A normal form for pure differential algebraic systems

Transcription

1 A normal form for pure differential algebraic systems Stephan Trenn December 2th 27 Abstract: In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes-Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and -observability. Keywords: differential algebraic equation, normal form, impulse controllability, impulse observability MSC (Scheme=msc2): 34A9, 34C2 Introduction Differential algebraic equations (DAEs) of the form Eẋ = Ax + bu, y = cx, () (c,e,a,b) R n R n n R n n R n, n N, play an important role in systems theory. Those equations arise when modelling for example electrical circuits, mechanical systems, or, in general, dynamical systems with additional algebraic conditions. Interconnected ordinary differential equations (ODEs) can also be desribed as a DAE. There is a wide range of literature for DAEs of the form (), i.e. linear time-invariant DAEs, e.g. (Campbell 98, Campbell 982, Dai 989, Rabier & Rheinboldt 22, Kunkel & Mehrmann 26). Normal or condensed forms for DAEs have always been a research topic and the most famous normal form is the Kronecker normal form or, if one considers a special class of DAEs, the Weierstraß normal form. The latter is basically a decoupling into an ODE and a pure DAE. Most normal or condensed forms concentrate on the two matrices E and A and not on the input and output vectors b and c. But for control problems normal forms must incorporate the input and output. For ODEs the Byrnes-Isidori normal form (which focus on the relative degree (Isidori 995, p. 65), see also (Ilchmann, Ryan & Townsend 27, Lem. 3.5)) and the Kalman-decomposition (which focus on controllable and observable sub-states (Kalman 962)) are examples of such normal forms. This paper gives a normal form for pure DAEs which can be seen as a generalization of the Byrnes-Isidori normal form combined with a Kalman-like decomposition. In fact, the state space is separated into impulse-controllable and -observable sub-states, see Theorem 24. Compared to a similar decomposition proposed in (Dai 989, p. 52) (without proof) the normal form from

2 Theorem 2 is more specific and allows for a better analysis. There are already results on normal or condensed forms of DAEs available, e.g. (Loiseau, Özçaldiran, Malabre & Karcanias 99), (Rath 997), (Kunkel & Mehrmann 26). But none of these result focus on the relative degree or on impulse-controllable and -observable states. In addition they partly use a different concept of equivalence which leads to other normal forms. On the other hand some of these results go much further as the results in this paper because rectangular (in particular non-regular) DAEs with time-varying coefficients are considered. This paper is structured as follows. First, some preliminaries (Section 2) are given, in particular the subtle difference between DAEs and differential algebraic systems (DASs) is explained. Section 3 deals with the transfer function of DASs and realization theory, in particular some specific minimal realizations of pure DASs are given. Before stating the main results in Section 5, impulse-controllability and -observability are revisited in Section 4, the invariants impulse-controllability-index and impulse-observability-index are defined. The main result is the normal form given in Theorem 2. This normal form can be used to give new characterizations of impulse-controllability and -observability, see Theorem 24. The following notation will be used throughout this paper. N and R are the natural and real numbers, R[s is the ring of polynomials and R(s) is the field of rational functions with real coefficients. For a polynomial p(s) R[s the degree of p(s) is denoted by deg p(s). The matrix I R n n is the identity matrix[ of size n N being clear from the context. For two square A matrizes A,B let diag (A,B) :=. For n N row vectors c B,c 2,...,c n R m of the c c 2 same length m N let [c /c 2 /... /c n :=. Rn m. The rank, image, and kernel of matrix A is denoted by rka, im A, and ker A, resp. c n 2 Preliminaries: Differential algebraic systems (DASs) In this work only differential algebraic systems (DASs), i.e. matrix-tuples (see Definition ), are considered and not differential algebraic equations (DAEs) like (). The reason is that for the latter one has always to define what the variable x should be. In particular it would be necessary to specify an appropriate solution space. Since the results of this work are independent of the chosen solution space, any discussion about solution spaces will be avoided by considering DASs instead of DAEs. Definition (DASs, regular and pure DASs, ODSs) A differential algebraic system (DAS) with state space dimension n N is a tupel (c,e,a,b) R n R n n R n n R n. The space of all DASs with state space dimension n is Σ n := R n R n n R n n R n. The space of regular DASs with state space dimension n N is Σ reg n := { (c,e,a,b) Σ n det(es A) R[s\{} }. 2

