Decoupling and pole assignment of singular systems: A frequency domain approach

Auromafica. Vol. 33, No. 8, pp. 1555-1560. 1997 1997 Elswier Science Ltd. All rights resewed Printed in Great Britain om5-lo!%/97 $17.00

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PII: sooo5-1098(97)ooo14-5

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Brief Paper

Decoupling and Pole Assignment of Singular Systems: A Frequency Domain Approach* D. VAFIADIST Key Word-singular

and N. KARCANIASt

systems; state feedback; decoupling; minimal bases.

sufficient condition for the solvability of the probIem. The condition involves only the numerator matrix of an appropriate coprime and column reduced MFD of the TF of the system and is much simpler than the conditions obtained by Ailon (1991) Minamide er al. (1991) and Paraskevopoulos and Koumboulis (1991) since the only transformation we need in order to test it is the reduction of a given MFD of the system to an appropriate coprime and column reduced form. Another generalization obtained in the present paper is the characterization of the fixed poles of the decoupled system in terms of the numerator matrix of an MFD of the original system. It is shown that the fixed poles of the system are characterized in exactly the same way as was done by Koussiouris (1980) for the case of state-space systems. The proof of the above suggests a way of choosing feedback for arbitrary placement of the assignable poles. The structure of the paper is as follows: In Section 2 the problem is formulated in the frequency domain and the main existing results for state-space systems are briefly discussed. In Section 3 the effect of state feedback and regular input transformation on the MFD of the TF of a singular system is studied. This is used in Section 4, where the necessary and sufficient solvability condition and a constructive way of choosing the control law are obtained. Finally, in Section 5 the issue of pole placement is considered, and the characterization of the fixed poles is established. Two examples are given in order to illustrate the approach.

Abstrati-A new approach to the row by row decoupling of singular systems via state feedback and regular input transformation is presented. The proposed method is based on the matrix fraction description of the system and is an extension of the frequency domain method for strictly proper systems. The solvability condition of the problem is readily obtained from the ‘numerator’ matrix of a coprime and column reduced matrix fraction description of the given system. Pole placement and decoupling is also considered, and the characterization of the fixed poles is given. 01997 Elsevier Science Ltd. 1. Introduction Input-output decoupling of square systems has been an attractive problem for researchers during the last three decades. Falb and Wolovich (1967) first solved the problem for square state-space systems, and since then many papers have appeared, giving characterizations of invariants (Commault et al., 1986), structural properties (Descusse and Dion, 1982), and covering different aspects of the problem, such as block decoupling (Wonham and Morse, 1970). Although the original treatment of the problem was done in the time domain, frequency domain methods also appeared (Koussiouris, 1979). The latter methods are based on the close relation of the matrix fraction descriptions (MFDs) of transfer functions (TFs) and their realizations in state-space form (Wolovich, 1973). The advantage of these methods over the state-space approaches is that they are much simpler. On the other hand, since decoupling is a problem concerning TFs, it is more natural to work in the frequency domain. The problem of decoupling by state feedback and regular input transformation was extended recently to the case of singular or generalized state-space systems in (Ailon, 1991; Minamide et al., 1991; Paraskevopoulos and Koumboulis, 1991). These papers use only generalized state-space tools for the solution of the problem. The aim of the present paper is to give a solution to the row by row decoupling of singular systems by state feedback in the frequency domain, and thus to generalize the frequency domain results for state-space systems (proper systems) to the case of singular (nonproper) systems. The approach used here follows similar lines to the one used by Koussiouris (1979) and leads to a simple necessary and

2. Statement of the problem and preliminaries Consider the singular system Ei=Ax+Bu,

y = Cx,

(1)

where E may be singular, with n states, I inputs and I outputs. The problem of decoupling is the problem of finding a control law of the type u = Fx + Gv,

det (G) # 0

(2)

such that the closed TF is diagonal. System (1) has the TF H(s), which is, in general, non-proper. If H(s) is written in MFD form i.e. H(s) =N(s)D-l(s), then the problem of decoupling may be formulated as follows (Koussiouris, 1979). For a system described by an MFD H(s) = N(s)D-i(s) having a realization of the type (l), find a control law of the type (2) such that the TF of the closed loop system H,(s) = N,(s)D;‘(s) is diagonal, i.e.

