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Multivariable Control Systems with a Structural Steady-State Signal Invariance
MULTIVARIABLE CONTROL SYSTEMS WITH A STRUCTURAL STEADY-STATE SIGNAL INVARIANCE
F. Nicol6 Istituto di Automatica Universitl di Roma Roma, Italy
0. M. Grasselli Centro di Studio dei Sistemi di Controllo e Calcolo Automatici Consiglio Nazionale delle Ricerche Roma, Italy
thesis procedure of the controller follows. T h i s ganer d e a l s l i n e a r , tize-invariant
feed-hack c o n t r o l s y s t e n s in v h i c h t h e c o p t r o l l e 6 ?lant is affected hy u n n e a s u r a k l e s t e ~ - w i s ed i s t u r b a n c e s . Necessary and sufficient conditions are given for the existence of a controller such that the control system is ssymptoticall>stable and steady-state output-invariant with respect to disturbances.
It is clear that any controller which, in the steadystate, compensates the disturbances may need some integral actions. The minimum number of these is also given. Lastly, on the basis of the given conditions, it is easily seen that parameter variations may not affect disturbance rejection.
A synthesis procedure of the controller follows from the sufficiency proof. STEADY-STATE D-INVARIANCE Moreover, the minimum number of the integral actions with are needed is also found.
In this section a disturbed feedback system S is considered, which can be assumed to be represented by:
INTRODUCTION This paper deals with linear, time-invariant, conti nuous and finite-dimensional systems affected by unmeasurable step-wise disturbances. For this kind of systems, the synthesis of a con- troller such that the output goes asymptotically to zero has been widely investigated in the last few years. Many authors treated this ~ r o b l e mfrom the optimal control point of view (7) ( 9 ) (ll). . Others considered the classical approach to be still important because of the significance, for the pra ctical applications, of the solutions thus obtained. Conditions for the existence of the controller were given for some special cases (3)(4)(12). In a larger framework, that is for disturbances be longing to larger classes,conditionsfor theexistence of the controller were iven for other important cases (1) (2) (6) (10) (13)?14). Here necessary and sufficient conditions are given for the existence of a controller such that the control system is asymptotically stable and the output goes asymptotically to zero in the presence of step-wise,unknown, and unmeasurable, disturbal ces. This result holds under no additional assum ption on the controlled plant. The sufficiency proof is constructive; hence a syn This work was supported by C.N.R., Roma, Italy.
where x, u, d, y are respectively the state vector of dimension n, the feedback input vector of dimel sion q , the disturbance vector of dimension r, the output vector of dimension q (see fig. 1) A, B, C, D, M, N, are constant matrices of the proper dimel sions, satisfying the condition:
-
det (I
q
where I
D) f 0
(4)
is the identity matrix of dimension q. More
4
over
l(t)
is assumed to be given by:
where a i s aconstantvector and 6-l(t) is the stepfunction. More specifycally the steady-state behaviour of the output of S will be studied. For this purpose the following definition is given. Definition 1: The system S is said to be steady-st2 te d-invariant (SSDI) if lim y(t) t"
=
0
V x
V a
That is S, whict is assumed to be affected by a stepwise disturbance, is SSDI if, for any initial state and for any disturbance, the steady-state value of the output is zero. The conditions under which the system S is SSDI will now be given. Since we are looking for applicatior. to thesynthesis of feedback control systems,it is natural to assume that S is asymptotically stable. Denote with R [ ] the range space of the argument matrix. The following theorem can now be stated. Theorem 1: If the system S is asymptotically stable, it is steady-state d-invariant if and only if tr;e following condition holds:
depends only on the sub-system of S which is controllable from the disturbance input and observable, and that it c ~ n n o tdepend on B and D. These matrices affect the steady-state d-invariance only trough the stability of S. Remark - 2 - The property defined by condition(7),
as well as stability,can also remain valid if large pa rameter variations occur. Thus steady-state d-invariance can be called a "structural" invariance,since it is independent of parameter variations which do not alter the above mentioned properties. Remark 3 - A definition of steady-state d-invariance similar to Definition 1 can be given if d is assumed to be a polynomial vector. Theorem 1 can be easily extended to cover the new definition. SYNTHESIS OF STEADY-STATE D-INVARIANT CONTROL SYSTEMS
Proof. The asymptotic stability implies that the steaty-state behaviour can be studied in the zero initial state. Thus, defining:
In the previous section the steady-state d-invarial ce of feedback systems has been studied. In this section the problem of the synthesis of asymptotically stable and SSDI feedback control syst ems will be dealt with, assuming that the disturbance d acting on the plant to be controlled, as well as the state of the latter, are not accessible to measurement. Let the plant P to be controlled be described by:
it follows that (6) holds if and only if: lim
C(SI
n-
Zi)-'fi
+
fi
=
;h =
Apxp + Bpup + Mpd
o
O'S
where s is the Laplace variable. Moreover (12) holds if and only if:
Substituting ( a ) , (9), (lo), (11) in (13), the latter is found to be equivalent to (7). This completes the proof.