3 The space of pure DASs is Σ pure n := { (c,e,a,b) Σ n deta A E is nilpotent }. The space of ordinary differential systems (ODSs) is Σ ODS n := { (c,e,a,b) Σ n det E }. Remark 2 (i) Every pure DAS and every ODS is regular, i.e. for all n N Σ pure n Σ reg n and Σ ODS n Σ reg n (ii) No pure DAS is an ODS and vice versa, i.e. for all n N. Σ pure n Σ ODS n (iii) For invertible A R n n and some E R n n, n N, the matrix A E is nilpotent if, and only if, EA is nilpotent, hence = Σ pure n = { (c,e,a,b) Σ n deta EA is nilpotent }. Definition 3 (Equivalence) Two DASs (c,e,a,b ) Σ n and (c 2,E 2,A 2,b 2 ) Σ n2, n,n 2 N, are called equivalent, written (c,e,a,b ) (c 2,E 2,A 2,b 2 ), if, and only if, n = n 2 =: n and there exist invertible matrices S,T R n n such that Note that is an equivalence relation. (c 2,E 2,A 2,b 2 ) = (c T,SE T,SA T,Sb ). Remark 4 Every regular DAS (c,e,a,b) is equivalent to a DAS in Weierstrass form ( [ [ [ ) I J b [ c c 2,,, N I where N is a nilpotent matrix ((Weierstraß 867), see also (Kunkel & Mehrmann 26, Thm. 2.7)). Clearly, this is a (unique) decomposition into an ODS (also called slow system) and a pure DAS (also known as the fast system). b 2 Proposition 5 For all n N { Σ pure n = (c,e,a,b) Σ n (ĉ,n,i,ˆb) Σ pure n : (c,e,a,b) (ĉ,n,i,ˆb) }. Proof. By definition, every pure DAE (c,e,a,b) Σ n is equivalent to (c,a E,I,A b) = (ĉ,n,i,ˆb) Σ pure n. If (c,e,a,b) Σ n is equivalent to (ĉ,n,i,ˆb) Σ pure n then there exist invertible matrices S,T R n n such that A = SIT = ST and E = SNT. In particular, A is invertible and A E = T NT. By assumption, N is nilpotent and hence A E is nilpotent which implies that (c,e,a,b) is pure. Proposition 5 (see also the forthcoming Proposition 5) justifies that every pure DASs can be considered to be in the standard form (c,n,i,b), where N is a nilpotent matrix. qed 3

4 3 Transfer function and minimal realization In this section transfer functions of DASs and minimal realizations are studied. From the theory of ODEs it is well known, that the transfer function is a useful tool to study the input-output behaviour of a linear system. Furthermore the definition of the (negative) relative degree is based on transfer functions and the negative relative degree is important for the normal form of pure DASs given in this paper (Theorem 2). It will also turn out that one of the given minimal realization is a part of this normal form. Definition 6 (Transfer function and IO-equivalence) The transfer function of a regular DAS (c,e,a,b) Σ reg n, n N, is the rational function g(s) R(s) given by g(s) = c(es A) b. Two regular DASs (c,e,a,b ) Σ reg n, n N, and (c 2,E 2,A 2,b 2 ) Σ reg n 2, n 2 N, are called IO-equivalent, written (c,e,a,b ) (c 2,E 2,A 2,b 2 ), if, and only if, their corresponding transfer functions g (s) and g 2 (s) are equal. Remark 7 For two regular DASs (c,e,a,b ) Σ reg n, n N, and (c 2,E 2,A 2,b 2 ) Σ reg n 2, n 2 N it follows from the definition that (c,e,a,b ) (c 2,E 2,A 2,b 2 ) (c,e,a,b ) (c 2,E 2,A 2,b 2 ). The converse is in general not true, but this question is strongly related to minimal realizations, impulse-controllability and (impulse-)observability, which are studied later in this work (see Corollary 26 and Remark 27). Remark 8 (i) The transfer function of an ODS is strictly proper (i.e. the degree of the numerator is smaller than the degree of the denominator). (ii) The transfer function of a pure DAS is a polynomial, in particular if the pure DAS is in standard form (c,n,i,b) Σ pure n, n N, the transfer function is given by n g(s) = cn i bs i. For convenience, if n = the transfer function is defined as g(s). Definition 9 A realization of a transfer function g(s) R(s) is a regular DAS (c,e,a,b) Σ reg n, n N, with g(s) = c(es A) b. A realization is called minimal if, and only if, there exists no other realization with smaller state-space dimension. For a given transfer function it is an interesting question how a realization might look, what the minimal dimension is and if there are some standard realizations. For ODEs these questions are studied in realization theory and most relevant questions are answered. The next propositions shows that for the realization theory of DASs one can basically concentrate on pure DASs. i= 4