*Received 7 February 1996; revised 25 September 1996; received in final form 17 March 1997. A short version of this paper was presented at the 13th IFAC World Congress, which was held in San Francisco, U.S.A., during 30 June-5 July 1996. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 lGB, U.K. This paper was recommended for publication in revised form by Associate Editor F. Delebecque under the direction of Editor Ruth F. Curtain. Corresponding author Professor N. Karcanias. Tel. +44 (0) 171477 8133; Fax +44 (0) 171 477 85678; E-mail [email protected]. t Control engineering Centre, City University, London EClV OHB, U.K.

H,(s) = N,(s)D;‘(s)

= diag {h,(s)}

or

(3) N,(s) = diag {hi(s)}&(s),

where hi(s) are rational scalars. The above yields that H,(s) is decoupled if, and only if, n:(s) = h,(s)&(s), where n:(s), respectively. the TF for closed loop 1555

(4)

#L(s) denote the ith row of N,(s) and D=(s), In the case of singular systems, the existence of the closed loop system is not guaranteed. The system can have a TF if, and only if, F is such

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that sE -A - BF is invertible. Thus, the solvability of the problem implies the existence of F such that (3) is satisfied and sE -A - BF is invertible. Throughout the paper it will be assumed that the system (1) is reachable, i.e. [sE - A,B] has no finite zeros and [E, B] is epic. The reachability assumption does not affect the generality, since the unreachable part of the system does not contribute to the TF and reachability is not affected by control of the type (2). Furthermore, it will be assumed that the pencil [sET AT, PIT has neither finite nor infinite zeros. This assumption is not restrictive because finite and infinite poles can be shifted to observable locations by state feedback, because the system is reachable. For the two classes of strictly proper and non-proper systems, it is well known that MFDs of the form H(s) = N(s)D-‘(s) can be realized as state-space and singular systems respectively. For strictly proper systems it is known (Wolovich, 1973) that if the triplet (A,B,C) is a controllable realization of the column reduced MFD (N(s), D(s)), then application of control law (2) does not affect the numerator matrix N(s), i.e. N,(s) = N(s) and the denominator has the form D,(s) = G_‘[D(S) - V(s)],

(5)

where &,(V(s)) < g,, and a, denotes the ith column degree and ai the ith controllability index (CI). Note that in the case of state-space systems we have ui = a,@(s)). In the following section we investigate the effect of the control law (2) on the numerator and denominator matrices of a MFD corresponding to the TF of (1). The solution of the problem is based on the results of this section. The following result gives the solvability conditions of the decoupling problem for strictly proper systems when [NT(s), DT(s)]’ is a minimal basis (Forney, 1975). Theorem

2.1. (Koussiouris, 1979). Let o, denote the CI of state-space system with the strictly proper TF H(s) = N(slDm’(sl and u = max la.}. Consider the matrix N,‘(s) = n;(J) diag is”-“1) i = 1, 1,‘i with N, the row high order coefficient matrix of N,(s). Then the system is decoupable if, and only if, N, is invertible.

the

3. The MFD of the closed loop system In this section the effect of control law (2) is considered. The results here are similar to the classical results for strictly proper systems. The system matrix of (1) may be transformed by restricted system equivalence (RSE) transformations (P, Q) to the form (Vatiadis and Karcanias, 1995)

r ~6) I 0

1

where L(s) = block-diag {L,(s)} and L,(s) = s[l,,, O,,,] Ex1, I,]. Then the input-state TF of (6) is H,(s) = where S(s) = block-diag &SEA)-‘B = S(s)D-l(s), {[l, s,. . . , s’l-’ IT) and r, = ei + 1 are the reachability indices (RIs) of the triplet (E,A,B). From (6) it is clear that D(s) = [SK - A]S(s),

N(s) = U(s).