where xp and up are respectively the state vector of dimension n and the manipulated input vector of dimension p; , Bps Cp, Dp, Mp ald Np are constant matrices of {he proper dimensions. Obviously y is the output of the control system to be synthesized as well as of the plant.
1
Since the disturbance and the state are not accessible,the controller K can be assumed to be represented by:
Remark. - 1 - The equivalence between (7) and shows that the steady-state d-invariance of
(13) S can be determined in the same way by means of the sets of matrices which describe the disturbance-output relationship either in the closed-loop system S (see 13)) or in the open-loop sub-system S described by (1) and (2) (see (7)). Indeed it can be shown that if S is asym~totically stable it is SSDI if and only if for any a there exists an initial state x such that the output y is identically zero. ~oreove? it can be shown that the latter property holds for S if and only if it holds for S , irrespective of the stability of either sys tem; 2nd that (13) and (7) are respectively the relevant conditions. In this framework it is clear that condition (7)
where z is the state vector of K, of suitable dimension, and F, G, H, L, are constant matrices of the proper dimensions. Now let S be the feedback system described by (14), (15), (16), (17) (see fig. 2). A first problem to be solved is: under which conditions on P is it possible to find a controller K such that S is asymptotically stable and SSDI? If such a controller exists, it is intuitive that in it some integrating actions may be needed. A second question is: which is the minimal number of integrat ing actions? In order to give the theorems which
answer these questions it is useful to give the following definition and lema. Definition 2: A system is said to be internally stable if, given a canonical decomposition with re? pect to the controllability from the manipulated input and observability, the eigenvalues of the noncontrollable and non-observable sub-systems have negative real parts.
This completes the necessity proof.
The lemma concerns two systems Si (i = 1,2) represented by:
Sufficiency - The proof will be deduced from Lemma 1, assuming K to be made up of the series connection of two controllers KS and KI. The latter will be chosen to be constituted by integrators in such a way that the series connection of K and P satisfies I condition (22), while K will be chosen such that S S is asymptotically stable.
with dim(x.)
Let h be the smallest integer such that there exists a p - h matrix E satisfying
=
n.
1'
dim(ui)
=
pi, dim(y.) = qi. As!
-in? M, = 0 , N1 = 0, p2 = ql, q2 = pl, and
-
D2D1) # 0 , let Sl2 be the feedback system q descri6ed by (18) and (19) (with i = 1,2) and by
det(I
By definition E has rank h; hepce there exists a p.(p-h) matrix 2 such that [E E] is non-singular. having y2 as output. A
Lemma 1:
Let KI be given by
If SI2 is asymptotically stable and
A
then S12 is steady-state d-invariant. Proof. The statement follows directly applying Theorem 1 to S12, bearing in mind that the open-loop system is the one described by (18) and (19) (with i = 1,2) and (20).
u1 where zI is the state vector of dimension h, a ~ d is the input vector, of dimension p. If h = p , E and the zero-matrix in (26) vanish. 'L
Call S2 the series connection of KI and P (seefig.3). Writing condition (22) for 22:
The following theorems can now be stated, which give the answers to the above questions. Theorem 2: There exists a controller K such that the system S is asymptotically stable and steadystate d-invariant if and only if the plant P is internally stable and
Proof. Necessity - P must be internally stable, otherwise S cannot be asymptotically stable.
As regards (23) assume K to be acontroller such that S is SSDI. Then, applying Theorem 1, it follows from (7) that:
it follows from
(25) that (28) holds.
