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A note on the design of industrial regulators: Integral feedback and feedforward controllers
Automatica, Vol. 10, pp. 329-332. Pergamon Press, 1974. Printed in Great Britain.
Correspondence Item A Note on the Design of Industrial Regulators: Integral Feedback and Feedforward Controllers* Une Note sur la Conception de Rrgulateurs Industriels: Contr61eurs ~t Rrtroaction et Action Avancre Intrgrales Eine Bemerkung zum Entwurf industrieller Regler: Regler mit integraler Rtickftihrung und mit St~Srgrrl3enaufschaltung 3aMe~iaHvie no npoeKTripoBaamo npoMt,IUtrieam,IX pery.rlarOpOB cri n'rerpa.rn,iao~ o6paTrlO~ CBa3I,IO rI BO3)XefiCTBVIeMno BO3MymeaI,IK) E. J. D A V I S O N t Summary--This note uses some of the ideas of a previous paper [1] to obtain explicit feedforward-integral feedback controllers for a linear multivariable time-invariant plant, subject to both measurable and unmeasurable constant disturbances, so that the outputs are regulated to preassigned set points.
a n d H. W . S M I T H ' f and let
/tEuEmo/II / w=
1. Introduction THERE has been recent interest in the last few years in dealing with the multivariable servomechanism problem for constant disturbances, e.g. Refs. [1-4]. It is the purpose of this note to extend the results of Ref. [1] to obtain explicit feedforwardfeedback controllers for the following system:
~=Ax+Bu+Ew S: e=Cx+Du+Fw
(1)
where x is the n-state vector, u the r-plant input vector, e the m-error vector, in the sense of being the difference between the actual output and desired output, and w is the q-disturbance vector which consists of unmeasurablemeasurable constant disturbances and reference inputs. The present work differs from previous work, e.g. Ref. [1--4] in that explicit feedforward-feedback controllers are obtained for (1), which were not developed in Ref. [1] and which have not been obtained before, e.g. in Refs. [2--4]. Following Ref. [11, assume, without loss of generality, that (1) has been reduced so that it is in minimal form, i.e.
= n+ m
Fm
wm
(3)
where wt,, w,~ are unmeasurable and measurable constant disturbances respectively and Yret, assumed to be constant, are the reference inputs to the system. It is assumed that rank B = r , rank C = m . Let yACx+Du+Fuwu+Fmw,~ be the m-output vector of the plant so that e = y - M y r e r . It is desired to find a linear time-invariant controller for (1) so that e ( t ) ~ 0 as t--* oo and so that the closed loop system is stable (controllable). Note that in practise M will normally be chosen so that M = I . It is assumed that w,n and yrer can be used in the controller but that w~ cannot be used.
2. Development Necessary and sufficient conditions [i] that there is a solution to the problem are given by:
(i) D
Fu
(A, B ) is s t a b i l i z a b l e ( c o n t r o l l a b l e )
(2)
(it) * Received 4 May 1973; revised 28 August 1973; revised 19 November 1973. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by Associate Editor I. Rhodes. t Department of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada. This work has been supported by the National Research Council of Canada under Grants No. A4396 (EJD) and A4165 (HWS).
rank
C
=n+m.