(7)

Now, applying law (2) to the system with system matrix as in (6) we get D,(s) = G-‘[(SK - h)S(s) - FS(s)]. (8) The numerator matrix of the input-state TF after the transformation (6) is S(s) = Q-‘N,,(s), where N,Js) is the numerator of the input-state transfer function before transformation (6), and thus N,(s) = CQS(s) = CQQ~‘N;.Js) = N(s),

(9)

which means that the numerator is not affected by control law (2). Thus, the problem of decoupling may be viewed as the problem of modifying the denominator matrix of the MFD as in (8) such that (4) holds true. Taking into account that a CI ai of the triplet (E,A,B) is equal to the RI r, if a, is

proper and oi = r, - 1 if ci is non-proper (Karcanias and Eliopoulou, 1990; Malabre et al., l!BO), we have the following: Proposition 3.1. Let Q(s)

denote the denominator the closed loop TF of the system (1) under the law D,(s) = G-*[D(s) + V(s)], where V(s) = [_vI(s), . . a polynomial matrix determined from the feedback deg vi(s) 5

ui - 1 if ui is proper, if u, is non-proper.

cr,

An obvious consequence Corollary

matrix of (2). Then , y,(s)] is F with (IO)

of the above is the following:

3.1. When

all the controllability indices are the system is decouplable, since the equation D(s) + FS(s) = N(s) has always a solution with respect to F. 0

non-proper,

4. The solution of the problem In this section the necessary and sufficient condition for the solvability of the decoupling problem is obtained and the construction of the feedback pairs (F, G) which decouple the system is described. The following notation will be used for the rows of N(s) and D(s): n’(s) = [n{(s), . . . , n:(s)] and g(s) = [d’,(s), . , dxs)], respectively. Throughout the rest of the paper it will be assumed that the system matrix is as in (6). Note that this form can be obtained as a realization of a normalized MFD of the TF from (7) without resorting to complicated RSE transformations. A notion which is important for the solution of the problem is that of pivot indices (PIs) of a minimal basis (Forney, 1975). The PIs are invariants of the minimal bases of a rational vector space. In the present case the minima1 matrix T(s) = basis consideration is the under [NT(s), DT(s)]‘. Next we recall the definition of PI. Dejinition 4.1. (Forney,

1975). Let M(s) be a minimal basis with ordered column degrees c, I-C* I.. . SC,. The PIs of M(s) are obtained by the following procedure. Let n, be the number of columns of degree c,. Find the first (lowest index) ni rows of the high order coefficient matrix of M(s), such that the n, X ni submatrix so defined is non-singular. The indices of these rows are the n, pivot indices of M(s). Delete these n, columns and rows from M(s) and repeat the above 0 for all distinct values of c,. The following definition classifies the PIs into two types. Definition 4.2. Let [NT(s), DT(s)IT. Then non-proper if 9i 5 1.

9,) . . ,9, denote 9; is called proper

the PI of if 9i > I and 0

Throughout the rest of the paper it will be assumed that the given system has r non-proper PIs and $ non-proper CIs. Next we define some sets of integers which are related to a singular system and its normalized MFD T(s). Definition 4.3. The integers p,, pi, pi, p,, q,, (p, and 6; are

defined as follows: (i) pi are the column indices of T(s) corresponding PIs 9p, are proper.

such that

the

(ii) j& are the column indices of T(s) corresponding PIs 9p, are non-proper.

such that

the

(iii) pi are the column indices of T(s) such that corresponding CIs u?, are non-proper and corresponding PIs 9p, are proper. (iv) pi are the column indices of T(s) such that corresponding CIs U~, are proper.

the the the

(v) cpi(i = 1,. ,I - 7) are the indices of the rows of N(s) not containing a pivot element of T(s). (vi) Cpi= %, - 1, where a,,, is a non-proper

CI.