Now it will be shown that there exists a KS which ma kes S asymptotically stable. Consider a representation of P which is canonical with respect to the controllability from the manipulated input and observability. Let Pb be the control lable and observable sub-system and Abb, Bb, Cb respectively its dynamic, manipulated input, and output matrices. 'L
The sub-system S2b made up of the series connection of KI and Pb is both controllable and observable. The controllability will now be deduced from Theorem A1 (see Appendix). Suppose Pb to be in the Jordan form representation and call h j (j=l, m) the m
...,
distinct eigenvalues of Abb, and ri the number of its Jordan blocks associated wlth .. Moreover suppose Bb to be partitioned into blocis according to m define the Jordan blocks of Abb; for each j=l, Bbj as the matrix made up of the r. last rows ofthe blocks of Bb associated with Aj. ~ i e nthe matrix (Al) becomes:
rank
...,
[:111
=nP+qv
where q' is the rank of
I Cp
Dp 1 .
In a similar way it can be seen that, if Mp = 0 and rank Np = q, then, by virtue of internal stability, condition (23) in Theorem 2 can be substituted with: rank
/
A~
B~
]
=np+q
which is readily seen to have full row-rank. This, together with the ,qjontrollability of KI, provesthe controllability of S2b. Lastly,it is obvious that if (I2) The observability will now be deduced from the ore^! A2 (see Appendix). The matrix (A2) becomes:
II" In order to show that the matrix (30) has full co lumn-rank consider condition (25), written for the above mentioned canonical representation of P; it takes on the following form:
condition (33) is equivalent to condition (23). Remark 5 - In this section the case in which the state of the plant is not accessible to measurement has been studied. This is because in such a case also the second question above formulated can be completely answered. If there is an output accessible to measurement other than the regulated output y, then the proof of Theorem 2 can be easily extended: the necessary and sufficient conditions for the existence of a controller are found to be the validity of (23) and the internal stability with respect to the manipulated input and the measured output. Define h as in the proof of the previous theorem.
where the meaning of the undefined simbols is ob vious. By virtue of the observability of the pair (Abb,Cb) and of the non-singularity of Aaa (implied by the internal stability of P) the matrix made up of the first two block-columns of the first argument in (31) has full column-rank. Hence by definition of E the matrix made up of the first, second and fifth block-columns also has full column-rank. It follows that the matrix (30) has full column-rank. This, together with thezobservability of Pb, proves the observability of S2b.
Theorem 3: If S is asymptotically stable and steadystate d-invariant, then the dynamic matrix F of K has at least h Jordan blocks associated with zero eigenvalues
.
Proof. Assume K to be in the Jordan form representation. Condition (7) takes on the form:
It is easily seen that t h e r ~exists a canonical dc composition of S2 in which S2b constitutes the con trolltble and observable sub-system. This implies that S2 is internally stable,and therefore there exists a controller KS such that S is asymptotically stable. This completes the proof. Remark 4 - It can be easily shown that, in the s ecia1 case in which rank Mp = np and Np = 0 (3)(4P, the condition (23) is equivalent to:
...
where Fo has only non-zero eigenvalues, F. (i=l, are the 4? Jordan blocks associatedlwith zero eigenvalues and Hi (i=O, are the corresponding blocks of H.
...,e)
...,e )
Let Hf be the matrix made up of the 1 first columns of the matrices Hi (i=l,...,1 ) . Condition (35) is
?1
equivalent to:
number of Jordan blocks of Ai associated with Xij (i=l,2). ' I ?1
Suppose B2 (C1) to be partitione,$ into blocks acand for cording to the Jordan blocks%of A2 each j=l, m2 (ml) define B2j (?ii) as the matrix made up of tke ~ 2 last . rows (rij first columns) of the blocks of B ~ ( cassociated ~ ~ with X2j (ilj).
(I1).
...,
By definition of h, it follows that e 2 h . Remark 6 - The sufficiency proof of Theorem 2 is col structive,that is it gives a synthesis procedure. Moreover the controller K there utilized has the minimal number of integrating actions. Remark 7 - Assuming K to be chosen such that S is asymptotically stable and SSD1,theseproperties remain valid if parameter variations occur in P such that S remains stable and (36) still holds. Thus, for the latter condition, it may be convenient to choose larger than the value of h corresponding to the nominal values of the parameters, provided stability can be preserved. CONCLUSION In this paper the problem of regulating a linear multivariable plant, affected by unmeasurable stepwise disturbances has been considered. The objective was to get conditions under which the steady-st5 te value of the regulated output can be made zero via a linear feedback controller in the most general case. Main results are given by Theorem 2 and Theorem 3. Theorem 2 gives the necessary and sufficient condi tions for the existence of the controller, in the case inwhich the stability of the control system is required, as it is natural. These conditions are the internal stability of the controlled plant (see Definition 2 and Remark 5) and (23). This latter can be specialized in previously found conditions (see Rernark 4). The approach followed in the proof of the Theorem can be utilized forthe synthesizingof the controller with the aid of any standard technique.