5 Proposition Let g(s) = p(s) q(s) R(s), then there exists unique p (s),p 2 (s) R[s with deg p 2 (s) < deg q(s) such that g(s) = p (s) + p 2(s) q(s). If (c,e,a,b ) Σ pure n, n N and (c 2,E 2,A 2,b 2 ) Σ ODS n 2, n 2 N are realizations of p (s) and p 2(s) q(s), resp., then (c,e,a,b) := ( (c,c 2 ),diag (E,E 2 ),diag (A,A 2 ),(b,b 2 ) ) Σ reg n +n 2 is a realization of g(s). Conversely, every realization of g(s) is equivalent to (c,e,a,b) = ( (c,c 2 ),diag (E,E 2 ),diag (A,A 2 ),(b,b 2 ) ) Σ reg n +n 2, where (c,e,a,b ) Σ pure n, n N, is a pure realization of p (s) and (c 2,E 2,A 2,b 2 ) Σ ODS n 2, n 2 N, is an ODS realizations of p 2(s) q(s) (n = or n 2 = is possible). Furthermore, in both cases (c i,e i,a i,b i ), i =,2, are minimal realizations if, and only if, (c, E, A, b) is a minimal realization. Proof. The unique decomposition g(s) = p (s)+ p 2(s) g(s) with deg p 2(s) < deg q(s) is a well known algebraic result (Euclidian algorithm for polynomials). From the definition of the transfer function it easily follows that (c,e,a,b) is a realization of g(s) if (c,e,a,b ) and (c 2,E 2,A 2,b 2 ), resp. Conversely, observe that every regular DAS is equivalent to a DAS in Weierstraß form (see Remark 4), which yields the assertion of the proposition. It remains to show the minimality property. Clearly, if (c,e,a,b) is minimal then (c i,e i,a i,b i ), i =,2, are minimal, too. To show that (c,e,a,b) is minimal if (c i,e i,a i,b i ), i =,2, are are realizations of p (s) and p 2(s) q(s) minimal, consider any realization (ĉ,ê,â,ĉ) Σreg ˆn with ˆn N. Then this realization is equivalent to a DAS in Weierstraß form with dimension ˆn, whose pure DAS and ODS parts have dimensions ˆn N and ˆn 2 N, resp. Let the transfer function of the pure DAS part be the polynomial ˆp (s) R[s and let the transfer function of the ODS part be the strictly proper rational function ˆp 2(s) ˆq(s) R(s). Since ˆp (s)+ ˆp 2(s) ˆq(s) = g(s) = p (s)+ p 2 q(s) it follows that ˆp (s) = p (s) and ˆp 2(s) ˆq(s) = p 2(s) q(s). This implies, by the minimality assumption, that ˆn i n i, i =,2, hence ˆn = ˆn + ˆn 2 n + n 2, i.e. the given realization of g(s) with dimension n + n 2 is minimal. qed The foregoing proposition justifies that for a realization theory of general DASs it is sufficient to consider pure DASs and ODSs separately. Realization theory of ODSs is well understood, hence it remains to study the realization theory of pure DASs. Proposition Consider a polynomial transfer function g(s) = r i= α is i R[s for r N, α r. Then the following DASs are minimal realizations of g(s) with state space dimension r + : (i) R-form: ( [,...,,,[,I, [ ) αr Σ pure r+, 5

