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Design of a minimal precompensator for the decoupling problem
Systems & Control Letters 10 (1988) 325-332 North-Holland
325
Design of a minimal precompensator for the decoupling problem S, I C A R T and J.F. L A F A Y Laboratoire d'Automatique, U.A. CN.R.$. 823, E.N.$.M., 1 rue de la I¢o~, 44072 Nantes 03, France
Received 25 November 1987 Revised 13 February 1987 Abstract: For fight invertible line,'u" time-invariant systems, which cannot be decoupled row by row using regular static state feedback laws, we propose a new algorithmic procedure for designing a minimal precompensator leading to a compensated system which can be decoupled by regular static state feedback. Keywords: Linear control systems, Decoupling, Structural approach, Precompensator.
1. Introduction The decoupling problem for linear time-invariant systems has been plentifully studied during the last 25 years. This paper concerns the row-by-row decoupling problem (named Morgan's problem) for linear systems by a dynamic precompensator. We propose here an easily implementable algorithm: this algorithm designs a precompensated system whick :esults in a new system that is decoupled, or can be decoupled by regular state feedback. In the sequel, we always consider right invertible systems (this condition is necessary for solving Morgan's problem). The precompensator to be obtained will be minimal in the sense that a minimal number of integrators is introduced when defining its dynamics. Similar algorithms have been published, for example by Wang [12] or Descusse and Moog [5] in the algebraic approach, and Dion and Commault [6] or Williams and Antsaldis [13] in the transfer matrix approach. The method proposed here is also algebraic, but requires some structural properties used for solving Morgan's problem by general static state feedback [4]. A major interest is that, in this way, we can greatly reduce the number of steps with respect to Wang's (or Descusse and Moog's) algorithm. Moreover, most of the computations are made on the field of real numbers, in opposition to Dion and Commault's or Williams and Antsaldis's solutions which are based on the inversion of a part of the transfer (e.g. the computation of the interactor). Before presenting the algorithm, we briefly recall the main elements of the structural approach we need in the sequel. Let us remark that, as we just consider dynamic precompensation, we need here less notions than in the solution of Morgan's problem by static state feedback [4].
2. Notations and basic concepts of the structural approach We shall consider, in the sequel, the right invertible linear time-invariant system (C, A, B) described by :c = A x + B u ,
y = Cx,
(2.~)
where x ~ ~ R", u ~ ~ - - R m, y E 0~,~ R P. The transfer matrix of this system, denoted T ( s ) , is defined by r ( s ) = C ( , I _ A)-~B. 0167-6911/88/$3.50 © 1988, Elsevier Science Publishers BN. (North-Holland)
(2.2)
S. Icart, J.F. Lafay / M, Mn: ~l precompensator for decoupling
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We shall assume, without loss of generality, that B is monic and C is epic. The same symbol will be used to denote a map or any of its matrix represent~ ;ans in particular bases. Let Ir denote the (r, r) identity matrix. The i-th row of C will be written as ci. A set of p elements will be written as {.}p. The number of elements of a given list {. } is denoted card{. }. We ,~ill need in our approach the Toeplitz matrix Fn and its sub-matrices F~ defined by
CAB
cB
CB
CA ~'- 1B
CA ~- 2B
(o)1 .. •
(2.3)
CB
The importance of this operator has already been mentioned by Sain and Massey [9] for invertibility problems, and Silverman and Kitap~ji [10] for a characterization of the infinite zero structure. This operator can also be used to efficiently compute the basic geometric algorithms [2,14]. For designing our precompensator, we only need two different families of structural invariants: the row-by-row infinite zero structure (R.I.Z.S.), and the essential orders of the outputs. Let us ~nly recall their definition and main properties. All the properties are detailed in the referenced papers. The row-by-row infinite zero structure of (C, A, B) [2]: Let ne be defined by
n,=inf(l~NlciA~-lB,O}
V i ~ {1, 2 , . . . , p } .
(2.4)
The R.I.Z.S. of the right invertible system (C, A, B) is the list { n~ } which contains p finite integers• The essential orders of the outputs of (C, A, B) [1]: The notion of essential order comes from the concept of essential row in a matrix due to Cremer [3]. A row is essential in a matrix W if it cannot be expressed as a linear combination of the other rows of W. The essential order of the i-th output of a right invertible system, denoted nie, is defined as follows• Let
~i, = [ C'a~l 1e~''" ~c,B ~l'l (0) Ill 11
(2.5)
hie
(2.6)
Then =
inf{/~ ¢ N I ~,~, is essential in F, }.
For fight invertible systems, the list { n~e } has p finite terms and satisfies
n,c>~n,
Vie {1,2,...,p}.