(4)
It may be noted that condition (it) implies that there is no solution to the problem unless there are at least as many plant inputs as there are plant outputs to be controlled, i.e. it is necessary that r~m. The frequency domain interpretation of condition (i0 is that no transmission zeros [5] of (1) should occur at the origin. 329
330
Correspondence item
-~m
_1
+
X,.~,
tt I"
Kz tt'~"dr
I
FIG. 1. Feedforward--lntegral controller obtained assuming x is available for measurement; g~_A-e. Assuming these conditions hold, on applying the affine transformation* followed by the differential transformation of Ref. [1], the following control system is obtained as shown in Fig. 1 :
u--K~×+K2
f
o
e(~)d~+(K~,-I)G]"
F,,, Wm
has a satisfactory stable response by using either pole assignment methods, optimal regulator theory, frequency domain methods etc; some specific examples of using this method are given in Ref. [1]. If optimal regulator theory is used, the controller (5) corresponds to a solution to the following problem for (1):
(7) where G I'=G'(G
assuming that R is positive definite and that
G')-~
where is detectable.t and where If the state x is not available for measurement, the following controller, illustrated in Fig. 2, is obtained, assuming that (A, C) is observable [I]:
KI, K2 are found so that the following stabilizable (controllable) system:
i=
0 z+
D V, v = ( K I , K2)z
(6)
* x and y are mid to be related by an afflne transformation, i f y=x-/-ll where p is a constant vector.
Fie. 2.
t Let (A, C) be decomposed as
where (Ab Cl) is observable; then (A, C) is said to be detectable if the eigenvalues of A3 are in the open lefthand part of the complex plane. -~m Em
Feedforward--Integral controller obtained assuming x is not available for measurement; c A - e ; AIA3 A4
is found so that
0]
fast dynamics, w h e r e { ( A
\A 3 A4]k0 00)'(0C
has satisfactory stable
: ) } is observable.
Correspondence item
u=ii+~(K,,-I)Gt (~") w,,,
+(_.,,oo, (o),., fi = K t J ~ -t- K211
331
°:.[,,, (8)
02)
where t is the output of a minimal order observer with input
where all other quantities are defined in (8). It may be remarked, that in the case when D - - 0 , 6 T = I and the resultant equation (12) is of much simpler form. It should be noted that the controller (12) corresponds to a generalization of the 3-term controller o f classical control theory, the observer in this case corresponding to a phase lead compensator.
e, 1 1 A f '° e ( Q d ~ and l], which estimates the state
Remark 1 In the development of the feedforward controller in [1], it was shown that the requirement for ~(t)--*0, y(t)--*0 as t.-.~oo, Vo~R q is that (A, B) be stabilizable and that rank
of the following observable system:
(9) be equal to nq-m. This observation proves that conditions (4) are necessary for a solution to exist to the problem for the class of all linear or nonlinear feedback controllers.
It should be noted that a weaker condition of (A, C) being detectable is really only required for there to be a solution to the problem for the case of stabilizing the closed loop system. In the special case when F, =0, M = I and when
Remark 2 When the controller (5), (8), (10) or (12) is applied to (1), the resulting controlled system has the property that asymptotic regulation occurs for all changes, large or small, in the feedback gains K1, K2 provided only that the resultant system remains stable. This follows from Ref. [2], on noting that the extra outputs that have been eliminated in reducing (1) so that (2) is true, are either linear combinations of .2 or the outputs which have been retailed in (2). This observation is interesting, because it means that if there exists a solution to (1), then there always exists a robust controller for (1), so that regulation is achieved for changes in the feedback gains of the controller provided the resultant system remains stable. If the original system's parameters before the reduction to (2) have the property tha~
{I+(K,,-I)G, (O)} is nonsingular, the controller (5) becomes the following on simplification:
u = S / ' ( K 1, - I ) G ' [ "
f' ,,o,(O)]x
(:)
e+SPK2
o e(z) dz
rank(
(lO) where
Ae=[I+(K,,-I)G, (O)] -'
A
B)=n+m,
then the controlled system is also robust with respect to perturbations, large or small, in A, B, C, D, E, F provided only that the resultant system remains stable [2]. It can be shown that the above condition is also a necessary condition that must hold for a controller to be robust, so that asymptotic regulation occurs for all perturbations in the parameters
(zl)
and controller (8) becomes, as shown in Fig. 3:
I
1
I
t/o\
"i ~'c-~''l~ -~ ~} I
I
,
2,,fo
I • ~
I -',: ........
I ,'--
k._ L. . . . . .
Fxo. 3.