6 where I = α α r α r α r. (ii) O-form: ( (iii) C-form: [,...,,,[,I, (c,n,î,b) = ([ α,..., α r,[ [ ) αr α Σ pure r+. ),I,[ Σ pure r+. Proof. A simple calculation invoking Remark 8 (ii) shows that the tranfer function of the O- and C-form coincide with g(s). With S = α r I and T = α r I the C-form and the R-form are equivalent and hence the transfer function of the R-form also coincides with g(s). To show that these realizations are minimal consider any realization (c,e,a,b) Σ reg n, n N, of g(s). The ODS part and pure DAS part of the Weierstraß form (see Remark 4) have state dimensions n,n 2 N with n + n 2 = n and transfer functions g (s) R(s) and g 2 (s) R[s, resp. By Remark 8, g (s) is strictly proper and deg g 2 (s) n 2. Together with g (s) + g 2 (s) = g(s) R[s this yields g (s) and r = deg g(s) = deg g 2 (s) n 2. This shows n = n + n 2 r +, i.e. the minimal state-space dimension for any realization of g(s) is r +. qed First results on realization theory can be found in (Dai 989), in particular Proposition is a constructive version of (Dai 989, Lemma 2-6.2) and Proposition is implicitely used in the proof of (Dai 989, Thm ). 4 Impulse-controllability and impulse-observability From the theory of linear ODEs it is well known, that the controllability- and observabilitymatrices play an important role for controllability and observability as well as for the construction of normal forms. It is possible to define analogous matrices for DASs, which play similar roles. Furthermore one can define impulse-controllability- and impulse-observability-indices which are invariants with respect to equivalence transformations. This is important for the normal form and can be used for characterizations of impulse-controllability and -observability. Definition 2 Consider a pure DAS (c,e,a,b) Σ pure n, n N. The impulse-controllabilitymatrix of (c,e,a,b) is B imp := [ b,n b b,n b 2 b,...,n b n b, where N b := EA. The impulse-controllability-index of (c, E, A, b) is d b := rk B imp. 6

7 The impulse-observability-matrix of (c, E, A, b) is C imp := [ c/cn c /cn c 2 /... /cn c n, where N c := A E. The impulse-observability-index of (c, E, A, b) is d c := rk C imp. Definition 3 A pure DAS (c,e,a,b) Σ pure n, n N, is called (i) impulse-controllable (in the sense of (Cobb 984)) if, and only if, im N b = im B imp, where N b = EA. (ii) impulse-observable (in the sense of (Cobb 984)) if, and only if, ker N c = ker C imp, where N c = A E. Remark 4 It might seem artificial to define impulse-controllability and -observability in terms of algebraic conditions. A natural definition should be based on reachability of certain impulsive states and deduction of impulsive states from the output. The problem is that these definitions would require a complete distributional solution theory leading to an unnecessary overhead for the purposes of this paper. For this reason, the definition of impulse-controllability and -observability is based on characterizations given in (Cobb 984, Thm. 4 and Thm. 9). Proposition 5 Impulse-controllability and -observability as well as the corresponding indices are invariant under equivalence transformations. Proof. Let (c,e,a,b ),(c 2,E 2,A 2,b 2 ) Σ pure n be equivalent via S,T R n n. Let B imp,, C imp,, B imp,2, and C imp,2 be the corresponding impulse-controllabel- and impulse-observablematrices. From the definition it follows that B imp,2 = SB imp,, hence the corresponding inpulse-controlability-indices are equal. Furthermore, im E 2 A 2 = im SE A S = im SE A, which yields that (c,e,a,b ) is impulse-controllable if, and only if, (c 2,E 2,A 2,b 2 ) is impulsecontrollable. Analogously, C imp,2 = C imp, T and ker A 2 E 2 = ker T A E T = ker A E T, which show that the impulse-observability-index and impulse-controllability are invariant. qed 7