(2.7)
Two important properties of these lists are given in the following remarks [1]: Remark 2.1. Let us denote h e = sup(n~e). The right invertibility of (C, A, B) is equivalent to the fact that the last p ,o'vs of rne are essential ;.n r n. Thus we just have to consider Fne iastead or rn in our study of (C, A, B). Remark 2.2. The two lists { n~ }p and { n ~e}p are not modified by bicausal operations on the transfer matrix T(s).
3. Structural approach of the decoupling problem Falb ad Wolovich's [7] or Morse and Wonham's [8] results on necessary and sufficient conditions for the existence of regular static state feedback solutions can be expressed in structural terms by several equivalent conditions [1]. The one we will use here is the following: Theorem 3.1. There exists a regular solution to Morgan's problem if and only if
ni=nie
Vi~{1,2,...,p}.
(3.1)
S. Icart,J.E Lafay/ Minimalprecompensatorfor decoup#ng
327
So, when this condition is not satisfied and because of the assumed right invertibility property, there always exists a dynamic precompensator [12,6] which will have as a result to modify the structure of the initial system (C, A, B) in order to obtain a compensated system for which (3.1) is verified. Such a precompensator modifies the structure as follows: (1) Infinite zeros and essential orders can only be increased. (2) If any control law decouples the system (C, A, B) row by row, then the orders 8i of the row infinite zeros of the decoupled system satisfy
8i>~nie Vi~ {1,2,...,p}.
(3.2)
Morgan's static problem has been fully solved by Descusse, Lafay and Malabre [4l, and dynamic solutions have been given by Dion and Commault [6]. One very important fact is the following: among all the possible decoupling solutions, there is at least one for which essential orders are not increased: such solutions are called 'minimal'. To achieve their result, Descusse~ Lafay and Malabre [4] had to modify the initial system by a technical procedure named 'shifting procedure'. We will also use this procedure, and this will allow us to reduce the number of steps with respect to the other algorithms. Let us briefly describe this shifting procedure: Let (C, A, B) be the initial system, and n e = sup{n~e }. The shifted system, denoted (C, A, B), is such that the essential orders of the outputs are all equal to n e. This transformation is always possible by addition of ne - n~e integrators on each output. From the Toeplitz matrices associated with (C, A, B) and (C, A, B) it is clear that, for each outpuL the difference n~e - n~ is not modified by this transformation. Note that the transfer matrix of the shifted system (C, A, B) is directly obtained from the one of (C, A, B) by a left multiplicatiox, by a diagonal matrix: indeed, let { n ie }p be the essential orders of (C, A, B), neff sup{n~e} and fl~ = n ~ - n~e. Let T(s) be the transfer matrix of (C, A, B) and T(s) that of (C, A, B). We then have:
T(s) = Diag{ s -#' } T(s).
(3.3)
Both systems are equivalent with respect to the decou_plingproblem: in fact,let G(s) be a precompensator which decouples (C, A, B): the transfer matrix T(s). G(s) is diagonal, and T(s). G(s) too. Thus G (s) decouples (C, A, B). This shiftingprocedure is a key point of the algorithm we present now.
4. Design of a minimal preeompensator
Suppose here that the essential orders of the outputs of the right invertible system (C, A, B) are all equal to ne (the last remark proved that this assumption is not restrictive). The proposed algorithm amounts to compute a dynamic precompensator able to increase the R.I.Z.S. {ni}p of (C, A, B) up to { ne }p without increasing the essential orders which is always possible because (C, A, B) is right invertible [4]• Thus the compensated system is decouplable with regular static feedback. Let the integer d~ be defined by
d, = inf{
(4.1)
N ICA '- B * O} = inf{ n, } p.
Consider now the useful part of the Toeplitz matrix related to (C, A, B), say
IICAd,1B _
CAa~-~B .
L CA ne- 1B
CAne- 2B
F-[y~] = [ CA:IB
where y~- CA~-IB.
"'"
(0) 1 CAdl- 1B J
(4.2)
S. Icart, J.F. Lafay / Minimal precompensator for decoupling
328
Step 1. Let r 1 = rank( CAd'- lB ). Then r 1 < p (otherwise d 1 = n~ and (C, A, B) is decouplable via regular static feedback). It is always possible to find linear transformations L~ and P~ (using Householder's method) such that
(o) (o)
|rl
Y~ = L1Y1P1 =
~2,d~
v
(4.3)
Y m
L~ just corresponds to a renumbering of the outputs without any combination. Let us denote by ~.*,/~ >i d~, the other blocks of the Toeplitz matrix of the system modified by L 1 and PI"
~:,.' I', z~ }r,
y: = L,y,P, = ~,~-l---h-_, I~3,"
, P,> d1.