~11
J
I
Sim ~lified fecdforward--lntegral controller obtained when F , = 0 , M = I and f
/__%%
when ~ I + ( K I , - I ) G ' [ ' ( ~ 0 1 ~
(
\u/j
isnonsingular;~
332
Correspondence
(A, B) [8]. In this case it can be shown that any robust controller must consist of an error feedback compensator consisting of m integrators [8]. Remark 3. The controllers (5), (10) have minilnum order and the controPers (8), (12) have minimum order for unspecified gains, if rank M - m ; if M = 0 , rank E n, F = 0 ; or i f M - ~ 0 , rank F - m , E 0 [2]. The following lemma is required for Remark 4. Lemma 1. Given (A, B), assume that rank (A, B ) < n ; then this implies that (A, B) is not stabilizable. Proof. Assume that (A, B) is stabilizable; then this implies there exists a K so that ( A + B K ) is stable, i.e. so that rank ( A + B K ) - - n or alternately that rank (A, B ) = r a n k ( A + BK, B ) = n . Therefore, (A, B) cannot be stabilizable if rank (A, B) < n. Remark 4. There is no loss of generality in assuming condition (2), because
rank
A C
B D
E) F
/
< n + m
if and only if any one or combinations of the following occur:
(i)
rank (C, D, F ) < m
(ii)
some rows of (C, D, F) are linear combinations of some rows of (A, B, E).
Off) rank (A, B, E ) < n . In the case of (i) the redundant outputs can be eliminated; in the case of (ii) some combination of outputs are linear combinations of the state derivatives and hence a fewer number of outputs can be chosen to satisfy the problem statement; in the case of (iii) (A, B) is not stabilizable from Lemma l, and hence there is no solution to the problem. Remark 5. The controller (5) is the same as the feedforward-integral controller of Ref. [6] for the special case when all disturbances in the system are assumed to be constant. Remark 6. There will be a solution to the problem posed for almost all (A, B, C, D) systems given by (1) provided only that rank B->rank C, i.e. the class of (A, B, C, D) systems in which equation (2) and conditions (4) do not both hold, lies on a hypersurface in the parameter space of (A, B, C, D). This follows immediately from the development of [7] on noting that the system
item
if the number of outputs to be regulated is greater than the number of controlled inputs, then these is almost never a solution to the servomecha,aism problem. (See Ref. [.~] for a detailed discussion of this generic problem.) References
[1] H. W. SMITH and E. J. DAVISON: Design of industrial regulators: integral feedback and feedforward control. Proc. lEE 119, 1210-1216 (1972). [2] E. J. DAVISON and H. W. SMITH: Pole assignment in linear time-invariant multivariable systems with constant disturbances. Automatica 7, 489--498 (1971). [3] C. D. JOHNSON: Optimal control of the linear regulator with constant disturbances. IEEE Trans. Aut. Control AC-15 416-421 (1968). [4] P. C. YOUNG and J. C. WILLEMS: An approach to 1he multivariable servomechanism problem. Int. Control 15, 961-979 (1972). [5] S. H. WANG and E. J. DAVlSO~: Canonical forms of linear multivariable systems. Dept. of Electrical Engineering, University of Toronto, Control System Report No. 7203, April 1972; presented at 6th Asilomar Conference on Circuits and Systems (1972). [6] E. J. DAVlSON: The systematic design of control systems for large multivariable linear time-invariant systems. Automatica 9, 441-452 (1973). [7] E. J. DAVISON and S. H. WA~CI: Properties of linear multivariable systems subject to arbitrary output and state feedback. IEEE Trans. Aut. Control AC-18, 24--32 (1973). [8] E. J. DAVlSON: The Feedforward, Feedback and robust control of a general servomechanism problem, Parts 1, II. Dept. of Electrical Engineering, University of Toronto, Control System Report No. 7305, 7306, April 1973; presented at l lth Allerton Conference on Circuit and System Theory.