The following theorems can now be stated. Theorem Al: S12 is controllable from ul if and only if S1 is controllable from ul and for each j=l, . ,m2 the (n + r2j). (nl + pi) matrix:
has rank nl+ r2j (i.e. full row-rank). Theorem A2: S12 is observable if and only if S2 ml the (n2 +q2) observable and,for each j=l, (n2 + rl.) matrix: 3
Consider the systems S; (i=1,2) described by (la), (19). Assuming p2 = ql, define S12 the series connection of S1 and S2, described by (18) and (19) (with i=1,2) and (20). %
'
-
13
is
REFERENCES (1)
Bhattacharyya S.P., Pearson J.B., "On Error Systems and the Servomechanism Problem", Int. J. Control, Vol. 15, n. 6, June 1972.
(2)
Bhattacharyya S.P., Pearson J.B., Wonham W.M., "On Zeroing the Output of a Linear System", Information and Control, Vol. 20, n. 2, March 1972.
(3)
Davison E.J., "The Systematic Design of Control Systems for Large Multivariable Linear Time-Invariant Systems", Preprints IFAC V, June 1972, paper n. 29.2.
(4)
Davison E.J., Smith H.W., "Pole Assigment in Li near Time-Invariant Multivariable Systems with Constant Disturbances", Automatica, Vol. 7, p. 489-498, 1971. Grasselli O.M., "Controllability and Observabil ity of Series Connections of Systems", to appear in Ricerche di Automatica.
(5)
(6)
Grasselli O.M., Nicol6 F., "Modal Synthesis of Astatic Multivariable Regulation Systems", 2nd IFAC Symposium on Multivariable Technical Control Systems, Duesseldorf, October 1971.
(7)
Johnson C.D., "Accomodation of External Disturbances in Linear Regulator and Servomechanism Problems", IEEE Trans. on AC, Vol. AC-16, n. 6, December 1971.
(8)
Johnson C.D., "Accomodation of Disturbances in Optimal Control Problems", Int. J. Control, Vol. 15, n. 2, February 1972.
%
Further call Ai, gi, Ci respectively the dynamic, manipulated input, output matrices of S; in the Jordan form representation; let Aij be the mi dis..,mi) the tinct eigenvalues of A;, and r.. (~=l,.
.
has rank n2 +rlj (i.e. full column rank).
APPENDIX Two theorems are here reported from (5) which are needed in the proof of Theorem 2.
...,
.
Theorem 4 gives the minimal number of integral actions which are needed in the controller. Note that the analysis condition preliminarly given(see Theorem 1) can be usetully applied in the problem of synthesizing a feedforward controller, that is when thedisturbancesare measured.
...
..
(9)
Latour P.R., "Optimal Control of Linear Multivariable Plants with Constant Disturbances", JACC 1971, paper 8-E3.
(10) Nicol6 F., "Sul Legame tra Comportamento a Ciclo Chiuso ed a Ciclo Aperto nella Sintesi di Sistemi di Controllo a Molte Variabili", Rendi conti della LXX Riunione AEI, 1969. (11) Parker K.T., "Design of Proportional-IntegralDerivative Controllers by the Use of OptimalLinear-Regulator Theory", Proc. IEE, Vol. 119, n. 7, July 1972. (12) Smith H.W., Davison E.J., "Design of Industrial Regulators: Integral Feedback and Feedforward Control", Proc. IEE, Vol. 119, n. 8, August 1972. (13) Wonham W.M., "Tracking and Regulation in Linear Multivariable Systems", University of Toronto, Dept. of Electrical Eng., report CS 7202, March 1972.
Fig. 1
(14) Wonham W.M. , Pearson J.B. , "Regulation and Internal Stabilization in Linear Multivariable Systems", University of Toronto, Dept. of Electrical Eng., report CS 7212, August 1972.