(4.4)
Step 2. According to (3.1), the final precompensator has to increase the R.I.Z.S. up to { n e }p. To do so, we need i-1 banks of (ne - d~) integrators, one connected with each of the r~ first (new) inputs of (4.3). They also allow to cancel each block Z~, of ~* (# > d~), thanks to the same rl inputs. This cancellation is achieved with an intermediate precompensator denoted H~(s). The computation of H~(s) is described and illustrated in an example in the Appendix. So, the intermediate system, precompensated by H~(s), has as Toeplitz matrix blocks:
Y"" =
i
L---" .. ] Z2,., . ..~3,~ /'/I
--
,
/~>d,.
(4.5)
r!
IIl(s) is a bicausal matrix of the form
[|,,', • ]
(4.6)
Since L 1 and P1 are regular and Hl(s ) is bicausal, the essential orders and the R.I.Z.S. of the system are not modified by steps 1 and 2 (L~ ":~a permutation matrix). We can now repeat the above procedure (steps 1 and 2) in the following way: Let d2 = inf{# ~ N IZ3,~ ~ 0}; the properties of essential orders yield that d2 ~
Ii:,,°,! I. i,°,i,°,, F;::T
+
")'he-1
II"i(°) :T
(4.7)
S. lcart, J.F. Lafay / Minimaiprecompensatorfor decoupling
329
Remark. Note
that all the transformations performed up to now are regular transformations which never increase the infinite zero structure or the essential orders. Step 3. Define now the precompensator R(s):
R(s)-Diag{s-(",-d~)I,,}.
(4.8)
This precompensator consists in the addition of integrators on the inputs, and allows to bring the R.I.Z.S. up to ne. We finally obtain the system precompensated by G(s), with G(s)- P~HI(s)... PvIIv(s)R(s). Its Toeplitz matrix has all strips equal to zero, except the last, which is
1
i
:(o)
•
".
•
*
1
•
(0)
-.-].
(4.9)
"
m-p
This system is really d~,~ouplable via regular static state feedback. It is clear that this precompensator is minhnal because the essential orders are not modified.
5. Comparison with other algorithms The above algorithm can be directly comp~ed with Wang [12] or Descusse and Moog's one [5]. (The latter is an extension to the non-linear case of the first.) The initialization of these algorithms is the matrix B*, defined by Falb and Wolovich in [7]:
[ ] c1A"1- 1B
B* =
i
•
(5.1)
epAnp-lB
The first step is the same as in our algorithm, but the second step can not be achieved using this initialization. In the procedure by Wang, just one ~tegrator can be added on the inputs at each step. So Wang's algorithm requires exactly sup(hie- ni) steps, all using a Householder or an equivalent transformation. Rec~l that because of Remark 2.2, sup(n~c-n~)ffi n e - d l . We have been able to reduce this number of steps thanks to the knowledge of essential orders and to a well chosen shifting procedure. Remarks. The knowledge of essential orders does not require a lot of computations: they are given by the left kernel of F, [1]. All computations can be me:le on the field of real numbers, in opposition to solutions that require the computation of the interactor [6,13]. The algorithm provides directly the infinite zeros structure of the shifted system. Unfortunately, the one of the initial system can not be deduced from this information. The method for obtaining the finite zero structure is very close to Verghese's one [11]•
S. Icart, J.F. Lafay ,/Minimaiprecompensator for decor,piing
330
Appendix: Computation of Ill(S) Let us consider the following blocks of the Toeplitz matrix defined by (4.3) and (4.4):
.•d 1,r !
i +1 + 1
II ( 0 )
~*= i
!
1,m e
.~l,dl+l
iio; I
~l,d
.~d~ +1 " " "
rl ,
.~d*l+ 1 .-
•~rdi +1 l.ri+l
~/l,m
! +1
• . .
~3,d ! +
1
,,d,+~ To replace a by zero, it is enough to make the following operation: Let us denote a =/~.,,+1. {(r t + l~-th column of ¥~ + 1} becomes { ( - a ) x 1st column of ~ + (r a + 1)-th column of 3'~ + 1}. This operation just corresponds, for the transfer, to the precompensation: 1
0
...
-as -l
O
0
0
i ..°
n~(~-')
'..
=
1
(o)
0 I I
|m
O
,J
0
. . .
--
i
o
0 0
rI
The procedure will be the same for the .¢d1+1 with first 1 ¢ i ~< rl and afterwards rI + 1 ~
2
(0) .
+ I
.
.
.
'~3,d I + I
m -
rI
Operations between columns of ')'a, and Yd,+k correspond to a precompensation rIk(s-k). By thiS iterative procedure, "t~' (of the precompensated system) has the following form:
"
[+,,,, i - - - - 1
,
. . . .
~ >dl,
V
m--r 1
and the precompensator associated to this realization in the product of Ilk(s-k), k >i 1:
[_bi
"
+,(+) = L(o;,:-]:::
] •
It is clear by construction that H 1 is a bicausal matrix. Remark. We stress upon the order of operations: first treat Ya,+l and afterwards Ya,+2. Actually, the cancellation of Za~+ 1 (defined in (4.4)) implies modifications on the submatrix 3'a,+2. So, treat next Ya, + 2, and so on. In this way, the structures obtained in (4.3) and (4.5) are not modified. We will show that in the following example.