R6sam6--Cette note utilise certaines des id6es d'un texte pr6c~dent pour obtenir des contrfleurs explicites ~ ri:troaction et action avanc6e intdgrales pour une installation lindaire multivariable inddpendante du temps, et soumise h des perturbations constantes mensurables et non-mensurables de sorte que les sorties soient rdgldes ',i des niveaux prdvu~. Zusammeafassung--Diese Bemerkung geht von einigcn 1Dberlegungen einer friiheren Arbei [1] aus, um explizi:c Regler mit St6rgr/Sflenaufschaltung--integraler Rtickfi.ihrung fiir eine lineare mehrvariable zeitinvariante Anlage zu erhalten, die sowohl mal3bare als auch nichts mellbaren konstanten Stt~rungen unterworfen ist, so dab Ausgtinge auf vorgegebene Sollwerte eingeregelt werden. Pe31oMe--B ~aHm)II 3aMeTre HOIOnbByK)TCa )leroTOpble n~eH npeab~Luett CTarbn [1]. 3aaa~a 3azmo~aeTcs n
satisfies (2) and conditions (4) if r>_m. If rank B < r a n k C and if condition (2) holds, there is no solution to the problem. Thus there is almost always a solution to the servomechanism problem if the number of controlled inputs is greater or equal to the number of outputs to be regulated;
no;lyqeHl4FI TOqHblX pa3OMKHyTO3aMKHyTblX I~FyJ'I~ITOpOB J1JI~! ~HHe~IHblX MtIOFOhaOaMeTpttqeCKHX ),cTaHOBOK C
HOCTO~IHH~IMI4 KO3qbqbntD[enTaMi4,noaBepraeMblM KaK H3MeplleMbIM, TaK 14 He H3MeplleMMM HOCTOIIHHblMBO3Mylil.eHHIIM. Pel-yJIHpoBaHne BC~eTCfl TaK, qTO BblXOJ],,I CTa6n.qH3HpyIOTC~ Ha 3apaHee 3aaaaHl,lx 3naqenH~x.
Correspondence Item A Note on the Design of Industrial Regulators: Integral Feedback and Feedforward Controllers* Une Note sur la Conception de Rrgulateurs Industriels: Contr61eurs ~t Rrtroaction et Action Avancre Intrgrales Eine Bemerkung zum Entwurf industrieller Regler: Regler mit integraler Rtickftihrung und mit St~Srgrrl3enaufschaltung 3aMe~iaHvie no npoeKTripoBaamo npoMt,IUtrieam,IX pery.rlarOpOB cri n'rerpa.rn,iao~ o6paTrlO~ CBa3I,IO rI BO3)XefiCTBVIeMno BO3MymeaI,IK) E. J. D A V I S O N t Summary--This note uses some of the ideas of a previous paper [1] to obtain explicit feedforward-integral feedback controllers for a linear multivariable time-invariant plant, subject to both measurable and unmeasurable constant disturbances, so that the outputs are regulated to preassigned set points.
a n d H. W . S M I T H ' f and let
/tEuEmo/II / w=
1. Introduction THERE has been recent interest in the last few years in dealing with the multivariable servomechanism problem for constant disturbances, e.g. Refs. [1-4]. It is the purpose of this note to extend the results of Ref. [1] to obtain explicit feedforwardfeedback controllers for the following system:
~=Ax+Bu+Ew S: e=Cx+Du+Fw
(1)
where x is the n-state vector, u the r-plant input vector, e the m-error vector, in the sense of being the difference between the actual output and desired output, and w is the q-disturbance vector which consists of unmeasurablemeasurable constant disturbances and reference inputs. The present work differs from previous work, e.g. Ref. [1--4] in that explicit feedforward-feedback controllers are obtained for (1), which were not developed in Ref. [1] and which have not been obtained before, e.g. in Refs. [2--4]. Following Ref. [11, assume, without loss of generality, that (1) has been reduced so that it is in minimal form, i.e.