(vii) & = qp, - I, where a,,, is a proper CI. Note that the set of q, is the union of the sets of & and $,.

0

1557

Brief Papers Lemma 4.1. Let (N(s), D(s)) be a coprime and column reduced MFD of H(s). There always exists a unimodular matrix U(s) such that the matrix T(s)U(s) = [fiT(s), bT(s)lT = t(s) is column reduced and has the following structure. IfJ& d:J’ is the column high order coefficient matrix of T(s), then all the entries in the pith column of Nhc (qp, is a proper PI) are zero. Minimal bases with the above structure will be referred to as normalized 0 minimal bases. Proof: Vafiadis and Karcanias (1995) have shown that the echelon form (Forney, 1975) [NT&), of,,(s)]’ of [Nr(s), DT(s)lT has the above property. Other bases with this property may be obtained by post-multiplication of [N’&,(s), D:&s)IT by appropriately structured unimodular 0 matrices (Wolovich, 1974). Remark 4.1. From (8) it follows that application of a regular

input transformation G may result in’a change in the values of the proper PI 9,,!. For example, if G-’ is a permutation matrix interchanging rows of D(s) containing pivot elements, then the proper PIs are changing accordingly. The non-proper PI and CI remain invariant. q

column reduced composite matrices corresponding to the same transfer function. Then they induce the same cl decoupling indices. Proof. Since T(s) and T’(s) are minimal bases of the same rational vector space, they are related by structured unimodular transformations (Wolovich, 1973) as follows: T(s) = T’(s)U(s). Let a(s) = ni(s) diag (s-7) and g:(s) = $(s)U(s) diag(s-“I). Next it is shown that they have the same zero structure at infinity. Consider the vectors ~,(l/w) =&l/w) diag (w-i) and &(1/w) = $(l/w)Ll(l/w) diag(w”/). From the structure of U(s) it follows that the entries of U(w) =_U(l/w)diag(w?) are homogeneous polynomials and &,U(w) = a,. Since the diagonal blocks of U(s) are constant and inve$ible, there exists a unimodular matrix R(w) such that U(w)R(w) = diag (w’q). Thus a(l/w) = a#/w)R(w). Since R(w) is unimodular we have that g(l/w) and #(l/w) have the same Smith-MacMillan form, and thus they have the same zero structure at w = 0, i.e. si(s) and g;(s) have the same infinite zero structure. Now, from the definition of fy,- it follows that T(s) and T’(s) induce the same f,,?. The invariance off, readily follows from the fact that T(s) and T’(s) are related by unimodular transformations. 0

The following result is obvious and is given without proof. Lemma

4.2. Let

a(s) = [(I,(S), . . . , q(s)] and /3(s) = [p,(s), . . , pk(s)] be polynomial vectors such that a(s) = [p(s)/9(s)]p(s), where p(s) and 9(s) are relatively prime polynomials. Then: (i) 9(s) is a common divisor of and (ii) deg {q(s)} - deg {pi(s)} = B,(s), “. 7A(S); 0 deg {p(s)} - deg {9(s)}, Vi E 1,. . , k. Let T’(s) denote the composite matrix of the closed loop system, i.e. T,(s) = [NT(s), D:(s)]~. If T(s) is column reduced, then T,(s) will be also column reduced. The following result characterizes the possible linear dependences of the rows of N(s) and D(s) of a decoupled system.

Consider now the matrices N,(s) and N, of Theorem 2.1, where now ai are the controllability indices of the singular system (1). Denote by ng the entries of N, and by NP, the (I- 7) X (1 - +) matrix:

Next, we define a system obtained from T(s):

I-

1,.