S. lcart, J.F. Lafay / Minimai precompensatorfor decoupling
331
Example. Let the A, B, C matrices of the system be
1 0
0
A=
(0)
c:[i
1
B=
9
0 0 0
(o)
(o)
0
9
(o) L
1 0
The structure of this system is n 1 = 1, n 2 -- 1,
n 3 =
2 (so dl = 1),
nl,--
n2e-
n3e -- n e =
3.
So the system is already shifted, and the useful part of the Toeplitz matrix is
F=
1
0
0
1 0 0 = 0 0 0 CB ] 0 0
0 0 3 2 1 0 0 0
0 0 0 1 -1 -1 0 1
(o)
CB
CAB
CB
CA2B
CAB
(o) 1
0
o~ 0 0 0
o
3 2 1
0 1 -1
Step 1. Y1 is already in the desired form (so, y - "h*).
Step 2.
v] =
=
°°]
Z2,1
(0)
[l-3s-1
,
v2 =
Ii 3 2
1
To cancel 3, we realize the following prec~mpensation: /'/](s-l)
=
1
,
0 which lets Y1 unchanged and brings Y2 and Y3 respectively to
[il i
°
Now, to cancel - 1, we realize the precompensation:
n~(s -~) =
1
o
0
1
,
which lets Y1 and ¥~ unchanged and brings y] to
~,1
_
[il ° 0
0
:] •
1
1 1 0
0 0 0
0 0 0
3 2
0 0
-1 0
1
0
1
0
1 -1
332
S. Icart, J.F. Lafay / Minimal precompensator for decoupfing
We resume now the procedure and obtain for the precompensator:
H(s)=
[1[0o
-1
½s-' + ½
0
½
1 •
The Toeplitz matrix associated to this precompensated system is (we only give the last strip as in (4.7)) 0 0 0
0 0 0
0 0 1
0 0 0
0 1 -1
0 0 0
1 1 0
0 0 0
0 0 0
[S:20 0]
Now by adding the precompensator R(s) defined in (4.8), R(s)=
s
O, 1
0
we obtain the final precompensator G(s):
[
s-2 38-2 - s-l-s -2]
o(s)=
o
-s-'
o
o
+½
.
½
The system precompensated by G(s) is decouplable via regular state feedback.
References [1] C. Commault, J. Descusse, J.M. Dion, J.F. Lafay and M. Malabre, About new decoupling invadants: the essential orders, lnternat. J. Control 44 (3) (1986) 689-700.
[2] C. Commault, J. Descusse, J.M. Dion, J.F. Lafay and M. Malabre, Influence de la structure h l'infini des syst/~meslin6alres sur la solution des probl~mes de commande, A.P.I.I. Commande des Systt,mes Complexes Technologiques 20 (3) (1986) 207-252.
[3] M. Cremer, A precompensator of minimal order for decoupling a finear multivariable system, Internat. J. Control 14 (6) (1971) 1089-1103.
[4] J. Descusse, J.F. Lafay and M. Malabre, A survey on Morgan's problem, Proc. of the 25th IEEE C.D.C., Vol. 2 (Dec. 1986) 1289-1294.
[s] J. Descusse and C.H. Moog, Dynamic decoupling for right-invertible nonlinear systems, Systems Control Left. 8 (4) (1987) 345-349.
[61 J.M. Dion and C. Commault, The minimal delay decoupling problem: feedback implementation with stability, S I A M J. Control Optim., to appear.
[7l P.L F~b and W.A. Wolovich, Decoupling in the design and synthesis of multivariable control systems, IEEE Trans. Automat. Control |2 (6) (1967) 651-669.
[8] A.S. Morse and W.M. Wonham, Status of non interacting control, IEEE Trans. Automat Control 16 (6) (1971) 568-581. [9] M.K.Sain and J.L. Massey, Invertibility of linear time invariant dynamical systems, IEEE Trans. Automat. Control. 14 (2) (1969) 141-149.
[io] LM. Silverman and A. Kitap~i, System structure at infimty, Outils et Mod~deles en Automatique, Analyse des Systdmes et Traitement du Signal, Vol. 3 (Editions du C.N.R.S., Pads, 1983).
[11] G.C. Verghese, Infinite frequehcy behaviour in generalized dynamical systems, Ph.D. Thesis, Stanford University (1978). [121 S.H. Wang, Design of precompensator for the decoupling problem, Electron. Lett. 6 (1970) 739-741. [13l T.W.C. Williams and P.J. Antsaldis, A unifying approach to the decoupling of linear multivadable systems, Internat. J. Control. 44 (1) (1986) 181-201.