= n+ m
Fm
wm
(3)
where wt,, w,~ are unmeasurable and measurable constant disturbances respectively and Yret, assumed to be constant, are the reference inputs to the system. It is assumed that rank B = r , rank C = m . Let yACx+Du+Fuwu+Fmw,~ be the m-output vector of the plant so that e = y - M y r e r . It is desired to find a linear time-invariant controller for (1) so that e ( t ) ~ 0 as t--* oo and so that the closed loop system is stable (controllable). Note that in practise M will normally be chosen so that M = I . It is assumed that w,n and yrer can be used in the controller but that w~ cannot be used.
2. Development Necessary and sufficient conditions [i] that there is a solution to the problem are given by:
(i) D
Fu
(A, B ) is s t a b i l i z a b l e ( c o n t r o l l a b l e )
(2)
(it) * Received 4 May 1973; revised 28 August 1973; revised 19 November 1973. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by Associate Editor I. Rhodes. t Department of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada. This work has been supported by the National Research Council of Canada under Grants No. A4396 (EJD) and A4165 (HWS).
rank
C
=n+m.
(4)
It may be noted that condition (it) implies that there is no solution to the problem unless there are at least as many plant inputs as there are plant outputs to be controlled, i.e. it is necessary that r~m. The frequency domain interpretation of condition (i0 is that no transmission zeros [5] of (1) should occur at the origin. 329
330
Correspondence item
-~m
_1
+
X,.~,
tt I"
Kz tt'~"dr
I
FIG. 1. Feedforward--lntegral controller obtained assuming x is available for measurement; g~_A-e. Assuming these conditions hold, on applying the affine transformation* followed by the differential transformation of Ref. [1], the following control system is obtained as shown in Fig. 1 :
u--K~×+K2
f
o
e(~)d~+(K~,-I)G]"
F,,, Wm
has a satisfactory stable response by using either pole assignment methods, optimal regulator theory, frequency domain methods etc; some specific examples of using this method are given in Ref. [1]. If optimal regulator theory is used, the controller (5) corresponds to a solution to the following problem for (1):
(7) where G I'=G'(G
assuming that R is positive definite and that
G')-~
where is detectable.t and where If the state x is not available for measurement, the following controller, illustrated in Fig. 2, is obtained, assuming that (A, C) is observable [I]:
KI, K2 are found so that the following stabilizable (controllable) system:
i=
0 z+
D V, v = ( K I , K2)z
(6)
* x and y are mid to be related by an afflne transformation, i f y=x-/-ll where p is a constant vector.
Fie. 2.
t Let (A, C) be decomposed as
where (Ab Cl) is observable; then (A, C) is said to be detectable if the eigenvalues of A3 are in the open lefthand part of the complex plane. -~m Em
Feedforward--Integral controller obtained assuming x is not available for measurement; c A - e ; AIA3 A4
is found so that
0]
fast dynamics, w h e r e { ( A
\A 3 A4]k0 00)'(0C
has satisfactory stable
: ) } is observable.
Correspondence item
u=ii+~(K,,-I)Gt (~") w,,,
+(_.,,oo, (o),., fi = K t J ~ -t- K211
331
°:.[,,, (8)
02)
where t is the output of a minimal order observer with input
where all other quantities are defined in (8). It may be remarked, that in the case when D - - 0 , 6 T = I and the resultant equation (12) is of much simpler form. It should be noted that the controller (12) corresponds to a generalization of the 3-term controller o f classical control theory, the observer in this case corresponding to a phase lead compensator.
e, 1 1 A f '° e ( Q d ~ and l], which estimates the state
Remark 1 In the development of the feedforward controller in [1], it was shown that the requirement for ~(t)--*0, y(t)--*0 as t.-.~oo, Vo~R q is that (A, B) be stabilizable and that rank
of the following observable system:
(9) be equal to nq-m. This observation proves that conditions (4) are necessary for a solution to exist to the problem for the class of all linear or nonlinear feedback controllers.