8 Proposition 5 again justifies that one can assume that every pure DAS is in the standard form (c,n,i,b). In fact, this simplifies the Definitions 2 and 3 because then N b = N c = N. The next proposition highlights an important property of the impulse-controllability- and impulseobservability-matrices. Proposition 6 Consider a pure DAS in standard form (c,n,i,b) Σ pure n, n N, with impulse-controllability- and impulse-observability-indices d b,d c N. Then B imp = [ b,nb,...,n db b,,..., and C imp = [ c/cn/... /cn dc //... /. Proof. Let d N be the smallest number such that N d b = (which exists since N is nilpotent). In terms of (Lang 97, XII.7) the vector b is N-cyclic with period d. Now (Lang 97, Lemma XII.7.) states that [b,nb,...,n d b has full rank which yields d b = rk B imp = rk [ b,nb,...,n d b,,..., = d, this is the assertion of the proposition. The same argument applied to N and c shows the analogous property for C imp. qed Remark 7 For (c,e,a,b) Σ pure n let (b,e,a,c ) Σ pure n be the dual system (see e.g. (Dai 989, 2.4)). If d b and d c are the impulse-controllable- and impulse-observable-indices of (c,e,a,b), then it is easy to see that d c and d b are the impulse-controllable- and impulseobservable-indices of the dual system. 5 A normal form for DASs In this section a normal form for pure DASs is given. For the derivation of the normal form the following definition of the negative relative degree is needed. Definition 8 Consider a pure DAS (c,e,a,b) Σ pure n, n N, with (polynomial) transfer function g(s) R[s. The negative relative degree of (c, E, A, b) is By convention, if g(s) then r :=. r := deg g(s). Remark 9 (i) For an ODS with transfer function g(s) = p(s) q(s) the relative degree ρ is defined as the difference between the degrees of the denominator and numerator, i.e. ρ := deg q(s) deg p(s). This definition is consistent with Definition 8 and r = ρ. 8

9 (ii) By Remark 7 the negative relative degree is invariant under equivalence transformations. Furthermore, for a pure DAS in standard form (c,n,i,b) Σ pure n the negative relative degree fulfills (see Remark 8 (ii)) r = max { i N cn i b }, where by convention the maximum of an empty set is. It is now possible to formulate the main result of this paper. The main advantage is the possibility to observe easily the influence of the input on the states and the influence of states on the output. Theorem 2 Consider a pure DAS (c,e,a,b) Σ pure n, n N, with negative relative degree r, impulse-controllability- and impulse-observability-indices d b,d c N. Then (c,e,a,b) is equivalent to (ĉ, ˆN,Î,ˆb) Σ pure n, where ĉ = [,...,,, ˆN = E N E 2 E 3 } dc r } n d c d b + r + } db r } r + Î = I I I I, ˆb =, γ where γ := ca (EA ) r b = c(a E) r A b, [ [ =, = [, I =, and N R (n dc d b+r+) (n d c d b +r+) is a nilpotent matrix (in Jordan normal form). Proof. Without loss of generality, assume that the DAS is in standard form, i.e. (c, E, A, b) = (c,n,i,b) for some nilpotent matrix N. In this case γ = cn r b (see Remark 9 (ii)). The proof consists of two main steps. The first step is the construction of the transformation matrices S and T, in particular the construction must ensure that S and T are invertible. In 9

10 the second step it is shown that indeed (c,n,i,b) (ĉ, ˆN,Î,ˆb) via S and T. Step. The construction is based on the five matrices L R n (dc r ), L R n (n dc d b+r+), B R n (d b r ), B R n (r+), and Î Rn n, which define the transformation matrix S and T by S := γ [ L, L, B, B, T := γ[ L, L, B, B Î. Step a: The matrix Î. Let where Obviously, Î is invertible. I := cb γ [ I Î := cnb γ I cn r b γ R n n, Step b: The matrices B and B. Let B := [ b,nb,...,n r b R n (r+) and By Remark 9 (ii) CB = C := [ cn r /... /cn/c R (r+) n cn r b cn r b. cb cn r b cn r b R(r+) (r+). (2) R(r+) (r+) (3) is invertible and hence B and C must have full rank. In particular this implies d b r + and d c r +. Let B := [ N r+ b,n r+2 b,...,n d b b R n (d b r ), then by the definition of d b the matrix [ B, B has full column rank. Step c: The matrix L. If d c = r +, then L is the empty matrix. Otherwise let C := [ cn dc /cn dc 2 / /cn r+ R (dc r ) n.