Fig. 2
Fig. 3
F. Nicol6 Istituto di Automatica Universitl di Roma Roma, Italy
0. M. Grasselli Centro di Studio dei Sistemi di Controllo e Calcolo Automatici Consiglio Nazionale delle Ricerche Roma, Italy
thesis procedure of the controller follows. T h i s ganer d e a l s l i n e a r , tize-invariant
feed-hack c o n t r o l s y s t e n s in v h i c h t h e c o p t r o l l e 6 ?lant is affected hy u n n e a s u r a k l e s t e ~ - w i s ed i s t u r b a n c e s . Necessary and sufficient conditions are given for the existence of a controller such that the control system is ssymptoticall>stable and steady-state output-invariant with respect to disturbances.
It is clear that any controller which, in the steadystate, compensates the disturbances may need some integral actions. The minimum number of these is also given. Lastly, on the basis of the given conditions, it is easily seen that parameter variations may not affect disturbance rejection.
A synthesis procedure of the controller follows from the sufficiency proof. STEADY-STATE D-INVARIANCE Moreover, the minimum number of the integral actions with are needed is also found.
In this section a disturbed feedback system S is considered, which can be assumed to be represented by:
INTRODUCTION This paper deals with linear, time-invariant, conti nuous and finite-dimensional systems affected by unmeasurable step-wise disturbances. For this kind of systems, the synthesis of a con- troller such that the output goes asymptotically to zero has been widely investigated in the last few years. Many authors treated this ~ r o b l e mfrom the optimal control point of view (7) ( 9 ) (ll). . Others considered the classical approach to be still important because of the significance, for the pra ctical applications, of the solutions thus obtained. Conditions for the existence of the controller were given for some special cases (3)(4)(12). In a larger framework, that is for disturbances be longing to larger classes,conditionsfor theexistence of the controller were iven for other important cases (1) (2) (6) (10) (13)?14). Here necessary and sufficient conditions are given for the existence of a controller such that the control system is asymptotically stable and the output goes asymptotically to zero in the presence of step-wise,unknown, and unmeasurable, disturbal ces. This result holds under no additional assum ption on the controlled plant. The sufficiency proof is constructive; hence a syn This work was supported by C.N.R., Roma, Italy.
where x, u, d, y are respectively the state vector of dimension n, the feedback input vector of dimel sion q , the disturbance vector of dimension r, the output vector of dimension q (see fig. 1) A, B, C, D, M, N, are constant matrices of the proper dimel sions, satisfying the condition:
-
det (I
q
where I
D) f 0
(4)
is the identity matrix of dimension q. More
4
over
l(t)
is assumed to be given by:
where a i s aconstantvector and 6-l(t) is the stepfunction. More specifycally the steady-state behaviour of the output of S will be studied. For this purpose the following definition is given. Definition 1: The system S is said to be steady-st2 te d-invariant (SSDI) if lim y(t) t"
=
0
V x
V a
That is S, whict is assumed to be affected by a stepwise disturbance, is SSDI if, for any initial state and for any disturbance, the steady-state value of the output is zero. The conditions under which the system S is SSDI will now be given. Since we are looking for applicatior. to thesynthesis of feedback control systems,it is natural to assume that S is asymptotically stable. Denote with R [ ] the range space of the argument matrix. The following theorem can now be stated. Theorem 1: If the system S is asymptotically stable, it is steady-state d-invariant if and only if tr;e following condition holds:
depends only on the sub-system of S which is controllable from the disturbance input and observable, and that it c ~ n n o tdepend on B and D. These matrices affect the steady-state d-invariance only trough the stability of S. Remark - 2 - The property defined by condition(7),
as well as stability,can also remain valid if large pa rameter variations occur. Thus steady-state d-invariance can be called a "structural" invariance,since it is independent of parameter variations which do not alter the above mentioned properties. Remark 3 - A definition of steady-state d-invariance similar to Definition 1 can be given if d is assumed to be a polynomial vector. Theorem 1 can be easily extended to cover the new definition. SYNTHESIS OF STEADY-STATE D-INVARIANT CONTROL SYSTEMS
Proof. The asymptotic stability implies that the steaty-state behaviour can be studied in the zero initial state. Thus, defining:
In the previous section the steady-state d-invarial ce of feedback systems has been studied. In this section the problem of the synthesis of asymptotically stable and SSDI feedback control syst ems will be dealt with, assuming that the disturbance d acting on the plant to be controlled, as well as the state of the latter, are not accessible to measurement. Let the plant P to be controlled be described by:
it follows that (6) holds if and only if: lim
C(SI
n-
Zi)-'fi
+
fi
=
;h =
Apxp + Bpup + Mpd
o
O'S
where s is the Laplace variable. Moreover (12) holds if and only if:
Substituting ( a ) , (9), (lo), (11) in (13), the latter is found to be equivalent to (7). This completes the proof.