It should be noted that a weaker condition of (A, C) being detectable is really only required for there to be a solution to the problem for the case of stabilizing the closed loop system. In the special case when F, =0, M = I and when
Remark 2 When the controller (5), (8), (10) or (12) is applied to (1), the resulting controlled system has the property that asymptotic regulation occurs for all changes, large or small, in the feedback gains K1, K2 provided only that the resultant system remains stable. This follows from Ref. [2], on noting that the extra outputs that have been eliminated in reducing (1) so that (2) is true, are either linear combinations of .2 or the outputs which have been retailed in (2). This observation is interesting, because it means that if there exists a solution to (1), then there always exists a robust controller for (1), so that regulation is achieved for changes in the feedback gains of the controller provided the resultant system remains stable. If the original system's parameters before the reduction to (2) have the property tha~
{I+(K,,-I)G, (O)} is nonsingular, the controller (5) becomes the following on simplification:
u = S / ' ( K 1, - I ) G ' [ "
f' ,,o,(O)]x
(:)
e+SPK2
o e(z) dz
rank(
(lO) where
Ae=[I+(K,,-I)G, (O)] -'
A
B)=n+m,
then the controlled system is also robust with respect to perturbations, large or small, in A, B, C, D, E, F provided only that the resultant system remains stable [2]. It can be shown that the above condition is also a necessary condition that must hold for a controller to be robust, so that asymptotic regulation occurs for all perturbations in the parameters
(zl)
and controller (8) becomes, as shown in Fig. 3:
I
1
I
t/o\
"i ~'c-~''l~ -~ ~} I
I
,
2,,fo
I • ~
I -',: ........
I ,'--
k._ L. . . . . .
Fxo. 3.
~11
J
I
Sim ~lified fecdforward--lntegral controller obtained when F , = 0 , M = I and f
/__%%
when ~ I + ( K I , - I ) G ' [ ' ( ~ 0 1 ~
(
\u/j
isnonsingular;~
332
Correspondence
(A, B) [8]. In this case it can be shown that any robust controller must consist of an error feedback compensator consisting of m integrators [8]. Remark 3. The controllers (5), (10) have minilnum order and the controPers (8), (12) have minimum order for unspecified gains, if rank M - m ; if M = 0 , rank E n, F = 0 ; or i f M - ~ 0 , rank F - m , E 0 [2]. The following lemma is required for Remark 4. Lemma 1. Given (A, B), assume that rank (A, B ) < n ; then this implies that (A, B) is not stabilizable. Proof. Assume that (A, B) is stabilizable; then this implies there exists a K so that ( A + B K ) is stable, i.e. so that rank ( A + B K ) - - n or alternately that rank (A, B ) = r a n k ( A + BK, B ) = n . Therefore, (A, B) cannot be stabilizable if rank (A, B) < n. Remark 4. There is no loss of generality in assuming condition (2), because
rank
A C
B D
E) F
/
< n + m
if and only if any one or combinations of the following occur:
(i)
rank (C, D, F ) < m
(ii)
some rows of (C, D, F) are linear combinations of some rows of (A, B, E).