1141 W.M. Wonham, Linear Muitivariable Control: A Geometric Approach, 2nd edition (Springer, Berlin-New York, 1979)
325
Design of a minimal precompensator for the decoupling problem S, I C A R T and J.F. L A F A Y Laboratoire d'Automatique, U.A. CN.R.$. 823, E.N.$.M., 1 rue de la I¢o~, 44072 Nantes 03, France
Received 25 November 1987 Revised 13 February 1987 Abstract: For fight invertible line,'u" time-invariant systems, which cannot be decoupled row by row using regular static state feedback laws, we propose a new algorithmic procedure for designing a minimal precompensator leading to a compensated system which can be decoupled by regular static state feedback. Keywords: Linear control systems, Decoupling, Structural approach, Precompensator.
1. Introduction The decoupling problem for linear time-invariant systems has been plentifully studied during the last 25 years. This paper concerns the row-by-row decoupling problem (named Morgan's problem) for linear systems by a dynamic precompensator. We propose here an easily implementable algorithm: this algorithm designs a precompensated system whick :esults in a new system that is decoupled, or can be decoupled by regular state feedback. In the sequel, we always consider right invertible systems (this condition is necessary for solving Morgan's problem). The precompensator to be obtained will be minimal in the sense that a minimal number of integrators is introduced when defining its dynamics. Similar algorithms have been published, for example by Wang [12] or Descusse and Moog [5] in the algebraic approach, and Dion and Commault [6] or Williams and Antsaldis [13] in the transfer matrix approach. The method proposed here is also algebraic, but requires some structural properties used for solving Morgan's problem by general static state feedback [4]. A major interest is that, in this way, we can greatly reduce the number of steps with respect to Wang's (or Descusse and Moog's) algorithm. Moreover, most of the computations are made on the field of real numbers, in opposition to Dion and Commault's or Williams and Antsaldis's solutions which are based on the inversion of a part of the transfer (e.g. the computation of the interactor). Before presenting the algorithm, we briefly recall the main elements of the structural approach we need in the sequel. Let us remark that, as we just consider dynamic precompensation, we need here less notions than in the solution of Morgan's problem by static state feedback [4].
2. Notations and basic concepts of the structural approach We shall consider, in the sequel, the right invertible linear time-invariant system (C, A, B) described by :c = A x + B u ,
y = Cx,
(2.~)
where x ~ ~ R", u ~ ~ - - R m, y E 0~,~ R P. The transfer matrix of this system, denoted T ( s ) , is defined by r ( s ) = C ( , I _ A)-~B. 0167-6911/88/$3.50 © 1988, Elsevier Science Publishers BN. (North-Holland)
(2.2)
S. Icart, J.F. Lafay / M, Mn: ~l precompensator for decoupling
326
We shall assume, without loss of generality, that B is monic and C is epic. The same symbol will be used to denote a map or any of its matrix represent~ ;ans in particular bases. Let Ir denote the (r, r) identity matrix. The i-th row of C will be written as ci. A set of p elements will be written as {.}p. The number of elements of a given list {. } is denoted card{. }. We ,~ill need in our approach the Toeplitz matrix Fn and its sub-matrices F~ defined by
CAB
cB
CB
CA ~'- 1B
CA ~- 2B
(o)1 .. •
(2.3)
CB
The importance of this operator has already been mentioned by Sain and Massey [9] for invertibility problems, and Silverman and Kitap~ji [10] for a characterization of the infinite zero structure. This operator can also be used to efficiently compute the basic geometric algorithms [2,14]. For designing our precompensator, we only need two different families of structural invariants: the row-by-row infinite zero structure (R.I.Z.S.), and the essential orders of the outputs. Let us ~nly recall their definition and main properties. All the properties are detailed in the referenced papers. The row-by-row infinite zero structure of (C, A, B) [2]: Let ne be defined by
n,=inf(l~NlciA~-lB,O}
V i ~ {1, 2 , . . . , p } .
(2.4)
The R.I.Z.S. of the right invertible system (C, A, B) is the list { n~ } which contains p finite integers• The essential orders of the outputs of (C, A, B) [1]: The notion of essential order comes from the concept of essential row in a matrix due to Cremer [3]. A row is essential in a matrix W if it cannot be expressed as a linear combination of the other rows of W. The essential order of the i-th output of a right invertible system, denoted nie, is defined as follows• Let
~i, = [ C'a~l 1e~''" ~c,B ~l'l (0) Ill 11
(2.5)
hie
(2.6)
Then =
inf{/~ ¢ N I ~,~, is essential in F, }.
For fight invertible systems, the list { n~e } has p finite terms and satisfies
n,c>~n,
Vie {1,2,...,p}.