,I -

7).

s a,,,. Then, since the system is from Lemma 4.2 that degdz(s) - deg rig(s)) 2 1, which conthat 9i, is a non-proper PI of T(s), 0

The above result has the following important interpretation. Corollary 4.1. When the system is decoupled and its TF is expressed as the column reduced MFD, G(s) = N(s)D-r(s), a row of N(s) containing a non-proper pivot element of T(s) cannot be linearly dependent (parallel) on a row of D(s) containing a proper pivot element of T(s). Thus, when a system is decoupled, the rows of N(s) containing pivot elements are generated by rows (in the sense of (4)) of D(s), which do not contain pivot elements of T(s). 0

Another

set of integers related to the system comprises the decoupling indices, which are defined below. Definition 4.4. The proper and non-proper decoupling indices, fq, and fy,-, respectively, are defined as follows: l

l

fq, are the orders of the infinite zeros of the rational vectors n’+(s) diag(s-“I), j = 1,. , 1. f,+ are the degrees of the greatest common divisors of the

entries of #5,(s).

cl

The decoupling indices are common to all column reduced MFD representations of a TF, as shown below. Proposition

..

T*(s) = d;;(s)

0

Proof. Let degdz(s) decoupled, it follows deg nz(s) = deg dz(s) tradicts the assumption and the result follows.

4.2. Let

T(s) and T’(s) be two coprime

and

1

matrix form,

n::(s)

n?,-*(s)

Proposition 4.1. Let T(s) = [NT(s), D’(s)lT be the normalized composite matrix of a diagonal TF G(s) = N(s)D-l(s). Then deg dz(s) < (+Pt (i = 1I..., r; j=l,...,l; k=

in composite

d?,-‘*(s)

r&$(s)

[ 1 d&:_,(s) =

N*(s) D*(s)

.

(11)

T*(s) is obtained from T(s) by taking out the columns and rows containing non-proper pivot elements, the columns corresponding to non-proper CIs and proper PIs, the rows of D(s) and N(s) corresponding to those proper PIs 9,,. for which u,,,,are non-proper CIs, and the rows of D(s) which do not contam pivot elements. Definition 4.5. Let [NL DL]’ be the column high order coefficient matrix of INS. DhlqT. We define the matrix D,, as the submatrix ‘of ‘D,‘, ’ formed by the columns @r.. . ,jT_, and the rows cp,, , q,-,. 0

We proceed now to the main result. Theorem 4.1. Let [NT(s), DT(s)lT be a normalized MPD of (1). Necessary and sufficient condition for the solvability of the decoupling problem is that rank Nz = 1 - I+?. cl Proof

Without loss of generality, we assume that the system is in the form (6) and the corresponding MFD is obtained by (7). Let (1) be decouplable. Then, there exists a state feedback F and a regular input transformation G such that n’(s) =$(s)/&(s), where D=(s) = [$(s)~, . . . , $(s)‘7’. Since the composite matrix T(s) of the original system is normalized, T*(s) has column high order coefficient matrix where det D& #O. Since the system ]O~/-+~x+,), (~iFJTIT~ is decoupled, it follows that n@{(s)= 8(s)h+,(s) and thus n,+(s) = &+(s)h,(s), where n:(s) and- a:(s) are the ith rows of N*(s) and D*(s), respectively. Clearly, the TF

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G*(s) = N*(s)(D*(s))-’

is strictly proper.

Let arj be the

maximal proper CI of (1). From the solvability condition of the decoupling problem for state-space systems, it follows that N*(s)diag(s”P%) is row reduced and thus (s”-“fi)N*(s) diag (s”6p^-o~,) is also row reduced. The non-singularity of D& yields that there exists k such that deg d$(s) = u$~. Then from Lemma 4.2 it follows that and, since deg r_l;(s) - deg a;(r) = deg n$$s) - ofiI, deg $$(r) 5 a~~, we have that deg n;(s) + (a - ~$5 degn>k(s) + (u-u~J and thus deg Q “-“M;;(s)} 5 deg{s”-“Fkn2k(s)}. Similarly, it can be shown that above The deg {s“-“tin;(s)} 5 deg {s0-rr$tn2k(s)}. mean that the row degrees of the matrices

n+qs) 1 diag (s”-“I),

deg p+,(s) - deg v+(s) = u6, - deg n& = f+,.