11 Then ker [ C/C is an (n d c )-dimensional subspace of ker C (where dim ker C = n r ), i.e. there exists a full rank matrix L R n (dc r ) such that im L ker [ C/C = ker C. In particular im L ker C = {} and im L ker C. Let L := γl ( CL ). It remains to show that, firstly, L is well defined, i.e. that CL is an invertible matrix, and, secondly, that [L, B, B has full rank (otherwise the matrix S is not well defined). Assume that CLm = for some m R n. Then Lm im L ker C = {}, hence CL has only a trivial kernel which implies invertibility. To show that [L, B, B has full rank, observe that im L = im L and, by the definition of the relative degree, im [ B, B ker C. Hence {} = im L ker C im L im [ B, B, which implies that [L, B, B has full rank. Step d: The matrix L. If d b = r +, then L is the empty matrix. Otherwise choose, analogously as in the previous step, a full rank matrix K R (d b r ) n such that im K ker [ B, B = ker B. Again the matrix B K is invertible. Let K = (KB) K, with an analogous argument as in Step c it can be shown that [ K/C/C has full rank, hence it is possible to choose a full rank matrix L R n (n dc d b+r+) such that im L = ker [ K/C/C. It remains to show that [ L, L, B, B has full rank (i.e. is invertible). To show this, first observe that, by the definition of the relative degree, im B ker C = {} and recall that im L ker C = {} and analogously im K ker B = {}, the latter is equivalent to im B ker K = {}. Altogether this yields ker [ K/C/C im [ L, B, B = {}, which implies that the square matrix [ L, L, B, B has full rank which completes the first step of the proof. Step 2. Step 2a: ST = Î. By definition ST = Î. Step 2b: Sb = ˆb. Let e r = [,...,,,,..., }{{} R n then Sb = ˆb = γe r if, and only if, b = γs e r. The latter is r fulfilled since γs = [ L, L, B,b,Nb,...,N r b. }{{} =B Step 2c: ct = ĉ. Choose a full rank matrix K R (n dc d b+r+) n such that im K = ker [ L, B, B.

12 It can be shown analogously as in Step d that the square matrix C := [ C/K/K/C has full rank (i.e. is invertible). Writing B := [ L, L, B, B the matrix T can be written as T = C γ CBÎ. Since cc = [,...,, it remains to show that [,...,,γ CBÎ = [,...,, = ĉ, or, equivalently, that the last row of the product CB equals the last row of γî. It is easy to see that the last row of γî is [,...,,cb,cnb,...,cn r b. Observe that CB = CL CL CB CB KL KL KB KB KL KL KB KB CL CL CB CB. (4) The matrices L and L are such that im L and im L are both subspaces of ker C, hence CL = and CL =. From the definition of the relative degree it follows that CB =. Together with (3) this shows that the last row of CB is [,...,,cb,cnb,...,cn r b. Step 2d: SNT. Invoking the notation of Step 2c write SNT = (CB) CNBÎ. Note that the product CB in (4) can further be simplified by the following observations, CL = γi, C[L, B, B =, K[L, B, B =, K[L, B =, and KB = I: γi CB = KL KL I. CB Hence By Proposition 6 furthermore CNB = CB [ (CB) = γ I (KL) γ KL I (CB) [ NB = B and CN = C,, KNB = and CNL =, hence [. [ γ KNL KNL CNB = [. KNL KNL KNB [ CNL CB 2

13 Therefore, SNT = (CB) CNBÎ = [ Ê ˆN [ E 2 Ê 3 KNBI [ (CB) CNL where Ê = (KL) KNL, ˆN = (KL) KNL, E 2 = KL [ given by (2). Note that CNL = hence (CB) CNL =, and cn r+ cn r. cn 2 cn L = γcn r+ L(CL). I, + KNL, Ê 3 = KNL, and I is = γ, KNB = K[Nb,N 2 b,...,n r b,n r+ b = [,,...,,(BK) KN r+ b = hence KNBI =. Clearly, [ I = [, and it remains to show that ˆN is nilpotent. This follows from the fact that SNTÎ = SNS is nilpotent and because of the special block structure this implies that ˆN must also be nilpotent. Without changing the block structure it is possible to transform ˆN to Jordan form N, this changes Ê and Ê3 to E and E 3. qed Remark 2 The proof of Theorem 2 is constructive. In fact, for a given DAS in standard form (c,n,i,b) Σ pure n with negative relative degree r and impulse-controllability- and impulse-observability-indices d b,d c N the specific matrices in the normal form are given as 3