(2.7)
Two important properties of these lists are given in the following remarks [1]: Remark 2.1. Let us denote h e = sup(n~e). The right invertibility of (C, A, B) is equivalent to the fact that the last p ,o'vs of rne are essential ;.n r n. Thus we just have to consider Fne iastead or rn in our study of (C, A, B). Remark 2.2. The two lists { n~ }p and { n ~e}p are not modified by bicausal operations on the transfer matrix T(s).
3. Structural approach of the decoupling problem Falb ad Wolovich's [7] or Morse and Wonham's [8] results on necessary and sufficient conditions for the existence of regular static state feedback solutions can be expressed in structural terms by several equivalent conditions [1]. The one we will use here is the following: Theorem 3.1. There exists a regular solution to Morgan's problem if and only if
ni=nie
Vi~{1,2,...,p}.
(3.1)
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327
So, when this condition is not satisfied and because of the assumed right invertibility property, there always exists a dynamic precompensator [12,6] which will have as a result to modify the structure of the initial system (C, A, B) in order to obtain a compensated system for which (3.1) is verified. Such a precompensator modifies the structure as follows: (1) Infinite zeros and essential orders can only be increased. (2) If any control law decouples the system (C, A, B) row by row, then the orders 8i of the row infinite zeros of the decoupled system satisfy
8i>~nie Vi~ {1,2,...,p}.
(3.2)
Morgan's static problem has been fully solved by Descusse, Lafay and Malabre [4l, and dynamic solutions have been given by Dion and Commault [6]. One very important fact is the following: among all the possible decoupling solutions, there is at least one for which essential orders are not increased: such solutions are called 'minimal'. To achieve their result, Descusse~ Lafay and Malabre [4] had to modify the initial system by a technical procedure named 'shifting procedure'. We will also use this procedure, and this will allow us to reduce the number of steps with respect to the other algorithms. Let us briefly describe this shifting procedure: Let (C, A, B) be the initial system, and n e = sup{n~e }. The shifted system, denoted (C, A, B), is such that the essential orders of the outputs are all equal to n e. This transformation is always possible by addition of ne - n~e integrators on each output. From the Toeplitz matrices associated with (C, A, B) and (C, A, B) it is clear that, for each outpuL the difference n~e - n~ is not modified by this transformation. Note that the transfer matrix of the shifted system (C, A, B) is directly obtained from the one of (C, A, B) by a left multiplicatiox, by a diagonal matrix: indeed, let { n ie }p be the essential orders of (C, A, B), neff sup{n~e} and fl~ = n ~ - n~e. Let T(s) be the transfer matrix of (C, A, B) and T(s) that of (C, A, B). We then have:
T(s) = Diag{ s -#' } T(s).
(3.3)
Both systems are equivalent with respect to the decou_plingproblem: in fact,let G(s) be a precompensator which decouples (C, A, B): the transfer matrix T(s). G(s) is diagonal, and T(s). G(s) too. Thus G (s) decouples (C, A, B). This shiftingprocedure is a key point of the algorithm we present now.
4. Design of a minimal preeompensator
Suppose here that the essential orders of the outputs of the right invertible system (C, A, B) are all equal to ne (the last remark proved that this assumption is not restrictive). The proposed algorithm amounts to compute a dynamic precompensator able to increase the R.I.Z.S. {ni}p of (C, A, B) up to { ne }p without increasing the essential orders which is always possible because (C, A, B) is right invertible [4]• Thus the compensated system is decouplable with regular static feedback. Let the integer d~ be defined by
d, = inf{
(4.1)
N ICA '- B * O} = inf{ n, } p.
Consider now the useful part of the Toeplitz matrix related to (C, A, B), say
IICAd,1B _
CAa~-~B .
L CA ne- 1B
CAne- 2B
F-[y~] = [ CA:IB
where y~- CA~-IB.
"'"
(0) 1 CAdl- 1B J
(4.2)
S. Icart, J.F. Lafay / Minimal precompensator for decoupling
328
Step 1. Let r 1 = rank( CAd'- lB ). Then r 1 < p (otherwise d 1 = n~ and (C, A, B) is decouplable via regular static feedback). It is always possible to find linear transformations L~ and P~ (using Householder's method) such that
(o) (o)
|rl
Y~ = L1Y1P1 =
~2,d~
v
(4.3)
Y m
L~ just corresponds to a renumbering of the outputs without any combination. Let us denote by ~.*,/~ >i d~, the other blocks of the Toeplitz matrix of the system modified by L 1 and PI"
~:,.' I', z~ }r,
y: = L,y,P, = ~,~-l---h-_, I~3,"
, P,> d1.
(4.4)
Step 2. According to (3.1), the final precompensator has to increase the R.I.Z.S. up to { n e }p. To do so, we need i-1 banks of (ne - d~) integrators, one connected with each of the r~ first (new) inputs of (4.3). They also allow to cancel each block Z~, of ~* (# > d~), thanks to the same rl inputs. This cancellation is achieved with an intermediate precompensator denoted H~(s). The computation of H~(s) is described and illustrated in an example in the Appendix. So, the intermediate system, precompensated by H~(s), has as Toeplitz matrix blocks:
Y"" =
i
L---" .. ] Z2,., . ..~3,~ /'/I
--
,
/~>d,.