(16)

Furthermore, we have that deg n?(s) < oj, because otherwise the (p, th row of N(s) would contain pivot elements. Thus f+;= min, {aj - deg n,?(s)} and (13) is solvable only if deg P+(S) - deg ~6)

'f+,.

(17)

Summarizing, we have that if P(s) is chosen such that

N*(s) diag (sl’Kur,)

n‘+ ‘( s) [_-I

are equal. Let &’ be the row high order coefficient of N*(s) diag (s Vfi-‘%). The necessity follows from the fact that fi< is a submatrix of N:. For the proof of sufficiency a regular feedback pair yielding diagonal closed loop TF has to be constructed. Since rank N”, = I- I& we can choose a non-singular (I - $) x (I - $) submatrix of NP,. Call this matrix NE. Clearly, fi: is a submatrix of N,. Let NE be formed by the intersection of the cpk,, . , (P+, rows and pi, , PI_+ columns of N,. Then choose a permutation matrix G, such that gd, = (pk,+ 1. Let D,(s) = G;‘D(s). For the system (N(s), D,(S)) we have (pk = Gi where 8, refers to (N(s), D,(s)). The matrix [(N*(s))~, (D~(s))~]’ formed from [NT(s), D:(s)]’ as in (11) has column high order coefficient matrix [Ot,_,,,t,_,,, (DQTIT with Dfh, invertible. The next step is to use an input transformation Gz on the system (N(s), D,(s)) such that D*(s) = G;‘D,(s) has the following structure. All the entries with coordinates (?,+p^,) have degree less than oB,. This is always possible smce Dh is invertible. Note that D* -D* Consider now the matrix G; = D$,JfiE)-‘. Clear;:,-thi?matrix is invertible, since fiP, is invertible and (G;)-‘D& = AZ. This means that we can use an invertible input transformation G3 such that the matrix D3(s) = G;‘DZ(s) has corresponding Qa, equal to &P, and D3 = NP, (see Definition 4.5). Since NP, is invertible, there lxists k E 11,. , I - JI} such that n$,,I,,#O. The following may be easily verified: ufilir- deg n$$s) = f+,,

where n$ ,Bk# 0.

(12)

It is shown next that the system (N(s), D3(s)) is decouplable by state feedback only, or equivalently the following equation has a solution with respect to &S(s): D3(s) - &S(S) = P(s)N(s)

g W(s),

(13)

where P(s) = diag MS)},

respect to &S(s) because of the limitation on the column degrees of S(s). On the other hand, since fiirp,is invertible, there exists always an entry @(s) of @‘i(s) satisfying (12). Taking into account that the CI are feedback invariants and Lemma 4.2, it follows that (13) may have a solution with respect to &S(s) only if

pi(s) = ti Y(S) ’

i=l,...,/

(4 v,,Js)IL&)

and v~,(s)I&,(s), where hb)

is

the

greatest common divisor of the elements of the kth row of the N(s) and a 1b means ‘a divides b’, and (iii) p,,(s) and vV,(s) are manic, then (13) is solvable with respect to &. Note that if (i)-(iii) above are not satisfied, then (13) is not solvable. Since F7 is a solution of (13). the svstem (N(s). D,(s)) is decompouplable by state feedback alone, namei; &I ‘The this realization of system is quadruplet the (E, A, BG,G,G,, C). This system is decoupled by state i.e. feedback &, the system (E, A + BG, G,G,F;, BG,G,G,, C) has a diagonal TF, i.e. the pair (F, G) = (G,G,G&, G,G,G,) decouples the system (E, A, B, C), 0 which proves the sufficiency. Remark

4.2. The TF of the H,(s) = P-‘(s). The invertibility regularity of SE - A - BE

closed loop system is of P(s) guarantees the 0

Corollary 4.2. When the system is decouplable we may always choose regular feedback such that the diagonal closed loop TF is proper by choosing deg {vyll,(s)}= deg {~~l’r(s)}. 0 5. Pole placement and decoupling