where xp and up are respectively the state vector of dimension n and the manipulated input vector of dimension p; , Bps Cp, Dp, Mp ald Np are constant matrices of {he proper dimensions. Obviously y is the output of the control system to be synthesized as well as of the plant.
1
Since the disturbance and the state are not accessible,the controller K can be assumed to be represented by:
Remark. - 1 - The equivalence between (7) and shows that the steady-state d-invariance of
(13) S can be determined in the same way by means of the sets of matrices which describe the disturbance-output relationship either in the closed-loop system S (see 13)) or in the open-loop sub-system S described by (1) and (2) (see (7)). Indeed it can be shown that if S is asym~totically stable it is SSDI if and only if for any a there exists an initial state x such that the output y is identically zero. ~oreove? it can be shown that the latter property holds for S if and only if it holds for S , irrespective of the stability of either sys tem; 2nd that (13) and (7) are respectively the relevant conditions. In this framework it is clear that condition (7)
where z is the state vector of K, of suitable dimension, and F, G, H, L, are constant matrices of the proper dimensions. Now let S be the feedback system described by (14), (15), (16), (17) (see fig. 2). A first problem to be solved is: under which conditions on P is it possible to find a controller K such that S is asymptotically stable and SSDI? If such a controller exists, it is intuitive that in it some integrating actions may be needed. A second question is: which is the minimal number of integrat ing actions? In order to give the theorems which
answer these questions it is useful to give the following definition and lema. Definition 2: A system is said to be internally stable if, given a canonical decomposition with re? pect to the controllability from the manipulated input and observability, the eigenvalues of the noncontrollable and non-observable sub-systems have negative real parts.
This completes the necessity proof.
The lemma concerns two systems Si (i = 1,2) represented by:
Sufficiency - The proof will be deduced from Lemma 1, assuming K to be made up of the series connection of two controllers KS and KI. The latter will be chosen to be constituted by integrators in such a way that the series connection of K and P satisfies I condition (22), while K will be chosen such that S S is asymptotically stable.
with dim(x.)
Let h be the smallest integer such that there exists a p - h matrix E satisfying
=
n.
1'
dim(ui)
=
pi, dim(y.) = qi. As!
-in? M, = 0 , N1 = 0, p2 = ql, q2 = pl, and
-
D2D1) # 0 , let Sl2 be the feedback system q descri6ed by (18) and (19) (with i = 1,2) and by
det(I
By definition E has rank h; hepce there exists a p.(p-h) matrix 2 such that [E E] is non-singular. having y2 as output. A
Lemma 1:
Let KI be given by
If SI2 is asymptotically stable and
A
then S12 is steady-state d-invariant. Proof. The statement follows directly applying Theorem 1 to S12, bearing in mind that the open-loop system is the one described by (18) and (19) (with i = 1,2) and (20).
u1 where zI is the state vector of dimension h, a ~ d is the input vector, of dimension p. If h = p , E and the zero-matrix in (26) vanish. 'L
Call S2 the series connection of KI and P (seefig.3). Writing condition (22) for 22:
The following theorems can now be stated, which give the answers to the above questions. Theorem 2: There exists a controller K such that the system S is asymptotically stable and steadystate d-invariant if and only if the plant P is internally stable and
Proof. Necessity - P must be internally stable, otherwise S cannot be asymptotically stable.
As regards (23) assume K to be acontroller such that S is SSDI. Then, applying Theorem 1, it follows from (7) that:
it follows from
(25) that (28) holds.