(4.5)
r!
IIl(s) is a bicausal matrix of the form
[|,,', • ]
(4.6)
Since L 1 and P1 are regular and Hl(s ) is bicausal, the essential orders and the R.I.Z.S. of the system are not modified by steps 1 and 2 (L~ ":~a permutation matrix). We can now repeat the above procedure (steps 1 and 2) in the following way: Let d2 = inf{# ~ N IZ3,~ ~ 0}; the properties of essential orders yield that d2 ~
Ii:,,°,! I. i,°,i,°,, F;::T
+
")'he-1
II"i(°) :T
(4.7)
S. lcart, J.F. Lafay / Minimaiprecompensatorfor decoupling
329
Remark. Note
that all the transformations performed up to now are regular transformations which never increase the infinite zero structure or the essential orders. Step 3. Define now the precompensator R(s):
R(s)-Diag{s-(",-d~)I,,}.
(4.8)
This precompensator consists in the addition of integrators on the inputs, and allows to bring the R.I.Z.S. up to ne. We finally obtain the system precompensated by G(s), with G(s)- P~HI(s)... PvIIv(s)R(s). Its Toeplitz matrix has all strips equal to zero, except the last, which is
1
i
:(o)
•
".
•
*
1
•
(0)
-.-].
(4.9)
"
m-p
This system is really d~,~ouplable via regular static state feedback. It is clear that this precompensator is minhnal because the essential orders are not modified.
5. Comparison with other algorithms The above algorithm can be directly comp~ed with Wang [12] or Descusse and Moog's one [5]. (The latter is an extension to the non-linear case of the first.) The initialization of these algorithms is the matrix B*, defined by Falb and Wolovich in [7]:
[ ] c1A"1- 1B
B* =
i
•
(5.1)
epAnp-lB
The first step is the same as in our algorithm, but the second step can not be achieved using this initialization. In the procedure by Wang, just one ~tegrator can be added on the inputs at each step. So Wang's algorithm requires exactly sup(hie- ni) steps, all using a Householder or an equivalent transformation. Rec~l that because of Remark 2.2, sup(n~c-n~)ffi n e - d l . We have been able to reduce this number of steps thanks to the knowledge of essential orders and to a well chosen shifting procedure. Remarks. The knowledge of essential orders does not require a lot of computations: they are given by the left kernel of F, [1]. All computations can be me:le on the field of real numbers, in opposition to solutions that require the computation of the interactor [6,13]. The algorithm provides directly the infinite zeros structure of the shifted system. Unfortunately, the one of the initial system can not be deduced from this information. The method for obtaining the finite zero structure is very close to Verghese's one [11]•
S. Icart, J.F. Lafay ,/Minimaiprecompensator for decor,piing
330
Appendix: Computation of Ill(S) Let us consider the following blocks of the Toeplitz matrix defined by (4.3) and (4.4):
.•d 1,r !
i +1 + 1
II ( 0 )
~*= i
!
1,m e
.~l,dl+l
iio; I
~l,d
.~d~ +1 " " "
rl ,
.~d*l+ 1 .-
•~rdi +1 l.ri+l
~/l,m
! +1
• . .
~3,d ! +
1
,,d,+~ To replace a by zero, it is enough to make the following operation: Let us denote a =/~.,,+1. {(r t + l~-th column of ¥~ + 1} becomes { ( - a ) x 1st column of ~ + (r a + 1)-th column of 3'~ + 1}. This operation just corresponds, for the transfer, to the precompensation: 1
0
...
-as -l
O
0
0
i ..°
n~(~-')
'..
=
1
(o)
0 I I
|m
O
,J
0
. . .
--
i
o
0 0
rI
The procedure will be the same for the .¢d1+1 with first 1 ¢ i ~< rl and afterwards rI + 1 ~
2
(0) .
+ I
.
.
.
'~3,d I + I
m -
rI
Operations between columns of ')'a, and Yd,+k correspond to a precompensation rIk(s-k). By thiS iterative procedure, "t~' (of the precompensated system) has the following form:
"
[+,,,, i - - - - 1
,
. . . .
~ >dl,
V
m--r 1
and the precompensator associated to this realization in the product of Ilk(s-k), k >i 1:
[_bi
"
+,(+) = L(o;,:-]:::
] •
It is clear by construction that H 1 is a bicausal matrix. Remark. We stress upon the order of operations: first treat Ya,+l and afterwards Ya,+2. Actually, the cancellation of Za~+ 1 (defined in (4.4)) implies modifications on the submatrix 3'a,+2. So, treat next Ya, + 2, and so on. In this way, the structures obtained in (4.3) and (4.5) are not modified. We will show that in the following example.