Off) rank (A, B, E ) < n . In the case of (i) the redundant outputs can be eliminated; in the case of (ii) some combination of outputs are linear combinations of the state derivatives and hence a fewer number of outputs can be chosen to satisfy the problem statement; in the case of (iii) (A, B) is not stabilizable from Lemma l, and hence there is no solution to the problem. Remark 5. The controller (5) is the same as the feedforward-integral controller of Ref. [6] for the special case when all disturbances in the system are assumed to be constant. Remark 6. There will be a solution to the problem posed for almost all (A, B, C, D) systems given by (1) provided only that rank B->rank C, i.e. the class of (A, B, C, D) systems in which equation (2) and conditions (4) do not both hold, lies on a hypersurface in the parameter space of (A, B, C, D). This follows immediately from the development of [7] on noting that the system
item
if the number of outputs to be regulated is greater than the number of controlled inputs, then these is almost never a solution to the servomecha,aism problem. (See Ref. [.~] for a detailed discussion of this generic problem.) References
[1] H. W. SMITH and E. J. DAVISON: Design of industrial regulators: integral feedback and feedforward control. Proc. lEE 119, 1210-1216 (1972). [2] E. J. DAVISON and H. W. SMITH: Pole assignment in linear time-invariant multivariable systems with constant disturbances. Automatica 7, 489--498 (1971). [3] C. D. JOHNSON: Optimal control of the linear regulator with constant disturbances. IEEE Trans. Aut. Control AC-15 416-421 (1968). [4] P. C. YOUNG and J. C. WILLEMS: An approach to 1he multivariable servomechanism problem. Int. Control 15, 961-979 (1972). [5] S. H. WANG and E. J. DAVlSO~: Canonical forms of linear multivariable systems. Dept. of Electrical Engineering, University of Toronto, Control System Report No. 7203, April 1972; presented at 6th Asilomar Conference on Circuits and Systems (1972). [6] E. J. DAVlSON: The systematic design of control systems for large multivariable linear time-invariant systems. Automatica 9, 441-452 (1973). [7] E. J. DAVISON and S. H. WA~CI: Properties of linear multivariable systems subject to arbitrary output and state feedback. IEEE Trans. Aut. Control AC-18, 24--32 (1973). [8] E. J. DAVlSON: The Feedforward, Feedback and robust control of a general servomechanism problem, Parts 1, II. Dept. of Electrical Engineering, University of Toronto, Control System Report No. 7305, 7306, April 1973; presented at l lth Allerton Conference on Circuit and System Theory.
R6sam6--Cette note utilise certaines des id6es d'un texte pr6c~dent pour obtenir des contrfleurs explicites ~ ri:troaction et action avanc6e intdgrales pour une installation lindaire multivariable inddpendante du temps, et soumise h des perturbations constantes mensurables et non-mensurables de sorte que les sorties soient rdgldes ',i des niveaux prdvu~. Zusammeafassung--Diese Bemerkung geht von einigcn 1Dberlegungen einer friiheren Arbei [1] aus, um explizi:c Regler mit St6rgr/Sflenaufschaltung--integraler Rtickfi.ihrung fiir eine lineare mehrvariable zeitinvariante Anlage zu erhalten, die sowohl mal3bare als auch nichts mellbaren konstanten Stt~rungen unterworfen ist, so dab Ausgtinge auf vorgegebene Sollwerte eingeregelt werden. Pe31oMe--B ~aHm)II 3aMeTre HOIOnbByK)TCa )leroTOpble n~eH npeab~Luett CTarbn [1]. 3aaa~a 3azmo~aeTcs n
satisfies (2) and conditions (4) if r>_m. If rank B < r a n k C and if condition (2) holds, there is no solution to the problem. Thus there is almost always a solution to the servomechanism problem if the number of controlled inputs is greater or equal to the number of outputs to be regulated;
no;lyqeHl4FI TOqHblX pa3OMKHyTO3aMKHyTblX I~FyJ'I~ITOpOB J1JI~! ~HHe~IHblX MtIOFOhaOaMeTpttqeCKHX ),cTaHOBOK C
HOCTO~IHH~IMI4 KO3qbqbntD[enTaMi4,noaBepraeMblM KaK H3MeplleMbIM, TaK 14 He H3MeplleMMM HOCTOIIHHblMBO3Mylil.eHHIIM. Pel-yJIHpoBaHne BC~eTCfl TaK, qTO BblXOJ],,I CTa6n.qH3HpyIOTC~ Ha 3apaHee 3aaaaHl,lx 3naqenH~x.