Now it will be shown that there exists a KS which ma kes S asymptotically stable. Consider a representation of P which is canonical with respect to the controllability from the manipulated input and observability. Let Pb be the control lable and observable sub-system and Abb, Bb, Cb respectively its dynamic, manipulated input, and output matrices. 'L
The sub-system S2b made up of the series connection of KI and Pb is both controllable and observable. The controllability will now be deduced from Theorem A1 (see Appendix). Suppose Pb to be in the Jordan form representation and call h j (j=l, m) the m
...,
distinct eigenvalues of Abb, and ri the number of its Jordan blocks associated wlth .. Moreover suppose Bb to be partitioned into blocis according to m define the Jordan blocks of Abb; for each j=l, Bbj as the matrix made up of the r. last rows ofthe blocks of Bb associated with Aj. ~ i e nthe matrix (Al) becomes:
rank
...,
[:111
=nP+qv
where q' is the rank of
I Cp
Dp 1 .
In a similar way it can be seen that, if Mp = 0 and rank Np = q, then, by virtue of internal stability, condition (23) in Theorem 2 can be substituted with: rank
/
A~
B~
]
=np+q
which is readily seen to have full row-rank. This, together with the ,qjontrollability of KI, provesthe controllability of S2b. Lastly,it is obvious that if (I2) The observability will now be deduced from the ore^! A2 (see Appendix). The matrix (A2) becomes:
II" In order to show that the matrix (30) has full co lumn-rank consider condition (25), written for the above mentioned canonical representation of P; it takes on the following form:
condition (33) is equivalent to condition (23). Remark 5 - In this section the case in which the state of the plant is not accessible to measurement has been studied. This is because in such a case also the second question above formulated can be completely answered. If there is an output accessible to measurement other than the regulated output y, then the proof of Theorem 2 can be easily extended: the necessary and sufficient conditions for the existence of a controller are found to be the validity of (23) and the internal stability with respect to the manipulated input and the measured output. Define h as in the proof of the previous theorem.
where the meaning of the undefined simbols is ob vious. By virtue of the observability of the pair (Abb,Cb) and of the non-singularity of Aaa (implied by the internal stability of P) the matrix made up of the first two block-columns of the first argument in (31) has full column-rank. Hence by definition of E the matrix made up of the first, second and fifth block-columns also has full column-rank. It follows that the matrix (30) has full column-rank. This, together with thezobservability of Pb, proves the observability of S2b.
Theorem 3: If S is asymptotically stable and steadystate d-invariant, then the dynamic matrix F of K has at least h Jordan blocks associated with zero eigenvalues
.
Proof. Assume K to be in the Jordan form representation. Condition (7) takes on the form:
It is easily seen that t h e r ~exists a canonical dc composition of S2 in which S2b constitutes the con trolltble and observable sub-system. This implies that S2 is internally stable,and therefore there exists a controller KS such that S is asymptotically stable. This completes the proof. Remark 4 - It can be easily shown that, in the s ecia1 case in which rank Mp = np and Np = 0 (3)(4P, the condition (23) is equivalent to:
...
where Fo has only non-zero eigenvalues, F. (i=l, are the 4? Jordan blocks associatedlwith zero eigenvalues and Hi (i=O, are the corresponding blocks of H.
...,e)
...,e )
Let Hf be the matrix made up of the 1 first columns of the matrices Hi (i=l,...,1 ) . Condition (35) is
?1
equivalent to:
number of Jordan blocks of Ai associated with Xij (i=l,2). ' I ?1
Suppose B2 (C1) to be partitione,$ into blocks acand for cording to the Jordan blocks%of A2 each j=l, m2 (ml) define B2j (?ii) as the matrix made up of tke ~ 2 last . rows (rij first columns) of the blocks of B ~ ( cassociated ~ ~ with X2j (ilj).
(I1).
...,
By definition of h, it follows that e 2 h . Remark 6 - The sufficiency proof of Theorem 2 is col structive,that is it gives a synthesis procedure. Moreover the controller K there utilized has the minimal number of integrating actions. Remark 7 - Assuming K to be chosen such that S is asymptotically stable and SSD1,theseproperties remain valid if parameter variations occur in P such that S remains stable and (36) still holds. Thus, for the latter condition, it may be convenient to choose larger than the value of h corresponding to the nominal values of the parameters, provided stability can be preserved. CONCLUSION In this paper the problem of regulating a linear multivariable plant, affected by unmeasurable stepwise disturbances has been considered. The objective was to get conditions under which the steady-st5 te value of the regulated output can be made zero via a linear feedback controller in the most general case. Main results are given by Theorem 2 and Theorem 3. Theorem 2 gives the necessary and sufficient condi tions for the existence of the controller, in the case inwhich the stability of the control system is required, as it is natural. These conditions are the internal stability of the controlled plant (see Definition 2 and Remark 5) and (23). This latter can be specialized in previously found conditions (see Rernark 4). The approach followed in the proof of the Theorem can be utilized forthe synthesizingof the controller with the aid of any standard technique.