S. lcart, J.F. Lafay / Minimai precompensatorfor decoupling
331
Example. Let the A, B, C matrices of the system be
1 0
0
A=
(0)
c:[i
1
B=
9
0 0 0
(o)
(o)
0
9
(o) L
1 0
The structure of this system is n 1 = 1, n 2 -- 1,
n 3 =
2 (so dl = 1),
nl,--
n2e-
n3e -- n e =
3.
So the system is already shifted, and the useful part of the Toeplitz matrix is
F=
1
0
0
1 0 0 = 0 0 0 CB ] 0 0
0 0 3 2 1 0 0 0
0 0 0 1 -1 -1 0 1
(o)
CB
CAB
CB
CA2B
CAB
(o) 1
0
o~ 0 0 0
o
3 2 1
0 1 -1
Step 1. Y1 is already in the desired form (so, y - "h*).
Step 2.
v] =
=
°°]
Z2,1
(0)
[l-3s-1
,
v2 =
Ii 3 2
1
To cancel 3, we realize the following prec~mpensation: /'/](s-l)
=
1
,
0 which lets Y1 unchanged and brings Y2 and Y3 respectively to
[il i
°
Now, to cancel - 1, we realize the precompensation:
n~(s -~) =
1
o
0
1
,
which lets Y1 and ¥~ unchanged and brings y] to
~,1
_
[il ° 0
0
:] •
1
1 1 0
0 0 0
0 0 0
3 2
0 0
-1 0
1
0
1
0
1 -1
332
S. Icart, J.F. Lafay / Minimal precompensator for decoupfing
We resume now the procedure and obtain for the precompensator:
H(s)=
[1[0o
-1
½s-' + ½
0
½
1 •
The Toeplitz matrix associated to this precompensated system is (we only give the last strip as in (4.7)) 0 0 0
0 0 0
0 0 1
0 0 0
0 1 -1
0 0 0
1 1 0
0 0 0
0 0 0
[S:20 0]
Now by adding the precompensator R(s) defined in (4.8), R(s)=
s
O, 1
0
we obtain the final precompensator G(s):
[
s-2 38-2 - s-l-s -2]
o(s)=
o
-s-'
o
o
+½
.
½
The system precompensated by G(s) is decouplable via regular state feedback.
References [1] C. Commault, J. Descusse, J.M. Dion, J.F. Lafay and M. Malabre, About new decoupling invadants: the essential orders, lnternat. J. Control 44 (3) (1986) 689-700.
[2] C. Commault, J. Descusse, J.M. Dion, J.F. Lafay and M. Malabre, Influence de la structure h l'infini des syst/~meslin6alres sur la solution des probl~mes de commande, A.P.I.I. Commande des Systt,mes Complexes Technologiques 20 (3) (1986) 207-252.
[3] M. Cremer, A precompensator of minimal order for decoupling a finear multivariable system, Internat. J. Control 14 (6) (1971) 1089-1103.
[4] J. Descusse, J.F. Lafay and M. Malabre, A survey on Morgan's problem, Proc. of the 25th IEEE C.D.C., Vol. 2 (Dec. 1986) 1289-1294.
[s] J. Descusse and C.H. Moog, Dynamic decoupling for right-invertible nonlinear systems, Systems Control Left. 8 (4) (1987) 345-349.
[61 J.M. Dion and C. Commault, The minimal delay decoupling problem: feedback implementation with stability, S I A M J. Control Optim., to appear.
[7l P.L F~b and W.A. Wolovich, Decoupling in the design and synthesis of multivariable control systems, IEEE Trans. Automat. Control |2 (6) (1967) 651-669.
[8] A.S. Morse and W.M. Wonham, Status of non interacting control, IEEE Trans. Automat Control 16 (6) (1971) 568-581. [9] M.K.Sain and J.L. Massey, Invertibility of linear time invariant dynamical systems, IEEE Trans. Automat. Control. 14 (2) (1969) 141-149.
[io] LM. Silverman and A. Kitap~i, System structure at infimty, Outils et Mod~deles en Automatique, Analyse des Systdmes et Traitement du Signal, Vol. 3 (Editions du C.N.R.S., Pads, 1983).
[11] G.C. Verghese, Infinite frequehcy behaviour in generalized dynamical systems, Ph.D. Thesis, Stanford University (1978). [121 S.H. Wang, Design of precompensator for the decoupling problem, Electron. Lett. 6 (1970) 739-741. [13l T.W.C. Williams and P.J. Antsaldis, A unifying approach to the decoupling of linear multivadable systems, Internat. J. Control. 44 (1) (1986) 181-201.
1141 W.M. Wonham, Linear Muitivariable Control: A Geometric Approach, 2nd edition (Springer, Berlin-New York, 1979)