In this section the problem of pole placement while decoupling is considered. It is shown that a number of poles of the closed loop system is fixed and unobservable when decoupling control is applied, and a characterization of these poles is given. From (13) and (14) we have that the rows of the denominator of the closed loop (decoupled) system are

CL,(s) v,(s)

(14)

and (pi(s), v,(s)) are relatively prime polynomials. The appropriate selection of P(s) will give the feedback matrix. Equation (13) has a solution with respect to &S(s), only if v,(s) is common divisor of the entries of ni(s). Let WPI.F,(S)

(i) 0 5 deg v -(s) - deg p -(s) sf _, deg CL&) deg v+,(s$ f+ 05 de&$,(s) ?‘deg v6,(s) 5 f+,.

d’(s) = --C(s)n’(s),

". %I,fi,-&) w&S) .1'

where wij(s) are the entries of W(s)= P(s)N(s) in (13). Then, since &S(s) =r, - 1, &,D3(s)sr, - 1 if a, is non-proper and &,D,(s) = r, if a, is proper (a,, denotes the ith column degree), it follows that W(s) must be such that W& = D,. Indeed, if ~Js) and v&) are chosen to be manic, the equality of the above column high order coefficient matrices is obtained. The polynomials p,,?(s), vp&s) in (14) must satisfy the degree condition 0 5 deg v,,(s) - deg pLyi, 'f,,,

(15)

otherwise the polynomial ~,(s)/v,,(s)n~(s) has degree greater than aF, and (13) does not ‘have a solution with

where S’(s) = (l/&(s))&(s). Since v&) 1t,(s), we have that deg {v,(s)} 5 deg {t,(s)}, and therefore deg {p+(s)} 5 Similarly, deg &Jr)1 + f+, and degIcL&N 5 deg&&)I. since v~&s) is a divisor of &,,(s), we have that deg {v,(s)} 5 deg {&,,&s)}.The TF of the closed loop system is ’

f&(s) = diag {h,(s)}, hi(s) = z. , We have now the following characterization poles of the decoupled system. Theorem

i = 1,.

(19) of the fixed

5.1. Let A(s) = [AIL,. . . , $(s)~]~, where 3i(s), , I, are defined in (18). Let i(s) = det (N(s)). Then

the roots of i(s) are (i) the fixed poles of the decoupled system and (ii) unobservable poles of the closed loop system. 0

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From (14) and (18) we have that for any decoupling

law D,(s) = diag {p,(s)}N(s) = diag {&(s)}fi(s),

(20)

where det (Q(s)) = det R(s) +i(s) = Pi(S)&(S). Thus n!=, Jll(s) = i(s) n!=, t,k,(s), which means that the zeros of e(s) are always poles of the closed loop system. Now, if we choose v+,(s) = t+,(s) and let p+,(s) be any polynomial satisfying (16) we readily see that GG,(s)= p+,(s) with ~Js) is an arbitrary polynomial of degree f+, + deg {&,(s)}. If we choose vyri(s) = &,&s) and let ~,,Js) be any polynomial of degree &I, we have that the degree condition (15) is satisfied. Finally, if we choose v,,(s) = &(s) and if &s) is any polynomial of degree fZ,, we have that the degree condition (17) is satisfied. Then det (Us))

respect to F7. The feedback and is 2 0 -1 F=G&= I -2 -4 -7 -3 -4 -9 -1 -3 -7

diag EhNfi(s) diag Ms)Ifi(~)

and thus the zeros of i’(s) are the common zeros of N,(s), D=(s), i.e. they are output decoupling zeros of the closed loop system, which proves (ii). Example 5.1. Consider the singular system as in (6). with