14 follows: E = J (KL) KNL R (n dc d b+r+) (d b r ), [ E 2 = KNL KL R (d b r ) (d c r ), E 3 = KNLJ R (d b r ) (n d c d b +r+), N = J (KL) KNLJ R (n dc d b+r+) (n d c d b +r+), = cb γ cnb γ = (CB) γ I = cb γ cnb γ cn r b γ R (db r ) (r+), R (r+) (dc r ), cn r b γ R(r+) (r+), where B := [ b,nb,...,n r b, B := [ N r+ b,n r+2 b,...,n d b b C := [ cn dc /cn dc 2 / /cn r+, C := [ cn r /... /cn/c, ( [B K := [,I /B ) [ [B, B B /B R (d b r ) n L := γ [ C, C ( [C/C [ C, C ) [I/ R n (d c r ), K R n (n dc d b+r+) is a basis of ker [ L /B /B, L R n (n dc d b+r+) is a basis of ker [ K/C/C, and J R (n dc d b+r+) (n d c d b +r+) is a basis transformation such that N is in Jordan normal form. If the DAS (c,e,a,b) is not in standard form, then either N and b in the above formulae must be replaced by A E and A b, resp., or N and c must be replaced by EA and ca, resp. Remark 22 If the negative relative degree of a pure DAE is maximal, i.e. r = n, then the normal form above coincides with the minimal realization in R-form as given in Proposition. Corollary 23 All minimal realizations of a pure DAS are equivalent. The normal form of Theorem 2 can be viewed as a specialization of (2-5.4) in (Dai 989, p. 52): it is more explicit and simpler, the size of the different blocks is explicitly given, and the influence of the input on the states can be seen more directly as well as the influence of the states on the output. Furthermore no proof is given in (Dai 989). 4

15 6 Impulse-controllability and -observability revisited With the normal form from Theorem 2 it is now possible to give characterization of impulsecontrollability and -observability. Theorem 24 Consider (c,e,a,b) Σ pure n, n N, with negative relative degree r, impulsecontrollability- and impulse-observability-indices d b,d c N and let N R (n d b d c+r+) (n d b d c+r+) be given as in Theorem 2. Then the following characterizations of impulse-controllability and -observability hold: (i) The DAS is impulse-controllable if, and only if, d c = r + and N = (ii) The DAS is impulse-observable if, and only if, d b = r + and N =. Proof. Let (ĉ, ˆN,Î,ˆb) be the normal form of (c,e,a,b) from Theorem 2. (i) It is easily seen that (ĉ, ˆN,Î,ˆb) is equivalent to ( c,ñ,i, b) with E N Ñ = E 2 E 3 where the matrices E,E 2,E 3,N, are the same as in Theorem 2 and has the same structure as from Theorem 2, in particular and have a one in the upper right corner. The vector b is given by b = [,...,,γ,,..., with γ at the (n d b +)-th position. It is easily seen that im [Ñ b,ñ2 b,..., Ñ n b = im, which implies that impulse-controllability for the given DAS is equivalent to the condition im E N im. E 2 E 3 A necessary and sufficient condition for this is that the matrix is not existent (because it has a one in the upper right corner), i.e. d c = r +, and that N is the zero matrix. (ii) It is easily seen that (ĉ, ˆN,Î,ĉ) is equivalent to ( c,ñ,i, b) with [ Ñ =, E N E 2 E 3 5

