1 = 1. 8796 .10- 1 = -4.0707.10- 2 = 1. 4241· 10: = 2.9041·10 = 1. 3695 ·10-1

1

- 1.1119z +_~2 + 5.2224·10 z

9.3577.10- 1 z 8.5161.10- 1 Z + z2 4.4736.10- 3 Z + 9.3622.10- 3 z + 1. 0865 .10-l z + 1.4526 . 10- 2 z

-

-

poles

No

zeros

real part

imago part

real part

imago part

.459601 .459601 .193003 .851325

.194988 -.194988 0,00000 0.00000

-15.0388 -2.88374

0.00000 0.00000

1 2 3 4

Parameters, poles and zeros of the 4th order model

s~,vo

molor

actuating yalve

Fig. 6 shows step responses in the chosen operating point for step amplitudes u I = u2 = ±O,25. As seen before from the static characteristics according to Fig. 5, the stationary values are different for positive and negative step inputs. This is also true for different step heights. Even for small deviations from the operating point the stationary value of h21 (t), especially for negative steps, is evidently different from zero.

high pressure air

~

Schematic diagram of the experimental setup

Fig. 5 shows the static characteristics of generator voltage versus excitation current, which is depicted for four different openings of the valve as parameter. If different working points are inspected, it is obvious, that the static gain of the according transfer function changes sign.

Fig. 5 illustrates, that by the low order model the dynamical and static behaviour of the plant is well approximated.

h" 0.250

h12

-

0.50

., ... ....••.......

~ - .- .,.

..

........•.........-.- ..•....•.•...

0.125 O. 000 1+-----<--+--4---1-__

a

- 0.125 , 1.00 - - - - - --

- 0.250 0.0

2.5

[s}

5.01

-0.50 0.0

2.5

[sI 5.0 1

h22

h 2'

0.250

0.50

0.125

o. 000 ~~-==t=:::::;:::=+---<- 0. 125 -0 .250 0.0 .30 ~

.50

2. 5

[sI

5.0 t

-0.50 0.0

.,..•....._...._._- ..--2.5

[sI 5.0 t

[A}

Ei&...Q

Static characteristics of generator voltage versus excitation current

I~

I

Model and measured step responses for the turbogenerator set ( ... : measured; : simulated)

~

~ .20~------------~~~

.20

z

0.00

0.00~------~------+----L--~------4-----~

tlsec

tlsec

-.20

-.20 0.00 1.00

/W

3.00

0.00

6.00

3.00

.'l.

6.00

.'l.

1.00

.50

tlsec

t/sec

-1.00 0.00

6.00

3.00

0.00

-.50

B.OO

r

t/sec

t/sec - . 20

-.20

0.00

6.00

3.00

0 . 00

3.00

B.OO

Proportional-integral controller, setpoint behaviour The measured response to a step setpoint change ~ = [0.2 O)T agrees with simulation r~ult s. Whereas a step change of voltage se tpoint ~=[O 0.2) exhibits a heavy coupling to freQuency. Due to the cha nge of the operating point - the static gain changes sign - the disturbance is regulated to zero after 3 s, and the stationary actuating signals differ from the linear values. The good performance of the PI-controller is evident for disturbance behaviour. Already after 500 ms, output disturbances have decayed to a rate of 10%. For input disturbances the se ttling time is higher due to the filtering by the plant dynamics. Servo compensator for step and si nusoida l sig nals

The cycle of syn thesis first uses design program and si mulation, up to the point where an approriate answer of the closed loop system to reference and disturbance signals is present. Quadratic optimal controllers have been designed. The weighting matrices have been evaluated such that closed loop performance with respect to settling time, maximal overshoot and regulation of disturbance could be judged as sufficient . The proportional-integral controller PI controllers are known for their robust behaviour with respect to un modelled dynamics of the plant. Parameter un certainties can be interpreted for steady state as output step disturbances . If the control loop remains stable, the controlled value will reach the setpoint asymptotically, in contrary to controllers with purely proportional control ac tion .

The servo compensator was designed for sinusoidal disturbances and setpoints having a period of 2 secs. From the characteristic polynomials

Fig. 7 shows the response to a stepwise change of the reference signal, whereas Fig. 8 illustrates disturbance behaviour for step output disturbances.

PSl(Z) = Z - 1

z

.20

-y

O.OOr------4-------+--~---+------~-4--~4-------j~----~· r------~------44· --~~ tlsec

-.20 0.00 1.00

5.00

10.00

15.00

20.00

25.00

.".