-1 -3 -5 -3

-1 0 0 -1

1

1 0 0 2

-2 -2 1 0 -1 -1 -1 1

1

-1

-1

-1 -1 -1 -1 -1

-1 -1 1 -1

1

1 -5 -6 -4 -2 . 0 -10 21 -10 0 and the TF of the closed loop system is H,(s) = P-‘(s). 0 Example 5.2. Consider the system in the form (6). with

( i

sK-il=

= i(s) fi CL&) ‘fit,,(s) r=l j=l

which clearly shows that the poles of the closed loop system, which cannot be shifted, are only the roots of i(s) which proves (i). From (20), it follows that

matrix F is calculated from I;;

+

E ii;

-;

i

; 1-i

5 ;I;;

:a

p g

_Ej -I

:i

_;

p (,

ii.

The above has r, = 3. r, = 4, r, = 4, cr, = 2, cr2 = 3 and (T?= 3 (all CIs are non-proper). The normalized MFD is

T(s) =

I sZ+2r+1 SL+ 2s 2$+3si-1 ~

sz+s

-2-3 X2- 4s s*-6.-l

7s + 9 13s +6 -9

s3 + 2s

0

0

1

-sJ+8s

-3s

0

+ 21s + 6

I

39 +6 s’-3s2-b-6

J

In (13) choose D,(s) = D(s) and P(s) = IX. Then. we take

G =6, F=1 0 1 . 01 We have: r, = 1, r, = 4, r, = 5, r, = 5, (T, = 1, 02 = 3 (non-proper), (TV= 4 (non-proper), oq = 5, 5 = 1, * = 2, tj,=l,~z=4,cp,=2,cp,=3,cp,=4,~,=2,q,,=1,

- 0 1 0 0 T(s) = 2 s-l s-2 S

$+1 3s+4 s* + 1 1 1 0 0 2

S

s4 +s

s3 s’+2s 1 1 S

s4 + 1 s2 + 2

rankN’;=2=I=

0 s4 + s= sz + 2 s”+s+1 ’ 2 s+3 1 _

$,

Choose fiC = 1z (the first two rows of NC). With

we obtain a system with 4, = 2, & = 3, Cp,= 4, fi = 1. fz = 1, f; = 1 and f4 = 3. Now, choosing f(s) = diag{l,s + 1, s + 1, (s + 1)‘) it can be verified that (13) is solvable with

llO--3 0 2 1 0 1 3 2 --1

I I

10 09-l -6 1 -1 0 -6 1 0 12

IO 13 -3 23 -4

0 I, -1

and H,(s) = I+ The above system is unobservable at infinity, i.e. I,!?, CTIT is not manic. It can be checked that if it was minimal, decoupling would be impossible (in that case, or and (TVwould be a proper CI). 0 Remark

5.1. The system considered in the above example has a proper TF and is not decouplable when it is considered as a state-space system with feedthrough term in the output equation. However, when a non-minimal singular system model is considered, decoupling is possible. This can be explained as follows. The redundant states are x, and x,,, and we have x7 = & and x,, = i,,,. Thus, state feedback in the non-minimal system of the example can be interpreted as proportional plus derivative state feedback on the minima1 singular system or in the proper state-space system. When the system is considered in a proper state-space description, it is decouplable by proportional plus derivative state feedback, since the TF is right invertible. The above observation and Corollary 3.1 show that the decouplability of a singular system is not determined only from the transfer function, as in the case of state-space systems, but the internal structure also plays an important role. 0

6. Conclusions The problem of input-output decoupling for square singular systems via state feedback and regular input transformation has been solved by using a frequency domain method. The proposed approach has been shown to be a direct generalization of the frequency domain approaches for state-space systems. An easily testable necessary and sufficient condition for the solvability of the problem as well as a constructive method for the choice of control law have been obtained, and the complete characterization of the fixed poles of the decoupled system has been given. It has also been shown that for the case of singular systems, decouplability does not depend only on the structure of the transfer function, as in the case of state-space systems; it depends also on the internal structure of the system. Acknowledgement-This work was supported ESPIRIT III Project No. 8924.

by SESDIP

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