16 where E,E 2,E 3,N,O are as in the normal form in Theorem 2, has the same structure as from Theorem 2 and [ = The vector c is given by c = [,...,,,,..., where the one is at position d c. Easy calculations show that (here it is needed that has a one in the upper right corner) ker[ cñdc / cñdc /... / cñ = ker. [ Hence impulse-controllability is equivalent to the condition [ E N ker ker [ E 2 E 3. Because has a one in the upper right corner the inclusion holds if, and only if, does not exists, i.e. d b = r +, and N =. qed Corollary 25 A pure DAS with state space dimension n N is impulse-controllable and -observable if, and only if, it is equivalent to (c,n,î,b) = [,...,,,, I [,...,,γ,,...,, }{{}}{{} n r r where the diagonal square blocks of N and Î have size (n r ) and (r + ) for some r {,...,n } and γ. Corollary 26 (i) Two pure DASs with the same state space dimension which are impulsecontrollable and -observable are equivalent if, and only if, they are IO-equivalent. (ii) If the negative relative degree of a pure DAS (c,e,a,b) Σ pure n, n N, is maximal, i.e. r = n, then (c, E, A, b) is impulse-controllable and -observable. (iii) All minimal realization of a pure DAS are impulse-controllable and -observable. Remark 27 Note that an impulse-controllable and -observable DAS which is a realization of a polynomial transfer function need not to be minimal, because one can add arbitrarily many trivial state equations z =, z 2 =,..., z N = without loosing the property of impulsecontrollability and -observability. 6

17 7 Conclusion For pure differential algebraic equation a normal form is derived which shows very clearly the influence of the input on the states as well as the influence of the states on the output. The relative degree appears very clearly in the normal form. The normal form also separates the states into impulse-controllable and -observable states and easy characterizations of impulsecontrollability and -observability based on the normal form are given. Some specific minimal realizations of pure DAEs are given and connections between the relative degree and the normal form are highlighted. In combination with distributional solution theory the normal form might be used in future research to study the influence of inconsistent initial values on the output and the influence of non-smooth inputs on the states and the output. The normal form might also help for synthesis of controllers for specific control tasks, e.g. impulse elimination. Finally the proof of the normal is constructive, i.e. it is possible to calculate the transformation matrices and the normal form explicitly, nevertheless the given formulae are not studied with respect to numerical feasibility. Acknowledgment I thank Achim Ilchmann who carefully read the manuscript and who gave valuable hints and suggestions which helped to improve the clarity and quality of the paper. I would also like to thank Vu Hoang Linh (Hanoi Univ., Vietnam) whose visit in Ilmenau initiated the work on this paper. References Campbell, S. L. (98), Singular Systems of Differential Equations I, Pitman, New York. Campbell, S. L. (982), Singular Systems of Differential Equations II, Pitman, New York. Cobb, D. J. (984), Controllability, observability and duality in singular systems, IEEE Trans. Automat. Contr. AC-29, Dai, L. (989), Singular Control Systems, number 8 in Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin. Ilchmann, A., Ryan, E. & Townsend, P. (27), Tracking with prescribed transient behaviour for nonlinear systems of known relative degree, SIAM J. Control Optimization 46(), Isidori, A. (995), Nonlinear Control Systems, Communication and Control Engineering Series, 3 edn, Springer-Verlag, Berlin. Kalman, R. (962), Canonical structure of linear dynamical systems, Proc. Nat. Acad. Sci. U.S.A. 48(4), Kunkel, P. & Mehrmann, V. (26), Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Publishing House, Zürich, Switzerland. 7

18 Lang, S. (97), Linear Algebra, 2nd edn, Addison-Wesley, Reading, Massachusetts. fourth printing 973. Loiseau, J., Özçaldiran, K., Malabre, M. & Karcanias, N. (99), Feedback canonical forms of singular systems, Kybernetika 27(4), Rabier, P. J. & Rheinboldt, W. C. (22), Theoretical and Numerical Analysis of Differential- Algebraic Equations, Vol. VIII of Handbook of Numerical Analysis, Elsevier Publications, Amsterdam, The Netherlands. Rath, W. (997), Derivative and proportional state feedback for linear descriptor systems with variable coefficients, Lin. Alg. Appl. 26, Weierstraß, K. (867), Zur Theorie der bilinearen und quadratischen Formen, Monatsh. Akad. Wiss. pp