• 50

t/sec

0.00 -.50 -1. 00

0.00

5.00

10.00

15.00

20.00

25.00

'L .20

0.00

t/sec

-.20 0.00

5.00 ~

15.00

10.00

Proportional-integral controller, output disturbance

125

20.00

25.00

1. 00

Z

4J

t/sec

-1.00 0.00 le 00

4.00

2.00

B.OO

B.OO

10.00

~

tlsec

-1.00 0.00

2.00

4.00

B.OO

B.OO

10.00

'L .20 O.OO~~-->.J

tl sec

-.20 0.00

2.00

4.00

EiL2 Ps2(z) = z2 - 2z cos (wO T) + I ,

Wo

=

6.00

B.OO

10.00

Servo compensator for step and sinusoidal signals

n s-I, T = 0.1 s .

tude of the manipulating signal by enlarging the controller order.

the discrete disturbance model is evaluated as

The performance of the closed-loop system has been demons trated for setpoint changes as well as for load and input disturbances. Due to a special feedforward of the set-point signal the respo nse to set-point changes is fast with small overshoot. By incorporating a disturbance model for sinusoidal and step disturbances, constant offsets and sinusoidal input disturbances of known frequency are rejected. Even for variation of the operating point, which leads for example to a change of sign in the steady state gain of the transfer function between excitation current and line voltage, the robust servo compensa tor performs its control task.

~(z)=diag[(z-I )(z2-1.902z+l ),(z-l )(z2-l.902z+l »). During simulation studies sinusoidal test signals, entering the control loop as setpoint, input disturbance or output d,sturbance, were regulated to zero in less than one period. This ideal behaviour can not be observed for plant-model mismatch or during control of the real plant. Fig. 9 shows the reponse of frequency and voltage for a sinusoidal disturbance of the excitation current with amplitude 0.5. Sinusoidal disturbances at the input of the plant will be asymptotucally fully compensated. The large settling time is introduced by excitation of sinusoidal output signals of higher frequencies, due to the nonlinear characteristics of the plant.

REFERENCES

CONCLUSION

Bengtsson, G. (1977): Output regulation and internal models A frequency domain approach. Automatica, 13, pp. 333-345.

The polynomial matrix controller presented here parallels the robust servo compensator proposed by (Davison & Goldenberg, 1977). The evaluation of the controller design using the frequency domain approach has several advantages:

Davison , E.J. and A. Goldenberg (1975): Robust control of a general servomechanism problem: The servo compensator. Automatica, 11 , pp. 469-471.

For a specified class of reference signals (step, ramp, sine, .. ) there exist no steady state control deviation. This is also the case for parametric changes in the plant model, if the closed loop system remains stable.

Guidorzi, R. (1975): Canonical structures in the identification of multi variable sys te ms. Automatica, 11, pp. 361-374.

The entries of all polynomial matrices possess minimal polynomial degrees .

Keuchel, U. (1988): Two numerical methods for solving polynomial matrix equations. Prepr. of the IFAC Workshop on State Space and Pol ynomi al Methods, Prague .

No system representation augmented by the disturbance system has to be calculated, the design starts directly from the characteristic polynomial matrix of the 1isturbance model ~(z) and the plant model ~(z) ~(z).

Kucera, V. (1979): Discrete Linear Control. The Polynomial Equation Approach . John Wiley, Chichester. K wakernaak, H. , Sivan R. (1972): Linear Optimal Control Systems. Wiley Inte rsc ie nce, New York.

The characteristic polynomial matrix of the disturbance model ~(z) may be calculated easily.

Wolovich, W. A. (1977): Multivariable system synthesis with step dist urbance reject ion. IEEE Trans. on Automatic Control, AC-23 , pp. 127-130.

The initial value of the actuating signal for a step se t point change may be specified. For step output disturbances the initial value of the actuating signal can be specified, if a full order observer is used.

Youla, D. C. and N. N. Kazanian (1978): Bauer-type factoriza tion of positive matrices and the theory of matrix polynomials orthogona l o n the unit circle. IEEE Trans . Circuits Syst., C AS-25 , pp. 57 -69.

The polynomial approach turned out to be very flexible for our design purposes: the weighting matrices for the manipulating signal, the internal state, the output deviation and the compensator output allow to influence the relevant properties of the closed loop system; the nonuniqueness of the so lution of the polynomial design equations allows to constrain the ampli-

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