The following theorems can now be stated. Theorem Al: S12 is controllable from ul if and only if S1 is controllable from ul and for each j=l, . ,m2 the (n + r2j). (nl + pi) matrix:
has rank nl+ r2j (i.e. full row-rank). Theorem A2: S12 is observable if and only if S2 ml the (n2 +q2) observable and,for each j=l, (n2 + rl.) matrix: 3
Consider the systems S; (i=1,2) described by (la), (19). Assuming p2 = ql, define S12 the series connection of S1 and S2, described by (18) and (19) (with i=1,2) and (20). %
'
-
13
is
REFERENCES (1)
Bhattacharyya S.P., Pearson J.B., "On Error Systems and the Servomechanism Problem", Int. J. Control, Vol. 15, n. 6, June 1972.
(2)
Bhattacharyya S.P., Pearson J.B., Wonham W.M., "On Zeroing the Output of a Linear System", Information and Control, Vol. 20, n. 2, March 1972.
(3)
Davison E.J., "The Systematic Design of Control Systems for Large Multivariable Linear Time-Invariant Systems", Preprints IFAC V, June 1972, paper n. 29.2.
(4)
Davison E.J., Smith H.W., "Pole Assigment in Li near Time-Invariant Multivariable Systems with Constant Disturbances", Automatica, Vol. 7, p. 489-498, 1971. Grasselli O.M., "Controllability and Observabil ity of Series Connections of Systems", to appear in Ricerche di Automatica.
(5)
(6)
Grasselli O.M., Nicol6 F., "Modal Synthesis of Astatic Multivariable Regulation Systems", 2nd IFAC Symposium on Multivariable Technical Control Systems, Duesseldorf, October 1971.
(7)
Johnson C.D., "Accomodation of External Disturbances in Linear Regulator and Servomechanism Problems", IEEE Trans. on AC, Vol. AC-16, n. 6, December 1971.
(8)
Johnson C.D., "Accomodation of Disturbances in Optimal Control Problems", Int. J. Control, Vol. 15, n. 2, February 1972.
%
Further call Ai, gi, Ci respectively the dynamic, manipulated input, output matrices of S; in the Jordan form representation; let Aij be the mi dis..,mi) the tinct eigenvalues of A;, and r.. (~=l,.
.
has rank n2 +rlj (i.e. full column rank).
APPENDIX Two theorems are here reported from (5) which are needed in the proof of Theorem 2.
...,
.
Theorem 4 gives the minimal number of integral actions which are needed in the controller. Note that the analysis condition preliminarly given(see Theorem 1) can be usetully applied in the problem of synthesizing a feedforward controller, that is when thedisturbancesare measured.
...
..
(9)
Latour P.R., "Optimal Control of Linear Multivariable Plants with Constant Disturbances", JACC 1971, paper 8-E3.
(10) Nicol6 F., "Sul Legame tra Comportamento a Ciclo Chiuso ed a Ciclo Aperto nella Sintesi di Sistemi di Controllo a Molte Variabili", Rendi conti della LXX Riunione AEI, 1969. (11) Parker K.T., "Design of Proportional-IntegralDerivative Controllers by the Use of OptimalLinear-Regulator Theory", Proc. IEE, Vol. 119, n. 7, July 1972. (12) Smith H.W., Davison E.J., "Design of Industrial Regulators: Integral Feedback and Feedforward Control", Proc. IEE, Vol. 119, n. 8, August 1972. (13) Wonham W.M., "Tracking and Regulation in Linear Multivariable Systems", University of Toronto, Dept. of Electrical Eng., report CS 7202, March 1972.
Fig. 1
(14) Wonham W.M. , Pearson J.B. , "Regulation and Internal Stabilization in Linear Multivariable Systems", University of Toronto, Dept. of Electrical Eng., report CS 7212, August 1972.
Fig. 2
Fig. 3